[PDF]

! " $#&%' (*),+- (.)/+10341
2 5674 +8 )/+
9;: :
<>=@?BADCFEGIH'?JLKM?BNPORQBHFSUTWVYX@Z\[^]M_3`bac[d]Y]
e Of?hgQiEQhOIjkOIQBEhOl?mVn]Y]M_poFqsr e tvFuH ?BAxwzy|{
}F~/7 ~
, d~ 'n 7
c ¢¡/¡¤£'¡¤¥¦¨§©ªd«¬¦®D°¯±¥¡©²$«¦®³¬7¡¤¯´®Dµ³¬¦¶¥®Yªd¡¤¬¦®D·¥¦¨¬¬¦®Dª¸ /¹º¤¡»«¼µ²½®D®Dª¾«¼ª¿U¬¦®M«¼¶¬¦¡¤¥À
ªÁ©Â©Â©Ã/Ä
Contents
Contents
3
1
Introduction
%&('*)+,-./
!
1
243)5! ! ! !
1
6789: ! ! !
1<;
6789,=)>?! ! !
1BA
CDE:)+FGHFI)J!
1L0
M4NO)PNQ ! ! !
1<R
ST:)/F&! ! ! !
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
9
$#
(0
(0
(0
(@
$K
$K
$U
2
Foundations
1 V*WFG)X6ZY Y[/
! ! !
1
\H)O]-93F^_/)`F+ !
1
67)F7/3aNb,=)c !
1<;
de9`>,39 3`:)fbF"! ! !
1BA
gh!__PN,9`>)î ! !
1L0
jk,Fh/3a>lm>`>)F ! !
1<R
gn1E>FX/3aPN,`>)9/)op!
1L@
qr)^>/F ! ! ! ! !
1<K
jkONs9F ! ! ! ! !
1)W9>^F&/3a9/>F !
1$#tCDs9uv`>)Fp ! ! ! ! !
1w2x]19^F^F^Fy ! ! ! !
1 zq{f//,FZ/3^f//,F ! !
1$;tCDE:)F|! ! ! ! !
1A}\))>)F ! ! ! !
1(0zgh"WFI)Zc^>fHFG)3
1$R~ST:)/F&! ! ! ! ! !
1(@~ b/F ! ! ! ! ! !
1$K~2x]H>`FGF ! ! ! ! !
1$Ut679/F ! ! ! !
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
21
!
!
!
!
!
!
!
!
;
;
/A
0
R
@
@
@
@
K
K
U
;#
;#
;;
; 0
b
; @
b
;U
Ab#
A
A
1
1B;
1 A
1<0
de)>F ! ! ! ! ! ! ! ! ! ! """""""""""""""
1 1
VXF^F^`N) fH)o_/3>`3`!P E>)^Fz"""""""""""""
1 1
2x]19^F^Fâ)GF ! ! ! ! ! ! """""""""""""""
1 1<;
2 f//,9)9+:]19^F^F^F_ ! ! ! ! """""""""""""""
1 1BA
V&)>)^`XE>)Fh)T)vE ) """""""""""""""
1 1L0
jk,T/ ! ! ! ! ! ! ! """""""""""""""
1 1<R
gh")oE",F^3 ) ! ! ! ! """""""""""""""
1 1L@
]=3:]HFGF^FX/3a`>fb>FGF*! """""""""""""""
1 1<K
k9^_>FG)O) ! ! ! ! """""""""""""""
1 1`FGF ! ! ! ! ! ! ! ! """""""""""""""
1 1$;t679/F ! ! ! ! ! ! ! """""""""""""""
k$/FD ! ! ! ! ! ! ! ! ! ! """""""""""""""
1B;H
k1$PN,`>)F ! ! ! ! ! """""""""""""""
1B;H
gh")oE sm1 ! ! ! ! ! ! """""""""""""""
1B;H<;
\19G)&` `,)h>f//,9) ! ! ! ! """""""""""""""
1B;HBA
ST:)/F&! ! ! ! ! ! ! ! ! """""""""""""""
1B;HL0
b/F ! ! ! ! ! ! ! ! ! """""""""""""""
1B;H<R
2x]H>`FGF ! ! ! ! ! ! ! ! """""""""""""""
1B;HL@
679/F ! ! ! ! ! ! ! """""""""""""""
6 )^ÊFG))>)F ! ! ! ! ! ! ! """""""""""""""
1 A=
\1`>)^
4P /3aP eF^ FG)O))F7! """""""""""""""
1 A=
d )>)^
4PNZFG)O))F.! ! ! ! """""""""""""""
1 A=<;
k1` HFZ/3aF^`>Ey! ! ! ! ! ! """""""""""""""
1 A=BA
d )>)^
'&DFG)O))F ! ! ! """""""""""""""
1 A=L0
d )>)^
43 FG))>)F?! ! ! ! """""""""""""""
1 A=<R
/,8FG))>)F?! ! ! ! ! ! """""""""""""""
1 A=L@
2 ,9 f/`"P )>O)
FG))>)F!! """""""""""""""
1 A=<K
671FG+)
I) )>)^ FI))>)F """""""""""""
1 A=`FGF ! ! ! ! ! ! ! ! """""""""""""""
1 A= ~679/F ! ! ! ! ! ! ! """""""""""""""
b) E)Z,+sE:F ! ! ! ! ! ! """""""""""""""
1<01
gh")oE
F bO)&9339,=s9 ! ! ! """""""""""""""
1<01
]=3:]HFGF^F 9`fb>^F^F 9/3)F """""""""
1<01<;
2x]1 `)h`fb>F^F ! ! ! ! ! """""""""""""""
1<01BA
jk,T/ ! ! ! ! ! ! ! """""""""""""""
1<01L0
b)9.E)71,.sE>hFGHFI)F! ! """""""""""""""
1<01<R
gh d 24242yFI)/393-@0/A ! ! ! ! """""""""""""""
1<01L@
6789,=) 'X )!
9b)9 E)h1,.sE>F*""""""""""""""
Ab;
AA
A R
b
A R
b
A @
0H
0
0;
0/A
0R
0@
R=
R
R;
Rb0
Rb0
Rb@
@#
@#
@
@
@/A
@R
@R
@@
K=
KR
KK
KU
U#
U;
UA
UK
UU
$#;
$#A
$#b0
$#b@
$#K
$#U
$#
(0
$R
1BR
0
1<01<K
ST:)/F&! ! ! ! ! ! ! !
1<01`FGF ! ! ! ! ! ! !
1<01 679/F ! ! ! ! ! !
V&GHFZ/3E)>F ! ! ! ! ! !
1BRH
V&^_)o1EF ! ! ! ! ! !
1BRH
de9 )N/ ^$1FT! ! ! ! !
1BRH<;
&/3J/``FGF&)+)FJ! !
1BRHBA
V&^HF& )ZFGP 3FG`>s9Q !
1BRHL0
d )>)^(fb>D.`)/> ! !
1BRH<R
M )^>Z)oEFX93PN,9`>)^9/)ot!
1BRHL@
V&^ ) em):Z`>fb>FGp! !
1BRH<K
M )^>XO)>)` ! ! ! !
1BRHF ! ! ! !
1BRH ,9 )3F^9/ ^$1F ! ! !
1BRH
k>b3 ^HFZ/3E)>^F*! !
1BRH$;~ST:)/F&! ! ! ! ! ! ! !
1BRHA} b/F ! ! ! ! ! ! ! !
1BRH(0z2x]H>`FGF ! ! ! ! ! ! !
1BRH$Rt679/F ! ! ! ! ! !
3
Functions
;H V*WFG)X6ZY Y PN,9`:) ! ! ! ! !
;H
M /3mFG)`3)F ! !
;H
,`>)a3>W99 )F ! ! !
;H<;
,`>)a`$/F ! ! ! ! !
;HBA
gh")oE fb3-! ! ! ! !
;HL0
,`>)9FZ/3aF^`>Ec! ! !
;H<R
M `3,=/{/ !
;HL@
V&^HF&/FhPN,9`>)â^,9>)F ;H<K
13,N )c ! ! ! !
;H)9Fh ! !
;H$#~ST:)/F&! ! ! ! ! ! !
;H b/F ! ! ! ! ! ! !
;H z2x]H>`FGF ! ! ! ! ! !
;H$;t679/F ! ! ! ! !
;H
h`,=F^ ! ! ! ! ! ! ! !
;H 1
Vw'h^ ,= ! ! ! ! !
;H 1
gh!`/EFG)` ! ! ! ! !
;H 1<;
/F^`T/`>)`_! ! ! ! !
;H 1BA
Z`,=FGfb>FG,9FZ)>)z !
;H 1L0
M )fbX>`,=F^y ! ! !
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
#
;
0
@
@
U
U
$;#
$;=
$;
$;b0
$;R
$;U
Ab;
A@
(0@
(0U
$R=
$R
$Rb0
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
"""""""""""""""
169
(@#
(@H
(@;
(@/A
(@/A
(@0
(@K
$K#
$K
$KK
$U#
$U
$U;
$UK
#
#
#;
#A
#R
#b@
R
;H 1<R
\1^)
! ! !
;H 1L@
qr3_b>hFGHFI)F
;H 1<K
ST:)/F&! ! ! !
;H 1`FGF ! ! !
;H 1 679/F ! !
4
! !
!
! !
! !
! !
! !
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
""""""""""""""" #U
""""""""""""""" H$R
"""""""""""""""
""""""""""""""" ;
""""""""""""""" ;
""""""""""""""" K
Compound Types
A=
\)^^,9`:)F XM N/aCD3-ST)_ ! ! ! ! ! """""""""""""""
A=
\)^,`>)X3>W9)9F> ! ! ! ! ! """""""""""""""
A=
\)^,`>)F&/3aF^È&! ! ! ! ! ! """""""""""""""
A=<;
+sm>X``F^FJ ! ! ! ! ! ! """""""""""""""
A=BA
de9 )N/ )_93/F^F^9>)p ! ! """""""""""""""
A=L0
&F^> e3>W99>3-E>)^F ! ! ! ! """""""""""""""
A=<R
ST:)/F&! ! ! ! ! ! ! ! ! """""""""""""""
A=L@
b/F ! ! ! ! ! ! ! ! ! """""""""""""""
A=<K
2x]H>`FGF ! ! ! ! ! ! ! ! """""""""""""""
A=^`!)o1E>F ! ! ! ! ! ! """""""""""""""
A= 1
6h/{s_f/,9"/3a`$ms1
PN>^`c! """""""""""""""
A= 1<;
Z>),= s1
f/,9"/3 ^>),=s >PN>` """"""""""""""
A= 1BA
!,F^> 3>W93-E>O)F
! ! ! """""""""""""""
A= 1L0
67FG) )o1EF ! ! ! ! ! ! ! """""""""""""""
A= 1<R
9)Z:]=/`>)
Fh`FG)) 5 ! ! ! """""""""""""""
A= 1L@
67FG) PN>^`F ! ! ! ! ! ! """""""""""""""
A= 1<K
67FG) )o1EF&/F7>),=)oEF ! ! """""""""""""""
A= 1`FGF ! ! ! ! ! ! ! ! """""""""""""""
A= 1 ~679/F ! ! ! ! ! ! ! """""""""""""""
A=B;
6 N/F^FGFT! ! ! ! ! ! ! ! ! ! ! """""""""""""""
A=B;H
24`$9F^,NO)
! ! ! ! ! ! ! """""""""""""""
A=B;H
M4,=s9` /3f/)J ! ! ! ! ! """""""""""""""
A=B;H<;
+sm>&PN,`>)9F ! ! ! ! ! """""""""""""""
A=B;HBA
67FG)^^,9`:)F ! ! ! ! ! ! ! """""""""""""""
A=B;HL0
ST>P /,)h`FG)^,`>)Z ! ! ! ! ! """""""""""""""
A=B;H<R
&F^> e3>W99>3-`fb>^F^F_! ! ! ! """""""""""""""
A=B;HL@
+sm>&m>)^F ! ! ! ! ! """""""""""""""
A=B;H<K
ZFG)>3a)vEF ! ! ! ! ! ! ! """""""""""""""
A=B;HW9)9F ! ! ! ! ! ! """""""""""""""
A=B;H$# &/3c1,+sm>Fh! ! ! ! ! ! """""""""""""""
A=B;HwST:)/F&! ! ! ! ! ! ! ! ! """""""""""""""
231
;
;b0
;R
;R
;b@
;U
/A
/A
/Ab;
/A0
/AbK
/AbK
/AbU
0#
0H
00
0@
0K
0U
R#
R=
R=
RA
RR
RR
RU
@#
@
@;
@/A
@0
@@
@K
@U
KR
@
A=B;H ~ b/F ! !
!
!
!
!
!
!
!
"
"""""""""""""" KR
A=B;H$;~2x]H>`FGF !
!
!
!
!
!
!
!
"
"""""""""""""" Kb@
A=B;HA}679/F ! ! !
!
!
!
!
"
"""""""""""""" KU
A C++ Operators
291
Index
294
K
Chapter 1
Introduction
U
$#
1.1 Why learn programming?
!"$#&% '()*+,)-#./0*)12,)3+465798:
;<=
> <:<)?1"@'A+657/'A+012<./-#B/CD)E5CF&G;<//HI)J
3
#KILM://N:/' O'KLM7/:!-P!57Q.QB657RL=<:6ST/!01"'I
#USWV$% F&XNY9657B1"!Z1[+012=-#\5CF&]N&9657^/:!1['I
_`!a8M57<L@bcVd./-RefLYL=:g0h=10012<.ji/k&ldmCn/o
p 5&10.A.Q/!01qr,sgLtU1[M.Iuv5C+jw/']LM7<:ELM://N:/' 1[./-E)xuv5&Vy1[z10.1['E{
L=:g2MST:|Vh}EM~V ,9FG /']LM7<:?LM:</N
:/'E./-?;!CE+.Q@ 'Z)
657<'
9>|E9
9 PN 8uIs )F^
39$1F ).F^,)o19`$ `>)/9F.-F^>`>)
`$/3 /'KLMC<:^.<Q10". d )F&FG)>3 )>^+)X9`>,39 :/) 4 s$/< r _Ru`<:L=1"M g79>F^ ^ )^9 9/F_P !LYLtZ12,+012}LM:</N:/'j. X/F8)^ O) 9fb sE 'Z)^)^ s
`>G)/_E9X
e_) sE(fbT`$/F^F O)`>F^P)h`GE)C )+sm ,F^>3s1_)>7E> ePNZ:]8 =.F/F7>>F^))fbg gh 'KLMC<:W.<Q1[[~. F^`>))n/FG!FI)k)^F bR
p| M F c
R]"4g7F^
)
9/F P LM:</N:/'^'j1"7NU7N*NY<. n/, F ,F^3|) FG)^^,9`:) +/ `89,=)>$ ZF^ "N9,9/ Vh `$/'Z)X)D/F )^ O) 'X F^,=s FG 1,)_sE ,F^3as1
)>^F Zs
b,=FGP V `>8 ,=):
/ F FI)_P FG)^,`>)^F_)^ sE /,=)_)^`$/ `F^F^>3 s1 `89,=)>$ g7+`>8 ,=):T)F^P4FDFG),=9d
3 / )^9 )` `>FDP ) /-
de )^9F F^F^ k-/ PN.)
`>8 ,=): F. b_-` 1sE ` E_PN FG8'X
`$/)D` >fb> '&)afb>â)>3 F 1F 8>F^F^fb F^,)F&`$/-sE s)/9>3 )=,
.FG)> es eFG)> FG)G,`>){
FG)DE9"F^89_,F^ F uI,FG)X b!)>,F^ ` 1sE HFhV F/F&>>F^) )fb rPN":]=/89 E3F O99`$)a/FT/FT)1FDPN"FT'7 m.g7 P /`:)")9)Tb,
h$/3 )Fn>)Fk,F4sE:fbh)9)4b,8E)>)N sE>")!)^9Z`$)>^+PmE97'X
/FG 3a)^ '&^)D/F
gh>^TX_/ ^$/F^F PN 'Z) 9^/F \1Z89(b> )n 8PN7 ) FG
s /`Z`,=F^!) , ) =s9,)Z,)_O) )>"FZ 3>E:Z/FâsE3 )Z)^ O)
'7Z9N ) :]19N/8:]1)$ g79&,9F^)nFn)^ O)k('h/39$1F b,`/9)nsEX F^>,9Fn> >)k/Z"F^:,FnFG`)FG) b'X ),)n)n$FG) F^ s /F^`Z/9&F HFx2xfb_8>F^F
F^>^,FX`)^:]H)F m'7 `$/ `>93 )$^ / 1sE`/,F^.)X`$ sOsE,=)
.)hP PN, /3aF)^F^P /`>)^{
de )^98)o'7)>)^ `>),=^ x`>8 ,=):F 9fb8>fb,=) 3 )8'h F^`>9`>_/3 >9 3|g sE
_``>:) k'7_'X ,93>G )F 'X )|5:]8 P
_)_O)`FM
4,9^s Os 3M
) ]HE`:)Z_O) ) sEX>)38WFI)7_`9>`>)'X )
`89,=)>/F 39>
3 _/ _O)_)`>N/F!FG)^ ,F^ m>/3|m`7y 39/ s /F^F
k,=) uv5C+ )>'&)^ 3O'& 9/FD`9/3{ k>PN.`89,)^>F&'7>^ f//NOs )D'h/FDP)
9 v
E `FGF^ )D'Z)n39('X /`>)^, 11,.sE>F /3+)^XE>PN `/`,N)^Fr'&) ) s 9/3r
g79F!9E3 ) FG- ,` ?'Z)_1PNF PN 9:'t)^F s9,=) ) 1`>F^F P
)1[7N )^9>F^X)F>ng7Fk`F^F7P)
,9 F )..Ofb:Z_8`>9`:>)T:]=/89F h`,)::]8 F 1 3>7)+F^T`:^)/
)^)>^9F H7).39F^`(fb>Z)^ O)7F^XFG)O))
F P /F^yg7a`89,=)>+9/F )^^3,9FG /``>>)^35)F `F^F s1 ) H9 Ofb:8)
,=)!'7 mT b,? -) 8_O)_)`>N/M
BF7)> /3-)3 mb, FG) W93 > >)G)>FZ/3-/ 13FZP FG)^/ FGH.sEF 1s9,)ZFG)h b> ,9.sE>^Fh>h)9/ )^{
gh>^ F
a)9à)^ O)_9`|,FG)^)F )^9-FG),9)
h/3y)FF8)-F^$O` PN
) <:g.Q/=XLM:!1"'I!. Tde|$RAA 9)
=^` a/3 _)_O)`N a^ >F^
FG)s9F^3
)"PNO'&+`>N/-
:<?LM:!1['A
p 57|=h'A;/<:g. S|657GSQ:!' n
Mersenne’s Conjecture.
=h'A
;</:g.aSQ:
- ;!C SQ2:z−
I1657<:z=C'j{
;/<:
n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257
n < 257
>F^>9
`^>FGE33 '&) _/ P7)$/3a_)^9>_)`/F"O) )9) ) F^
b,_`$/fb>^P8)n'X7b,$/3)F Fk`>buv`:),=XsE`$Z'X39>HO'&{ Z_)
n=7
/3a?$RAA H)"`>buv`:),="'h/FZ/^$/3=fb>W93,)
n = 19
d ) ) +)9/ 1,3^3 b$F ,)^n)8:]1).:]Hm9>) >FG9QBF.FG)!`,3
sE fb>W93{zde* :)^)>_) k>,X9,=s9F^3 (@@ &qr>99^3 2 ,9> (fb3 )^ O)
F -_1,.sE>$ k,=) $Kb@R k/)> ,93=>3|b$F N): 231 − 1 = 2147483647
>FG9 EFG),9,F^ a`>fb3 $f1 s9(' 243,93 q{,`$/F"(fb3 )9) 267 − 1 =
F Xk1,+sm>
q{,9`/FxFÔ'7>3 Fx /F^F^+PN4N , 147573952589676412927
sE>^Fx/F^Z'X. fb>)3.) p u`<:W S 1 , g :xq{,`$/<F F 1P=3F ) '7 ")
'h b,'7,38 ]HE`:) )k3F :]H9 s ) T9^ZP `>) P 67
)&FG)n3`:)k'h
2 −1
P (fH )^ O)X+1,+sm>&FZ)&g Hs9,=)&)&,F^FX+`>fb:D3`>)&,)hfb)3
s1-qr,`$/FD )^9!F/!b$O$Tgh P /`:)FO)-P 67
>_/3-, HO'& PN /)>
2 −1
0.bF
de $U#; H ZFG 67"'h/FZF^`9>39,>3 )^8fb +`>)^,^") )^9!VX>`/ a)^9 _)`$/{\1`>)o ='XF^T))T'hF LCTa) 9`>)F^)
Pxqr Z,+sE:<F n6 T'7)h)
) s9N/` 1smb
3 E/3 'X),=)XF^$1
8F^ '7
3 m WFG)D'&^) 3O'&?8`$`,N))
s)/ 67
s_>E$O)3a+,)9`)+'X)
)o'7=k%&TW99/9/3
)!,9.sE>
2 −1
147573952589676412927
)hs /` 1sEb^3r ghZ'Z)h3O'&/)>e ,` ^Z)>>FG)+ `$/`>,9)+PN
)T9^13,`>)&Pr)o'7 1,.sE>F>
761838257287 x 193707721
-----------------------761838257287
6856544315583
2285514771861
5332867801009
5332867801009
5332867801009
1523676514574
761838257287
--------------------147573952589676412927
6 X9/38Ofb>3
)9) 67
7
/_ 1")hF^,) P{qr,`$/F
2 − 1 = 761838257287 193707721
sE>f/s9")_>fb:^1sE3= 67
FX) "1,+sm>T%& >`fb3 FG)93(fO)F
2 −1
PNh)F7/``89F^)k/3NO)>Z/3)^)>38)9)7"9/3
'7 b3
W993 )F^ P /`>)^F
>fb>G-\1,39$ PNh)"NFG)7)="bF
g 139 b, `/-FG)G)T `89,=)>&/:s9+/J-b,=D`89,=)>P
+E9,NZ
F

9)o1m"
Maple
ifactor(2^67-1);
/3'&)_F^Fh)9/a.F^`>93-:)Z) ,=)^9,)
(761838257287)(193707721)
g FG,9_O Z1,3=3?b$OF / rs/)&_)_)`N/a>39>3 )=.b$FT)
`,= 'X) aFG,9 ) )9). ,` F^F sN/) E9? '7__) )/1 sE,=) b,h
`,3 >) F^F )9/ F^`3 )139 ,F^ `89,=)> /3 ) )!/- g79F
F^>F83FG),=^s9 O) WFG)FG) /3y)^9 1 sm,) ) `>,F )
PDF8PNa67
3>fb)3 )-)9^s b,|_?>fb5PN> F^^^ PN+-E4, FG9,3M )<4O)> x)
FG)G 9/F7)="8E^)/) F^F^FhFG)^
q{F^F^+hF )9)46 Fx`$/`,NO)Fr'7:7:]H)^^!398`,) fb.) )1F
Tool skills.
"9/3A E> HE` H/3s s98fb>^ 13 >)/{O)>)`g:xTfb
) )F Vh
9fb3
) `89,=)>D93 8`89,=)>X/>s./ `/3
r6 BF&`$/`,NO)F
Maple
$/FI ^,)^? F^,=_)9) 67_'7,3 PNhF^^G PN /b9
,F^ )F^
9:'
)Fn.)">13,`ZF^Z,93=>3 b$ 38`$/`>,9){ ZF^PN, >'y)Fn$/38)"9:'
EF^FGs9)F 93+`9/F> CT+)79 9/
3 )F +" uv. b,.) 39Tn)9/+b, `,3
3 sEPN^
)^9")>X9/3-)&/FG SQ://!. b,-) 38 P b,a'h)h) b> ,=-'X )
)!3>fb>9)F> 9)^>fb>&b, 3 99('h/39HFhb,- ,FG)X/` 1,^!/3-_/)/)&$/FI)
F^Ds /F^` H('X>39TP `>8 ,=):Fh/3a9`$O)/F
q{F^F^ )v'7+F7)9) ){F HF7/D'7,3 )h9fb"m3 6 D) P `>)
Problem skills.
D
6
/
FG 'h/FD813-_)_O)`N_'& H>' 8)XP )G- `,3 ,F^!)
267 − 1
Ffb!`/`,N)^Fxg79FZF7)T$/F^ 'X
G9 +9>33a)=Tb$F>
2 fb>?O'h3 1F E`89,=)>FD93 9`)/FD )T>fb:^1)= /`>)^
F_$/FGsE`$,9FG )F
F
F^_/X,+sE:
sy)1397
F
FI)/393F k,=)P /`:)
267 − 1
N^ 1,.sE>]
F 1000 F!`>9FG3>3 N^+)39$C
x `>,9 P b$F x)!)TsE 2000
2
2
FZFG)r.fb>â3 `,)h9^s PN&'Xà8 `)DF^,=)FZ HO'&{7gh"s9>
9 v
E $;
P4P /`>)9 1,.sE>F&FX).FG)X)&s9 PNT'&9` FG)Xm9 +,9FI)
/`>)^, E_)9) `)8F^,=) '& >fb: sE
PN,93{ g79
$/FG5F )9)+_/
`>^)FGHFI)Fk)9) D,F^W ) P{F^`>,^D):>) s / H+ X9,= s /F^3_)
/`>)^`$/{8EFGF^s9)o P P /`>)^ NO!,9.sE>^F g7>PN )"'7^FG)XF^`>98PNZ)
>)o'7 b3+'73.'7,93+sE )^ O)4)^9 Is /3 ,=HF 3F^`(fb> WFG)4O'5)^"P /`:)4N ,9.sE>^F
`)b
gh>^ O _/)>Xs9Fh)9) .FDP TPp8F^,=) /FX)>a'7>^ `89,=)>"39$1F 679_s /` ?) >F^ '78FI) `$/) `9`>)> )8 ]HE)F
PND'X`-) 1,.sE> n
FD 1,+sm>$&? 39M
)D>fb H('['X>)^9:D)>
n
2 −1
T=W9) 8_/F^,` >FG9 F deP{b,9N/_)._ bT `)^s9,=)_> =b,
F^,3+9) s ,=+XP /FI)>4`89,=)> '&) ) N)>FG)xfb>F^ P
bs9,) FI),3._O){ 2xfb
Maple
).`$/F^+P4s9>FXPNT'X` `>8 ,=):FD`$/ $/ -`)Gs9,=) ) e"/`>),9/W9h
3
)+F^,=)
mb,?)o19`/->3?)9fb8_3> ,3>FG)/399
Pn)^9.s9p 3>
) 19(' 57u )-,F^8)`89,=)>$deP7b, 'h/) )-sE` / > +FG`)FG) b, +,FG) ` 1, /3 _/)/ _PN,3 H('X>39 sE,=)T) s9>FXb,?'X sE
3$/9 '&){ g79F P /`>)7'h/F )^,T1,3=3
b$Fh/ =93 )7F FG)^ )G,, `89,=)>F 9fb
)Zb:)&3) F^fb )^>FG)^+s9F s_)F^fb>F
q{F^F^.)= F 9 )^ O) 6 33.)xfbk)^DF^ ('h/39HF /s9> Programming Skills.
FGE>`W9` HO'&3.`$/ sE+)^,^9>3?)
s9> FGE>`W9` `89,=)>DF g79)T'h$ ).FG))^+P4)+^) ``>!>FG9+FX9/F /3=f//`3 ,) P $.d )T),=3 ,=)
/3
:F^
BF"FG)T3 9)
)9) >FG9 9/3?_/3 Wfb+FG) b>F n = 67
n = 257
$35)?
1,.sE><F Z5)a)>9/
3 >FG9-9/3 GPN^)^) ?)a ]HE)F
/3
n = 61, 89
107
VXF PX\1>).sE> ##K n'7 HO'JAbR >F^>F 4)^9
N^FG) PZ'X`59/F /
:]1E)"P
k,=)!3M
) sE>fb )9)T)F".'h/F PN,93?'&) l e)^9 n = 43, 112, 609
F^Pr/F
M ^s F 1``>,^ )"39/PNTP /:&& F^`)FG)Z^"P)>a)Z/FG_)
F^fb >fb 'X ) `>8 ,=): /3 FI)/393 F^P)o'h8)!9/3{_de 3>!)^-)^)` ) b,-3 .<Q1[[~. PN&)",=) `$/`>,9)F /3 LM:<&;! <' .<Q1[[~. )^8,93>^FG)/3-/3
:]1)^/`>)D)^9.FGE`>)^F Px) s9J)9)T`$/ -` /rsE F^fb3 s1 8`>8 ,=):$ k,=)
)"
3 =b,-3 LM:<N:/'j'^1[7N ./Q10". ) /`>),9/3 )
g ``,3+)F F^`:) >) ,9F sE+9FI) !PN"_/
The art of computer programming.
E9e`,938) /,=)FTPk)FTsE1 Y E)+`F^F P4'& )9^/FT9/F"FG
fb>^ 9 e,)^)ON/ /FGm`>)F./F"'7e8? fb8>)3 )v'7P7)^9> sEPN^ .PN,9 /3
F)^F^P /`>)^{ ?!`,3/33a_)_O)`$/sE$,)o /3a+sE1FI)$kde9 'h &/)> >fb>G /F^FG9)D/> PNF&)&$/FG)Z.)^)Xs )Z b /-^)^FG)$
gh8 /3=fb1`) P )F fH:' /F ST9/3 2h
D,=){a%&8F!)
/,=) Ph1,)/b/3 F^ FG>F PHF^>fb.sE HF >))^3 p 57R:gf S 'KLMC<:
_:</N:/'^'j1"7N \)^)D'X )+j4,kd $URK )=7P )hF^>fb fb,F 9,=s F^3
www.mersenne.org
A
s1!O'!xSX/P)F PHjn,94d j|``,NO)kF^9`> ##b0 /3 )k9N93 ^$/FGk39) PHj4,
j F #=(0^ )&FG9,3 sE /333a)9) X1,=)-'h/F7sE- $U;K /3a-Fh'7>s /
http:
48) /FG) 89` )
)^F
//www-cs-faculty.stanford.edu/~knuth/taocp.html
)TEF^F^s9 )9)hj4,FkjTdn/3jTdId '&m)hsE 'Z)^)â/H! :
qr:) X1,=)-9fbT) W99/{F>^K

1,) Pc) sm+P jn,9Xd p 57LM:<
/<.<.K SfLM:<qL7:!1[7NPLM:</N:/'j. ST:]^)1"N1[2+ /']LM7<:P10.X!.qL=<1q+"~V
+60:/J<01[F&<-`WMZV ;<C.Q^1[W ;< /'j1q+[ZV )A.Q<1q<M01 ,+[ZVI:<<{
u:/)1"7N-;<7j+~.QH;<7.T1[? ;<@ J!.<657<012,sgL7<:!12< '^ !5 Z10
'KL=./1"7NBL=Q!0:!V :B'jC./12T
&
wX v
(0
1.2 How to run a program
# _`:!1[.x657<V <C.<A.1['KLtZV <L=/) 657/1[: /Vh!. )
./2:<)Guv57<|u`W.qL7dc3657/'(1[ M:/<!5 \<F&<:
)1 )3./*//)1"I'A+T1[7N3657.T\1q)1q6.Rh)J<:g.<2)G657/1[:
uU7NtNYQ
:g p u1[=- p 57K# =Q/<M6. X;!:/QY) i/kmCndl!o
p 5&10..Q/!01q ,sgLtU1[M.9uv5C+10:<,+[ZVB'I,M. [u:!10cLM:</N
:/' g- )]57u
V$x<Y;gDz657j'KLMC<:3j:!h 1[2 :z65&1[./-uR)J!.Q<:!1 ;<B657K1"7N:<)1q<M6.
1"=F&~Fd)1[ 6579LM:<Q/!.<. \657 3) - 657 `89: -657 `89,=)> 106.Q! S/-v)
657 E>O)"FIHFG)> ']LM7<:!-v/']LM10D<:B)E<L7<:/+01[7N3./Vd./<'(NY!657<:
ST:!' 657 9N)^PN uv5&12!5Vh:<Ku:!1001[7NKLM:</N:'E.T
1.2.1 Editor
?^)"TtF4)kFG 3lm>^)nPt'Z)9 T>)^)^>$4CD9&`8EF^F4T):]H) )9)nF eEPN, J
E
h/PN,9rF^ ,`.P4`9`>)>F> ZF^,9/ 9)>. `>>^)/ `>fb)F
('F^,` 8D)^:]H)nF FG)^,`>)^,^31/3+)h9,=ÊF^7P9)h) ]H)nFx) )G/FGE^)4PN_O){
O)Z9/F sE>aF/3 F^+P &9Fk)^+sE)^>)G)>F7/3
9^/Fk,=)7'X
'&)^ / )^9:&Fn/)>k/FGE>`>)k)9)49/F4)"sEh) b )!/``,) xV/z Fn) sE
I$3 8s- `>8 ,=): / )9)&)X+,FG)XsE f//NOs )8)^9!`89,)^>X->`>)^^9`
PN-_de )8PN,=),= '78) sm8s9+)a/ 39`>)).) ) )8`89,=)> s9,=) O'h3 1F )Z` 'h Fn) ,F^X b>sEb3
/3F^89 )oE&)k{xVX )1[: F
/ `$)_/ )9)Z/('XF b,
) 3FI $ 13P 1/3a`:)^`$/8FG)^TF^,` )o1E>3 e ) ]H)$ g7Z,F^ZP 3)^FnF4) FG)^^`>)3 )" Pm`>,^F^x) FG
FG)=:]=FG)"^_/)` :]=`:9)^F :fb>)^)^>Fn&`>8EF^3 ,9FG!3)^FnF^,`_/z
F :) 1.2.2 Compiler
a 1a9^/ $f//s9 ) ) `89,)^> >`>)^`+PN F",F^,9/ 9)!,r
g 9 'J!5&1[]U7N*NY `89,=)> `/,3>FG)93 3`>)^F fb>G
)fb&/31,)^
7
3lm:)ZPJ9),=/EN9,9/>F
? )+)^9T/F a_/`DN/,9/TFZ+fHNs9T)>9)^fb =F^`T)9)7'7,3
,-) s9^$ ) 9^/ ) -1,.sE>
PT9^)fb FG)^,`>)^F8)9))
`89,=)> `/|,3>FG)93{?ghF F b)9 b,+P3 ) `(fb>+PN 3> s
).9:Z'&9` ,F^`>F7) (fb 3>h) >)Z)^ b,=Z9N`
(fb> _` N9,9/>FTf/^ `F^3>s9asE>)o'7> 39 lm>>)"`89,)^>F+g7 O)
F x 3> )^ ,F^
/i'& )^) PN+FGE`>W9``89,)^> V a3lm>^).` 9,=)> m W9^FG) FX)
)G/F^N)^!) 9^/ P )._/` N/,9/ P4V ))
$R
// Program: power8.C
// Raise a number to the power eight.
MZ ÿÿ@
U‰å]é×È
U‰åƒì ƒäð¸
‰Eô‹Eôè'Ë è‚×ÇD
0C Ç $ÀeC è.ë
Eü‰D$ Ç $ eC èKàD$ 0
Ç $ÀeC è ë
Eø‰D$ Ç $ eC è$à ÇD$ <0
Ç $ÀeC èàê ‰Â‹Eü‰D$ ‰ $è
ÇD$ H0C
‰ $è¿ê ‰Â‹Eø‰D$ ‰ $èn
ÇD$ N0C
‰ $è ê ‰Â‹EøEü‰D$ ‰ $èJ
ÇD$ S0C
‰ $èzê ÇD$ óB ‰ $èZúÃ U‰åƒì } ÿÿ
u ƒ} u Ç $ `C èÑ
} ÿÿ u ƒ}
u Ç $ `C èF
ÉÃU‰åƒì ÇD$ ÿÿ Ç $
è¨ÿÿÿÉÃU‰åƒì ÇD$ ÿÿ Ç $
ÿÿÉÃ
U‰åSƒì ‹] ‹M Ç
˚JC ‹@ôŠ p9C ˆ h
‹ R S RèÖ Æƒ
ƒÄ ‹]ü‰ì]Ã ´&
¼'
U‰åSƒì ‹] ‹M Ç
˚JC ‹ @ôŠ p9C ˆ h
‹ P S RèÏÕ Æƒ
ƒÄ ‹]ü‰ì]Ã ´&
¼'
U‰åƒì ‹U Ç ˚JC
RèûY ƒÄ ‰ì]Ã t& U‰åƒì ‹U Ç ˚JC RèÛY
ƒÄ ‰ì]Ã t& U‰åSƒì ‹] Ç
˚JC SèºY ƒÄ ‰] ‹]ü‰ì]éyX
´&
U‰å‹E ƒÀ ]Ã t& U‰åWVSƒì<ŠM ˆMÇÇEàp @
ÇEäP C Eô‰EèÇEì @ ‰eð UÈƒì RèWÉ
ƒÄ }Ç •Â âÿ
‰UÀÿMÀ eÀ
}Ç „
#include <iostream>
int main()
{
// input
std::cout << "Compute a^8 for a =? ";
int a;
std::cin >> a;
// computation
int b = a * a;
b = b * b;
// Output b * b, i.e., a^8
std::cout << a << "^8 = "
<3 L=:g01[7N 7`$ysmfb>^ `>,9.sE>^F^ P&)
_/`_N/,9/F+PXV 93 tFG,s9FG)/)/ 3l >)$5VXF^ nEG) `/ sE
3"'X )a.3>)/!
3 19('X3"P )^9TE`>,9N)^F P ) fb fb3a`89,=)>F> k,=)Z)^9F
)o1EZP H('X>397F4) >/ '7^)^'X ) /` 1, /Fn)4F4)^3 )"Zfb>^ FGE>`W9`
`89,=)>$ VXF F^5/F )F `>8 ,=): F >9N/`>3 s /)>. 4_/uI! O^)F PhF^,` `89,=)> eFGE`>W9` 1O'&3DsE`D'7^)>F^F&93 9$fb ) sm :s , )7P FG`>)`{
g a3,`8)F",93FGs9 )9)"P `89,=)>F!/3 /F /3 )-/O'
,F) 'Z) 9^/F8FGF_)fb N/,9/ è :fbD + N9,9/>F
9fbXsE_3>fbm3{ng7F^T^DFG)/393 >3N/,9/Fn)^ O) PNQ 13_PE`8F^
sE>)o'7>
9)^,/ /, F /3_/`hN/,9/xde3
3 )^9&,9FGXPm)&'73 G`8 F^ F{uv,FG) W >3 sE`/,F^Z)>^&^7)o'7"` `>)Tb/F CT )Zh 9
3 '7 '7,3 b
).'& )D9^/F +/, X)9)7FZ/F7`F^T)+ O),=/EN/,9/T/F7EF^FGs94CD)
)>"9/3 E'7+9$fb.) _ b )+`>8 ,=):FT,93>FI)/3?) /8/, ./F
'7[ )F )F _F7s1f1,F^8+,` $F^>hPE)D/ N/,9/XF7`F^X)+_/`99
N/,9/
O)n3F ) $/ I)T_ b )7`89,=)>Fx,3>FG)93.) /ZN/,9/ de8)&>9
3 1/+/~9/Fn) sE&)^/F^N)>3 )!_/`7N/,9/xg7&1`>F^FkPm)^9F
)^9FGN)FX`$/3 /']LM10U+012 ZO' b,-'X{s s9/F I> Fh) sE>W)DP
)F '&9X9^/!/, X``>) -de^39:h)^ 3.)T)G/F^N)^_d4FI)m9fbD)
H(' /9)F^T`89,=)> eFGE`>W9`X3>)/F 13M )7do Z)$ deP{b, '7,3_9$fbX)^.)G/F^N)^
)Z/~b,=FGP g79
b> F 4,h) F^,=EF^38)^")^9FGN)h)4b,^F^P 4deFG)$3 >)Z /c3+)hPNZb,{k\1,9`- /QF7^PN>^>3a) F& /']LM10D<: 9 )h)G/F^N)^F7
fb- /.N/,9/ 9) .Qh:<//Q)J r)_ / _/`99
N/,9/ ) ,sJ/<C2Y;g 7\1 ,=
PND/,FG)^){
de FG,9_O^ g7 s9 sm>W)K
P 9 e>fbU ./9 N9,9/>F8F8)9)8)^9:
7
8( %9#"5
/7(J
.+(*0,2
!
(7
9"K.*L1.*H7
($
#"
%$&'()*'(
%:$&'(8;<>=?"@.6#
M( 9-BI.*
+( #, -.*
A$&
6N7O"K(P A
0/
1*
%*CB>#EDF2./
% 9 H7
(:
2
$3$&
4
%*.BGH.I2/@7
#"5
% %(
Q :*
2RS
46 %(
9*
"K*TU 2/
(
J
&
wX v
(@
s9FG)^/`>).Pi)`$O Os )>F"PhFGE`>W9``89,)^>F Mx/F"'&)G) -9 e>fb
N/,9/D`$ sED,/ 139F7Pr`89,=)>F H/FZ9+/Fh `89: PNZ)^9TN9,9/&F
fNs9D)^9T ^)`,N7`89,=)>$
1.2.3 Computer
ed Pkb, ^ ) )>^FG)3??'&^)_`89>^F E)"F")"`F^F^^?)â,93>^FG)/3 ?39 )/kO'Qa`89,)^>!'7 1F k,=) )>
F^ s F^` ` F!sE3 )3F^ P
FG)k`89,)^>F )9) :< 8E^)) ) ,3>FG)93{4g79>F^X`9F4PN})^9 F& B<C{
'A= :<!5&1[/!0h:< /3 )> 8E^)/) F^` /FG) / N9,9/>F
T)/3_) )^9Tfb Z,_/ `)`:),=
VX!`89,=)> '&)!fb Z,_/ O`9 )`>),=n9/Fx :/)J' J/!.<.9'I/'I:!V ZV .F^89 _/ ^J
/3 //M0:/+fLM:<
/<.<./1[7Nh=10 o67M .F^89 1`>F^F^, g79h_/.G FG)>F4)7/ ) sE7, s9,=) F^T39)X)9) )h/ 1,^F
/FD=9,=) E/3 39) )9)D)^9 / 13,`F /FD,=)^9,=)$ g79 `F^FG"FX) Is Pk) `89,)^> T)":]=`>,)^F!).9^/ {$/)9)") `$^F!,=) ) F^ ,`8P
FG)^^,9`:)F 9^F^`> sE3s
)^9T/c_/`99D/, 4,_`$/) 8PE)D`89,=)><BFk_^ /F7!9 Ô' P FG'&)`F Main memory.
$/`5Ph) sE )^9: +l SD,=-9^/ :]H`,=)4FG'X)^` F+^ 9E3{
V )_/5) n)a^ `)) )a`,=^)EF^)^F P /ZFG'X )`9>F 3>W99>F )
LM:</N
:/' .<2+ g79T/JFG)O)"`>8 >)839:)>F '& O)Z9E>9Fh:]H) 6 9`>> ),9/ '7
/F^ `F^3>+,F^>+=9,=)+/3 / ,=)^9,)+F. G).Ph)/ FI)) >fb ), )8`^>FGE3 GFG'X )`9>F -)TsE )8,F^>! F s9/
)3
E>
\1`h13>+`89,=)>F47`$O Os hP 9 9DFG>fb>/9FG'&)`
F e; F^$Y
x)n)^97F
) `F^>`,=)fb FG'&)`FT./`>`3 ,=E3 )^ 'I<'A:!V /<[~. g7 EF^)9F
PD/hFI'X)`F |)`>73>W99
) M<M P&)`>0 75 s9FG)^/`>)+)>F 4)
FG'&)`F `$/>3 ;!106. $/` P )^9>i`$ s9 P FG)a8P )8,+sE:F
de
{0, 1}
)FhF^F^ 9b,-`$/-)>^:)&)!`>))XPx+^_`> FX+s99G_1,+sm>Z'&)
PN
:]=/89 =; .39)F ?!/F^ F_)9)h'7"9fb s9)h_/`99 HX 32 { ;!10./Vd./<' 32
2n` ^+`mF , 1, 83)W938s1_ )F Y)Y):<!./. n,_`$
) Pr)^9T393=>F^F
F^89/Fh)^9TEF^ )
P ) ^`>{) FG)7PxE^`>F
g +1 _,=s9)7f//,F =h)
9 s9)^F7'&)
FGE`>W9`"G8` 1) È9/F7).sE
J/!.<.Q,) )=, )F&393=>F^Fngh _Px sE)ZQ'&)a; W9>F7)9)Zb,`$/)>m)
(fbT)^ ^8`{1,.sE>"(@1
gh )^> :/)J' J/!.<. ^PN>F )a 9HFG`$/ E>^)o?P )8`89,=)>! F"G ))) ) bF8) ``F^F
?`>\
) GOfb
) )F s )F ++F )-F/ PN
/h`/F X
^)`,N m )"3F 3>E3 ) 393=>F^F"P4).è" b,?) )>FTP
)Z^sE)n )4sE`F4`n)^ O)n93~/``>F^F `$9)4sEZ) bPN /)>3r
d ) Fk9) 9>`F^FO^) 3F^`,F^F )^9Z HF^`$/ /Fks1 'X`/3 /``>F^F7Fk/ >t
3 )
8EG)/)kE)7>^!F )9)h/3J/``FGF&P>F&,FZP^J) 1 OsE,=)h'&>T) FG)^
$K
.3 O) )ca^39:Z) /``>F^F&)h `)b
4,-`$/a) P )!`>8 ,=):!BF71`>F^F^T/F&+sE$] )9)&FXs9 )8b/3/3
Processor.
) :]H`,=)8) _` +N/,9/ 9FI)^,`>)F P7 / 3>g789^1`FGF^
9/F+F^_^ `>F.P&)^F ('Xk`/3 FG)>F /3|)+`$|)^/F^PN:839)-P )
`89,=)>! FZ_ >^)8)^Fh>FG)>^F 93af1` fb>F1hg7"FG)>&`)>)FDO /F^
^)"Pk) /pFG))/ FG) 8E^)/) 9) 1`>F^F^!`$/ E>^PN _W=]=3 F^:) P
F^89 E:)
F e bn/33Z F^,=s)^/`>)^ZFG)> `)>)F 3`>)^ `^FIE399h)
)&_` N9b,97FG)G,`>)Fxg7F4Fn'&9:h)^9ZPN,9`:)9/)o P )Z'X `FkPQ)D3{n2xfb
fb>^`89`$)38/3
,F^>PN,9m/Fk`$
sEX9,=) ):)>
P +FG89DF^>)ZP _/`DN/,9/TFG)^,`>)^F
VcF^_FG)^,`>)^ /`>)F+ b_ _)_)^`$/ PN,`>)
fb )_`,=^) FG)O) f/35FG)^^,9`:)|>9:)F >' /35'7 3>W935:]H) / FG)O)yg79F
89>F")9) / FG 1,` P&FG)^^,9`:)F 4/35 ^)`>,9 )
'&8/ 9/F+
'7 3>W93 sE> f1 =3>E3+)"9 )N/99^/cFG))^
1.2.4 Operating system
_
9fb FG|)9). 3>.)-'Z)_a/ 93 ,| ) xb, WFG)+9fb_)^ FG)^)+/
3) /)o1E7.) / O) `$1) `89: )D)^9FGN)k)^9 / )^D_/`99
N/,9/ /3 W99/ )=)&`>8 ,=):n)":]=>`,=)X)$ de_/=)F GFG)^)^ G`/ T/3
I) b,?^ )^9+`89,=)>! F <L=<:/+01"7NO.Vd.</' oC"\Y 8 F^ s /F^`.)^ O)
b, _ )!:fb E>`>fb )!F 9^/-.M 9,N"E>O)_FGHFG)^F!O@ 1")Ju. =1Us 1"=Js /3 Jje8 =D:]8 H'X>)>Zb, FG)^)Z)!3)^Zsa`` 1 -F^ ` 9&'&9:)>&b,
)o1E `_93 PN )^9F"F^:'X> {) m>)9
FG1FG) _ bF!FG,^ )9) ) 3)^
/ F+b/335) ) _/|^ 4/3|)^ O) )
9^1`FGF^8FG)^)F :]=`>,)^?)$
\1 E'X b, FG)8b,= '&^)^) / )E>)aFG1FG) 1`)F FI `
PNZ)& )"93 3F /3-/F^FG1`NO)Fh)7'X)_)TW "9/Xb,-9fb (fH39>3r
V `>8 ,=):x'X),=) E>)TFGHFI)~F bh `n'&),=) )^F /3 FG) `89,=)>^F
b,`/
s9,=_`X'&)
! 9FI)/>3
m>)9 FIHFG)>-nd )7F 8E^))4) ,3>FG)/3 ),
b)^ O)4)^9hE>)TFGHFI)}F4)n:]H)G`$s9!)3+)")h`89,=)> rb,8`$/ ) b
b,= I3('XFhMk6 /3
FG)E)h,93>Xqr1,H]E
1.2.5 Platform
g79 `89,=)> )F m>)9 FGHFI) 935)^9 `89> ^ )>)^9: PN:^3y) /F+)
LtU+ SQ:!' .'&` b, Ok'&^)7b, 9^/F gh 3)^ F 9) ^)xPH)k9NO)PN F^` )h31FZ)Z 9,`T) sm9$f1hP )T9^/-
de/839/H'73)>ZF !38PN4b, ) H(')^979N)PN^ 'X b,Oh'Z)9
/F7a+ >fbE/ /, h`$/r)9)7) /aF7) 3b)T)
9N)PN^ FGE>`W9` /FGE`:)F&)8)"`89>$ Vw)o19`$/rF^,` 9NO)PN FIE`W9` /FGm`>)DFh)
&
wX v
$U
F^ "P .>^8`> He x)T1,+sm>hPrs9)F )9)h`$ sE _/9,N)3 )>)>$ng79F F
FG) 8; )F^T39HF Hs9,=)7PNhF^X`89,=)>F ) F7RA /3
PNhfb:^) fbZ`89,=)>^F
e bT)^9:O!,F^3F^_O^)h`$3F FJ 9 )Z`/sE" ,`>F^Fh)^ -; 1
9> b,* ,F^ _1 _/`99 >)3yPN$O),=F
P") 9^/
N/,9/ )PN eFGE` W `hsE9fHh)ksET)DF^,)
a
>fb>m9^/
N/,9/F 9fb&F^,`(
' e>fbPN$)^,^Fn)"P /`)) )^97)^9FGN)+) <12/M _/`99
N/,9/
4,Zb/{F^,3a'hHF7sE )^ '&^) LtD+ ST:!'E{ 1[)J2L=/)J<M `13 F^9`>!)>^'&F^ = )
_8sEDfb>^
3 `,) )+>)7b,=h/ ) ,/)>7`89,=)> H:fbaPEb,a9fb
"`>8 >4PNk)9)k`89,=)>$xg7Fk8 F )9)k`>^)8PN$)^,^F F^,38sE&$fb39>3 >fb
),)h)7F^ /3=f)/>,9F7)+,F^")J-.FGE`>W9`T9NO)PN-
1.2.6 Details
j4 Z,_/MBF43$!Pr!` ^+PN )X/ ) )D3 O)"F^>F s1f1,F
P )39$CBF8E)PDfH>' 7s9,))^9->FG)`89,=)>F b X=/3E,F^QBFE; 33M )
'7 )9)&'hb de-){; 9PND:]=/89 H)"^_PNX) 9^/c'h/FX. ,`-)E 3`,=93aP^J) =9,)Z/3 ,)G ,=)Z3>fH`> 9/3Pc)!_/
>^b
VX )>>FG)&PN),=kP)nfb h,_/ `)>`>),=4Fr)^ O) ) /O'&F{F^Pe13PH9
/F> ghF^!TE9,NZ/ )"3F^9:FZP `89,)^> fH^,9FGF PNX:]=/89
gh
fb Z,_/ `)`>)^,^
'&)| )F )v'7 >fbF PX>^@
e_ >^ /3
1`>F^Fâ^FG):F
Fa/*3$ 3 13 X/3 '7 89` ) '7 H ,93>a)^9F
13m)^^,9,=)7)"`>,^F^
gh /)o
1 1F&"`89`$O)3{
13> `89,=)>^FX/FG89$fb J!57 m`$
sE99 )^+)T_/_G s9,=)Z/('X +,` P /FI)>&/``FGFh)+^ `R
F e) )
`hPE 7NsE)Z/38:]HmF^fb&3FGh
: ghZ3$DFn)9)nP ,) 3339)
FG)^3a)^9 `$/`")^ FGE>3a,=a)T9^/-
9-`$/` F`>G)/5 113 ) 7)_ bF )-PN-P ?> ^
3 8`>,9 ) b,_`$/_ >n )XP /`>) )^ O) /``>F^F ) ) 39)"Fn39:E3)kP
'X>^Z)>8XFG)^3{ de_P /`>) ) >)n)&PN, m>PN_/`7sE>W)n)9) `$` `/lm> ) /: 9/F )T_ bkF^,= )9)x39)DO7/``>F^F^3 + J!57!{,/57<:/<M 'hbxSD
)F ('7>fb> 1,FxFG `89,)^> FIE`W9
` HO'&37sE,=) )^9 `/`9 1O'&3 '7
'7>!9/ 8)^^1 )fb3asa,F^ 9 e>fb{/9 /, F7q{,9` 1 ='7
`$/ P) 9^ )FTF^F^,+/3 eFG,9`>`F^F^PN, J
X ) ,)^_)`"`$/` _/9/>)
sE9 lm:3{4gh>ZFn/F^"T)^9>>)`/=13=PNkF^ `$3 J!57!{,&;gZ1[F
1q7. / )F 8'&9` _ )z:]1 `) 3F ) 19(' )^9Z O/>):FkPm)X`/`9/nVX)F
'X` ^8 8`>) ,93> )^9F!13 Ê
e `>>^)/ F^F^g ! `) PN./ ``>^>)
`$/`!FG de ^$/ ePN9`)F r'7_/F^ s9F^>^fb)89 )9) )39) )^ sm89^ `FGF^3y_) NO) W) )^ )^9
`89,=)>! F._/ ^b CDE:) FGHFI)F.`$/
/,=)_)^`$/ 3$/ '&)a)F&sa`$/
:]1)38) _/->^_)8) 93 3F m
%&('7>fb> ) ./u!LYLM1[7N )^ O)X) bFX9N`!'X 93 3F 39) )8sm./``FGF^3 O!)^9F PN>>3?)_)+_-^`,=FTF^>fb>+E>PN^_/` E9/)o m+,` '7FG+)9/ E
#
`$/` ,F8de?)^9F"F^ ),9)E) F P) ,F^F^F")^ )8/,=)_)^` `9/F^F
(fH33as) E:) FGHFG)^F /3a)!/>7FX` 3)^8`!,= 'X )
1[&LMC47 LMCR <12<M /F
2 fb>+'&9>.'7k:]H)>93.)kfb Z,_/ O`9 )`>),=n)X`,3kFG>fb>/=$b:F P= ^ E)>+^ `89,=)>^FX)9) 39M
)DW9) {
FI)")Os m)>+ L7:/+" !9'KLMC<:g.
'X )T)9/-"`F^FG$ ? ) 8`>)&/F7PNDF^,` `89,=)>ZFZ+)/F ) 3lm>^)7PQ"PN7)Dfb Z>,9_9_O`9 )`>),=/ng +) b PN,m/3 f//) P{)D /F^ /F 9fbT)^+sET3`8mF^3__/1,9/ ).3>E3)
^)F m$/` Px'X` F&)-, s1- Px) 1`>F^F^F>Dde _ `$/F^F m)F&FD)TO)
/{.FG)^/)^PN^'h3
)/F 93 FGE`>N/ 3 N/,9/F 9fb")+sE ,9FG3{
V `>)hF^,``FGF^PN, / )>9) fbZ) /9`89,)^>F ^ >)o'7 1F P FG e`F^FG
`89,=)>F> 4,`$->fb `$/E)FZ.`89,)^>XO`9 )`>),=/ 9/ 1)>!
*M0h'
/']LM7<:g. )9)hXs /F^>3
`89>) 3l >) HF^`$/ ` Fk)9/)Dfb h, _/_`)>`>),= Z/ 1,9/),Q`89,=)>F `/9) smXs , ) b>) 1s9,=)h/Fh!):)`$/
13 1,9/)^,9 `89,=)>^F+ ]=FG) k/3|/)^9F! //3= sE 39:fbE3 )^9F
F^9 3EP `89,=))^{
Chapter 2
Foundations
H
& ]av 2.1 A first C++ program
p 57E;/+./1q3
hST:G657z'A=1~LM7U+012 SK:<+Z100VA10.z6573'=1LM7{
U+012 SRuR:)+.TR#US\V$ ,/M0:<657\'I=1"7N SPu`:/)+.-fVh
/M0:<657PL=<Lt Kuv57j'jC.<C.QK657Bu`:/)+.T
_v5&10Z1L3 W
> 12<Q- u Eb9h10U)I =1[F&/:g.Q p 5C+
>GQ!./ % +[ 9LC:g p uRE>3V. +<:?i/k&lm o
p 5&10.K.Q/!01q|LM:<!.Q<M6.j :g.<c/']Lt ! `K LM:</N
:/' )?1[M0:/Q) !.G657
) F^_)`/ <:!'E.X/!.<.T:!VAjh)J<:g.<2)O+" 106.L7:g6.T
FG1)`>)`$
%&> F ,=+WFG)_6hY Y - d )8/F 1F PN 1,.sE>
/F 9,=) /3 ,=)^9,)^F )^F
a
) EO'7: 8 deP7b,59fb >fb> F^> 6ZY Y /isE>PN k:fb|)F.F^^)+
a
)X -F^`$G rFG` )T`)/FT)DP4FG)^9 HFG1+sEFD/3 '7^39FD)9) O
)DPN,3 9),= N/,9/XCT ) )>D9/3 )F&FX13 9:'XF Z/FDF^^)TFD)DF )F7/J/^$/3=_`)/FZ__8EG)/) PN$),=FZP ) 6ZY Y N/,9/nCT` '7
9fbD9Z)=,_)^9> )FkFG`>)
1)^9Fn/ /3_>fb
)> 1s9>n/F
'7M
)h _F^`$^/H
Á
Ã
// Program : power8 .C
// Raise a number to the eighth power .
# include < iostream >
Â
Á
Ã
Á©Â
int main ()
{
// input
std :: cout < < " Compute a ^8 for a =? ";
int a;
std :: cin > > a;
// computation
int b = a * a ; // b = a ^2
b = b * b;
// b = a ^4
// output b * b , i .e . , a ^8
std :: cout < < a < < " ^8 = " < 8 ,=):&93 )a,a)!:]H`,=)s9TW9T9^
3 ,`3 s1 ) `89> Eb, W993 )^9 PN('X
+ ) FG)939^3 ,=)^9,)_gn19`$
) FG)/393a,=)^9,=)hFh)^)` 3 ) F^D'&39('*b,=&`89,)^>7F^`>>{
Compute a ^8 for a =?
`$/ ('~): /?)>> rB= r,9FG
) b
>1sEbO3{ VXP)^>">F^F^
) ,)G ,=)hb,=XFG`>a^$/3F&FZPNO'&F
4,
Compute a ^8 for a =? 2
2^8 = 256.
kPNT3F^`,F^FG!)X/
_3>) b>) ,F (fb>7) ` ,`Hb g7
power8.C
FkFG)^)^ 'X))o'7.F^N/FG9>F
/'^'I<M6. H)>_31`,) )X/QFG,9`
)9)
//
)Z`$/a/F^_sE",93>^FG)13 sae1,_/h
n$39:$ q{"A `)F&/ 1"!Z*)J!{ )1[:<!01[F&
)F7`$/F^ )h3`$O)F )9)7)T9^/Q,F^Fh)"=9,=) /,=)^9,=) sG
kg7
'A1"XS/C<012 'X`.F ) $^) P>fb:^6hY Y/ FI 9FxFxR #Hiostream
xg7FxPN,9`:)
F `$3 s1?)
m>)9FGHFG)^ 'X )89^/iF FI)^)>t
3 n) 3F 'X)^ :/!0C:!
.<2+<'A<M _$UH g7 f//, FT>),=3 ) ) E>)^
FIHFG)> r'&9` s1 ` 0
fb)aF^ F )9)h) c)^>9)>3
FG,9`>`F^F^PN, b
gh"_/
PN,9`>)^Fh3fH39>3)+)^^" ^)^F
FG) FhK )^9 =9,=)Z,+sE:
F.$/3{ q{ U ,=)^9,)^F F^F^/) )
,F^> )^ O)+)^F > 'X` 13|P&9,=).)
/ :]Hm`>)FXde 8$#8 F:!1qY;gD
FX3`>N3-)9)X/`>)FX/FD 9N/`>3>&)FI)
a
) 9,=) 1,.sE>$ gh b>1'73
93`)FT)9)
F )^>$ de W99/ int
a
)Tf^Ns9
>`fbF&)F f,"Pc) = ,=)$
a
gh ?F $; (08)8/`>)^, 4`>8 ,=)O)-) bF 9N/` de 9
A >'zf/NOs F 3`3 'X` /`>)^F./F. 9N/`>939:!) FG)^ )8F^,) P7)^9`>8 ,=)O){
gh
b
f/Ns9
Fk9 )N/ 3 )^!)&13,`>)
4q{ (0"`>8 ,=)>Fn)&39,`>)
)9)
b
a * a
b * b
F 4 93FG)F7)F7F^,)Z/b a
b
gh )3 ^)ZaF!(@ $K.Of13F7)"/J,=)^9,=)$ M O^)&P )&FZ) `89, ))^ P ) 39,`>)
=)9)hF a8 b * b
2.1.1 Syntax and semantics.
ed a3>h)+,93>^FG)/3)T/
3>)/ =/3aD8m^)/) b)+'Z)
power8.C
/F"b,^F^P7N)^> xb, >3 ) H(' )8,F.``3-)-'X` /F O
'&^)^){ ghFGn,F PN ) ./V+M2Qs Pr6ZY Y.4,"PN,G)> 3 )^H(' 9(' )&)>^:)
9^/ I'X9)h31>FZ)T9^/J39 + /3)^9FZFh3>)>^3
s )^9 .Q<'M012!. P
6hY"Y k2xfb
/t)9)kFn'7> PN^38/`>`3!)!)^9X6ZY YwFIH)O]_.sE 1"=Fd+Z1q)
P F^_)`/XE)P!f1>'! V F+~1 ) / F
9 )9)aF FG1)`>)`$ )
F^_)`/ `^`:)$
d !) BFZ) F/ 'X) O),=/rN/,9/ _ )FZb,-'X9)&F^)^`FT s9,)Z)
)>^>))yP F^>)` e O^)`,NO 'X>)>
)
_ bF_F^>9FG?) /D ,9 F
``>)ZP $/=
9>-./cF7f//
3 )"`89> __,=)^9,)ha>^ZF^F^/ /3 )^9F7'&
3>W9)!9E '& ) / `>)/9F .VM2Qsj<:!:<:g. f1N)9F P9)^9hFG1)/`:)`$/
& ]av /A
,F>
V~/)9) F"F^_/)`$-f3 _ `89 'X ),)">^^F s9,=)!'7O
) ('73 ) _ b /?/F^F^,8)FTsm,) )^F sE> f1! )./ /CD) , W PNa ]/89-P") F^>_/)`$/D>^^ 1,FG) 9/F 5`>9FG 1,`F* G)`,N
9N)PN^--CD )> 9NO)PNF r)89^/i_?sm9$fbFI)^/ x.`:/F^{ 2xfb|
)+F/ 9N)PN E )")D'7 F^>)^F s9,=)"P /4) )> )F ? F )9)T)
/?
FmsE9fH F C)J ,) 6 $ kFG9,3!fb3!'&^) /FE)9) :]=s9)
,39:W 3 sE9fH$
gh FG1)(].P 6hY YF FIE`W93.PN_ +&_)_O)`$//N/,9/ g7 3F^`>^)
PE)TF^_/)`FnF F^F FI)^`><) H) O)> F^>+s9F )^9X)^:]H)7Pr N' 1/3F F^,` ) F^,=l >F
P F^F^Fh/3aEF^FGs9TF^)>^>))Fngh" 8`N N'wPZ6ZY Y `(fb>9 sE)
FG1)O]/3aFG_/)^`F HFh)"do\=
C d 2n6 FG)/393 AbKK .P $UUKH
9DF^,`- PN_/ FIE`W9`$O)_F73FGEFOs c
e)>^'&F^ O' F^,3 `89>
H(' 'X>)^9: b,=T/ )^:]H) FT/`:),9/a_6ZY Yt/ E93 'X9)T )TFDF^,=EF^3
) 3 7)8F)FG,9 )s9
PN_$ 6hY"Y g7^,9,=) )^9F sE1 '7a:]H9N5)
>f) FGH)/`>)`$/ /3 F^_)`/X)>F_ O),=/TN/,9/ /3 s :]=/89 =
)aF baPX$/39s9)o 4'7'& P)>) FG)^`:) 39FG),F^ sE>)o'7 FG1)`>)`$&/3
F^_)`/m)>F 4FG ):Fh!FI)7 O),=/_)^3,9`>3a/Fh9fH sm)aFG1)/`:)`$/
/3aF^>_/)`$/E/FIE`>)F 93)h39:E3F&) `)^:]H)h'XàFGE`>)ZF7:f/)$
\1>)F >fb>?f3-/F&sm9$fb
Unspecified and implementation defined behavior.
3lm:)^8_3lm:) 9NO)PNF<b)F F XPE)D&,9 /FGm`>)F P 6ZY Y )9)k'7
B3
PN>k)!FG'7>>
,93>n)Z,= &PN^),9)> '7&`$t ) 7)hF^F^,&`>8 >) FG`
)h1`>`$/F^9/8E9Fh,a I/{PN =
gh>^ )v'7 H3F&P )PN e39:E3)7sE9fH$ g7!`>&!Fh`$/3 1[']Lt <{
'I/M2+012)J ,) sE9fH$
9>9:fb>a)?6ZY Y FI)/393 `/F_F^ /FGE>`>)P") N9,9/ G89>)O)
3>W93 b,`$:]HE>`>) b,=k9N)^PN~)^!`)/83`,)O).)9)kPN, FIE`W9F4)
/FGm`>)$ g7h)o19`/=:]8 PNkFG,9`8/ 89))"3>W99>38/FGm`>)nFx)h,+sE:
PEs9)Fn)^ O) _ b&,= !G `> F^ \1`>)^ 1<;H de_`$F^DPE89))+3>W93
/FGm`>)Fk/3 F^,)^ sE> f1 )D6hY"Y*FG)939^38/38)7 )PN )>)^9: `89>)^
3>)>^X) /`>)^, msE9fH$
gh F^F`> H93*F`$/3 hM.qL=//1 ) ;/!5CF
1q: `> P F^?,FGE>`W93
/FGm`>)+Ph)^9/, %&>b,|`/ ^ -'7> 3>W93 /3 ,9FG, .'A+[ F^>)+P
EF^FGs9TFGE` W `)F 1s9,=)7) 9NO)PNQF7)Z^ 1,>3 ) `)a PN,{FGE`>W9`$)
P
)./FGm`>)$TV )o19`$ :]8 !PN F^,`?/ ,FGE` W >3?/FIE`>)TFX)^9 :f/, O)-3> P
E>/39F7'X ):]19^F^F^
F^ \1`>) 1
de '&^)D/F /,9FGm`W93FGE`>)^Fn`$9)n/'h1F4sEhfb3
3 bs ,=)4,F^,9/ FG
`$^"F^,=>F&)9)h ,FGE` W >3X:fb-,93>W93
sE9fHhF^,)^F
2.1.2 Comments and layout
2xfb>^ 113
/c`)/Fh`)^F 1
PN&:]/89
& 6 v
0
// Program : power8 .C
// Raise a number to the eighth power .
Vy`)xFG)G)Fx'X).)o'7"FGN/F^F
/3 `>),F ,9)H)738P9)h 6 )F
38)Z(fH39"/PN,`>)9/)o H
//$/ )9)h)"/Q'7,3a3 :]/`:)_)!F
'X ),)h)-7-FX /J'X),=)Z`)^Fhs /3 )^9> 3 '&)^
/F PNn)&`89:x)!)^9FGN) )t)":]=>`,=)s9F bs9,=)k'7XF^"'Z) )tPN
)>hE9X
e`,93.,=F^ fbF ) /
3 93P 1`G`>)&&:]1)3)-
),=) `)F {)8N)G)> )/F 1F!sE`+fb>^ )^3,F"'X )^9 9^/ F )
`89>)^ )^f1N/e g ,FG).,F 2 fb b, '&4) sE_s9 ) ,3>FG)93 b,=+O'& 9^ /Fh/P)^>X.`,=9TP '7 1F ='X)^9,=)7`>)F g7>"F79 .<2)Y:/) 'h$
P 'Z)9
`)F bs9,) '7D'& PNO' F^X` eF^F^X,39>F4CTTP{)QF )9)7>fb:^
/K
:fb PZ).F.fb:^|FG89, F^,3 FG)^).'X)^ ^F.PZ`>)F
)9) >) )X9^/?
BF49/X93
F8'& O) ) 3F4de ,= `/F^ 1)^9TOsEOfbX)o'7 9>F
PN,_F^, `
VX)> b> PN$),= P. $/39s9 9^/ Fa)F Nb,<) `F^3>a)?fb:F^ P
FG9('X M / 1w? 9fb-(fb3`)^F h/3 / G,9`>F^FG power8.C
Nb,=)h)F7 b FG /`F 9Ts HF =s9N/ _9>F /3a3))F
Á
# include < iostream >
int main (){ std :: cout < <" Compute a ^8 for a =? ";
int a; std :: cin > >a ; int b= a*a; b=b *b; std :: cout < <
a < <" ^8 = " <&FT`89>)>a)Tsm,)D)^9>F^ `9/F Es9,=)"m>F^ $/3)
/t'XW93_)^9F `39>9FG3
fb:F^ 1,)&39 8`,) )^ ,93>FI)/3{ng7X9,GEF^DPr
13 Nb,)XFX)fHFG, ) 9^/ FG)G,`>),= g79FDPNT:]/89!$/FZ)9)T`$/
s91`1F P{)^9X/ F^,3sEDF^> )3_s1s9N/ F )9) DXP{F^,=`>`13
F^,3sE&>FGEF^s9XPN7 X)= # )J<M2+012 = b
9/F )ksm>)o'7_)
power8.C
/.Ph`,= s`F 4F+/)>.3FGE>9F^s9 >39).Ph13 $b,=) 4/ ),9 b,
'Xm
N):hsm!s9D) PN,
`)T)F
gn19`/ x`NsEO)fb8F^P)o'h^_uv`>)^F+9fb
Nb,).,3F _ H9F^,=
)^ O)
>fb>G1sE3y)-uI>`>)
`$/ /F^|$/3 :fb>^1sm13=yF^Q BF
`13 V )_) >fbTPT)
F^89 9^/F 3F^`,F^FG35 )F sE1 kF^,`5PN_/ ,3F )+9>`F^FO^7
7'7
F^89 /39:&)"FG)/393,39>F )^ O)k9fbZ(fb)!'7 +'7 9/`>)`> 1938)^ O)
TsE9+,F^3-/FG)h
)>hsE1 _ 6hY YzFZ'7 2.1.3 Include directives
2xfb>^ ,9FGPN, /c`)/F7"Z
# include < iostream >
include
)1":</<01[F&!. F^,` /F
& ]av R
&F^,9/ x)FâO9m$ ).) fb>^ sE9 PZ)^9
/-
3^`>)fb>F ^
#include
33 FG` E 6ZY Y{_/-8EG)/)&PN$),=>F"O.9)T ^)TP4).`.N/,9/"de FG)/
3 n)> 89>)3 ) FG `$/3 ./2)Y:/)Z1q;!::!V 'X`|F. ^) P&>fb:^
6hY"Y 89>))^{ V Z1q;!::!V F+ `$/ ,)+,F^3|) ,=5`:^)/|PN,9`>)^9/)o
/3 )^.(fH39T) )^.)^9T,F^>Z F^,```>)hPN-4de
P /`>) =)DFG)/3 O3 s9^8`>9FGFG)F7P
F^>fb:/rsF !P '&`F7)"9,=) /,=)^9,=)7sGb
V s^ F^)Fh)F PN,`>)9/ )v ) )T,9FG>&
)TPNcPr9ThF^>fb:/ 57Y)J<:g. 24/` F^,`?$39:T`)FXPN_))9)DF&9>33 sa) `89>Xde 3>D)_,F^
`:^)/ PN$O),=P s9^ 9/F )-9`>,39+)`^>FGE3-9/3> )a)
/ s1 $/F PZ/
3^`>)fb/ de
x'7'h/) ) ,F^=9,=)./3
#include
power8.C
,=)^9,=)4'&`R
e_1sE7F^,=^F^J
)4 ^)4P )h`&/, xg79h`^^FGE3
$/3>ZP ) FG)/393 s^_Fh`/3
iostream
V '7 3F^9>3 6hY"Yys^D9,=)F )FrPN,9`>)^9/)oT)X 'I!.qLCJ/ g7k9/FG /`>
P{)DFG)/393_sO^8Fk`$/>3
4gh
=_3> )^ ``F^FZ!PN$O),=TP )DsG std
'7 9fb ) t+~1 S/V )F 9/4'X)^ ) 9/FG /` bn
)F F )^9kPN$),= )9)
std::cin
/O'&Fr,F{)&/3 =9,=) P^ ) b>1sEbOh
3 :xg79F `9/F^* F{)Dfb3 'IP!D+./57!.
'X`?3l >)"PN),=F!/``3)/ a>) ) F^/ 9/ V )T) F.)^ m:]19`)
1,9/W9`$) `>$F^FD) $/39s9)o P4 /FD )DFX>39) O)XP
'X` sO^-8fb> PN$O),=.`FDV 9/!)^ O)TFX) 1,9/ W >3 FX`$/3 C t+~1 )
/3a,F^,9/ `^>FGE3FZ) +PN$O),="3>W93aa,=Z('X 9^/-
2.1.4 The main function
2xfb>^ 6hY"Y*9^/}+,FG) 9fbT
int main () { return 0; }
main
PN,9`:){ ghDF^^)^FG) t$39F /F PN('XF>
g 9FZ/ 31FD)=7g79
7
PN,9`:)-FX`/3 s1a) m>)9 FGHFI) '&9>
main
b, )^D)) ,) /?&s9,=)
'&yF )
SC!01q X/3'X9)F return 0;
F^,=EF^3*)|$/ ,9FI)ab?|_)_O)`$/ZPN,9`:)D)
PN,`>)*`/ fb
main
,)Ffb) )_,=E :]H`,=) P")-/ Z/3 )^9 `89,=))^F 'X )
)X`,= s/`>F H39Fk!PN,`>)f//,&)9)kF fbs /` ?ek>),=3h4) )^9&E>)^
FG1FG)-xde+,= `$/F^ '7h9fb '&^)^)+
PN,`>).)^ O)x31Fx) :]1E`>)4/+,)F
main
)FhFh3`$)>3 s1 )">89)o_s/` b>)F () sE3 main h/3'XF^ >),=af//,"Fh)^9
)> g79hP /`>)n)9) )Z:),= f//,Z+,FG) sEZ/8)>nF439`$)3.s1+)7'73
0
int
sEPN^
n-`fb>)
)^Fh) E>O) FG1FG) )^ O)&)! 9/F
main
return 0
,-F^,``>F^F^PN,A
e&)^ O)Z'7"39M
)&`$O!'X>)^9:&)Z9/F 'X" )>&f//,! ]H9` )
F^9/FhP ,=
dea FG)^^`>)7_)_O)`$/F^>9FG =)
PN,`>) P
Fh,=)^)^>8sm= gh
main
power8.C
'XTPN,`>)9/)oP )T9^/J`FhPJ)^9 W! P )"PN,`>)rng79FZ>l `>)&F
)
$/3 _1,.sE>"P )+FG)/393?= ,=)!/3?'&^).)F )?E('7>!)^ ) FI)/393
,=)^9,=)$kg7"P `>)&)9)ZPN,`>)Fh`$/a9fb!>l `>)F&F^>)FX6ZY Y ^)ZP _ S/h!012+
LM:</N
:/'j'^1[7N U7N*N&!. & 6 v
@
2.1.5 Values and effects
g 94f,k/3 :lm`>) P=7PN,9`>)^ 43>)>^3"s")k6ZY Y F^_/)`F >> HO'&
7
)aFG1)`>)`$Z,FPX'Z)9 PN,9`:)F831F)8)^&,F_/1) OsE,=) f//,F/3
>l `>)Fkde)F7F^F^ 9f//,"/3a>lm>`>)D^ 9,=
F^>_/)`$/ ):F
=.:]/89 '79fb8) Qu )9)! 6ZY Yx)`9/`:)> 0 F )>^:)3 /F )
)>
/ ),9 )FkFn) 3 8`>,9 )n)!,9>F^F :4d ) Fk/F^"8E^)/) ) ,3>FG)/38)9)
0
f//,!93>lm`:)X3>E3a )"``>^>)"/cFG)) 'X` ) PN,9`:)aF7`$3{
2.1.6 Types and functionality
g 97'73
7
F )^9Z9/ PE 6ZY Y )o1E4g7Fx)vE&F4,F^>3F^`7)h
int
power8.C
3$/F '&) )>F> de _)_)`F ):F._133 s1 )8
g7F
( , +, )
/:s9/`8FG)G,`>),=
39:W F )
)^>F. )>F Ph) f//,/ )_F^>)
k/3
?)^>FTPk) PN,`>)9/)o| /33) /3 +,)9`)h:"de 6hY"Y )>^F!`$/ sE
133+s )^97)o1E
nq{ bZ
G_O)_)`/)vE !6hY Y*)o1mZ9/Fn 'A 1 Fd+Z :/7NY /3 SC!01qint
+~1[0V 39:W 'X9)!'7 `$/ 3'X)^ )$8 '7+PN> )^ )o1E '7T'X 39+F^+s1 )F79/ Z)D)9)7) 9/DFZ.FIH)/`>)`/r/FGE`:)ZPr) )oE ='X
f//, /!/3aPN,`>) )o^"P FG_/)^`$/m9)^,^
67fb) 6hY Yy`)/F Z,9.sE> P S/h)Y'I<M2+J0VgL=!. eF^>)FE`$/3!s9,) F7TP )- g7"_/uv 3l >`D)
)vEF PNh)o19`$/m `$)9F>xghT)o1E
int
)
G_)_)`$/)vE F7)9)
F& W99 )Df/,9T/ b
( , +, )
int
2.1.7 Literals
)^>/>F^>)FnD`FG)/)xf/,9 P9F^ )o1m = :]/89 +D$UTP9) Fn )>/=P9)o1E int b>>F^)^")7f/,9 0 =k/`PN,939/)/1)oE power8.C 0
).F.F^: O) 3>W99>3|O'c )F.)>/F!1 b k93 'X9) )^9>.f/,9>F O Vc)^>/
`$/_sEDF^>a/F )^9TFG1)`>)`$ `,)^>^ ^)7Pr!f//, ) _ b>F )Xf/,9 If1F^s9 )
/-
V
2.1.8 Variables
g79 int a;
F& )J!D:+012 P Fd:!1qY;gD Vwf^Ns9T>F^>)FD+)&`>F^F`>9FI)/)Zf,!P
F^ )vEZgh!f/s9!9/F& 'I 0VgL= E Fd+~* m93 / Y)Y):/!.<. )v9`$/ _a)
`89,=)>! F _/ >^7
xb,|`/ )^9 Ph)_/33=FGF F^8 /F )^9EFG) P7)
f/Ns9 )8_/ ^Y
:_gh 9,=ÊFG8P )/33=F^F F ) H('}'&9: )-FG)^
/3a _,=a) f//,4g7"$FâPNZ`$/9.F^,`-/) )v Fd:!1qY;! F7)9)h)F f//,
`$/ sE ` 9/3 s 13P1
)^9 ^ `)) )!)8`>^FGm93-393=>F^Fag7
/33=F^FZ)F^>P __`9/!Fh'7e4dea`)G/FG) H)^9"9/T/3)o1ET_
W=]H3{
& ]av K
9>y'7 PN>8)^ f/Ns9 n'7'&73 F^?s5)F 9/
fONs9D'X )
)^9"PN('X `9/`:)>FI)`F
)J !.
9/
)oE
f//,
,3>W93
g7a3`N)
int a;
393=>F^F
`9FGas1 `89> C"\
a
int
'73>4'X+)F4Fk`/3 )J =1[012 P >fb>_)^9,8 )k31>Fk)k3>W9
a
) f//, P k,)"`$4)^ O)")FTf//, 3>E>93F )^9+/FG))^ /3 )9) )
a
3>W9) PN,?FGE`>W9F 57u )8f,F s)/3+1 ?).,=5) F+/33^F^F?\HH
a
)9)nDf^Ns9 5C+. Df/,97F )>>PNZFG>'X9) 8>`F^ s9,=) '7
B=FG)`.) ) $uv,9FI)kb
_)_O)`N9F )/ OsE,=) S/h!012 Fd+Z '&
)^9:_/`>),9/ /)Xf/,9Ds)3
s1a>f,9) ) PN,`>) '&)-``>^>)+,)F7 >fb FG)> PN,=^)>D'X )
,=DF^9F^F P
9/F&f//, PND:]=/89 ='7 /F^8F)^ O) F =Zg7F&FZ)!'h
a
2
a 2
)9) />F ,F^,9/ )/ sE,=) f/NOs F /3 ) f//,F>a 8'Xn:) ) H('
`9/F^F PNX/F^F^99+/3a`9/.f/,9>FZP f/s9F7-\1>`>) 1$;H
4,8)
2.1.9 Identifiers and names
g79 9/.P / /f NOs ++,FG)TsE / 1q)J<M01 <: /``3
) )+PN('X_3>W9)
/3a)h+,FG)7sE"3l > )&P^ `:^)/ :<!.Q/:!F&) 9
/F7b
int
R
10. =V T. *</ S@!5C:/J<<:g./'KL7.Q) S|657} !6/:g.
)1"N1[6.
-W)O657Gh)J<:g.Q/:/Ii od p 57 :g./]!5C:J!<:
Definition 1

) Q identifier
-P657 k
a z
A Z
5C+.]A;<?D!6<:

0
9
_
/ _$ / F^ `)/ )> 9/F PN ]/89_) t+~1 ) 9/>F
/3
7
g _6hY"YQFGH)O] FGE` W >F"'X9) 9/+F r'X.)6hY"Y
std::cin
std::cout
F^_)`>F7)^F7,Fh'&9)7) >FGE`>)^fb!9/DPN:FZ) a.fb`) ]H)$
Vp6hY"Y
2.1.10 Objects
VXs9uv`>) F ^)kPm)&`89,=)><BFk_ G )9)kF ,F^3s1 )Z/~) FG)^
f/,9/+V& s uv`:)"9/F!/ 393=>F^F 8)o1m /3?_f//,+Pk)FT)o1Ee3>):3 s1 )
^8`)>)&)Z)^9 s9uI>`>)!BFh/33=FGF :
) )^9F43>W9) Tf/NOs `$/ sE7`F^39:3/FnD9/3+s uv`>) s9,=) '7Z_./F^
9fb+,9/3 s9uI>`>)F"VX),a'7.`$/M
)TF^('~/ :]/89!PN! ,9/3 s uv`:)!O)
)F E) H'7 `$/-^,9T)9)h, 3s9uv`>)FZ 8E^)/)$
de P /`>) 9P b,a'h/)&) 'Z) )>FI) /F =)XF&s9F^,=) _9>`F^FOâ) '7 'X )s9uv`>)F7)9)& 9)Z9/3
s1
f^Ns9Fng79FZ`$sE F^-s1_)"PN('X F^89
),) ]HE>^) F^,=EF^X)9) b,
9$fbX'Z)^)^
"/})^ O) FG)>FZ"F^ 1,` P
)>Fh) sE"$/3-P +W9B
PNX:]=/89 =)8F^G)&) /P)>^'hO3F : Z(' b,- aO)
b,=k}/3`,)k)X,9.sE>nP f/NOs F4)^ O) )n`)/F4\H )^9Fk1,.sE>nF ;=
k,=)& )F^";=Df^Ns9F Hb,`$/aFG)^"+D)9/a;= )^>F4deP b,=h/QFhP
/-9/`>)`/x,F^ {) `$/ `>^)-FG)^ F^ 1,` Pk; _):F Es9,)D)^9>?)>.+,9FI)
sE")Z$/FG)h )>>Z)^ O)Z`/9)hsE FG)>3a,93>X.f/s9D9/
v
& 6 U
2.1.11 Expressions
ed )n/
O)=k`9`>)>xFG 1,`F FG)/3 ,=) sm`$/,F^4)>!1 P /
power8.C
/3-^!` ^FGEF^s9 PN&)^9"PN,9`:)9/)oP )"/ )F^! )!`9/`>)^>
F^ ,`F
a9+A /3
F (0 93 $KH
a * a
b * b
VX ]HF^FG>>F^)^FX `89,=))fbf1+)^9:Z ]HF^FGF ^ `>F^ / ]HF^FG F )>! LM:!1"'A:!V,s!LM:<!.<./1q PN!:]/89+_):/x! 9/ {") F
'KL=./1[?,sgLM:<<.<./12 ZV `8EFG)T:]H>F^F^ F&s)3as1-`>+s9+:]HFGF^F
)=,`>G)/E:)F s 9,=)^) T / Pm ^)FGF
O,93_:]HFGF^{
()
ghD:]1F^F^
F / :!1065&'A!012 :]1F^F^ 1fbf1 =h'I<:!1q f/s9F /3
a * a
) +,)9`$)E>)^ uv,FG) b '7 HO't)"P _O)_)`>F.VX``3
)a,=
sE(fb&3>W9) F `8EF^)^h ]HF^FG
s9,)nPt)^9Z ,)^9`$)^ E>)
a * a
/3)T)o'7 _G8:]19^F^F^F
/3 a
a
VX``^399-) )
sE(fb_39:W )^
/ ]HF^FG|F+-FG1)/`:)`$/k) )v xs9,=)+).9/F
F^_)`/E/FGm`>)FZ/F7'7 /
:]HFGFâ9/Fh )vE 9 f,"Pr)F )vE /3
EF^F^s9
/y>l `>)$*g7a)o1E-F W]H
3 s ,=)8)^9 f/,9a/3y)^9-:lm`>)_ _):N/ 'X )
:]19^F^F^|>)F <Fd+Z*+,) k$/a)9).)_`89,=)) ) >FG)F F.`$^^3|,=)$
2xf/, O) / ]HF^FG_Fk)&FI) P ,)7`>)f1)o "_'&Z 6hY"Y*9^/~F
:]H`,=)M
3 m) >f//,9)a`89,=)F&)!f//, Px) :]1F^F^?/3 `$GFX,=)D )FX>l `>)
eP /Y :
gn1m /3af,!P _O^_:]H>F^F^ ^ 3>)>9>3s1 )FZ39:W )>/ =Xs
)o1Ea/35f/,9 PX)a))o sm35)F 39:W 9// M ^_^ :]19^F^F^F89fba
>l `>)$hgnE f/,9!/3a>l `>)DP +`8EF^)^D:]HFGF^ "39:)>3
s1 )!fbfb3
E>)
=3>E3+a) f//,F&/3a>lm>`>)FXP )^9 fbfb3aF^,=s ]HF^FGF M ,)G)
,3
/ ]HF^FG
13F :]1F^F^
'&)8)DF/Z)vE 1f//,D93
)F^F
()
>l `>)$
ghT:]19^F^F^
=PNh:]/89 1F P{)o1E int 93
)7,9 ]HE`:)3 =)F f//,DF
a * a
).F 1,9^ P4).f//,+P g7 ]HF^FG 9/FT >l `>)$+g7+ ]HF^FG
a
b = b * b
s9,)nP}) +.<.1"N='A<M9<L=<:/+: /3)&)o'7 :]H>F^F^F
/3
19/Fk)^9&F/
b
b * b
)o1E!/3af, /F
9s9,=)X)&9/FX/ 393)^9/m>l `>) )X/F^F^9Fh)!F ,9 P b s /` b * b
) b
"F )9)&/-:]1F^FâF /Fd+~t+) )>h)9/ ,s$<7,) sE>`$/,F^ _/_:]H>F F^Fh3 )&9fb >l `>) 9F^ )9)7) &PN,`>)9/)o8FZ/F^F^`N)^3'X )
)^9 f,"b
2xfbPNh:]19^F^F^F '&)_>lm>`>) =F^&sm1 HF ,F^D)X): ./1q)J3 W/! ).89 F^ &)^ O)
)&8E^)/)x)!F4)&f//, g79D6ZY Y >))^F4` 9+FIEF^s9&PNk>lm>`>)F7h)
.<2+<'A<M6. ) '&9` '7":)&sEO'!
7'h/)x)^ >_ )9) ) MZV 'h.P /`>`F^F^9!/ ]HF^FGM
BFxf,hF )^">f//,9)^
) 93 )Fh/F^+`$^^Fh,)Z)F7>l `>)$
4,`$/)Z>)Z)^9Tf//, '&),=) )":lm`>)
$
87
-
Q M
7O7
Q
Q
-
27
2*
>'( A07
8/
/
2A
22
82
.#*
B>J
* I
27 (,
b
J
:
7"
5
$&
(9
"
(7+(
-BK 2"O
,
+(*
9 297
'(97
(P
N*
%$
2 2/
(
.
N
b
.. B#,
;#
& ]av 2.1.12 Lvalues and rvalues
VX_f//,XF7
]HF^FG
)^ O) 9/F7/
/33=F^Fnde_)&/
1)^9Xf/s9 b
power8.C
FZ/af//, H/3a)FZ/33=F^FhF7)!/33=FGF&P )^9Tf/Ns9 b
gh!f//,"P /-f//,"F&3>W93 /FZ)^9"f/,9"P )!s9uv`>)DO)X)FZ/33=F^F>hV&-f,
`$/_)>>PNXsEXfH>'73
/F )DFG1)/`:)`$/ `,):^ ^) Pr/
s uv`:) x) fb>Fk)Ds uv`:)7
)>8E^Y k9/ /3_ bFh) IfHF^s9 .'X)^9 6hY"Y /- ? /FG F
)9)7)
f//, :<0ST<:g. )^+) s uv`:)X)Z)Fh/33=FGF
de ^)^`,N / f/s9F./ f//, k,=).f,F Of13_-$/F PN /``>F^F^
/3 `9/ s9uv`>) f,F r>fb 'X ),) 9fH `>^FGm93_fONs9.VXFT'7 '&
F^ ?\1>`>) 1$; sEO' 9) :]1F^F^
G33 8- $K8FXF^,`
std::cout << a
/a f/,9/
VX :]1F^F^ )9) FT)"/?f//,.F / ^f,
=" ]/89 )>/FT ^f//,F )>.FD
/33=F^F /FGF^1`)3 'X)^ )^9
)> 0 EFb!M ,=)^)
/DP )F^F
int
,3aaf//,DH39FZ/a f/,9 =/3aF^N PNh^f,F
gh&)^>F4 Fd+Z /38 Fd+Z //3= 93`) )9)4'7Z) sE,=)4)}) FG ,`
):F!P :]1F^F^9F s ,=) )> )>F"P ) f//,F_ 'X P)> 39>)P?/
f//,D'X)^ ) s uv`>)Z)7PN:FZ) 9/3- ^f//, F^898'&)
)F7f/,9/
2.1.13 Operators
q{ A P
PN&:]=/89 HPN),=Fh) s99^_+,)9`$) E>O) * power8.C
qrb PN,`>) / E>O)X:]1E`>)^FT,)^FPe>^ /F^ `$/>3 <L=<:/)+. ZPxFGE>` W93a)o1mF mP 'X` )X`>8 ,=)>FX :<<0C:!xFd+Z P FIE`W93-)o1m m/`>`3 ))^F
S/h!012+Z100V nde-/33))^9>F^!`89,=))^F __ fb!/>l `>)$
ghF 'h/F")^9 F^_/)`$ f1>\
' ) FG1)`>)`$n:fb )^9 E>93F!/F"'7k/F )
`8EFG)D:]19^F^F^|
s , )7PJ)^9 E>)Z/3a)FhE>/39F =F^.\1`:) 1g =O
:]19^F^F^<F )!m>)ZFGE>`W9F&PN&$/`-P )J'X>)>Z)ZFZ/-f//,TD/a^f//,
deP )h`8EF^ )7:]HFGF^8Fn/f//, )hE>O)kFnF3 ) >),= )^9Zs uv`:)n>PN>^3
)_s1 )+f, dePn)^9+`8EFG) :]19^F^F^ F"/ ^f//, E)+E>O) F^89^>),=F
)F f//,
gh
1,.sE>.P&m>/3F.F.`$/>3 ) :!100V Ph)
E>O)$ FI) E:)F.9fb
)v
e,9^
E>)F 2 s99G_E>)^F :
1
9>9:fb> /+^f//,7Fx:]Hm`>)38/Fk E:/
3 ) F4/F^DEF^FGs9 )T9Ôf13Z f//,
de )^9Fh`/F^ H)Tf//,D'X F^89 sET)^>^>)>3a/Fh/
^f, =$/ )9)7)^FZ393=>F^F
F ,F^>3|) 1 , )f, xs ,=) ) `9/_)$ g79F+F HO'&|F ZFd+Z !{q{
:!Fd+Z ]/=F&/:g./12 4de8FG))D)9)k/ E:/3 +,FG) sE& ^f//, )Zm>) )>^PN
,9/)F8)^ O))-E>/M
3 BF8f//,a>_/F ,` 9>t
3 Ds :]Hm`>) /f//, )
E>)h:]H9`)8F^9/F7)F7))_) `9/")Tf,
9>?8`8mF^)":]HFGF^ fbf18/ E>)
Evaluation of composite expressions.
>)F >f,9)3)^9kE>/3F O >f//,9)>3 W9^FG) >`$/b)9) )^9F /FGX`$^^F ,=) ) >lm>`>)F
PE)TE>/3F Pr/Y
: /F^3 )X>F^,)!f//,F H)^9Dm>) `89,)^Fk)Xf//,
v
& 6 ;=
P ) `8EF^)^!:]19^F^F^{"gh NO)^)>T`89,=))^FZ_a9fb+/33)9/r>lm>`>)F {/3
/E:lm`>)^F&)>)>hPNc)!:lm`>)ZP ) `8EF^ )D:]H>F^F^{
gh
3>+ 'X` )_E>/39F.PZ-`8EF^)^8:]H>F^F^|
>f,9)3 FG 'X )
^":]H`>)F ,FGE` W >3 =FG /F^\1`:) 1
gh>^PN mP ) >l `>)DPx!m>/3- 9,`FZf//,F&X>l `>)FDP )^9:DE>93F f//,./3 :lm`>) Px).`8EF^ )" ]HF^FG?_a3>m3 ) >f//,9) 3>"g7
`F^ ,`.FX)9)Xf//,./3 >lm>`>) P )^9 `>8EF^)":]H>F^F^ _sE ,FGE`>W93 /F
'7 \1`!) `>8 >ZF&)X 1,3-) F^F^, 'hO F^,` `$/F^F m)XF&)!FIE F^s9)o Pr)T/>4) fb3 /_:]HFGF^'&9FGTf//,Dh>l `>)Z39:E3Fh
)
>f//,9) 3>&P E:/3F
9)xF ) )9) FG>)FxE>)FxO O^)xP PN,`>)F CT.) 9/3
Operator specifics.
)>!F&+W9)"1,+sm>&P mF^F^s9"E>)^ d/M. F^,` F
a/P )F^
+
=
) bF73`:) `^FIE3 ) '7 19('X__)^9>_)`$HE>O)7FGH.sEFn3`$)9 )
PN,`>)9/ )v+PE)DE>)$ CD_)D)> 9/
3 H/3_FG) `fb) 1m>)k`$/F
XVXP)> '7 'h)
3_9)X fb )sE:a) ,F^,9/ PN,`>) `$/ )) b f(x, y)
) 'Z)
-+ /3a)
7de-F^,_G 1E>)FZ>)Z,9Fh'& )"^
a * a
*(a,a)
9),=/{93D$/39s9T`39/
=,!3lm>^)"E>O)B
F e/xPk)s99^J
X``,= 9/a)^9 ,)^ power8.C
9`$O)aE:) )./F^F^)&m>) m)^9!= ,=)Dm>)
m/3 )^9 ,=)^9,=)
*
=
>>
E>)
nq{>)&,Fh3F^`,F^Fh)c),={
<<
gh +,9 )9`$O) E>O)
:]HE>`>)F)o'7 Gf/,9-m>/3FP
Multiplication operator.
*
F^X)o1m p /3
) >),=F )D13,`>)7Pr)F )o'7+m>/3Fh/Fh/
Gf/,9/ng7"+,)9 `$)Ê:)h9/Fh9 >lm>`>)Xa )F7O'&r
ghD/FGF^)nE>)
:]1E`>)F /_f,&PEF^h)vE p /Fk)F
Assignment operator.
=
WFG)7E:/3=/3
^f//,DPE)TF/&)oET/F7)F F^>`3 E:/3{nd )hF^F^F )Xf//,
P )8F^>`3 E:/3 )a) WFG) E>/3 /3 >)^,^9F") WFG) E:/3 /F / f//,
dea,=h9^/
)^9":]H>F^F^ b = b * b )>>PN!F^:)FZ)Tf,"P b ) )^9
power8.C
F , O!P )F >fH,9F7f//, 9/3)^>),=F b
de P /`>) )^9!>)^): G 8a)^9 )^> f//,"FG)/3FXPND) P /`:)X)9)X) ]HF^FG _$
E$Ox ) [S< 93 FG3 P/ F^F^) E:)$x\1N^ )k)>^[^f//, F^9/F
/a ]HF^FG)9)h_
E$Z_) :!1"N5Y 93 F^3 Px/aF^F^)Zm>)
de
x)_`8EF^ ) :]1F^F^ std::cin >> a 9a_FG>)F
Input Operator.
power8.C
)Tf^Ns9
)+)":]1)&f//, PJ)^9 .<2)Y:/)j1[&LMC 9,F^,9/8) b:1sEb^3r
a
* 7*'( B>,
I'. B
=
,
O#*
7
-*
2P %*
=
27
# M2*+
1*
/
7?"K(7
*L &
%6#.*
*
"K(29
J
.6#*
"
%*
:=
,&2*
/
*
@7?"K(7
"K(2
& ]av ;
de_>/ b)&9,=)nE>)^
:]Hm`>)F F )F WFG) E:/3/f//,hPN>^^!) /
>>
gh.F^`>93?E>/3 FT/ f, PnF^ )oE p Eg7 m>)TFG>)FD)
F^`>93E>/3
)+)^9T:]1)Zf//,D$/3PQ) 9,=)hFG)^/J/3 >)^,^9F ) FG)^/J/F
/a f/,9/
VX8=9,=)4FG)^$z^>F^)Fk)hFG))^ZP F^7= ,=)43>fH` 7) .P )F 3>f1`Z/F
13,`. `)1,,F FG)^^$/QPr3 O)!)9)7`$/_sED)E>3
) (fH3D=9,=)7 3_93{
&3> )F E)!P fH>' r)8FG))^8P )8FG)^^$/i`GFGE3F!) )8F^ ,9>9`>
P 39)
) b>)7$/3{4deFG>)^) )Df//,DP )F F^`3 E>/
3 1)T=9,=) E>O) (fbF 39)! )}P^ )DFG)^$}) 9`:) )DP /`>) )9)n)F )> 9/F O' sE$3r
= )F )4F48E^)/) )^ O)n)hFG)^/t`>FkF /8 f/,9/ 67`:9)^, b/89,=)4FG)^/tFk/F^
`F^3>^3 ^)ZP )T/cFG)O)
%&('c ,`|P7) 39)aF $/3 F+
) 93 O'c:]=/`>) )+F.):^>)3|/F+
f//,8P )o1E p 3>E3F ) )o1E p P )8FG`3 m>/3{ =.O' )!F
, )^ 1O' )9)&)FD)^>^>)O) FX$39a3>W93 PN )^9 )o1m
/3 PNT)
int
)>ZPN,939/>)/m)oEF7)9)Z'7T'Xm`,)^>X
)"PNO'&.F^`>)9F>
PN>^FT)^8)!fONs9
3>W93 a) =9,=) /,=)^9,)
de 6ZY Y m)^9!f,
std::cin
cin
sO^ =/3)F f/NOs `>^FGm93F7) )^9 FG)939^3a=9,)7FI)^$/-
d )kF4,=8)^ )7/?
F `$/> ) W9H)^9hFG)939^3 =9,=) FG)^$~'&)+3 O)1
=n:]/ 9 mFG,EFG.)^ O)D) 9^/ 'h/FTFG)^)3 Pp8`_93-FG9>eTg79>?,F^,9/ 9'X
).9^/FD,9 r/ =9,=)T) )+`_/3 F^4F PN^'h33?)^ ).9^/?
BF
FG)939^3 =9,=) FG)^^$/- d )7F7/F^ EF^F^s9Z) 3`:)h!9^/?
BFkFG)/393_=9,=) FG)^/
)+$/3a39).P^ W9TFG)$/3{
ghnP /`>)r)9){)n=9,=)rE>)^r^>),=Fr)^94=9,=) FI)^$/ Fr) ``3)/ O/Fr) /('XF
)+s ,3 :]1F^F^9FZfb fH.` FhP = ,=)7E:)F =FG,9`-F
std::cin >> x >> y
T'XE3F^`,F^F7)Fh` 9F^Q3>)/EPN&)^9 ,=)^9,=)Zm>)hsEO'!
1[&LMCP./0:<'
de
T) `8EFG) :]HFGF^ std::cout << a w|$K
Output Operator.
power8.C
'&^)F )Tf/,9 P ) ) .<2)Y:/)|C LM7 =,F^,9/ )"`89,=)>7FG`>{
a
de >/ ) ,=)^9,=)&m>)
]HE`:)FD/FX )FZWFG)&E>93 /- f/,9 PN>G )
<<
/ C LM7 .<0:<' kg7ZFG`3E>93 Fk ^f//,ZP F^ )vE p ghhE>O)n'Z)>F
)Tf/,9 P )^9 F^`>93-m>/3 )^ ) ,)G ,=)hFG)^^$/J/3>)^,^9F7) ,)G ,=)hFG)^^$/ /F
/a f/,9/
VX8,=)^9,=) FG)^/~^>F^)Fn)^97FG))hP F^ ,=)^9,=)439:fH`/ 7) .P )F 3>fH`
/F"FG)
) `)1,,F FG)^$ P ,=)^9,=)!39)_)9)"F >)^3 s1 ) - de
'&^).) )!FI)^$/ =)",)G ,=)&E>)^Z)>^PN `9/FZ) FI)^$/ FG))^ 93a)^9F
_ bF&)X`>F^F^-)^89Ôf13!) FG)^/ /FD f/,9/D6 9`>>),9/ / ,=)^9,)&FG)^/
FZ/F^+`F^3>^3 ^)ZP )T/cFG)O)
d ) 39:E3FD ) )o1E p -'X` PN_)X).F^`>93 E>9M
3 FXf/,9 FX'&)G)-)
)
FG)^$?
PN )
)vE
/3|) )^9: PN,39/)/ )o1EF k)F.PN_)+F.$/3 int
3>W93{
6ZY Yi3>W99>F FG)939^3 ,=)^9,=) FI)^$/
93 ./2)Y:/) /:!:<: FG)^/
std::cout
)"=9,=) /,=)^9,=)7s^b
std::cerr
& 6 v
;;
d ) F ,=.)D)k/?
F `$> )D9^1`FGF )F^ ,=)^9,=)xFG)^/F= :]=/89 F^,=9mF^
)9)4)Z~'h/FnFG)G)38P}"`_93+F^ xgh,F^,9/ 'X )Z/zF
, sE) FG)/393 ,=)^9,=)TFG)^$p/3?FG)939^3 >G FG)^$p.PN^'h33 ))
`_/3
Fê k,)Z)&F&F^ mF^F^s9T) >39 `>)&9"&sE)âP )F^"FG)^$FZ)+'&)^
)."W9ZFG)$3r g7F `$_smX,F^PN, ) F^: O)&^,N ,=)^9,=
) eFG) )^
4P
std::cout
>^^&,=)^9,=\
) F^)h)

std::cerr
VXF&93`)3sE(fb"PN&= ,=)hFG)^^$/F )hF7EF^F^ s ) ,=)^9,)ZF^>fb:/{f/,9>FZ)=,
" ]HF^FG
FZ
std :: cout < < a < < " ^8 = " < +F )9/?
_
E>)T) b 93
<<
T)9/a)o'78E>/39F/ s9,=)X-_)^9>_)`F 1'7 /FG '&^)
/FD+F^^)`,=)DPN
a+b+c
)^9:
=sE`$,9FG /33)
FXF^F^1`>N)fb H'7!3M )Z>fb 9$fb )

(a + b) + c
a + (b + c)
FGE>`P
'X` f/N)7'7")3{
de_6hY"Y 1F^,`F^^)`>,)^Fk^&/F^!('73 3>4) fb3 `,=)^):",= )Z`3Z'X )
)F^F> k,=)D6hY YzE:)Fh^! >/r)X/FGF^1`)fb 1F^ '7 fb ) 19(' )
<`$/ ^)FG<F 93>h) ,93>FI)/3) $/.P ) F^^)`,=)$
93
^ D0S/6{,+.<.QQ<1q+01"F& {$/ )9)D)^9+smOfb :]1F^F^?F
gh.E>)F
>>
<<
`/ O)^9>F^ 3a/FZPNO'&F
((( std :: cout < < a ) < < " ^8 = " ) < `$/E)9)7) 9:FG)7:]19^F^F^
FZ/f//,DPN:^ )^ ) FG)/ std::cout << a
393 ,)G ,=)TFG)G$/-"%&` E)FT:]1F^F^?F^>GfbF /F b WFG) E>93 PN )+:]1)
,=)>&`8EF^)^ ]HF^FG
93-FG8{7gh!PN, :]1F (std::cout << a) << "^8 = "
F^ )>^PN_,=)^9,)^F!)8f,F P +[ :]HFGF^F.1``,=^^-/P)>+F^
P P)
<<
)a)$
g79 )^FG)!P )^9>F^:]H>F^F^F 3F 'X )
'X` `13F s \n
e>'&g :
2.1.14 Statements
GF )))XF& s /F^`"s ,399.s91`P 6ZY Y~ 93 )ZEF^F^ s 9/FD :lm`>)
g794>l `>) 3>m39Fr")^9 / FG))4/3 _):N/ F '&!)4FG))>){F ,sJ/<C) VXFk'&)8 ]HF^FGF '7XF^$ )9) "FG))>) )J
!. F^>)= V FG))^)n,F^,9/ 3F
'X ) FG`?/3 >FG)F. GFG)> aP )+9^/-_\)O))F ^ :]H`,=)3 ) e)^ esE)^)^ 3>$ g7+FG9G)FG)DEFGF^s9 FG)))DF&)^9 =C[./2+<'I<M `F^FI)
aP ) F^`M
m)D9/FD_>l `>)$!de? )o19`/ / FG)XFI))>)FD>f//,9)^
"ZF^>fb:/ ]HF^FGF
VzFG))>)DF )D>FG)^`:)3 )^
. P4F^,=``>13
r ).`)G^ m/39s9)o
P) 1,F )s$ ,= FG))^)^F")FG>fb>/kF P `13_g78`89> >F
)F^79 s HF /F T/Fx'773T9)x9,=) )z) ,=$/FG Os /`Fx b +)^9733
P .9/
de
=)> O )^^ 13FZP FI))>)F
power8.C
V
& ]av ;A
V&E3 F^` ) :]HFGF^y$/3F+) / : ]1F^F^
Expression statement.
FG)O))$ d )+>f,9)F.) ]HF^FG|s9,=) 31>F )+_ b_,F^PZ)F f, h
g F+F+
P ,)&PN P FG))>)F =/3a ,ZF^_/m9^/ )^9 FG)O))
b = b * b;
/F '7/Fn/=FI))>)FxFG)^)D'X )
std::cin
std::cout
OZ ]HF^FG8FG)O))F
\1,` "FG)O))k)^39,`>F !
>' 'I ) ) &/- ghF
Declaration statement.
`$/ sE )^9 9/TPx f ONs9TPx.fb
)vE b ) "3`) GF )))F
int a;
/3
int b = a * a;
V 3`N) FG)O)) `F^FG)+PZ )J/!U:/+012 /35-``,3-F^`{ de|,=
c
/
='7"3$r'&) Fd:!1qY;gD 39>`NO)F<1)>
`$sE P )"PN^
power8.C
'X>^ F+ )o1m *F+)
9/_PZ)
9:'Jf/s9 n/3
F /|Gf/,9_P&)oE
p Vtf/s9 3`) F 3l >) P ]HF^FGM4PN. ]/89 r)!`$/ 1``>,+O)
FGE>`W9`!9N/`>F&b k,)Z'X )&1``,=F m)&sm9$fb>FDb -:]HFGF^ a) F^F^ )^ O)
.39>`NO)
/F^ 9/Fh/a>l `>)D93.f//,4d )Fh>l `>)XFh) /1`$)^D^8PNh) 9:'
f/Ns9&)7F^&/33^F^F /3) 1"=1001q+Z1 + ) '&)8)Xf//,XP
HPEF^)$nd )Fkf//,
F )X>F^,) f//,DPr)^9D>' f^Ns9 g7D39>`NO)F F3
)^ )J )Df^Ns9
VXF )X`$/F^&PE:]HFGF^FG)O))F "3`NO) FG))>)n`$^^Fk,=)n)&>l `>)
P )^9"3`N) /3>FZ)^F7f//,
Return statement.
\1,` .FI))>)hFhP ) PN
M
'X>^
F / Gf/,9/.d )! ``,=FT'X) PN,`>){.g79.>)^,^?FG)O))
>f//,9)>F W9F^F )8PN,9`:)MBF"`89,))F r/3 9,=)F
t F!f//,
)nF^` )8m^Y /33^F^F )9)n)^9h`$/>4P )ZPN,9`>)^ `$/``>F^F4V&s9FG)^/`:)TP
)F^ )``$/ )F '7"FG89F
)9)7)"FG))) :<!0C:!M. ) ) `$/>$
return
89fb FG 9 :]8 F^P !) FG))>)
>),=F")^9+f,
return 0;
ePN_ )+)> 0 P4f//, 0 Z)) E>)FG1FG) 'X` 9/ FD`$/3-) main
0
PN,`>) P ,=Z-
& 6 v
;b0
,= F^,_^ Fx)DFG1)/`:)`$/ /3_F^_/)`$/=):Fk'7X9fbD)G13,`3H
'X )5)>N)F h
g aW9,=-899/F^ F )a`>)^X)9)8:]HFGF^F $|
6hY"Y +$,
- *.
()
#$
%$
&'"
*
$
&'
!"#
Figure 2:
8=V+M2J!012,+)z.Q/'AM012,+=<:!'E.W!LYL=,:!1[7NK1[C: :g.<=LM:</N:/'

power8.C
_vh:<!ZV .Q<'AM01q+<:!'j.G!LYL7:B1" uv5&10/-LMC:<!ZV ./V+M2J!012,+4<:!'E.X1[x)Y:g
N:/VJ 1Us$)<:!'j.j:<j):/ux1[@~1 N5YaN:/VJB8Z1q)|:!:/u.
:/3O;<
→B
:<Y)+.
5C+. g-uv5&10DG.QZ1q) Z1[!.z1" 657?,sgLM:<!././12|L7:gA`'I
,y65C+R657
LYL=<:B<:!A
' 10.X'IB:<]NY<<:/+465C657B u`<:^Q
2.1.15 The first program revisited
ed P{b, , )^9D ]=`,=)Os TW9X^F^,) PJMx/pTPNZ ` ,=9TP =9,=) f//,F Hb,
'X 1,`1+9)^` )9)4F^>)TF '73{=n:]8 )
7 )PNzP )^9Z/,=)F
) PN('X.9EF Compute a ^8 for a =? 15
15^8 = -1732076671.
& ]av ;R
CXs1fH,9FG ) ) EO'7:TPn mF^)fb!1,+sm>T`$/)XsE.b) fb mF^'X9)TFX
? !'&{3F^`,F^F&)^9F&-3>) a) :]H)DFG`>)s9,)&)!F^^)D ]H9N/9)aFZ)^ O)
)a)o1E
`$/9 3$/&'&)y1,.sE>F8,=) `>^)/F^ wdePT)-_)_)^`$/
int
FG,9 ) Pr!`89,=)) :]=`>3Fh)^9FkF^ 1) 6hY"Y*F^,) '&9`F^F^ 3lm>7P^})
_)_O)`$/FG,9 )$
ghF"F^,39F! b.s 3 9:'X<F 4/P):./ F") ./*!5 _s a1,.sE> /3 Pk'78`/ 158
).>fb `89,)^ 'X ) 1,+sm>F P7)F F^ x'X9) `/ '7`>8 ,=)8)./N}gh13
>'XF F )9) )8s9iF $/FG )^ W=]m gh_/,=)F 9fb89)3?)vE`$/3
)9) F `$ s9XP{3$/!'X)^)>>F P :
;!100::!V F^ W
e,= ) )^9D> ifm::integer
^ )^F PE`,=F^! : &F^")^9F4)o1m&F4fb>^8/FG H938)^9FnFn&P ) b:8FG)^^)FkP
6hY"Y '7 F^89
9fb!)8>9N`
s1
,=&/ /3- /33)
int
ifm::integer
`,3D)"3>W9) Pr)">' )o1E/k%&:!F7)"/``38` 9/3 9^/-
Á
Ã
// Program : power8_exact.C
// Raise a number to the eighth power ,
// using integers of arbitrary size
# include < iostream >
# include " integer .h"
Â
Á
Ã
Á©Â
Á
Á©Á
int main ()
{
// input
std :: cout < < " Compute a ^8 for a =? ";
ifm :: integer a;
std :: cin > > a;
// computation
ifm :: integer b = a * a ; // b = a ^2
b = b * b;
// b = a ^4
// output b * b , i .e . , a ^8
std :: cout < < a < < " ^8 = " < +PN+)F ) `89 b,|3 ) 9fb)W b,.'7 1
integer.h
3`:)^ ) 9T)9)Z/FG `)/F
:

power8_exact.C
&F^.) smOfbT/ 1b, `>8 ,=)D) `^>`>)&f//, P
158
Compute a ^8 for a =? 15
15^8 = 2562890625.
k,=)XF^+
,`NO>hf/,9>Fh'& '7
eP
b,a9E ) sE )>FI)3a
)
& 6 v
;b@
Compute a ^8 for a =? 1234567
1234567^8 = 5396563761318393964062660689603780554533710504641.
'&{)X3F^`,F^FZ)!)o1E
/PN,=^)>Da)^9FZsE m93a)^9:
BF&9
ifm::integer
3aPNh) HF^`T)xuv,FG)7'7 HF7b
e:]=`>>)h)^ O)7)731Fh)7 fbT)TF^ T ))F

int
P
: k,=)"'&>fb> b, 3xe / :]H>`FG `9/gT>"1,.sE>F rb,
int
P ) ,F^ ) )oE
ifm::integer
2.1.16 Details
gh>_F+'h P7'&)^-`)^F")9)+^
)+ )3?) _P
Commenting.
`39/XVX):]H)X`F^3 s1
eFI)^)DP `>), /3 */ e>93 P `), FX3
/*
s1?)8`89> g7)N4`>)"P ,= 9^/
`,3 /F^-9$fb8sm
power8.C
'&^)^)/F
/*
Program : power8 . C
Raise a number to the power 8.
*/
g 9F `9/F^ _?F^ ,F^PN, PN.> `>)F"FG /-F^>fb> F P7`39 7
s9,=) )Ts9> F )9) b,
39+)73N)^ `>9 X_)T339&PrF^,` `FG)^^,9`:) /FT`) 7b,?/'hHFD9fb )_1 aPN )^9.`F^
/3
)sE
/*
*/
F^,=
\1>)>F /3
O",F^>3aPN&fb>GF^^)Z`)F 'X )_FhP `39 9bT
/*
*/
c = a + /* don ’t subtract ! */ b;
=Z^$/39s9)o8^$/F^F ='7
39 )X/3=fb`$)T)F H
3 P `) 1 )>$
CT`>`$/F^9/ $ PN+6ZY Y `13.`)/F G3)^
Identifiers starting with an underscore.
W9>F FI)^)9"'X)8)^9&,93>FG`D`9/`>)^> =/ ),9 )FkFk9) /('73/`>`3!)
_
SD>W9)-4g7X)^,=)_Fk)9) 6579LM:</N:/'^'I<: F )7/O'738)^ ,9FG FG,9` G3)^W9>F Y
)> _FG>^fb3yPN8)>^ ,9FG s1 )
`89> 6789>F F^,3|F^F^,O) /FG) 'h H'&9>a)>3FG`Ofb:XF^,` +s /3
PN3 G3) W : Hs9,)ZP)a)^9: uv,FG)X>)& )
/F^F
de 6ZY Y )!F 13 >k/`>)` )-3>W9 f/NOs Define variables where needed!
3NO)TsEPN^Z) Fn,F^3 )7WFI)n) g77^$/39s9)o!PmD9^/ 8(fbF P PN
/ f/NOs h)9)kE$OFk F^&:]1F^F^ b)&`>^FGm93 3>W9) Fn$^s8/3
`$/ ` sm8PN,93 ,9` 1b_g79F ,3.F! `)^/FG) )F^ )>!9^/
N/,9//F xPN.:]/89 6z/4f^Ns9F" fb8) sE3`N^3 )!)8sE>99 P )
PN,`>)
'X> )> ,9FG3{
& ]av ;K
g7
PN,`>) F!/ ]=`>) 4PN,`>) F^:fb>/4'h$1FCD
The main function.
main
^)`,NrFGE`>N/)o F{)9) )^94>),= FI))>) `$/ sEn)^)3{{V
PN,`>)!'&),=)
main
":),=_FG))>) ) )X3sE9fbF `>F^8/F PE )k'7,33_'X)
4ghF
return 0;
3>W9) 9/F.sE5_39PN FG)^`$/k^$/F^F+FG)Ck).F+/5/_/ `8 3 )
)>kPN,9`>)^a
F '&9` 'XHsmX3F^`,F^F^>3N)> : gh>^PN '7XFG)`+)!)Z:]H9`>) >),=
FG)O))&93/F b,) 39+)"F^/
d )&FZEFGF^s9T)fb3 /
9^>W=]=F&)^^,9-!/339 )9/E P
Using directives.
std::
)1[:<!01[Fd dea`$F^!P
`39 =)F7'7,3
1 _b"M ^/cA=
using
power8.C
Á
Ã
// Program : power8 .C
// Raise a number to the eighth power .
# include < iostream >
Â
Á
Ã
Á©Â
Á
Á©Á
using namespace std ;
int main ()
{
// input
cout < < " Compute x ^8 for x =? ";
int a;
cin > > a;
// computation
int b = a * a ; // b = a ^2
b = b * b;
// b = a ^4
// output b * b , i .e . , a ^8
cout < < a < < " ^8 = " < )fb Fh.39>`NO)
FG))>)hP ) PN
using namespace

d ) /O'&F&,FX)_,F^+/ PN),=FTP 9/FI `
'X),=) ,9/P1 ) )=,-)
X
>W=]
xg7F `9/F^wF^>F 1,)^k9PN,1O) WFG)xFG) Os9,) ) 9/F F^>fb>^ 3$'Zs /`HF
X::
)9)7>fb)X,F&P^ ,Fê>)Z/9 /3=fb`$)+ )h
)F sE1 qr:!) BFrFG)G) '&)T)n_/uv{3='&s /`m &/FI `F{_T fbnh 1,.sE>rP1PN$O),=F
e8 G)`,N b)^9Z9/FG /` std 9/F '&)!&,+sE:kPE9/F> cin /3 cout )o'78F^,` F&PJ) 9/FG /`>
hd )DFZfb:â3 `,c
) 93a/F^8)&39>F^s9! )
std
v
& 6 ;U
H('z/r)F^
fb3` 9
`>)F
/F>TCT ) )>T9/3m) u`CD) sE 13-) 19(' ) 3>D)
9
'X ) )&9/>F u` )^3,9`> = :]8 bP '7&3>W9X f^Ns9h 3
F^>'&9:x MxwA ('7nn/F H9hPNr)^,=s9 m'X"'74)>r,F^ )n:]1F^F^
cout
)ZFh)&`$&'&>)>Z)7PN>FZ) ) FG)/393a,=)^9,=)ZFG)^$ =Z) ,=&f/Ns9/
cout
`/ /F^ fb3 ) f^Ns9 9/
P`,=F^ s9,=)x'7 _ /`>`3)/")G13,`
cout
/)>& T)^ O)D/FG8E$OF& )!9/FG /`>
7gh",9PN^)^,99)^!`F^ ,` F
std
)9). F^ ]HF^FG|P7,=.9^/ )F 9/ ) >PN>.) )^9PN),='7
)^13,`
3 s9,=)X)
8PN$O),=.P )^9 F^/ 9/ P )^9 FI)/393 s^bZ _a3
,= F^>)
,F^ PN$)^,^ P )^9"FG)/393-s^)9)h'7 3M
)Z>fb H(' 93a)^ O)
'7 >fb: )^39>3 )a,F^
g79+F^,)^FG)^/ sE9fH!P ,=!/ `/ sE fb:^
3 8`>,9 )h)^ )^` 3('X{
de ) 9/9/
=)^13,`+ 9/ cout eZ/_)>h T/F^
power8.C
E$O 9/FI `
n31F ) `/,F^D/9 '&),=)n) std:: ,9/W9`$)^ std
)h`$/9:fb> ^/`>`3)/ .^PN>&)^ F^>)^99.Pc) FG)/393 s^b
%&>
F.) FG`353='&s /` P
3`:)fbF VJN `)F+_
using
9/F /3 3>")
b: )G/` P ) F"3FÔs +)9) )89/ I)F ,F!'&> )
`FnP Fk) Z'7X9fbX)^13,`
3 31Fk) `ZP "s^=dePEF^ P
'X` 9$a)^
3`>)^fbF ='7 F^D)9)hPN_) $/ )9)7)T/
using
sE`>F7F^Fh$/39s9 /3aT3 `,) ) _){
2.1.17 Goals
V7)7)F7E) 1
b,aFG9,3?
Dispositional.
g,93>^FG)/3|)_s /F^`_FG1)/`:)`$/h/35FG_/)^`$/k)^>F.P+6ZY Y LM:<!.<./1q <L=<:/+: .<2+<'A<M ZFd+Z :!Fd+Z Z10/:/+ /3 Fd:!1 Y;gD
,93>^FG)/3aFG1)(]a93F^_/)`F7P )^9T/ power8.C Operational.
s{
G)`,N
de ^)`,N b,-F^,93 sE"s9D)>$
o+g_3FG`>sE
^) 9 s /F^` FGH)/`>)^`$/x93 F^_)`/ ):FXPT6hY"Y
b,=ZO'&'73F&93fb" ]/89F/
/FDFI)3
sE(fbg&
o )^m'X>)^9:X.fba`9`>)>ZF^ ,9>9`> FZ/3)W9>!
o ;
)^m'X>)^9:X.fba`9`>)>ZF^ ,9>9`> FZ+F^8 X:]H>F^F^9/F&3>W93sm('c
oTA+
W93,=)h'X>)>&.fbaF^8 X:]H>F^F^-Fh/f, &/Gf/,9Q
o"0_
:f/, O) +fb>aF^89D:]1F^F^t
o R
^$/3-/3'Z)DF 'X) PN,9`:)9/)o F^NO ) power8.C V ./1"'KLtDO,sgLM:<<.<./12 F. ]HF^FG '&` fbfb>F
e):/F 3) W :F
int
s 9^_+,)9`) 9
E>)^ H)!F^F^)Zm>) =/3 )F^F>
)
& ]av Ab#
2.1.18 Exercises
5&1q!5 S\657SQ[Du1[7N !5C:J!<:z.T * </!.3:<X R] 1q)J/M01 <:g./-)
uv5&V?
o
i;o
i/2o
i)!o
i!o
identifier
int
x_i
i/2o
iS o
i N<o
i25Qo 4x__
Exercise 1
A99_
_tmp
T#
x12b
&5 1q!5? S9657ST[Duv1"7N3!5C:/J<<:\.T *</!.c:< `K ,sgLM:<<.<./12M.-f)
uv5&V? <:</- ) :<]Fd:!1qY;! !. SX0V L7

o";
a
b
int
i!o
i;o
i/2o
i)!o
i/2o 1*(2*3)
i S o a=(b=5) iqN<o 1=a
i5Qo (a=1)
Exercise 2
(a=5)*(b=7)
(1
(a=b)*(b=5)
(a*3)=(b+5)
:I+" S^657I,sgLM:<!././12M.^65C+\Vh 5CF&^1q)J<M01 ) 1" sJ<:<<1[.Q Y -K)J<1q)J
uv57!657<:3657<.Q^:<B~F+~*!.:K:!Fd+Z !./-)I,sgLtU1[ V$C:)J/<1[./12M.
oTA+
Exercise 3
> !/:!'j1[ 657jFd+~*!.A S^657O,s!LM:<!.<./1qM.E65C+cV$5CF&j1q)J<M01 ) 1[ s{
3
<:</10.Q ) sgLtD1" 57u 657!.Q Fd+Z <.:/ &;!21[)Y 5&12!5 S|657!.TF+~*!. :<
CM. L=/<1 )O)I657/:<0SQ:/zP;<)J!<:!'^1[)EC=1 !ZV
o"0
Exercise 4
5C+`10.3657^./'A+" <.<+0C:/+=h'A;/<:E65C+`10.j)1"F
10./1 ;gDE;!V +[=C';<<:g.?;<!{
o R
Exercise 5
0u`/< )
2
$# sC # `g ;# g79TFG,9 )&FZ/FG 19('Xa/Fh)^9 D+.<\'j'A@'jC01~Lt P )^9TFIE`>) fb!1,.sE>F>
ghF :]=>^`F^a3F8)8 ,b, )?'Z) ks9,) b,y_$ ,F^-
Note:
/Q) b,- ) `89,))F
65C+:<,Y)+.65&:</1"MqNY/:g.
:!1[cLM:</N:/'
1[&LMCR)I7 LM76.]657<1":PLM:<Qmulthree.C
)*!

a, b, c
:!1[GLM:</N
:/' power20.C 65C+`:<Y)+.E 1[M NY<:
)OC LM76. 20 7.1[7N+'I./ F&]'j7~01LtZ12,+012M.T
S/:/' .<2)Y:/)1[&LMC
o R
Exercise 6
abc
Exercise 7
a
a
S:<' ./2)Y:/)
o R
> h:!1[7N|}! !0:<=1qE0:/M./'^10.<.12=-`657`SQ" u1[7N `] LM:</N:/' NYN&:g{
W
;gD)Y X.WVh,@.Q<-v657zUVhCNYv'I!./.Q)jLM-9;!CR+a657K.T'AK01"'I<-v.Q'AG<:!:<:g.
NYa1"M0:<Q)*/)|+.Wu`!"D
o A+,o "R
Exercise 8
# include < iostream > int main []{ int a ; int b; int c; std :: cin > > a ;
cin > > b;c = a * b ; std :: cout < < c *c; return 0;}
)D)T9^/J39('X a '7 PN_)G)3'hb
v
& 6 A
sC 7
g J`>)/9F )o'7 FIH)O] >^F> ] )^9> 3
9)Z31F7)TW]H3
`g g7E W=]=3hD9^/ `>)/9F",9.sE>!P `8EF^ ).:]HFGF^F qrFI)!) / /3 3`39nPNx$/` `8mF^) :]19^F^F^ '&9:)> ) F / ^f//,k f, h`$/
)9)X `8mF^) :]H>F^F^ _
`F^FG)&P 9^_^
:]19^F^F^F 9s9,)X/F^ Px)>
`8mF^)D:]1F^F^9F>
3h 2 9`> )T/Q'X )PN_O)fbD,=)^9,=)ZFG )9)h)hsE`>Fh/F^>h) ,9FG
g
VX33
F^F^s9T`>)Fk)^.)^9D?1FG) )Os 1
)>TF^,3_sE .`>)
) sm.)^ O)&F1Fh'& O)Z)T/c31F>
2.1.19 Challenges
p 57 0.<Du ]'A!657Q)PSQ:/'KLMC01[7N^657G<1"N5Y65GL=u`<: SB1[M NY<: /,)+.
.Q<Fd< 'j7~01LtZ12,+012M.T_v:<N:/' kW:< C1":<!.cMZVB65&://<- )Xu`\;<<~1q<F&R65C+65&10.a.<577U)
;<RST+./<:
p 57NYT+a SG65&10.?!5C+" <7N&^1[.3 ) 7 57u '^ !5GST+./<: 4:,sJ'j{
LtD<-1 S3u` /']LM7657 <1"N5Y65?L=u`<:A Sj
{,)1 N109=h'A;/<:^C./1[7N ;/65'A!657Q)+./uv5C+vu1[[;/B6573)1 W<:<<B1[ :!hM01['I 4./10,
1"7N000
657K0VgL=
-657hN5&-fu`G=R/:g{
int
:<!6~V /']LM7Bu1[65
{,)1"N
10=C'A;/<:g.ji,+.XV$ u10[ ,+./10ZV?012zuv57<@V$ ./2:g
10, 000
uv1[65.Q'I/u
5C+XU:2NY<:1[&LMC6./-W.Q 8/!01q 7[k&0k o& : 65&10.E:<,+.Q=-XVh
power8.C
.<57CD)^7.QK657K0V L=
ST:BV$C:/']LM72+012M.
ifm::integer
)
65C+9/'KLMCW6573<1 N5Y65
:!10X0u`cLM:</N:'E./L=u`<: S^x1[M NY<:^uv1[65 power8_slow.C
) '^7~01LtZ12+power8_fast.C
01qM./-9:<<.qL=/<01[F&!ZVxi/F&/:j657?,sYJ!P0VgL=
od 8=1"/uR 7 uM3y'A+./h:< :!CM01['A!./-j657<:< .<577U) ;<w7 LM7
ifm::integer
i2V$ )J % BuM3@:/Y)x100-^=VuVgo& # :/)J<:OH;< Y;gDO 7.QI657O.T'AOU:2NY
1[&LMC6.ST:j;<65jLM:<N:/'E.-a1091[.;/< <1 + 5CF&B657\LM:</N
:/'E.z:<,Y)657z1[&LMC S/:/'
T : ,sJ'KLtD<-`1 SBVh5CFdE D
u1065x/M<M6.
-Vh
!"657WLM:</N:'
:<,Ypower8.dat
)I1[6.z1[&LMC4S:<' 65&1[. ?;!1234567
VI.<2:g01[7N1[`u1[65
power8_exact.C
657j'j')
Exercise 9
./ power8_exact < power8 . dat
Bu 65C+XV$w5CF&|;<65|LM:</N:/'j./-j/:<+ 1"&LM7 -) [v10
u1065U:2NY<:|)U:2NY<: =h'A;</:g.Oi/J<501['IA)J*;gZ1[7N 657j=Cpower8.dat
'A;/<: S)1 N106.-fSQ:O s{
'KLtD2o& p 57< 'I+.C:<65701['A!.|2+d< ;!V ,J!5 SO657 LM://N:/'j.
)
657!.QK1[&LMC6.T @./1[']Lt~VO)Jj65&10.X7./1"7NjV$power8_slow.C
C:Ku`+<5i SQ:
./ <power8_fast.C
12/M6~V'A=V)1"N
106.;/65ILM:</N:'E.Eu10";/?.<Du <hN5Qo
-B:?V$},}.<2:gW657
LM:</N
:/'E.K~1[dz65&1[.
time ./ power8_fast < power8 . dat
u 10[:!h657\LM:<N:/' )I S<<:!u:/)+.jC LM7657z=C';<<:? S].Q/)+.
p &5 10.G/'j') 65C+a1[a
Ai657 g: ./=C';<<:!-657j^<)1[7N^1[ -v1[.X657B:<!D<FdMcqo&
u
& ]av A
5C+R)JVh&;g.T<:!F& A# . power8_fast.C :<C)E0u12/+. ST+.<+. power8_slow.C
i65&1[. 1[.uv5C+]Vh '^1"N5Yj sgL=/<9S/:<' 657=C'A;/<:@ S 'jC01~Lt~1q+01qM.qo e`: )J Vh
&;g.Q/:!F&O .qL=)+L S J!:O)1 W/:<<MS/:<' R) 10.j65&1[.\STJ<: .<2Y;gDI+.E657E1"&LM7
=C';<<:g.XNY!9U:2NY/:
5C+<F&<:3V$&;g.Q<:!Fd<-a0:!VA,sgLtU1[ V$C:^&;g.Q<:!F+012M.
;<657./'A+" <.<3=C'A;/<: SA'jC01~Lt~1q+012M. 65C+E:</,)J)H1"
!
`(n)
:/)J<:A /'KLMC 657 {q65 L7uR/: n SA 1"MqNY<: y8=1"/
'Vy)J2L=/)
n
a
a
`(n)
uv5C+*u`R/M./1 )J<:R+.9 2/']LM72+012 g-*u`v'A+d
u`![~{,)J )B;<VW:<!.<0:!1q!01[7Ncc657
SQ" u1"7NQ1[) SB'KLMC2+012M.T !
)J<]`(n)
657W1"&LM7 =C';<<: 'KLMC2+012
/M.10.<6. S .<2Lt./-Ruv57<:< 10..Q'Aa0+0h:/+ =C';<<:!-])|.<2L a
5C+.^657
t
t
i, 1 i t
SQ:!'
Exercise 10
ai = a j * a k
u1065
p 573/']LM72+012|1[.B/:!://!v1 S
65&:<3j,.<k2<
Lt.i i65&:<z'jC01~Lt~1q+012M. o+.X1[
:G,sJ']Lt /- /'KLMC
8
a t = an
-vu`^@C.QB657^/']LM72+012 a
power8.C
a1 = a 0 * a 0
a2 = a 1 * a 1
a3 = a 2 * a 2
Bu10.c)J ,)E+.`657c./'+[D!.< F+~*K S
/']LM7`(n)
2+01qjSQ: n
t
1[
./*!565C+657<:/z,s+1[.<6.X^/:!://! {2.<2L
t
a
!o@# 657^Y;<FdB'IT)J! S3/'KLMC2+012=-LM:<F&B65C+ SQ:+[
λ(n)
`(n)
n
1
-
λ(n) + ν(n) − 1,
uv57<:<
1[.E !./.365C657=C';<<: S3./1 N=1 ,M];!1[6.E S 1"657?;!1":!V
λ(n)
:<qLM:<!.Q<M2+01q S i.Q/B8/!01q 7 7 m o-)
1[.B6573=h'A;<n/: Sk% .]1[ 657
n
ν(n)
;!1":!V :/2LM:<!.Q/M2+012 S 4:| sJ'KLtD<-X657I;!1":!V :/2LM:<!.Q/M2+012 S
1[.
n
20
-9)?57<
)
-:<!.7~01[7NE1"

λ(20) = 4
10100
ν(20) = 2
`(n)
5
;o 10657<:LM:<F&c65C+657\LYL7<:K;/C)31" ! oK10.W+~uV.];<<.<*L=.<.1q;gD<-v:
. !565C+

n
`(n ) < λ(n ) + ν(n ) − 1
)EBFd+~*
& & Ab;
2.2 Integers
>W12jNd < +5Y <w5C+B)J/:AZ12T;<]6cNY/'AJ!5Y0-z+" !.)J<:<
10./ <M.Q<57<=u`<:g
://!d/:!-91[ ?D/!0h:<3E657Bbc<:gZ1[<:
<L=D) X
K+0h:6SQ:g.Q<57<:g{ <:g.T'^'EZC7NAi/kmYmCn/o
)
ST:j:<2LM:/!.Q<M01"7N
p 5&10.G.Q!012 )1[.Q<C.<.Q!.3657j0V L=<.
1"MqNY<:g.3)^+0C:/+ =h'A;/<:g./- :<!. int
L=/!01"F&!ZV$unsigned
Iuv1[[*int,:!@57u <Fd+Z7{
+z:!1[65&'I!01q3,sgLM:<!././12M.BFd<:3;<65 0V L7!.TEu10"4+~.Q3C)J/:g.<2)^657
Z1['^102+012M.3 SX657!.TB0VgL=!./-) w:<<D+,)?E65&1[.}57u 657/1[:BF+~*!.3y;/
:<qLM:<!.Q<M,)1[@657^/']LM7<:7% .]'A<'I:!VJ
% >.FT,= :]H)!6hY"Yt-"d ) F HFT).,9FG>T) 9,=)T8)8m>),=! 3^F
&
6 F^,F /38,=)^9,=)F4)k39>F9^9>)$ g79Z`fb:F^Fn3>W99>38s1+)&PN('X9
PN+,N1
SD>F /=)
Á
Ã
=
9
ST>FX67F^,F
5
+ 32.
// Program : fahrenheit.C
// Convert temperatures from Celsius to Fahrenheit.
# include < iostream >
Â
Á
Ã
int main ()
{
// Input
std :: cout < < " Temperature in degrees Celsius =? ";
int celsius ;
std :: cin > > celsius ;
// Computation and output
std :: cout < < celsius < < " degrees Celsius are "
< < 9 * celsius / 5 + 32 < < " degrees Fahrenheit .\ n";
return 0;
}
LM:<N. ST+5&:/<M57<102
Program 5:
deP!b, )^^,=)
) / ) =9,)
P(0 3 F-67F^,F &
b,
PN('X9 ,=)^9,=)$
15 degrees Celsius are 59 degrees Fahrenheit.
'&D>)
)
& ]av AA
ghF ,=)^9,=)4Fx9^13,`3 '& ) 7:]19^F^F^ GF ))>) +FhA (0XP )^9 Fh:]=>`,=)3{k%Z>"'7 PN1`>,9FZ) > f,9)
P )
")>)^`X:]19^F^F^
9 * celsius / 5 + 32
.D(01 h
g Fx:]1F^F^ `)/F ) _O^! ]HF^FGF b/3 celsius '&>
9 5 32
F !
f/NOs &PE)o1E
ng7F NP ,3 )9)o1mDF DPE) :!1065&'A!012X0VgL=!.
celsius
int
a6ZY Y.
/3 32 T3`_/ )>/F Pr)vE int H'&)_)7f,F7 Literals of type int. 9 5
3N)> O)nSD`>_/=)^>/F4PE)o1E
`F^FI) P{!F^ 1,`TPE3)F4P
) int
0 9
'X>^&)XWFG) 3)n+,9FI) ) sE 4g79Zf/,9&P "39>`_/9):/Fk)D3`_/1,+sm>
0
>>F^)^3 s1 ) F^ ,`8P 3)^F g7> a)>FDPN b)^fb ):FB4,
`$/>)Zf//,
s_'&)^
=s9,=)h)^9FhF&.`8EF^ )X:]19^F^Fâs9,)7P^c)^9 ,9^
−9
-9
F^,=s)^/`:)m>)R
o\1>`>) 1 1B+
A /3)^9 )^>/ 9
2.2.1 Associativity and precedence of operators
g79 >f//,9)^ P"/ :]HFGF^ F8)^| N^ ]H))
Ofb>^9>3 s1y) +.<.Q
/1q+01[FQ10012<.
/3 LM:</)J<!. PD)-fbfb3E>O)F deF^^) Z/F^F^`N)^fH)^F /3y>`3`F
3>)>^)-`$Z )F^F
/:]H>F^F^)9)_F) h_9 `89>)> )F^ 3{ -9fb //3=|),`3 F^F^1`>N)f1)o y`9>`>)5'&)5)-,=)^9,=)
E>)h \1`:) 1$;H
6ZY YQ/('XFT`89:)
)F^ >3 :]HFGF^F" 3> ) F^$fb O)^9>F^F O)
s1f1,F /`F g7F F+ b |_)_O)`F 'X>_'7
'Z)
'X '7 $/
3+4 5
x &/F^ '&)^
H:fb),9 )kFn) D`$k'&>)>4)Fn/F
3 + (4 5)
3+4+5
4%&: )^9nuI,FG)^W9`$) Fn)^ O) /33) F +.<.T
<1q+01"F& F^ )k39F
(3 + 4) + 5 3 + (4 + 5)
)&_O)^)> '&9` fON/)Z'7 $/{
gh&`Z) $ PN >F^Fn >)F^>F Fn)^ O)n'7Z fb&)^ H(' ) /N
12+ >)F^>F
k,=) )F Fh 13>)&`> =F^`X)D)o'7 ,F )9)hD,F^>3aFG) P 1,)^
1,)^
),)fb93 $/FI ) ^.sE>$ VXFG )^9:_F./'hHF )^9_) Ph:]H9`) /33
)F^Fh `$/F^Db,-^ )hF^,=T'X>"6hY"Y '7,3_9,=)h)^9>- qr>)h,FhFG)^) '&)_)
)o'7 FGF^)^N/E,FhPND^)>)`Z:]H>F^F^F
,)9`)fbZE>)^FZ9fb
Arithmetic Evaluation Rule 1:
/33) fbDE>)^F
99:Z`>3`")9/
g79":]H>F^F^
fbfbF7)!+,)9`) E>)^ =)"3f1
9 * celsius / 5 + 32
*
F^ E>O) /3?) /33) E>O) +VXx)=> ^+s99^ E:)F> de 6ZY Y
/
+
/F _)_O)`F {)+,9 )9`$O)fb E>)F
/3
9fb 5&1"N57<: `3`_)9/
*
/
) /339 )fbE>)F
/3
/F^ F )9)+ ,) `$) fb+m>)^F ;<1[) +
FG)^^9 )^ y/339 )fb
9>F ghF $/F ,= :]HFGF^ `)/F )^9 `$7 ^ *Q"
.(**
"K*C !
*P-T
6#. -0,
97
I2I- AP*
$&*? ,'.*H 7 -*6> *S*6"J
,4*
/
.*
& & A0
)F^>F
) F a`8EFG) ]HF^FG s9,)!Pi)_/33)
(9 * celsius / 5) + 32
E>)Z/3a)F7E:/3F
/3
9 * celsius / 5
32
k9^
^)>)`&E>)^Fh!P)Z/F^F^`N) fb/
Arithmetic Evaluation Rule 2:
de _)^9>_)`F )r31F 9) _)^)^>{O' )4F^,=s e:]H>F^F^
F{ ) 9 * celsius / 5
F^ 3{k,)4 6hY"Y ) Fx397P D0S< K:!1 N5Y )9) F ) )o'7TP)FG) F^,=s e:]H>F^F^F
.,=E>3 ):)>$ g7FTFT`F^ ,9>9`> P ).P /`>) )9)D) s O^-^)>)`" >O)FD^ 3>W99>3 )^8sE D0S< /F^FG1`NO)fbZgh ]HF^FG
F&)^9:PN
9 * celsius / 5
`/D >)F^ 3 /F
/3.,=x :]1F^F^+9/F )^XsEk$3
(9 * celsius) / 5
/F
((9 * celsius ) / 5) + 32
g7> F_9 FGF^, '7 9fbM )
3F^`,F^FG3 b>) Identifying the operators in an expression.
9/.)9) )1 W<:/<M 6hY YwE>)Fn_$ 9fb&) .T'A ) b{= :]=/89 `$/sE
+s99G m>)X/F&
9s9,=)X )&`$ /F^ sm ,9^ E>)&/FX
`-9
3 - 4
-5
Fh$/)Z+,FG)hsE"9PN:^3-PJ)"`>):]1)$ &FG, 1)FhFh`$O 9/3aa`$F^FZ'&>! )
F a
) s9,) /F^ )> `$/FGF H ) Fks s9 "133$ ) /33_F^& ]H)^ O)^9>F^F
) _ bD)":]1F^FâD$/39s9]
eF^>!/F^ )"SD>)/F F^`>)sE('` qr:) ,F `>9FG3>89)^9: `>9`:>) :]/89 4)
:]H>F^F^
d )+F `>$+)9)
-3 - 4
) WFG)
+,FG)DsE+,99G )>.FT>P)"9/3 F^3.E>/h
3 m'&9 ).F^`3 .F
s99^j
)>T^XE>93F _sE)F^39>F : k,=)7Fk)F ]HF^FG `$/ + ^)FG 3
/F
bn/F (-3) - 4a\1`7'7h:)n3l >)nf//,F +sE)+`$/F^>F '77sE>)G)>4_ b
-(3 - 4)
F^,= )9)7'7 19(' )!9FI'7>$
gh"`^`:)&`$/m )F^F&
(( − 3) − 4),
^F )Xf//,XPE)D:]H>F^F^
F
4ghF PNO'&FkP
-3 - 4 −7
, PNXO)>)`X:]H>F^F^F
&9^_E:)F&Y
Arithmetic Evaluation Rule 3:
) hs99G_`,)>G O^)F
)^9Z)3_FG) 8E^)/)
/3 k
9$fb >h9^`3` )9/
n
, FÂ :]H9`>),7 ^)FGF!F m>`3`F"`$/ sm
9 * (celsius + 5) * 32
(fb>^,3{ g >)T).`/ )F^F PN"F^,` G)N/ >)F^ 3? ]HF^FG
'7T9 )^9D,>F PcsE(fb H`F^39: )T//3= )F^ 3 ^)^F /F7E>93F
de) :]8 1)F7$39F7)+)"`$/ >)F^>F
(9 * (celsius + 5)) * 32
gh.SD>)F&FG`>) 3F^`>,9FGF^FDO' )8 )F^ . >/ :]HFGF^ fbf18 s9)Ô^6ZY Y E>)^F ,F^+)^9>Z)F H`>39>9`>FD/3aF^F^1`>N)f1)>F
& ]av AbR
2.2.2 Expression trees
ed / `8mF^) :]H>F^F^D)?`$/D )F^Fa3>)^> , 1, I) >fb> E>) 9/> )^9 )^ O).E$^F!'X)^9 F^_/FG)"1,+sm> P )F^F g7
:]19^F^F^*F ) `8EF^)^ ]HF^FGTs9,)
P ) ) e>fb"E>)â/3*)^F
E>/39F7)9)Z"/b/:]19^F^F^F
gh7`,=FGfb5ZFI)^,`>),=hPm/ :]H>F^F^8`/ `> sE fHFG, 3++)hPN^~P / s{
LM:<!././120:< kde ,=h; ) :]1F^F^ )^>hPN ) :]1F^F^
9 * celsius / 5 + 32
FhFÔ'&{
9
*
celsius
/
Á
Ã
9 * celsius
+
32
(9 * celsius) / 5
((9 * celsius) / 5) + 32
Figure 3:
R ,sgLM:<!././12 0:</KSQ:
) 106. D/N1q+L7:</M657!.Q!.
B?.T<F&<
9 * celsius / 5 + 32
BT)J!.G:/zUY;<!D)XS:<'
((9 * celsius) / 5) + 32
%&(' 3'7 >)D)F&)^^(|g7 :]HFGF^ )F^Px39:W FX)^9 :/
Px) )^^ {/3-)
E>/39FkPm)&) :fb m>)nsE`h)&)!BF !5&10U):<< 8)^9Z)^424/` m>/3
)yF^>GfbF_/F )^1)PT/)> F^,=s)^9>y'7 $/` _^ ]HF^FGh )
3>W9F& D S ) )G ='X ) PN,=^)^9:&`3={
2.2.3 Evaluating expressions
H
/ :]HFGF^ )G'7_`$/ $/F^ $/3 l )8EF^F^ s <Fd+Z*+012 .Q <!. PN
) )>)`_:]19^F^F^{ \1,` F^ 1,`-`>)/9F8/&FG,s e:]HFGF^F1``>,G ) )^> m3>>3 sa)Z)!P >f//,9)^{ =D)FZF^ 1,` )sE"f/3 '7 9fb )
_ b+F^,= )9) '78>f//,9)^8/ ]HF^FG 9 S/<: ) :]1F^F^9F!`^FIE399
)
+[ )FZ`3=>9fb"sE>->f,9)3{
n H+) ,=!; )^9FhsE>`F7`$O 4smPN
>f//,9)9
E'7+9$fb.)
>f//,9)
/3
)>G'XF^ m'7 3M ) 9fb
9 * celsius
9
celsius
,aPN_)^8) m>PNc)":f/, O){
9> '7&F^F^1`>N) )h>f//,9)+F^ ,`h'&).)Z`>^FGm93TF^ ,`&P 93F
.) )^ &f//3 13 F^ 1,` <L= N12,+[ZV3.Q:g6. ) )^> ghFx/F )9)4/ 13
)^9hF^ ,`X1`>`,=Fk+/P)^> /=)F4` 3^89fb&1`>`,=^3{ de ,^X; bPN :]=/89 ) 13"F^ 1,`
3,`FD.f/3a>f//,9)aF^ ,9>9`>&VXFGF^,+)9)
(1, 2, 5, 3, 6, 4, 7)
& & A@
)Zf/s9
Fkf,
b'7&s)/ )&PNO'& :f/, O)8F^ 1,`v de$/`
celsius
15
FG): =) F^,=s :]19^F^F^ a)^ sE >f// ,9)3 :]H)&F7_ b3
s1+F^,=^^,93.sE(]m
9 * celsius / 5 + 32
−→1
9
−→6
celsius
0 Y
9 * 15
Y ;
135 5
Y ;
135 / 5
Y
27
32
−→4
27 + 32
−→2
−→5
−→3
0 Y
;
;
−→7 59
g F^ ,9>9`>
h
F 9)^9:+f//3 39_F^ 1,` n39,`9 a3l >)
(1, 2, 3, 4, 5, 6, 7)
>f//,9)F^ 1,`Q)^9D>F^,) f//,DP
F )DF/4g7>TT+,` _&>f,9 59
) F^ ,9>9`>F 9P `,=F^ /3 )hFZ,FGE>`W93 s )^9!6hY"Y FG)939^3 'X` 9 F )+sE
,F^3{
de , F^_/ :]=/89 4mF^F^s9 >f,9) FG 1,`F '&xF^,)! f,
s9,=)
59
).F+/F^ 9)+93 ) '&)^39('X :]H>F^F^F 'XF^f//,F /3|>l `>)F+3>E3| )
i N<o 72 ]=>`>F^ $; i5Qo /3y)^9aST:)/F
>f//,9)5F^ ,9>9`> sm `F^w
eF^ 2 ]=:`F^
F^`:)?sE>O
' : Vz9^/ )9) `)/FTF^,` :]H>F^F^ )T:]H9 s )",FGE>`W93
sE9fH$ k,=)8)=,y113 FG)o1 7)F8F^FG,9-F$/FG5)^ fb
3 7F^`- )
)o19`$8_``,=F7'X 9D)^^F )+F 1, ") +,` PN,9`:)9/)o )+ F^D
P `>13
2.2.4 Arithmetic operators on the type int
de ) h9^/
b'7&9fbX/^$/3=.`,):3)Z+,)9`$)fbnE>O)F
fahrenheit.C
/3 /F '7EF7)^9Ds99^8/33)_E>) 4d )F7s1f1,F `,)>G O^)hF )Ds O^
*
/
+
F^,=s)^/`:)m>) g s9 FG)F&)>)^`TE>)^c
F /3-) 3> fb3 +././1"N='A<MW<L=<:/+:g. h)9)TO
fNs9ZPN )X)vE
1'X)^8)7^)F `>39>9`>F&93
/FGF^1`)f1)F> g79T`>),9/
int
1,.sE>Fh)9)DO9m$Xa)"`>39>9`> `,-!9)Z:f/) )&FZ) :/)J<: /
`>39>9`>F&)9)h_)^)^>F
qr:) ,F.3F^`,F^F.)_PN,9`>)^9/)F P7)F^
E>)^F. ),=
'X> /3
* +
F^P :]1 /9)^b k,=)X$/3=) 3fHFG m>)7 1,FZ.39F^`,F^F^{
The division operator.
F^FG
VX``^399
))+,F P _)_)`F
9 * celsius / 5 + 32
s1_)":]HFGF^
9 / 5 * celsius + 32
'7 `,3?>9N`+)^9 :]
& ]av AbK
Description
EFG) e`>)
EFG) e39>`>)
9^ `>^)
9^ 3`>^)
F^
F^
,) `$)
39 fHFÂ
e):,
13,,F
/339 )
F^,=s9)G/`>)
/F^F^)
,)h/FGF^)
39 fF^F^)
13 /F^F^)
/33-/F^F^9>)
F^,=s /F^F^9>)
Table 1:
Operator
++
-++
-+
*
/
%
+
=
*=
/=
%=
+=
-=
Arity
Prec.
(@
(@
$R
$R
$R
$R
A
A
A
$;
$;
A
A
A
A
A
A
Assoc.
P)
P)
^)
^)
^)
^)
P)
P)
P)
P)
P)
^)
^)
^)
^)
^)
^)
P:!1[65&'I!01q)}+././1"N='A<M <L=<:/+:g.GSQ:6570V L7
J!5w1[<:/<'I<M
int
:I)J<:<<'A<M<L7<:/+: ,sgL=!6.IwZFd+~*Q p 57A/']L=./10I,sgLM:<!././12 1[.
ZFd+~*i LM:<!{q1"<:<<'A<MO) LM:<<{,)J//:<<'I/M o- : :!Fd+Z yi"L=.<6{ 1[<:/<'I<M
) L=.<6{ )J/<:</'I<M o& J!5H+.<.1"N='A<M<L7<:/+:,sgL=!6. }ZFd+Z |+.? [S<
<L=<:) ) :!Fd+~* +.A:!1 N5YE<L=<:) ?657@/']L=./1[,sgLM:<!././12 1[.
ZFd+~*Q \"657<:E<L7<:/+:g.K1"=F&ZF&B:!Fd+Z <.MZVO)?5CF&KI W!6.T
& & AbU
'X ),) Olm`>)^)FDf//, /3?) PN,`>)9/)oaPk)./
k,=)!P
fahrenheit.C
'7T,)"c'&)_)")^)>7fb>^F^-P )^9 :]19^F^F^- )"=9,=)hP
39>F
15
6 F^,F H'7 >)h)"PNO'&.,)G ,=) 15 degrees Celsius are 47 degrees Fahrenheit.
g F F^,)F P /^|3l >)P ,=8:fH,FE 93y`G`>),.F^,)P
h
39>F
59
/=) EFG 'X9)TF 9 >( g7 9FI'7> FD)9)D)+s99G-3fHFG E>)^
5) )oE
89)F ) 1"MqNY<:)1"F
10./1q 5_O)_)`>F 39>9)^35s1|39 fm
/
int
g79F 31F )7`^^FGE3_) )X,N 39 fHF^8'X>X) ,9)^) P{)o'7.)>FkF > . )/EO)9/m,9.sE>
g79._39:TPk) ):!3f1F^ `$/?sm s)/3 'X ) )
The modulus operator.
s99^ 'AQ)7Z7. E:) =a_)^9>_)`Fn3)3s 3Ekg7"_O)_)`/9^,9
%
a = (a
3f
b)b + a
3
b
/ FG!3Fn_6ZY Y 4PN :]/89 P /3
^Zf/Ns9>FnP )oE
)hf/,9ZP b sE9
a
b
int
>^ )^9 :]19^F^F^
(a / b ) * b + a % b
9/Fx)7F/ f//,h/F ghh13,,F E>) F `F^3>^38/F4D+,9 )9`$O)fb E>O)
a
/39/Fk)XF^/Z`>3`WA+n93
/FGF^1`)f1)oGeP),4/Fk)X)^9: )o'7 ,)^9`$)^fb
E>)F
/3 *
/
dePZsE)^
/3
9fb_ eb)fb8f//,F n)
9/F+- b) fb8f/,9
/F
a
b
a % b
'7 xghFk89Fx)9)n)X)^> 3f1F^ ^,93F )Ju 8)Fk`$/F^4de4
P )k$/FI,) nXP
9/F& b)^fb"f//, 9)ZF&89>)O)3>W93a'X>)>&3fHFG ,3F&,= a
b
3O'&{ Z)7)9)4s1+)Z3))o
b)h^,93 137PN 3fHF^
(a / b) * b + a % b
/FG
3>)^>FX).PN,`>)9/ )vP4)^9.39,,FXm>) deP
9/FDf, E) f//,F
b
0
P
/3
O",3>W93{
a / b
a % b
679 s ` |) ,8:]/89I
/3 ) H9 9^`3`F /3 /F^F^`N) fH)F ) /` `,) h'7-:)
)^9-PNO'& f//3y>f//,9)^yF^ 1,` PN_,=
/):9)fb 6 FG,F e) /=)h`fb>F^{ 9 / 5 * celsius + 32 −→
−→
−→
−→
1 * celsius + 32
1 * 15 + 32
15 + 32
47
%&>^ '7"F^")^9G>^^ _/3Xs ) )^>Z39 fHF^
8
% 5 OK %"K(7PE.*?7
J
0L I/L*
6# .'(*
-*
/
(/E
I
%*
,S$
+7
M$&.)"
I *
*
$
9 / 5
M"KB
*
9/F7f/,9
-
%7
@K#*
1
/.*6
.
(7.#*
7$&(/
%C !. % J
& ]av 0#
?n9fb / $/3=X),`3!)4, O^
E:) /3")^9Frm>)
Unary additive operators.
31F '&9) :]1E`>)F ) f, P=) `8EF^)^n:]1F^F^ ,s!LM: F )^9 bO)fb P=)
f//, P sgLM: 4gh>7F X,9^
E>) PN4`89>)^F^F ),+ )F GPN,9`:)9/)o +
,sgLM: F7) F/"/Fh)^9 f,"P
Fh e:]=FG) x) f//, P )!`>8EF^)D ]HF^FG
+
,sgLM: 24/` Px)!) bF
/3
FD/F^F^`N)^3a'X)^)o'7
Increment and decrement operators.
-)10./01[! ,9^_m>)^Fh)9)h3l >& `>39>9`> /3-++
/F^FG1`NO)fH )vb
gh. 9`:)
/3 )^9 e3`>>)
) /F^FG1`NO)fb!gh.>lm>`>)!P
++
- sgLM: /3 ,sgLM: F7)^ `>^$/F^]e3`>/F^ =FIE`>) fbJk)
) `8EF^)^X:]H>F^F^F
++
-f//, P ,sgLM: s1 Xg7
) s9uI>`>)XPN:^3 )8s1 ,sgLM: FZ:),=3{ =X)FZ)^_ b
1
F^F^ ,sgLM: 9/Fn)"sED/f//, ?D/FG!F )^ O)k e`>)kF
LM:< s2+01q ++
/3aF^N PN
-gh
EFG) e`>)
/3|) EFI) 3`>^)
P) /F^F^`N)^fb VXF sE>PN ++
-)8>l `>) P )^9 `8EF^ ).:]HFGF^F ,sgLM:
/3 ,sgLM:
F")a9`:$/F^^
FIE`>) fb
-3`>/F^g ")+f/,9 P ,sgLM: s1
1 x/3 ,sgLM: ++
9/F!)sm/ f//, PN )^9F")'7 m gh
>)^,^|f, k),
kF /|^f//,
`^>FGE3 ) ) U) f//,
P ,sgLM:H;/0SQ:< )
`>>)+ 39>`>).)1 9N/`/ ?a/F^ F^$ )9)+EFG) `>>).F
L=.< s
++
2+012 9/3aFGN^ PN -gh"39 lm>>9`>"sE>)o'7>a)"`>^)hE:)Fh /3mFG)^W=]
)) Fh,F )Ô)3a
)"PNO'& :]8 T/-
# include < iostream >
int main () {
int a = 7;
std :: cout < < ++ a < < "\n " ; // outputs 8
std :: cout < < a ++ < < "\n " ; // outputs 8
std :: cout < < a
< < "\n " ; // outputs 9
return 0;
}
`>)h/33`>^)ZE>)Fh F^,=E> ,,F =F^9`> ) PN,`>)9/ )v!`$/+sE $/ 3.s1+`.s 99Z)ZF^F^)xE>O)o\1`>)^ 1$; 'X )
F7 1,f//)h
f//,
//339 )fbDE>)$4de33=P Fh f^Ns9 1)^9 :]19^F^F^
a
++a
/3y>lm>`>)_) )a:]H>F^F^
g79: F8-F^,=s)>)o ), P ,s!LM: F
,sgLM: F a=`>aF^F+1 ,9 f/) ) ,s!LM:
,sgLM:
> f, gh$F^
++
=
+ 1
,
g
s
M
L
:
F )9) )^9 PN>!:]HFGF^
F">f//,9)3 ` '&+ ) NO)^)> )"F
>f//,9)>3 0u12/ deP sgLM: F&/ >lm>`>) )F7`$/-_ b .39 lm>>9`>
CT ) )> 9/
3 k)F+F^,=s)>)o F+9) ) /F^5'X|9`:) 93 3`:)
E>)F& F^8m ,NOD/3a'X39>,F^>3 6ZY Y.ZVXP)>D/ 9) u`CD) sm $/FG )_fb3
) `>)`/Dgh )^^,)^ F&)9)D`>^)^ T3`>>)9 f/,9>FXs1
^ F^,`
1
P ,)&E>)^Fh )o19`$ 6hY Y `>13 )^ O)&)7 HFhl ) 9fb!F^^)^`,=)FZPNh)-
4,__^,9T)9)7)
& & 0H
gh-FG)O))F
93
^-s1fH,9FG
Prefer pre-increment over post-increment.
++i;
i++;
,9 f/) {FT)^9>T>lm>`>) FD)+F^/./3 )^9 f//, Pn)^9.:]1F^F^?FD)T,F^3{\4,
`$/_:]H` 9X)^9>t'X )$/`_)>7Gs )^ 'X)^9,=)klm>`>)!)&sm9$f1 PE)XFG, ,3T/- >fb>kb, fb7)F4` `> b,8F^,3 )nPNn)7 9`:)
E>)$ Mx e`>) Fn)&F^89:kE:) sE`$/,F^Z)&f,XP
`/_F^89 sE
++i
$3 l?)!fONs9 &de `>)^FG) )^9 mFG) 9`:)&9/FX) I^.sE> +) 9/
i
f//,"P nVXF7 e`>)hFZFG89> )Z/F^.)3FZ).sE"T 8`)$
i
R/'A:g h'&)^ I e`>>)k)^39F4)!sEZh 8`) sm`$/,F^X8_/+`$F^F
)8`89>T$/ F"'& ) f//,8P7/ ]HF^FG F!) ,F^>3r de F^,` `$/F^ )
`89:D_ ` F^ )FT('X?)_>9N`+)^9 EFI) `>^)T )+F^,=`+`13.s1 I e`>) _?_` N9,9/.F!/ ) O){ %&O'7:fb> {)>+F!Os FG,=)
sE>W)7
` F^.!EFG) e`>>) '&9:T! 9`:) '7,3_39+/F '7e de
)^9F
`$/FG =b,-F^,93 ) bT)Ts ,=3-P^J) `89> /3a) Zs1
b,^F^P VXF^ bEFG) `>>)n/38EFG) 3`>>)kh)&+, O^_6hY"Y*E>)F4)9) O
P)&/F^FG1`NO)fb4g79Fh_ bFh)^9>h,9F^/!O9m$ZF^>'X9)7`>,9)>),) fb
gh/FGF^)"E>)
F f//Ns9.PN.4)oEF FG_\1`>)^
Assignment operators.
=
1$;H k,=)T)>+^ FGm`W9` E>)FD)9)T`.s9!) )>)^`"m>)^FT'&) /
/F^FG)$wg7F^ -)-s99^ E:)F
-= *= /= /3 %= [gh :]19^F^F^
,sgLM:k
,sgLM: 9/FZ)">lm`:)DPx/33 ) f+=
/,9 P ,sgLM: /aGf/,9! ) )"f//,!P
+=
,sgLM:k f//,g ghTs uv`:)h^PN>^>3).s1 ,sgLM:k F ^>),=3{kg7F Fh >/ )
sgLM: F 1,f//)) ,sgLM:
PT)a9^ `>^) )-:]19^F^F^
V&F_smPN ,sgLM:k
,sgLM: F 1,f//) ) ++ sgLM:k
,sgLM:k
sgLM: |+=>1/ FG` )
+=
=
+
N)G)>h:]H>F^Fâ:f/, O)F ,sgLM:k )v'&`
ghTE>)F
/3 %= '7 )TF/&P /F^ 1s /F^3 )^9TF^,=s)^`>) -= *= /=
+,9 )9`$O)
3fHFG
H/3a39,,F E:) H^FGE`:)fbb
VX )^9TF^F^) E>)F 9fbD9^`3`TA He 1)> s 93_Z'7$ H 8)9/)
F $/F a=(b*c-d) )>&)^9:)`XE:)F>ng7Fh
F ,)T)^,9 )fb a=b*c-d
2.2.5 Value range
V*fONs9TP )oE
F&F^F^1`>N)3
'X)^ s$,) 1,.sE>hP >^8`>F /3
)>PN
int
/FG+'&)a W=]=31,+sm>hP s9)F =F^$
`$/E)Fh {,;<10:/2LM:<!.Q/M2+012 b
b
\1,` y->>F^)) 89F!)^ O)8/ s uv`:) Ph)vE
`$/ /FGF^, W9) int
_/a3lm:)Tf//,F!\1`+/-s )T`$/ 39:E3) a9fb+)o'7_FG))^F m)._O]= ,
1,.sE>hP ^>F^)s9 f/,9>FZF b 9/3 ) `>),9/mf//, /"Fh3>W93-/F7)"FG>)
2
{−2b−1 , −2b−1 + 1, . . . , −1, 0, 1, . . . , 2b−1 − 1} P b ,9.sE>^FW4,
`/ W93 ,=) )^9TF^_FG)k/3
NFI)
f//,F _b,h`89,=)>
2
int
,F^.)" s9^
4gh"`^FIE399+`39 FhfbaaMx/JRH
limits
* ( 7*
*
,6>
%*
/
(/O/
%
*
2
>
"
7
H7 ,
%* R."51
*SBO/
/@BL
% *H
-2*
/
/4J
2
.
R%*6
$
./O
#"
%$&'(
& ]av 0
Á
Ã
Á
Â
// Program : limits .C
// Output the smallest and the largest value of type int .
# include < iostream >
# include < limits >
int main ()
{
std :: cout < <
<<
<<
<<
return 0;
}
" Minimum int value is "
std :: numeric_limits < int >:: min () < < " .\ n"
" Maximum int value is "
std :: numeric_limits < int >:: max () < < " .\ n";
Program 6:
9>_b,8^,9 )Z/
PN('X9 ,=)^9,=)$
limits.C
LM:</N. t~1"'j106.
_ ; e s9)nFGHFG)^ b,8'&=FG)4b.>)n)
Minimum int value is -2147483648.
Maximum int value is 2147483647.
de3
3 /F
/b, `/.39>39,`n)9) )^9n1,.sE> PHs )F ,F^>3.)^&>FG)
2147483647 = 231 −1
/
f//,h8)FnFIHFG)>}F
xV )k)F E) b,_h) F^,=EF^3 ) ,39:FG)/3 )^9
int
32
:]19^F^F^
39:)/ s9,=)k'7hsE:fbh)9)4b,>)k)F
std::numeric_limits<int>::min()
3$1
d )hF `$Oh)^ O) )"O)>)`ZE:)
F :]=`:9)h)T, O^
/3 )Ds99^
/3
+
/
%
`$/) '7 :]=/`>)? b )^9> _)^9>_)`$ `,):^ ^)F >fb 'X ).O,)F
8^FG)^`>)3 )a:9^F^)s9
f/,9>F-g798^$/F^ F!)9)")8f,F P7`8EF^ )
int
:]19^F^F^FT`FG)^^,9`:)3?P^ )F^.E>O)FD`$/ ,39: hTOfb: 9O' ) f//, / P
) )o1E
.gh.FG)Ts1f1,FDF^,` :]8 FD)+ ]HF^FG
V&FT'7
int
2147483647+1
9fbZuv,FG)XF^>
)^Fh_O)_)`/+`^^`>)&f//, P
FZ)h>FG)Os "(fb>
2147483648
)T)vE
b,=XFG1FG) =F^+b,'XE>fH )s9>)ZF^T)>hf,
int
\1,` ,93> h/3 (fb> 9('XFT^. F^>fb>^+s9 _/
/`>)`$/ O99`$)F =s9,=)
)7'7,3 sm!/>fb-TF^>fb:!s9c)7) 19(' )9)7)>
`$/a``,=$
2.2.6 The type unsigned int
VX s9uI>`>) P )o1E
`$/ 9fb9>b)fb f//,F s ,=) P)^ '7 9 ?'7 'X) 9)^,/
int
,.sE>F> &F^9 )oE )9) >>F^)^F 9 e9>b)fb.f,F /O'&FD)a:]1)3 )
1
/TP{EF^) fbXf/,9>F7'&),=) ,F^+Xs9)F46hY"Y[(fH3F7F^,à )o1m =)hF `$/3
kCTa)^9Fh)o1m ='7"9fb /m) )>)`Xm>)^F7'7!/F^+9fb!PN
unsigned int
int
SQ
,
20*'(2)*S"O
%
-(7 $&
0
,
= {0, 1, 2, . . .}
J
& & 0;
'X )
)^9 F/ ^)F H9^`3`F&/3-/F^FG1`NO)fH )Fx fb es9)7>FG)O)H)
b
f//, /"P
F )!FG>)
unsigned int
{0, 1, . . . , 2b − 1}
P b 9),=/1,.sE>F de93>3 '&9>+b,+>9N/`7H1`>`,=^`F4P
s1
2
int
unsigned int
) )h_9^13,` )^9 PN('X9 ,=)^9,=)$
limits.C
Minimum value of an unsigned int object is 0.
Maximum value of an unsigned int object is 4294967295.
qr )>/F!P )o1E
?b8)^>/F P )o1E
xPN('73 s1 )> )^9
unsigned int
int
>)G)>
:]/89 93
)>/F P )vE
'& ) ) u
U
127u
0U
unsigned int
f//,Fh3N)> O)
2.2.7 Mixed expressions and conversions
) P7)o1E
2 ]HF^FGF+_ fbfbF^,=s ]HF^FGF+P7)o1E
int
unsigned int
:]=/89
F ab/k)>)^`.:]HFGF^ s9,) '&9) ^)F!)o1m_/3 f//,(~de
17+17u
F^,` 'j1DsJ)w,sgLM:/!.<./1qM. )E:/3F889`)^ =F&<:g,) )^?) 'A:<INY</:/+
)o1E/ n?)
6ZY Y FG)/393 ) >/k)oEF
g79:PN )
unsigned int
:]19^F^F^
FhP )o1m
/3a>)Fh>f//,9)3FG)>asFI)> /F
17+17u
unsigned int
17+17 u −→ 17 u +17 u −→ 34 u
g79F ){sE FG>'X9) `PN,F^ F^`k._)_)`F $)xF uv,FG)x) )> 'h ,3 ) F^>)XP )^>F F7T>{)^ ) F^>)ZP 9),=/E,+sE:F :x "")
'h4P=/!3>m> uv,FG)W9`$O) PN )n'h ) Fr39n.6hY"Y /s9,=) ) $FG) )^9n`fb:F^
F7'7 e3>W99>3 Z b)^fb
f//,F+^ G`fb>G)3 -)-)F/ f//,P7)vE

int
unsigned int
b) fb
f,FD"`fb:^)3 ) )
f,")9)ZF^,)FZP e_O) int
unsigned int
_)`$/Y
/3399 b g79F&, FG)s9F^FD s9Buv`:)-sE>)o'7 ) f//, /FTP
2
int
/3
unsigned int
de89`)X`fb>^F^F ?)^9 )> 3^`>) _?F^``,=!s9,=) O )!/'hHFT'7
3>W93{ 6 F^3>hPNX:]=/89X) 39>`NO)F
int a = 3 u ;
int b = 4294967295 u;
g 9.f//, P
7
F rF^`.)FDf/,9+F )^9./+Pn).)o1E
k,=)"Pn'7 / F^F^,
a 3
int
) ; es9)7FG1FG)JPJsE(fb H)Tf, P Fh8 )) 39:W 3/``^399 ) .)
b
6hY"Y FG)939^3 =FG` )^9"):/
Fh,=)F^3D) /"P
4294967295
int
& ]av 0/A
2.2.8 Binary representation
VXF^FG,99
b
s9)7>F^>))^H'7!/ $/3= H('
)9)7)")o1E
int
`(fb>Fh)"f//,F
−2b−1 , . . . , 2b−1 − 1,
'X
`(fb>F
unsigned int
0, . . . , 2b − 1.
de )^9F+FG,s9F^>`>) '7
'h/).) ) b-`F^>+1 O) ('Q)F^_f,F+_>F^>)3
^ ,F^9 )
f/Ns9 s9)F> g79F.'&k/F^ ^F 35^) |F^P7)
b
_):N/m )T9^>fH,FhF^,=s9F^`>)r
gh ;!1[:!V|,s!L7M./12 Px. O),=/E1,.sE>
h
F )^9 F^,
n
n=
∞
X
bi 2 i ,
i=0
' >^ )
X
9
, ,_3>)>^3 `>1 `)F&P
bi
)csE+ > = Z:]=/89 {0, 1}
H'X)
W9)8_/_P
13 = 1 20 + 0 21 + 1 22 + 1 23 .
g79_F^ 1,`_Ph)^9
:fb>F^
3>.F `$3 )8s99^ ^>F^)) P ?g7
bi
n
s99^:9^F^))P
F
=PN& ]/89
13 1101
Conversion decimal → binary.
n =
∞
X
g7"39>))o
bi 2 i = b 0 +
i=0
∞
X
bi 2 i = b 0 +
i=1
∞
X
bi+1 2i+1 = b0 + 2
i=0
∞
X
bi+1 2i
|i=0 {z
=:n
}
(fH3F FG89_/ )i)?`89,=))_s O^ >FG)O)5PD fb|39>`_/
1,.sE>
zgh $/FI) FGW9`$/)_`>1 `)
P ) s99^5:]H /F^ P
F
n
b0
n
3 4ghD)> `1 `)F
H`$/_F^,=s9F^ ,) 8sEX:]1)^/`:)3_s1_91")
n
2
bi i 1
F/D)>` ,9T)
=.:]/89 PNn n==(n14− b'70)/2
:)
13
93
?
b0 = 14
2 = 0
n = (14 − 0)/2 = 7
`)1,X'&)
/3
>)
13
/3
= n = 3 '7T>)
n=7
b1 = 7
2=1
n = (7 − 1)/2 = 3
13
/3
'X` $fbF&,F7'X)^
4deaF^,_^ b2 = 3
2=1
n = (3 − 1)/2 = 1
n = b3 = 1
)Ts O^>FG)O)aP
F
14 b3 b2 b1 b0 = 1110
g ` fb>^) fb s99G 1
,+sm>
) 39>`_/
Conversion binary → decimal.
b k . . . b0
>>F^))H'7 `$/-`"/b/ 9
, FG ) 3)^)o_P OsEO fb/
k
X
i=0
i
bi 2 = b 0 + 2
k−1
X
i=0
bi+1 2i = . . . = b0 + 2(b1 + 2(b2 + 2( + 2bk ) . . . ))
& & 00
=D:]8 9)^8`fb:^)D)!s99^ 1,.sE> b b b b b = 10100 )83`>_/E>
4 3 2 1 0
F^)) H'7 `
89,=)
(((b4 2 + b3 ) 2 + b2 ) 2 + b1 ) 2 + b0 = (((1 2 + 0) 2 + 1) 2 + 0) 2 + 0 = 20.
Representing unsigned int values.
n
\1`"/
unsigned int
f/,9
{0, . . . , 2b − 1}
9/F4Xs O^ >>F^))+P >9)^ :]=/`>) W9X,= '&).$399 >F b)F s99^
b
>>F^)) F `$/`$/nPN_) PN FG)
,Fâ)
f//Ns9.s9)Fqrb )
n
b
f//,D/T)F^P 1)F FG)/TPN_)7F )h:]H9`) F^`>^sE3
s1)"6ZY Y[FG)939^3 s9,=)Z9^39 /) F^"_ bF7F^F^! /`>)^` VXF7)>" b
f/,9>F 2 unsigned int
/3 )^9 F/T,9.sE>7P es9)7 )^)^>F =$/` )G)> `13Fh f//,=
H)F
b
b=3
HFh/FhPN('XF
n
0
1
2
3
4
5
6
7
>>F^))
000
001
010
011
100
101
110
111
V*` 'h_PE>F^>)
f//,F ,F^ )DF/
s9)^F
Representing int values.
int
b
1>F"F PNO'&F deP4)^9 f//,
F bO)fb '7.FG)^ ).s99Ga>FG)O) P
n
n
)FGP 1,.sE>hP
{0, . . . , 2b−1 − 1}.
g7 O)Z'h_'7",F^"/E) es9)7 )^)^>Fh)^ O)ZFI)^)h'X)^ b
0
dePh)^98f//,
F b)fb '7FG))8s99^ >F^>))^ P
na,+sE:
n
n + 2b
P
{2b−1 , . . . , 2b − 1}.
g79F H>3F ) FGF^ es9) )G)>F / )^9k9>F )9) FG)^) '&) )k>F^,)
b
1
b=3
>>F^))FZ
& ]av 0R
n
−4
−3
−2
−1
0
1
2
3
:9^F^))
100
101
110
111
000
001
010
011
ghFDFD`$/>3 )^9 0u`% . 'KLtD<'I/M >>F^)){ de )^9F&>FG)O){/33
)o'7
f,F
/3
FDfb>G /FG DFG89a393 ) >F^>))^FT/``3)_)
int
n
n
,F^,9/=,FnP s99^+1,.sE>n/33)93 h)^9h(fb> 9O's9)veP /Y: =k:]=/89 ) /33
93
`$/F^"P
H'7"`89,=)
−2
−1
b=3
+
110
111
1101
de-) P)FG).(fb> 9('Js9) 4)F.fb>F
n)_>F^>))^|Ph)
>F^,) −3
101
-)o'7YBFX`>8 )Zg7F&'7 1FDF^` ) s99^ 1,.sE>Xsm3a) `13P
F
n
)^9:
!g71,F m'&9> '7+/33 ) s O^-1,.sE>F&PN
/3
m) FG,9 )TF
n
n + 2b
n
n
`,) )
13, b /3_)^9:PN />F 'X)
)
)FG)4s9)F
n+n
2
n+n
b
&F^!)^9h)o'7Y
BFk`>8 )x^>F^))8'7X`$/(' sE:)^)>k,93>FI)/38'X9)n9 EF&'& b)fb
f//,
>)FX`>fb>^)^3 )^8)vE
&gh FG)939^3
int
n
unsigned int
FGE>`W9F )9) PN7)F 9/F ) sET`>>)>3_s b k,=)7,3>7)X)v'7Y
F7`89) n
2
)Tb)fb
f,
/3
)^9D>F^,) EF^ )fb
f//,
9fbD)
int
n
unsigned int
n + 2b
F/ :9^F^)) xg7F4$/F )9)4)h`>fb>FGFx9,^+``>),9/ b93 9"/`>)^, `89,=))_) bF7 /`
gh"6hY"Y FI)/393_31F7) F^`:sEX)T,F^DP{)X)v'7Y
F `89) s9,)k)T^,9
PNZ`fb>^F^-P
)
Fh`$O_)f/)>3
s )$
int
unsigned int
2.2.9 Integral types
g 9:&FnT1,+sm>4P )>kPN,939/)/1)oEFn)^">>F^)kF^393 ,9F^9>38)^>F 7
F^Z)DSD>)/F4FG`>){ g7F^&)o1mF _+39 lm>kP
/3
'&) ^FGE`:)
int
unsigned int
) ).f//,/ VX )F^_)vEF _`$/3 1[MqN
:/+`0V L=<. /3|PN+$` |PZ) /{) E>)FZ g s98zeM / Ab
K &O fNs9 'X )) F )^F `3`>F /F^FG1`NO)fH )F!/3|PN,`>)9/ )K
F e,=|) )_s1fH,9F )F"39`>))>3 s )
>FGE`>)^fb
f//, /F :
& & 0@
2.2.10 Details
g7> /F^ 3`_/k )>/F Ph)oE
VX <2+ ):/ FG)^)F 'X )
Literals.
int
)!3 ) 9PN('73sa+F^ ,9>9`> P 3)F7P
) 7g79 f,!Fh)!1`:)/{1,.sE>
0
0
7
>>F^)^3 s1?)^9 F^ ,9>9`>P 3)F PN('X)8/3 = :]=/89 E)8 )>/
0
9/F7f/,9
1
0
011
,sJY)J<1['9+=1):8/F +FG1)^8)h'X) 0x 9PNO'7>3
sa.F^ ,9>9`>!P 3)^F P 0 ) 9 /3
>)G)>FTP
) !gh.f, FD)+:]=/3`_ 1,.sE>X^>F^)3 s-)+FG 1,`
A
F
P73)F!/3 :)^)>F!PN('X)/3
. ]/89 ))>/
F f/,9
0x
0x1F
31 = 1 161 + 15 160
fb /y:]1F^F^ )9)`>9FGFG)F_P F^ Logically parenthesizing a general expression.
1,`!PxE>)^FX/3-E:/3F '7 'h/)Z)833,` )!`$/E )F^F=D$/`
E>) )DF^ 1,` H'7 19(' )F )v )Fk>`3`X !,9.sE>ksE>)o'7 D/3?$K F^ g Os D
M /DAbK PN )T)^9:)`7E>O)F 1/3_)F /FGF^1`)f1)ojeP) ),:
de `$/F^ P ,99GaE>) ) /F^FG1`NO)fH )v
FGm`W9FX-'&9` -FG3 P ) E>O)X)^F
E>/3F7) sm"PN,3{
qr:)7,9F7`F^39: )DPN('X s9FG)G/`>)7:]=/89Z)^+>8 9/FG &)9) '&9) '7D3 >^DF
`89>)^>/r/3a)h^FG)^`>)3)^8 )>)`X ]HF^FGF
:]19^F^F^
)v
`>39>9`>
/F^FG1`NO)fH )v
1
x2
2
x3
x4
4
2
2
2
1
4
13
13
16
r
l
l
r
%&> FXO' )! )F^F !s9)/3 PNT$/` E>)^ '7 39>)P)F ,Y)1[7N
E>/3 93>W93 /FZ)^9!P)X9/3aFG3 E>93PNDP)X/F^F^`N) fb"E:)F /3 /Fh)
)D9/3 F^3 E>93 )>^'&F^Xgh $399 E>/3 PNT 9`>,39>FX>fb>^)8)
).>f//) F^3.sE>)o'7 /3?) :]1) E:)"P Du`<: 9i ^`3` )^ 8de
)>Z'73F :fb>^1)^99 asE>i )o'7-)FG")o'78E>)FZF GOs9sm3 s1 ) GFG)Gi> E>)$
dea,=h:]/89 1)"$/3.E>/3 P Fh)TF^,=s9F^ ,9>9`>
)+) P)
x2
x3
3
2
P F^`T) 9 ]H)Xm>)hP O'7:Z`>3`!).)"P)&P F7 3 ) `$/F^ P s99^ E>)F 9'7./F^ W 3-) .Q/)Y:!V E>3/
de3 )^19!9")8)
)> FG3XPE)D$/3!m>/3{4ghXF^>`3 OÊ>/3
PN7 `,93Fn>fb>^) )
) :f/)!FG3.sm>)o'7? /3 ).:]H) E>) P 657^.T'Ii :^ u`<: >`3`
i )^.)^9T/3 E>/3
,DF )9)7)DF^`>939G m>/3
)9/
ng7"839 lm>>9`>T
i
/^$/3=_3FZ'Xa/E:)ZP ) .T'I `>3` E$OF
VX``^399 ).)F 3>W9)
b)TF^`39^_E:/3 P Fh
,=7:]=/89
9/ {'7+9,) / P >)F^>F ,3?) FG,s9F^ 1,3 `8`4Gx4FGE3 )^)
$399.E:/
3 H)!E>)h)F^>P /3) F^`39^
E>/O
3 eP /Y
:
%&> FX)!)s9 PNT,=T:]=/89 b/
>99/`>3 '&)a) F^,=s9F^ ,9>9`>F"P4/ PN,=
E>)F7)9)ZT9,)Z
O)^9>F^FX``3.) )T,F uv,9FI)&3F^`:sE3{
x1
3
& ]av 0K
:]19^F^F^
)v
`>39>9`>
/F^FG1`NO)fH )v
1
2
3
x1
( x1
1
2
4
r
1
x2
x2
( x2
( x2
2
2
13
l
2
2
x3
3
4
2
13
l
1
16
r
3
4
x3
x3 )
x3
x4
x4 )
x4 )
(
x4 )
4
4
Z(''7 F^89| ,=)
)>)>/D >)F^>F )9)
'7 9fb?s)3&) 1 ) +,9 )9` )F4)^ /`>`,)$4dea,=Z:]/89D'7 >)h)":]1F^F^
2
3
4
( x1
1 (( x2
2 x3 )
3 (
4 x4 ))).
n+F^h_/` /)^9F '7 b3,=) /3 '7Z9fbXTPN, >)F^ 38:]1F^FÊ )h,)^>
/ Pn ^)FGF!`$?sE+3^9m3 /brPn`,=F^g k,=) )+)^ O) '7+`$/)":]HE>`>)
F^,`-`TsE9fHh >/en6 F^3>h)"PNO'&+ ]/89
:]19^F^F^
)v
`>39>9`>
/F^FG1`NO)fH )v
1
x1
1
2
13
r
1
x2
( x1
x2
( x1
x2
2
1
g79TFG,9 )+ ^)FG 3a H
] F^FG-F
2
2
13
l
2
2
x3
x3 )
x3 )
(( x1
1 x2
2 x3 )),
'X` 31F.).FGE`>P )_>f,9) 3> 9).`F ) ,= F^`>,9
F )9) 6hY Y
_/O'&F :]19^F^F^FhPN7'X`
)^9T_/`&'7 HF7,=)4g79D>f1,F s /3`$F^"F 8EF F^s9 =PN&:]=/89 =F^9`>!/ms O^
E>)^F7P )"F^/D9^`3` F^+ fb )"F
/F^FG1`NO)fH )vb
?. fb 3F^`>,9FGF^3?('
f/,9>FT `fb>^)3 )
Unsigned arithmetic.
int
unsigned int
f//,F /3 fH`?fb>F^1Jgh?_ FGF^,@ 'X9)
)53|'&) e^>F^)s9 f/,9>F
/FG ``,=F&3,=+>f//,9)
Px^)>)`&:]HFGF^FZfbf1+ P )T)o1EF>
g79 6ZY Y FG)/3 O3*`)/F ?^,9?PN-)F = ",F^3*)/ )oEF T)
)>)`"E>)^FX'7 13, b {fb
es9)D^>F^))r!ghFD/FX)9)D)
2
b
f//,7Pm/+)>)^`kE:).'X ).E>/3F P )o1m
Fx'7 3>W93{ d )
unsigned int
31F&9)X9>`F^FO fb!) _O)_)`/ `^>`>)Df, s9,=)X) , 1, f/,9!a)
/X)9) F `,) ).) 13, b =7:]8 P F !f^Ns9XP
unsigned int
2
a
)o1E
'&)
> f/,9 H)
9/F7f/,9 b
unsigned int
-a
2 −a
Z
F^,` ,.:]=FG)FTPNT)+FG3?) )o1EF r/ )9)T(fb> &/3?,3> 9O'
3$/)7'&)aO)Z) 3F^`>:)Pr)"`>8 >$
& & 0U
k fbk,3+sE(fbn)9) ) F ,F^,9/ "`>$ 'X` E>)F 1``,=
Sequences of + and -.
-a:]H>F^F^9>fba),aF^ P )JF^9 )7) b{k,=)ZF^`")^9"` O/`>):F
/3
^ $fH (fb>,F^3 E>)^&) bF FGE`/ ,>FT 33 )8>F^fb )
+
$/F P F^ 1,`>FXP BF 9ZP BF>
=X:]=/89 P + ^)F H9- ^`3`FT93 F^F^1`>N)f1)>F )&F&)&`>$&9(' )
)>^>) )7:]19^F^F^F
gh WFG)4:]H>F^F^8`>,93 $/
s9,)4)
a+++b ---a
(a++)+b
`,3 /FX'7> $
D\1 1)^9 FG`3 ]HF^FG `,3 )>
a+(++b)
a+(+(+b)
$/
-(--a) --(-a)
-(-(-a)
ghX6ZY Y FI)/393 >F^fbFk)^9Fn3>_Zs1 3>W999T)9)k"F^ 1,`X`F^FG)!
P BF ET-P BF E9/FX)^_sm+^,m3 )8 /^FDP P)D)) '&)-EF^FGs9a
+
_
) )h3r g71,F $/F
/3 ---a $/F --(-a) Z)
+ a+++b
uRCD) _ b(a++)+b
)9)ZPNX:]=/89X)! ]HF^FG
F^F^"'& )F^ 3 /F
a++b
a+(+b)
s9,=)x/``>3Z)&)^94, uv,FG) FG)s9F^
3 ) F ) h'7 PN>3!:]19^F^F^
FG` h,99G
E>)
`$/)!9fb E:/3F" sE)^?F^3F>gh+:]H>F^F^
'X)^?)F `/
++
---a
)F^F
F f/38PN7/)>n$/FG )^9&E>/3PE)Z `>>)k ,FG)
--(-a)
sE"/af//, Hs9,=)h)":]1F^F^
FZ/ ^f//,
-a
6hY Y `)/Fn",9.sE>nPmPN,39/)/ ./1"N,) /3 hM./1"N,) ) Other integral types.
/m)oEFngh F^3 9>FZO
/3
ng79TFG)/ signed char short int int
long int
393-FGm`W9F&)^ O)X$/` P ) F&>>F^)^3 s-)&$/FI)D/FX__s9)FXF&)">f1,F
7+)^9 FG)$ gh71,.sE> Ps )Fx,F^3 )^T^>F^)
f,F 39:E3F4.) 9N)PN-
int
g79T`GFGE3 F^ ,`"Pr,F^3_)vEF F
unsigned char unsigned short int
/3
unsigned int
unsigned long int
ghFG)o1EF fb`89>F )aP>39 PXlm>^ )^>F+'X)|N> F^_>
/3
n\1_/> f//,Z/F &,9FGPN, 'X_G
f//,Z9>Fk)9/
int
unsigned int
`F^,8)-F 8``:
93 N:DF!O+)^)^`>)fb!'&?(fb> Z/3 ,93> 9(' 1` `,=FTg7 F^W9`$/`!P )^9>F^ )o1EW
F 'X` ./^$/3=FG)T-)+6*9^/
N/,9/g 9/F7P /33 a6hY"Y nghD$/FG F )^ O) '7T`$/ 1,)D$F^ 89)n,=h('X
)/ _39")^/ )o1mF" 6hY"Y Pn'7+9>3?)- de 6 )FTF"+,` .`,.sE> F^ 6 F^ ,9>) _/ 6hY"Y `>8 >F"F^89 _ b
93
/
short int
long int
//F PN
9/3 ) F/T3FZPNh)"`>^FGm93 ,9F^9>3 )o1E>F
int
de`>^)k/383`>^)kE>O)Fn/Fn'7 /Fk/F^F^ Order of effects and sequence points
) m>)^F+`>9FI)^,`>) :]HFGF^F '&) /5>l `>)$|\1,` |E>)^F 9fb_) sE
,F^3
'X )`$O"PNZ)o'7+/F^F
gh_s1f1,F!$F^ F )93
) /F '7_$/3=?$3 )3 PX\1`:) 1g )
>f//,9) 3> PN )^9 F^,=s :]19^F^F^F PX fb5:]1F^F^ F )+FGE>`W935| >eZ67F^ ,) f//, /3 >l `>)D_ 39:E3--) >f//,9)3>$&67F^3>Z)
:]19^F^F^
++ i + i
'X>^"'7 F^,=9mF^")9)
FZ+f/s9TP )o1E
deP
Fh)N/ 9F =)a)Tf/,9
i
int
i
5
Pm)X`8mF^)h:]H>F^F^__ 89/`>)`>ZsE
4g79ZF^,)k39:E3F 8'X>)>
11 12
& ]av R#
+)+)^9_>l `>) Ph)
P)+E>/3
P7)393)^ F 9^1`FGF^3|sE>PN
)^9)
++i
E>/3
Fh>f//,9)3{ngh f/,9 P ) :]H>F^F^
F7)>>PN ,FGE` W >3s1
i
++i + i
)!6ZY Y FG)/393{
g X:]19N/ )kF^`>933e/3 ,` FGF s1f1,F s9,=) PN^),9)"/F^&+,` !F^Fr>f//),
$F^
H>) ,F7`F^3>k)DPN('X 1`>) 1 H9!:]HFGF^_)9) fbfbF7"f/NOs P )vE
i
int
i = ++ i + 1
g 9F!:]19^F^F^ 9/F )o'7a>lm>`>)F !)^9 `>>)"P
7
/3?)8F^F^)") k`$/,F^
i
i
)+/F^F^9>)X`/?-9E>?/P)^>T)+m>/3FD9$fb sE>?>f//,9)^3E)DF^>FT)^ O)
)X3>7PE)&)o'7 >l `>)F F `$O )D`>>) `>FnsE>PNX)D/F^FG) /3)
(fb>/Ef//,"/3a>lm>`>)D^ '7 3>W93{
%&('7>fb> 9)FZFZ)&)G, PNZ$/FG9Fh)9)Z fb!)^ 3 'X ),=X,3>1 `>8 ,=):
13 ()^94fb Z,_/ O`9 )`>),=/
H )^94`89,=)>! Frm)rP1f1>' O)k>f//,9)^
P )^9"F^,=s :]19^F^F^
`F^FG)FhP ) PN('X FG)^>9F
++i
Z6 1_)Tf/,9 P
i
PJ)^9 _/
^8)+"Pr) 67M
>FG)>^F<
)+)F7f/,9 ) >FG)><
1
;H )D)TFG)^>&`>))&s /`_) _ >^ H)7)"/33=F^FZP
17VX33

i
6 $ 9)!WFI)&)o'7FG)>9FT^ `F^F^â)s))!f//, Px) :]1F^F^
/3
++i
` fb )7sEx9^1`FGF^3 sEPN^4)4/F^FG)$ k,=)r) )^9 3"FI)> 31>Fr)r`FGF 9fb )+sE `89>)3_sEPN^ )"/F^F^9>)$ de 3>Z)^8/O' )"`89> ) 9)^ ) )G/F^PN>-P!3 O) sm>)o'7w67M pFG)^>F-/3 _ G 'X` F fb>^ ,`
9N)PN^3>m39>), r)F 3>!9/F")"sE> FGE`>W93{8de P /`:) )"F )!,=$/FG Os ) /F^F^,8)9) )^9)^ 8`sE>)o'7> FG)>F+/3 _/ ^?F b/ 3 FG,9`|)^ O)
F^>fb:/r)FhT)^9FGPN>3 )h`" ,9` 1
/P):D/)> =,F^9+F^ e`$/>3 ;!C:g./6. \1,=9mF^ F sE>PN+)^ O)
)N FTf//, .dePn) /F^F^9>)TFDE>^PN3 P)>
i
5
) ^FG): `))!F"'Z)^)^?s /` )â_/ ^ F^:)F
) k,)!P
i = ++i + 1
i
7
)_/F^FG)! O9mF ;<0ST:< )N): )^/F^PN:.P7)^98^FG): f,
Ofb>G3F )
6
>f1,Fhf//, P 9/3
FZF^:)Z)
9FI)$/3{
7
i
6
gh.6hY YTFG)/3 O3 39:W FT .Q <XL71[M )8sE +E)X3,= )^9!>f,9)aFG 1,` P / :]19^F^F^ ) 'X` F ,9/)3 )^ O) /n>l `>)F!Pk>fH,9FG?:f/, O)3
eF^,=Cs I:]1F^F^9F.9fbsE `$^3 ,=)$?d ) 'h/F s s9?)_:]HFG)`_Ph ) >365`89>^F )9)k>)n)^9X6ZY Y*FG)/393 ^P/ P}3`T)X/F^FG)4/Fk
F^ ,`!E)$ de-)^9:Z'73F 'X ) /F^FG)7)
) bFZ /` )!:f/, O)
i
1 )kFn) FGE>`W938'X>)^9:k)X>l `>) Pm)^9h>fH,9FG :f/, O)3`>>)
i = ++i + 1
E>)89/F sE`$^^3 ,=))$ dey`)^/FG) )aF^>` )9) ):9)F /
:]19^F^F^FG))>)hFZ/'hHFh.F^ ,9>9`>!E)$
gh>^PN '7 9fb_/ F^F^,8'X)^ ]HF^FGF )9).9$fb^ )9/ :lm`>)
%&` xPkb, 9^PN>.)!)a'7G^ OsE,=) >l `>).3> F^,=8)9) /` :]1F^F^ )^ O)
b, 'Z)&>)>F&)7FI) :lm`>)k2 ]HFGF^F 'X)
&)^ T>lm>`>)h`$/_ b
& & R=
F^F^ H),9
=93_)>_OT =/F7./F FGXF^ ,9>9`> E)F FG> )D)D>l `>)Fh/3
9,=)Z)c)8 '7 3>W93 ^39:$kg7FhFZFG,9_O 3_
)"PNO'& ,
k>)o'7>a)v'7 F^ ,9>9`>"E)F =) >f//,9) P /:]HFGF^
Single Modification Rule:
__13P8)^9Tf//,"P /as9uv`>)ZP PN,3 )m)vE")&FI)Z`/
V :]1F^F^ b
X
)^ O)
f1N)^F8)F_, F `>9FG3>3 F^_/)`$/
i = ++i + 1
b/E/3a$/3F ) ,3>W99>3sE9fH
dePhb, E>^`fb8)F ]/89 /F.^) W `>N/ r>_F G O),=/ f1N)^ P )
F^T39 W `)8, 4P
Fh f^Ns9TP )o1E
9 )Z) F^> )9)
nextvalue
int
nextvalue = 5 * nextvalue + 3
`,3 T`8 /`>) sE 'Z)^)â/F
( nextvalue *= 5) += 3
g 9F4'&=`89 7
Fk/8f//, bF^"'7Z`$_/F^F^+)! )$4\) )ZNO)^)>
(nextvalue *= 5)
:]19^F^F^FZf//3
F^`")h13W9F
)o'X`
nextvalue
V )n)FxE) b/8)^))fb $39:kF^,3+'793>nO' / :]H>F^F^ )^ O)nfbfbF4F^>f > ,=)^9,=) E>)Fn`89F 'X)^ )T\1 13W9`$) Z,/ de93>3 1/_:]H>F^F^
b
std :: cout < < a < < " ^8 = " < / :lm`>)^FT/rP 'X`-13P_) f//,
9
&ghFh'7 HF&F^`!)!)oE
std::cout
P
'X`8'7h'X) 3FG`,F^F >g4F
PN,3 ) 939>9`> 1)D\19
1std::cout
3W9`$O) Z, 31Fh)X9_
)Fh`$/F^
2.2.11 Goals
V7)7)F7E) 1b,aFG9,3?
Dispositional.
g H('w)^9T)= V& )>)`D2xf//,9) Z,F/
,93>^FG)/3) ``>)FZP E>)7`3` /3-/F^F^`N)^fH)oC
; H('w)^9" )>)`Xm>)^FZPNh) )oEF
93

int
unsigned int
A+8
sE 'hX)9)7`89,=))^FnfbfH9 )^9X)oEF int /3 unsigned int _$839>fb>
`^`:)Z>F^,)F 3, )+EF^F^ s (fb>
/3a,3> 9
O'&F
de ^)`,N b,-F^,93 sE"s9D)>$
Operational.
o+g ^)FG /3 >f//,9)^ fb ^)>)` :]H>F^F^ fbf1 E>93F_P
)o1mF
/3
)s99^ )>)`_E>)F + - * / % 7/3
unsigned int
int
).,9^
) /O O)^9>F^ >/x ]HF^FG? )+SD>)/F
F^>`>)a> Os Fhb, ) 3 )^9FhPN&/r^)>)`&E>)^F
o _3>^fb
s /FG`"FI))>)FhsE,=)Z )>)`X ]HF^FGF<
& ]av R
o ;_`>fb>^)X+ fb3`_/E1,.sE>h)^+s99^^>F^))-/3 f1`Tfb>Fd
oTA+_3>^fbh)^9 0u`% .K'KLtD<'I/M >F^>))^8PETfb8,+sE:k bes9)4>>F^ )O)HPNZF^

b
o"0
'Z) /F7'XF^ ,=)^9,=)DFZ3>)>9>3as1-+W]H3 1,+sm>XP ^)>)`T:] FGF^FZfbf1+)>F7/3a=9,=)hf/s9F7P )o1mF
/3

int
unsigned int
o R_
3>)^>k) f//,h/hP9)/H)oEF 8Dfb _/`e ,F^ D/:
2.2.12 Exercises
_ :/<M657!./1 657XSQ[Du1[7N sgLM:<!.<.12M.I)657< <Fd+Z*+ 6 57<' .<2L ;!V
`
.<2L p 5&10.v'I,M.65C+0VgL=!.\)XF+~*!.c SP+"$1"M<:!'I,)1q+:<!.7~6.65C+ :/W/'KLMC)
)C:!1"7NE657^<Fd+Z*+01q@.<57CD);<RLM:<F
1q)J,)Y
o+g
!o
;o
2o
)!o -2-4*3 2o 10%6*8%3 STo 6-3+4*5
Exercise 11
5u+5*3u
31/4/2
-1-1u+1-(-1)
5&12!5 S|657SQ[Du1[7Nw!5C:J!<:@.T *</!.x:<^DqN&+z sgLM:<!.<.12M./)|uv5&V :?657I<.E65C+K:<<-PN1[F&E657? N12,+4L7:/<M657!.Q!. iq# :/)J<:?@F&1 )
i2'^10.<DY)1"7N qo 5&1[M6./-Wu` 5CF&?:/<'IF&,)@657 . L7J/!.?65C+Wu`?7.*+[ZV 1[!Z*)J\SQ:657
.T+d S3;<!6/:3:<Y)YY;!1[~1[0V$ o
o+g
!o
;o
2o
)!o c=a+7+--b 2o c=-a=b
STo c=a=-b
N<o a-a/b*b
5Qo b*=++a+b
1~o a-a*+-b
Exercise 12
7+a=b*2
a+3*--b+a++
b+++--a
g 9>F^ :]=:`F^>FT^ 1,"b, )8^$/3 )! /O `/ >)F^ 8>
7
:]19^F^F^ ) ST:)/FnF^`>)^{n2 ]=:`F^ 1o /F^ 1,F b,
) $/3
)X /O _
F^ ,`FXP
/3
)^9"SD>)/F F^`>)^{
+
:A+[DqN&+,s!LM:<!.<./1qM.PS:<' s$<:/<10.QAk Y-LM:<FQ1q)J .<2Lt{ ;!Vd{q.<2L /Fd+{
*+01q=-./+LYL=./1[7N?65C+`1"=1001 +[ZV 5C+.BFd+Z - 5C+.BF+~* -\)A6573Fd+~*? S 10.
C)J )Y 5&1q!5 S]657^,sgLM:<!././a12M.X:<!./C1[5 hbM.qL=<1 ,)|2:Bh)J )O;/!5CF
1q: c
o +g
Exercise 13
Exercise 14
(a
3f
_v:/F&z65C+ ST:+"
b)
3f
c=a
3f
a
0
)
b, c > 0
(bc).
>3
!.@65&1[. 1"'KLtZV 65C+ 6570u` ,s!LM:<!.<./1qM.
+[K./ <5 Fd+~*!.H S657xFd:!1 Y;g <. - - )
a b
o
unsigned int
Exercise 15
;</:g.T
!o
(0
-v657ST[Du1[7NA t +01257U)+.T
)
a/b/c
iqu5&1q!5
c
:/H h1[Fd+D<MWSQ:
:< +.<./h'I,) ;/} S@0V L=
o
a/(b*c)
']LM7;!VI5C)I;!1[:!V :<2LM:/!.Q<M2+01qM.E SB6579ST[Du1[7NI)J<1['+=h'E{
o ;
;o (@
2o ; U )!o $#
& & R;
']LM7;!VI5C)I)J/<1"'A+:<2LM:<<.Q<M2+012M.? S]6579ST[Duv1"7NI;!1":!VA=h'E{
;</:g.T
o ;
! o $#= ;o $#####= 2o $#=$##= )!o $#=$#=$#=$#=
Exercise 16
b9V?82Lt<';<<: m-657BD:2N&!.<9Qu <:g.T<=z_:!1['AB10. 43,112,609
5C+10.36573=C';<<:A S^)J<1['+)1"N1[6.z65C+R65&10.B=h'A;</:E5C+. s!LtD1"y257u V$ −NY1
$V C:^M.uR/:
o ;
, __9>3a) s /FG`D,FhP b )F /3- E1`b>)Z`$/`,N)$
n
Hint:
Exercise 17
.<./h'j1"7NI { ;!109:/2LM:<!.Q/M2+012=-X/']LM73657j;<1[:!VA0uR% .j/']Lt /'I<M
X
:<2LM:/!.Q<M2+01qM.j S]657ST4[Duv1"7NA)J<1['+=C'A;/<:g.T
oTA+
!o R ;o A 2o oK )!o U 2o o;
Exercise 18
65C+f/=F&/:g6.R/'KL=<:+0C:<!.S:<' )J N:</<. +5Y{
:!10\LM:<N:/'
:<<M57/10a1"MI)JqN://!. !~./1[C.Tcelsius.C

p 57|1"=1001 +]C LM7365C+cLM:<'KLt6.|657|C.Q<: w<M<: 657 <']L=<:/+0h:< 1[ )JqN://!.
+5&:<<M57<1[ .<577U) +~.Q /M21" ('7> ) ,=E> ;<C)+.^SQ:657H+" u`)w1"&LM76.T
p 57!.Q ;/C)+.A.<577U)y;/@!57.Q/ ./ <565C+BHFd<:g{) C)J<: Pu. Q/<h:|1[
657X./t;g.Q <M/']LM72+01qM./-N1"F&<65C+657W7.Q<:W:/!.qL=!6.K657z;/C)+.Tj'AVRSQ:
65&10.K,s$<:</10.Qz+.<.C'Ic65C+v657\1[MqN&<:)1"F
10.12|:<h)+.cu:/)+. <:/RST:z+[4<L=<:/)+.
SQ:^,sJ']Lt <657/@:<C)+.]657j sJJ!a:<<./7~

-5 / 2
−2.5
−2
p 57cLM://N:/' .<57CD) C LM7657 `G`>) :<<./7~1[ )JqN://!. <./1"7.^+.A'^1Ds$,):/+{
012+ =C'A;/<:I Sz657RST:!' y iq'I=1"7N
o-uv57<:<
)
:
x9
x + y/9
x, y
|y| 8
,sJ']Lt /- 4 /7U) ;< 7 LM7B./1[']Lt~V +.
+~.Q+" u ST:@,sJ']Lt |657
13 9
13 4/9
7 LMC
iq'I=1"7N
od
o !0
-1 -1/9
−1 − 1/9 = −10/9
Exercise 19
:!10E3LM:</N
:/' threebin.C 65C+9:<Y)+.ji)J<1['+ oE=C'A;/<: a 0 S/:<'
.<2)Y:/)E1[&LM7`)OC LM76.]657 /FG)h)= ;!106. S % .3;<1[:!V :<2LM:<<.Q<M2+012 10"4+L
u1065A ,Y)1[7
N +<:<.K1",+.QB657;!1[:!V :<qLM:<!.Q<M2+01qax5C+.X !./.K65C65&://j;!1[6.T o"0
Exercise 20
2.2.13 Challenges
.Q2Lt5&7.Wu+. <u10.<5I'^10Z102:!V DY)J<:31"657 <u1[.<5Y{ `'A@u:E SnYn{
/S <:657 R'M.O5CY)x1[=FY)J)y5&10.ANd:!:!10.Q u=-j6573ST<u .QD)1q<:g.@i'A7N
657<' .Q2Lt5&C.qo65C+K5CY) ./h:!F
1[F&,)657AQ10"~1"7N.O;!Vx657 R'M.|)J<1q)J,)x /'j'^10
./h12<1 )JQRb97f.T'I!57u- .QqLt5&7.W)?z SP5&10.X/'j:Y)J!.c'ANY,)G./C:!:/<)J<:X
657 `'A3SQ://!.cu10657Ca;<<1"7N^Q10" ,)i .Q2Lt5&7.PD+/:G;//,'I3 R'x<1001 <@)
u`![~{2Tuvx5&1[.<:!1q&od
p 5&1[.E5&10./:!12+`<F&/MK10.j657O;/J!TN
:<C)^SQ: 657 F^: ,F.M s9> 65C+3 W<:g.I
i2'^Vd65&1q+ o ,sgLtU+012 Y;<CP57u .qL=!5&C.Bu+.jY;gDG Fd1q)A./C1q<1q)JT <:<31[.3657
LM:<&;gD<'
o:< ./2)1[7Nx1[ w/1[:<!D<- )}<F&<:!V {265@L=<:g{
kOL=/<LtD i2=C';<<:/)
0, 1, ..., 40
.Q 10.xQ10" ,) CM01[z y./C:!FQ1[F&!. :
- 657yQ10"~1"7N :)J<:H10.xk657<:/0SQ:<
h
Exercise 21
k = 3
& ]av RA
57<:/E1[657I<1":<!DI)J
!. . L=!5&7.G5CF&^jL=.100125&1['j.Q! S1"
:/)J<:zA;<B657zU+.<a.C:!F
1"F&:Ei2uv57j657<H&;!F
1q7.<ZVI)J
<./ % /)jQ10"5&1['j.Q! STo
2, 5, 8, ..., 38, 0, 4, ...
!o :!10]\LM:<N:/' 65C+.Q~Fd!.P657 .Q2Lt5&7.fLM:<&;! <'
+.O1"&LM7657=h'A;/<:
) C LM7657 =C'A;/<:
k
./h:!F
1[Fd:

657LM://N:/' .<577U)z:////1[F&
S 657D+./
p(k)
{0, ..., 40}
;o ! C.X+././C'A\65C+ .QqLt5&7.910.`Y;gDcj!57.QP5&10. L=./1001qI1"|657B<1":<!D<-;<7
65C+c57I 1[:<!0C:! <57
.Q?657]L7:/'I<<:
I# .^1[L=.<./1 ;gD\SQ:
5&1"' j./C:!FQ1[F&K657<=-vj'A+6/:zuv57<:<357]1[=1[0k1q+[ZV?{1,.<2...)+, . 41}
0:!1LtD
S|L=./1[01[F& 1[M NY<:g. 10. +" ,) Mx1)^ $/w)^9 1 S
Exercise 22
(a, b, c)
:y,sJ'KLtD<10.w_Vd65CNY:/r0:!1LtDQ :!1[y@LM://N:/'
a2 + b 2 = c 2
65C+W+[Du.zV$(3,
Z14,0.<5)
W+[ _vV65CNY:<,H0:!1LtD!.SQ:Guv5&12<5
pythagoras.C
+b+c =
h% :<^])J<')1[7NI65C+\657XLM:</N:/' Z10.<6.G657<' )1[://!6ZV-];!7P6a57]
LM://N:/'
1000
.<57CD) LM:<F& 65C+cV$C:?Z10./\10./:!:<!2 u ' =V . !5@_V65CNY:<}0:!1~Lt !. :<
657<:< @i p 5&1[.c10.zj.<~1 N5YFd:!1q+01qy SW_:<&;gD<' l\S :<' 675 MxuI>`>)X2 ,> -f.Q/
http://
o
projecteuler.net/
&
4 2.3 Booleans
Rb0
p 57K0:!765+~uV.KZ12!.].Q'A<uv57<:<j<.QT
MQu
C.Q)@:<2LM:<<.Q<MG0:!765F+~*!. :
p 5&10..Q/<012 )10.T<7.<.T!. 657|0V L7
bc
DM.-vST: .<57:g2 uv1[[v.Q/bool
| =h'A;</: SI<L=<:/+01qM.Owbc
,M.
) uv5&V M~V?ST<u( S 657!.T <L=/:/+012M../ /Ay,s!LM:<!.<.|+[9657657<:g.T
?u10"= ,:!|57uwj<F+~t+B,sgLM:/!.<./1qM.R1"=F&~FQ1[7N3657\0V L7
-7./1"7N
bool
F^^) `^`,)h>f,9)
O)8F.) FG89FG) 6hY"Y )o1E_b, `$/ ) PGpdeP&'7
)^9 P&)oEF+|)>^F
P ) f//,8/F )^9> b, 'X4s s9?`8,=|'&) )o1E8'&F^8f,8/
FX>89)o D`F^FI)FXP !EF^F^ s "f//, bhV&,9s9 f/,9>FXPxF^,` )o1mFT!fb:^
$/FI )X>>F^) :fb+'&),=) FIE3T/!^TF^,=`>F4%&('7>fb> b/ ),9!F^,` )o1E>F&^ ,F^>PN,9{ `>^)/
``,FG)/`F 1b, `$/M
)h3 .)7P )>>FG).`89,=))F
'X ) )-xV&P)>&/ 1)^9:!Fh m>))c)>h)9/)^9 3))ob
\1 >) ,F : =F^n)nsE(fb
,9>FG)
E O) F{)kF^8 FG){ e)^ fHN/6hY Y5)oEnb,
`$/_) P^aVXP): )TsE(fbD3F^`,F^F^_'7&`>^)89fbDD`$/339) x )vEX'X )_
f//,"9 )9)X`>9FGFG)FZPx:]/`:)_)o'78>)F V )ZW9^FG)XF^) F^,` +)o1E!_$ /b
E$O&fb>^)^3{ h>fb>^)F^F 9'7"'XEF^!sm('w)^ O)X)&/('XF7PND__)>FI)
E>)F V&`>),9/ /F^,` _X)o1m&F F^, 8`>)k/FnTs /FGFnPNk/ 13FnP `89,))F b,
`$/ _/]
h`$/ EPN!:]=/89 )^ O)")/x1,.sE>FT`$/ sE.>F^>)3? s99^
PN_) 1)9)hF ,F^.) )o'7 f//,F
/3
b
0
1
2.3.1 Boolean functions
g 9 9/ k1$ FG)FDP )^9 n^)F^-_O)_)`>N/ kz
7

1815 1864
'X9>>3? FG)Os F^8``>)9FDsE:)v'7>?` /3?FG1+sm` >s1 n-)
)>^ bc
D S/h!012 '7 3) PN,9`:)
'&9: := {0, 1} 93 n
f: n→
h$/3 0 /F ST+~.Q /3 1 F 0:!*
67$ )8,9.sE>!P 3lm>^) k$/?PN,9`>)^F F!W9)+PN.>fb>^?W]H3
n2 ]=: n
`FG ; /F HF b, ) F^(' 'X9).:]=/`>) ).,9.sE> F g fb8b,5WFG)+) =
)>.!9 PN,= k$/-PN,`>)F 9)!)o'78`FG))DPN,`>)F
/3
n=1
c :x→
0
)^9 3)^)o?3
/3 )8b)^ XCTg
'X>^0
/3
c1 : x →
1
:x→
x
:x→
x
0 := 1
1 := 0
de )PN('X_'7 FG)G`>) ,= PN`,F )-,9^ 93 s99^ k1$/ PN,`>)F )^ O)
F xPN,`>)9F!P
2 )^
\1,` PN,`>)F OFG) `>fb)?39>F^`>sm3|F F^_/ )s9.)9) FG)^FT)^9 PN,9`:) f//,F"PN xEF^F^s9+^,9>)F.VX?:]/89.PN!
s99^ k$/ PN,9`:)|F+V &S
F^('X5 ,=_Y
A
d ) F 9/>3
: (x, y) →
x∧y
)
V &S sm`$/,F^
PT/3 P
4, _|,FGF8'X )

x∧y = 1
x = 1
y = 1
& ]av RR
PN,`>)
3>W93 ,=+AY sC F"`$3 C "_de P `>) )^9:8^ )o'7
f : (x, y) →
x∨y
F^FGs9a)>^>))F8PT)a'73 G A4,y`$/$/3)
/F ^)_$/FG)_-P hs9,=)
E
uv,FG)8F '77)+`$ $/ G)> $^ 4)9) F G ]/`>)^ 9
P =5g79
PN,`>)|)^ O)
`^^FGE3Fh)^.)^9 N)G)> )>^:))_F7F^('X ,^DAYe`g:nd )hF ,F^,9/8^PN>^>3)
/F ZC
,s$<~C./1[F&I: ,= Y
A eg "3>9`>)^F!) )s9 PN )8,9^
: (x, y) →
x y
PN,`>) XC gT
x
0
0
1
1
y
0
1
0
1
DFG
x∧y
0
0
0
1
x
0
0
1
1
J
x∨y
0
1
1
1
y
0
1
0
1
D+)G
x
0
0
1
1
Figure 4:
x y
0
1
1
0
y
0
1
0
1
AJ
D G
AJ
x
0
0
1
1
y
0
1
0
1
DU/
x↑y
1
1
1
0
G
x
0
1
x
1
0
D (G
M8J
J
sJ'KLtD!.ST:BbcQ ,ES/h!012M.T
,^?A F^('XF uv,FG)a PN>' :]/89>FQ%&('7>fb> T `>G)/*F^>9FG " )
Completeness.
F^('XF b,*:fb>^1)^995sE,=)_s O^ k1$ PN,9`:)FQ\1 P!)FG PN,9`:)FO
FâPN,39/)/x)9) /F&<:!V s99^ k1$?PN,`>) `$ sE8>O)3 P)- =
:]=/89 ZC
`/sE">)>3aPJV &S 9C
/3 &C g ZC
(x, y) =
V &S
(
C
(x, y),
XC g
(
V &S
(x, y))).
ed PN_/ G)^9:& $/F G s9,=)&) ^93 =^ ,N/F7b ) FZ"$/F^_`9>` b>3
s1
.)=,-/ PN, EFGF^s9T`.s O)F P O,)F>
\1NO ) PN,9`>)^ ZV &S
39>F^`>sm3 ,= AYe3h8`$/ sE
: (x, y) →
x ↑ y
>O)3aP XCTg*/3aV &S e`")^9 9/. ZV &S
(x, y) =
XC g
(
V ZS
(x, y)).
G>9:) =
qr:)&,F&3>W9T'X9)7'7"/ s
M./1 )J<:O .Q!
SA;/
D SC!01qM.T ;<1[:!Vy;/
DS/C<012
1[.
Definition 2
+" ,)
;!V
1 S ,F ;/H,sgLM:/!.<.Q,) ;<V |ST:!'jCDw65C+AM~V /M21"M.f657
generated
F
Fd:!1 Y;g <. ) -v657j/fM./2M6. ) -)?657vS/h!012M.S/:<'
x
:K3.T!
ySc;!1":!VS/h!012M0./-B.Q!1 SW;!1[:!VSC!01qM.10.R.TF1q)Kj;/
complete
1 S3)IM~V?1 S/F&<:!VXS/h!012
y;<KF N&<<:/+,);<V F
f
! 9(' > 3PNX.`89>)>9>F^F 9^1P p 57A.T! S\S/C<012M.
bc
DESC!01qM.T
Theorem 1
{
V &S
,
C
,
XCTg
}
10.O'KLtD!KST:657I.Q< SI;<1[:!V
&
4 Rb@
V&+s O^ k1/ PN,`>)
F4`89>) 3F^`> sE3 s )^F <5C:/J!<:!1[.<012cFd/!{
Proof.
f
:
= :]8 V &S 9/Fr` O/`>):FG)`4fb`:)
(f(0, 0), f(0, 1), f(1, 0), f(1, 1))
(0, 0, 0, 1)
PN F^^)$ q{>)
3).) k$/ PN,`>)?'&) ` O/`>):FG)`.fb>`>)
0001
f
Z:]=/89 HbV 1 bZS2 b3 b4
b1 b2 b3 b4
= f0001
de )7W9^FG)4FG)>P9)hP '7ZF^('y)9)n/H)F^hPN,9`:)F `$/ sE7>)3 '&F^
`9/`>)^>FG)^`Dfb`>)&`)FZ.F^ nde3
3 1
f0001 (x, y)
f0010 (x, y)
f0100 (x, y)
f1000 (x, y)
V &S
=
V &S
=
V &S
=
XCTg
=
(x, y),
XCTg
(x,
(y)),
XC g
(y,
(x)),
C
(
(x, y)).
g 5``*) PN+,N PN
PN-:]8 X'7 `$/ `>^$) )s9?PN) PN,9`:)
f0010
V &S
&C g
/FZ ,= A /3a`f1` ,^F^fbF7)9)7)TF^,)9.`9/`:)>FI)`
(x,
(y))
fb`>)^XF
0010
de )TF^`>93 FG)> H'7TF^('*)9)h/_PN,`>)
'&9FGD` 9/`>)>^FG)`Xfb>`>)hF7 >
`$/
sET>)^3{ng7F7F 3Ts_`.s 99")"$/3=>9:)3 GF^ PN,`>)F
1
)=,-C
H'&9` F^89393Fh,=a) 1 BF> =&:]/89 C
f1100 (x, y) =
C
f0111 (x, y) =
(f1000 (x, y), f0100(x, y)),
C
(
(f0100 (x, y), f0010(x, y)), f0001(x, y)).
+
s9FG)/ P 7
' H)FT^,9>)T,=)TPN_/ 3$1 1'7 >)^
/F
f0000
F^` '7.sE>fb )9)Db, >) )^F
f0000 (x, y) = 0.
2 ]H>`F^ R /F 1F4b, )^"F^('y)9)x)ZFG>)F V ZS XC g C
x
XCTg /38:fb )hF^>)
{
,
} {
,
}
)9)T`F^FG)FDP ) F^ PN,`>) ZV &S .
`89>)"PNT) F^>) 4
P s99G k1$/
{
}
PN,`>)F
2.3.2 The type bool
ed +6hY"Y k1/F ^k>FG)>3+s") PN,39/)/)o1E
xd )F f/,94/k`>9FGFG)F
bool
PE)D)v'7 >)F 0:! 93 ST+~.Q )9)hT/FGF^1`)3'X ))D)>/F
/3
true
false
FIE`>) fbb=X:]=/89 bool b = true ;
3>W9F& f/Ns9
Pr)o1E
93 )N Fk)7) 0:!* b
bool
=_/ O) )o1E bool F / )^/)vE b3>W93 )DsE7>F^F >/H)9/ int X
' `
),=aFhF^Fh>r)9/
F^ \1`>) 1 1L@ unsigned int
& ]av RK
g7"`>8 >)XF^:)&Prs O^ k1$/
PN,9`>)^F Fh$f//s9XfHN!)^9 D/N{
Logical operators.
12,+R<L7<:/+:g.
V &S` || C a /3 ! XC g:
6 8 3 )) )O)?,F^>3 &&
\1`>)^ 1B;H '7 FG89a3)P
'X ) 0:!* /3
'X) ST+~.Q k)
93
.s9
1
0
&&
||
9^-m>)^F m'X
F ,9^bTVX E>93FT^.Gf/,9>FDP4)o1m
{/3? `/
!
bool
E>)FT/F^8>),= ^f//,FTPx)o1E
q{ b `F s93FX!FG)â)^ bool
&&
/3
s93Fh^ FI)^)9/
||
!
&&
g7> F /F^&Z,9.sE> P=E>)F .)>)^` )o1EFr'XF^k^F^,)
Relational operators.
FXP )o1E
=T$/`?O)>)`T)oE )>.:]HFG)X) F^ ] :<!U+012+v<L=<:/+:g. bool
< >
{/3
g7F^+ s99Ê>)^FX'XF^ )o'7^f//, E>93FT^ PnF^
<= >= ==
!=
)>)` )o1E8/3 '&F^ FG,9 ) F / ^f//,8P )oE
gh8E:)F
/3
<=
>=
`^^FGE3-)8) _)^9>_)`$ NO)F
/3
^FGEbool
`:)fbb&gh m>)
)>FG)F
==
PNZ 1,9/ )v
93
)FG)FhPNX9 1,9/)ob
!=
\1`
Fx/.)^/)vE )k^N) E:)F _ /FGX9$fb m>/3F P=)o1E
bool
ng79TFIE`>) fb!`8 F^F7 39"/``^399+).)"`fb)^ ST+~.Q 0:! bool
<
VtP 1,)!sE>! F FG) b.FT),F^+) F^F^)"m>)
Watch out!
=
'X>T) 1,9/ )v_m>)
Fh$/)$
==
VXFX.>r, )>)^`XE>)F7s93 FG)^)^ NO)9/mF 9/3
)F^" )^,^s93TFG)^^9 )9/ )"`$/ E>)^F
k9^
^)>)`&E>)^F7 fb"99:Z`>3`")9/
Boolean Evaluation Rule:
)9/mm>)^F 9/3)F^"9$fb >h9^`3` )9/s99^_`$/mE:)F>
=X:]=/89 1)
:]H>F^F^
7 + x < y && y != 3 * z
Fh`$ ^)FG 3-F
((7 + x ) < y ) && ( y != (3 * z )).
kk`PN,'&)"_)_)`$/OF^^)`,=)
)O) F^,` +/F
a=b=c
VXF Z6hY"Y
:]H>F^F^
a == b == c
Fh)Z 1
,f//)Z)
a == b && b == c.
n
P) /F^FG1`NO)fH )vyP
D) : ]19^F^F^ a == b == c F `$ )F^ 3w/F
==
4dePm/=P /3
7fONs9F P9)vE
'X).f//, b)^9h>f//,9)
(a == b) == c
a b
c
int
0
H3F
(0 == 0) == 0 −→ true == 0 −→ 1 == 0 −→ false ,
uv,FG)Z)^9 EF^)^ P 'X9)7b,a,F^,9/_/
s
(.''(A*
2
%(7
.*
/
R"L
--*6>.BO7
*
a=b=c
*
27
&0 &
:
6>
%R
/
%*
J
&
4 RU
g7&'7 H('X8PN+,N/ZPE9(' )^ :]H>F^F7V &S ):FkPrC
93
De Morgan’s laws.
fH`kfb>F^Z'X)^!) .P XCTg 9/>3 /P)> ) n)^F^ _)_O)`NDVX,,9FI),F SD
b/?
^$K#R $Kb@Hg: %&h'h/F T9>nFGH.sE`7/>s /3`F> VXF^ )Z,F
PN+,N) P G_)_)^`$/H93,`>) !/Fk'7 1O' /3,9FGD)k)39$ 1>F s ` ) -
g79 3 ^b/ PN+,N/TFG))^ )9R
) e-6hY"Y N9,9/!
!( x && y ) == (! x || ! y )
/3
!( x || y ) == (! x && ! y ) .
g 9>F^!PN+,N/T`$/P)asE 7
, F^3a)+)^/F^PN bc
,H,sgLM:/!.<./1q
)o1E

) GF^89:. 1,f//)hPN-Z:]=/89 bool
/:]HFGF^-P
!( x < y || x + 1 > z ) && !( y <= 5 * z || !( y > 7 * z ))
`$/ 1
,f//)sET'&^)^)/F
x >= y && x + 1 <= z && y > 5 * z && y > 7 * z
'X` FZ`>$ 9^PN>s9 )^>F7P /39s9)ob
. 3>)F"sE,=)!`>3`F+/3 /F^F^`N) fH)FTP7)^9 `$/4/3 )9/
=
E>)F (F^>kg Os 1 4,"_XW93 )^9Fr9PN^_)X PN,"3>{)hF^fb42x]H>`F^ KH
Description
`$m)
F^F
$)^>
F^FhZ 1,9/
$)^>&Z , 1,9/)o
1,9/)o
`$E/3
`$m
Table 2:
Operator
!
<
>
<=
>=
==
!=
&&
||
Arity
Prec. Assoc.
$R
)
>P)
>P)
>P)
>P)
$#
>P)
$#
>P)
R
>P)
0
>P)
_v:</,)J</!. ) +.<.QQ<1q+01"F
1001q!.} SyD/N12,+) :/!D+01q+A<L=/:/+:g.T
<L=<:)+.G)j:<!0C:!F+~*!.G:/B:!Fd+Z !.
P[
d ) F"EF^FGs9 )9) ) )o'7aE:/3F!P7
^N) 4m>)
Conversion and promotion.
9fbX3lm>^) )oE g79F `$/F^&Fn)^^$)3_ )XF/h'h$ /F PNk)D)^9:)` E>)F
g797`8mF^) ]HF^FG8Fx>f//,9)^3 +)h^ >/H)o1m ) '&`+)7E>/3 P
) F^Fh>/E)o1m"Fh89`>)+`fb>^)3{nde ^)`,N E:/3FZO"`>fb>^)^3
bool
))!>FGE`>)^fb.)>/ )oE Px) )>TE>93{D%Z> m) f//, S +.Q FX`fb>^)3
)
T/3 0:! ) deP!) )>/D)o1E F
X)F
`fb>F^*F 3>W93 ) sE?
LM:<0'A012 xVy9^1).Fn FGm`N/`fb>F^8int
PNn'&` )X6hY Y*FI)/393 ,9/)>F
)9)h PN_O)_>)FhFG)$
& ]av @#
gh `fb>F^?1>FD)8)^9!)>T3`>)^ PNT`$/rE>)FTde ]H3-:]H>F F^F b)&)/E>/3FkP `$/E:)FnZ`fb:^)3)
8FG,9`
'h+)9)
bool
FZ`>fb>^)^3a) S +.T /3-/
)>hf//,"FZ`fb>^)>3a) 0:!* 0
ghFGk`fb>F^F /F^7) b 9N/` ")N )^F /3!/F^F^)F O/Fr")^94PN('X9
:]=/89F
bool b = 5; // b is initialized to true
int i = b ; // i is initialized to 1
2.3.3 Short circuit evaluation
g79&>f,9) PE:]HFGF^F fbfH"`$/938^N) 9m>)^Fn1`>3F /``>3 8)8) >/ ,F E/FD3F^`,F^FG3 ?\1>`>)F 1 1 /3 1 1<;H&%&('7>fb> m)>^ FX
8EG)/) 39 lm>>9`>"bO3 )!^39:& 'X`)!m>/3FZP /aE>)&!:f/ ,9)3{X >9:/ )FX^39:TFX,39:W 3m) s99^`$ E:)F
/3
&&
||
/'h1F ,9/)
)9) ) 8P)8E>/3 F >f//,9)3 WFG) Ôfb: ZP&)f//,aP
)_`8EF^ ) :]1F^F^|F+/^$/3= 3>W93 s )f/,9_P7)_P).E>/3 ) )
)7E>93
F R<Fd+Z*+) )h/e ghF :f/, O)_F^`TF HO'&F ./57:gR<1":<<h10
<Fd+Z*+01q %&('c`$/5)+9E )9).)_W99/ f//, F $/3= 3>):3 s1 ) P) m>/3
\1,EFGh)9)4 /
m>) )hP)4E>/38>f//,9)Fx) ST+~.Q b) 9"_)^):
&&
'X9) )^9T)7E>93 fbF H)^9D>F^,)7'Xm/'hHF sE S +.Q k%&>9`> H)> F7 3
)
>f//,9)+) )"E>/3 )!/ gh /9/,FTF^),9)?1`>`,=F P /
E>O)
||
) P)ZE>/3a:f/, O)F ) 0:!* V ) WFG) F^) )!1 1F /F!P F^^) ` `,) :f/, O) F">?_)G)>"Ph 8`>9`:b
k,=)")^9: F"9)^9:"sE>9:W9) d ) ``,=F"'&?3$/_'&) :]HFGF^F )^ O)!O 3>W93
PNZ`>^) >)>F bn6 F^3>hPN& ]/89X) 3fHFG_E>)^
)^ O)ZFh3>W93
PN&+ > 39 fHF^hb SD, )^ F^^)Z``>,9 )Z:f/, O)
1'7 `$/'Z)
x != 0 && z / x > y
/3-sE F^,= ^) O)X)F& ]HF^FG FX/'h1F&f3{&dePx)!^)&E>/3-'h/FX:f/, O)3 PN
) ) >F^,)Z'7,3
sE ,9 3>W93{
x==0
2.3.4 Details
g7 ZC
PN,`>) F.F^-P ,) `$/>3 M01[Fd+D</ /3 3)3 s1
Naming.
=
g79 hV &S PN,`>) F /F^ H('X /F +~<:!+)J/=1q+ 8M57W<:A.<0:<d g7N)^):
9/ F /P): )8VX>`$/ _)^9>_)`/ %&=^ ? \1>lm>G^$KK; $URA+D'&9 9Ôfb3
)9)Z/{)>Z`/mE>)F7`$/sm":]1F^F^>3 )>^FhP ZV &S!
.*6O
>'. BK2
%(2,>$
N*
*
$
A Q Q
277 2
x
0
&
4 @H
??9fb F^>**\1>`>) 1 1<K )9)
)^>F
`$/ sm :9^F^)3*
Bitwise operators.
s99^ PN_) 4)9) F k/F8 F^ ,àP&s9)F+$/` P&'&`5F+)>
k$/
0
1
PN,`>)F `$/ 9),=asE+:]1)33 ))/x)vEF s1 9H)ps9)o'XF^ )
)
s99^:9^F^))F
M./1 )J<: =qN&+01"F&O1"MqNY<: b ) 0u`@1[M NY<:g. x = Pb a 2i )
i=0 i
SQ:+[

{0, 1}
0 i b
:^^C:!V?bc
D^S/h!012
657
/:!:<<.qL=)+{
f
:
{0,
1}
→
{0,
1}
bitwise
operator
ϕ
f
Pb
1[7Nj 1[.3)J )O+.
i
: f ;<1[:!V bc
Dϕf (x)
@SC=!01qi=0
f(ai)2 2
657
:!:<!{
g : {0, 1} → {0, 1}
bitwise operator ϕg
Pb
.qL=)1"7NE 10.z)J )I+.
i
Definition 3
P
-MST:Buv5&12!5
y = bi=0 bi 2i
ai , bi
g
ϕg (x, y) = i=0 g(ai , bi)2
=",FG)^O)F^,=9mF^ )9)D'7 9fb ,F^3 )/ )vE 'X ) 8A es9)X:9^ F^))r_g79) F >F^>)F ^>F^)F x/3 F^- , )
'&9` 0000
0 0001
1
1111
>>F^)^F
15
ghab,`$/a`` )^ O)
/3 ϕ (2) = 13 ϕ (4, 13) = 13 ϕ (13, 9) = 6
\1>fb>!s9)o'XF^ E:)F 39:W 3*PN)?)/D)o1mFa 6hY Y.Jg79: F s9)o'XFG"V ZS
+s9)o'XF^ C | E/3 s9)o'&F^ &C ^ /F&'7 /FD+s )v'&F^ XC g ~ )9)
&
F",9FG, -PN:^3 )-/F /']Lt <'A<M VXF") )>)^` E>O)F r) s99^ s9)o'XF^
E>)F e ]=`>)nPN x9fbZT`>^FGm93 /F^FG)xm>)xgh `>39>9`>Fk93
~
/F^FG1`NO)fH )v8P )F^"E>O)Fh"FG)>3g s9 ;H
Description
s9)o'XF^D`89)
s9)o'XF^T/3
s9)o'XF^&]=
s9)o'XF^D
/3-F^F^)
]HX/FGF^)
&/F^F^9>)
Table 3:
Operator
~
&
^
|
&=
^=
|=
Arity
Prec. Assoc.
$R
)
U
P)
K
P)
@
P)
A
)
A
)
A
)
_v:///,)J</?)A+.<.Q
<1 +01[F
1[0V| S3;!100u10.T^<L=/:/+:g.T
Z) )^ O) )PN,9`:)9/)o P )F^_E:)F"F 89>))^ 39:W 3xF^9`> )
s9)o'XFG.^>F^))9F"P )/x)o1m f//,F O )"FGE` W >3?s1 )86hY"YQFG)939^3r
9fb?y3F^`,F^FG3 ) FG)_P ,9>)Oe/3 FG)
b>J F^,` :9^F^))F
\1`>)^ 1 1BKHW
n, F^,3 )>^PN MZV ,F^ )^9>F^ m>)^F '&9>?b, Qu ) ^> FG)O){h2 fb-)
:]HFGF^FXfbfH+)!s9)o'XF^ E>)F& 89>)O)
3>W93{
ghFDFXFG)DsfH,F&'&)-) s9)o'XF^"`89) >fb Px'7./F^F^,!)^9 FI)/393
s99^ :9^F^)) PT\1`>)^ 1 1<K )
f//,
PZ)
:]HFGF^
3>E>93F+|)
~0
1,.sE>
Pks9)FT?).>FG)O){ g7FDf, )>>PN `9/FT'X?b, FG'&)`
b
P es9)h_/`X)8
es9)h_/`
32
64
& ]av @
2.3.5 Goals
V7)7)F7E) 1b,aFG9,3?
Dispositional.
g H(' )!s /F^` )^>
^,93 k1$/ PN,9`:)FD/3 ,93>FI)/3-) ``>)
P `89>)>9>F^F<
H('

).)oE
bool
{)FXf//,
/ r
/3 )+`>fb>FGF!/3 E>)^FXfb fH
bool
;,93>^FG)/3-) > f//,9)aPx:]H>F^F^FDfbfH `$/r/3-)9/{E>)F ^)^`,N )^9 k1
$/ 2 f//,9) h,h/3 )7``>)nP F^^)4`^`,) >f//,9){
de ^)`,N b,-F^,93 sE"s9D)>$
Operational.
o+g(fb!Z3FG(fb s /FG `TFG))>)Fhsm,) k1$/
PN,9`>)^F<
o (fb"'&>)>h&)&+ fbaF^>)ZP s99^!k$/ PN,9`:)F7FZ`>8 >)

o ;_
:f/, O)h fb>8:]H>F^F^8fbf1"O)>)` /`$/ 93 N)^9/HE>)F<
oTA+
^$/3y/3 ,39:FG)/3y fb F^89 eF^>sE('` kfbfH s uv`:)F8P
^)>)`Z)vE]e `,3
/3-^)>)` `$/ 1/3 )9/mm>)^F
bool
g 9+)> ./1[']Lt XLM:</N
:/' >PN>F )
/p)9)"`F^FG)^F!P _/?PN,`>)?'X`
7
)^,^a`F^FG)^F&Px.F^ ,9>9`> P 3`NO)/3a:]19^F^F^-FI))>)F &)^,/ H
) PN,939/)/m)o1mF&93E>)^Fh3FG`,F^F^3a) >`3 F^>`>)FZ ,9FG3{
2.3.6 Exercises
:
Exercise 23
n
-v57u '=V|)1 W<:</M9bc
D?S/h!012M.
f:
n
_v:/F&j:)10.qLM:<FdB65C+ ST:^+"
Exercise 24
x, y, z
eeB/ ZC *FZ/F^FG1`NO)fbg
(x y) z = x (y z)
eB (V ZS , C ) FZ3FG)Gs9,=)fbg
sC
(x ∧ y) ∨ z = (x ∨ z) ∧ (y ∨ z)
!` (x ∨ y) ∧ z = (x ∧ z) ∨ (y ∧ z) eB ( C , V ZS ) FZ3FG)Gs9,=)fbg
h3 (x ↑ y) ↑ z = x ↑ (y ↑ z) e B ZV &SwFh/F^F^`N)^fbg
→
,s10.< o+g
o+g
:
x 1 , . . . , xn n
<:!'j.^ SX657
Exercise 25
xi 1
i
n
-\N1"F&I|F&/:
;/+R)J<.Q<:!1Lt01q S
x1
x2
... 1[
xn
o+g
M8 57u 65C+j657^SQ" u1[7Ny.Q<6. S3SC!01qM.x:< /']Lt <ESQ: 657 T. !I S
;!1":!V bc
,?S/h!012M.T
o
Exercise 26

{
V &S
,
XCTg
}
4 &
sC
XC g
,
}
!` {ZV &S }
h3 {XC } H'&> &C := &C g hC "
! &C
V &S
{
,
}
4, _ ,9FGn)kP `>) )9) {V &S , C , XCTg } F h`89>) F^>) P1s99^ k
1$/ PN,`>)F
{
C
@;
8=LYL=.T - -E)
:< +"`F:!1qY;gD!. SA0V L7
a b
c
uv5&12<5y657|,s!LM:<!.<./1qM.
)
Exercise 27
-3) SQ:
b
c
:<!.7~6.T
a < b < c
1")xFd+Z !.]SQ: V+12<D)x)1 W/:<<Ma
o ;
int
a < b && b < c
_`:/<M657!./1 W657ST[Du1[7N^,sgLM:<!././12M.XJ//:)1[7NE<L=/:/+:aLM:</)J//!.
)I+./.Q
<1 +01[F
1[012!.T
o ;
Exercise 28

sC
`!
x != 3 < 2 || y && -3 <= 4 - 2 * 3
z > 1 && ! x != 2 - 2 == 1 && y
3 * z > z || 1 / x != 0 && 3 + 4 >= 7
F +~t+a657P,sgLM:<!././12M. N1[Fd<E1[ sJ<:<<1[.Q hmX.<2Lt{,;<Vd{q.<2LM- +././C'^1[7N\65C+
:<+[ S]0VgL=
u1065
- )
9

o ;
Exercise 29
- -`)
x y
z
int
x==0 y==1
z==2
5C+],V$x.TV Y;<CP6577 LMC] SG657`ST[Duv1"7NGLM:</N:/'
<:!1 B1[R)J2L=/)1[7NO 657B1"&LM7R)I,sgLtU1[ Vhh:B:<,+.Q=1[7N&
Exercise 30
Á
Ã
Â
# include < iostream >
int main ()
{
int a ;
std :: cin > > a ;
std :: cout < < ( a ++ < 3) < < " .\ n" ;
bool b = a * 3 > a + 4 && !( a >= 5);
std :: cout < < (! b || ++ a > 4) < < " .\ n";
return 0;
}
5C:/J!{
o TA+
1[)?6573 N12,+ML7:/<M657!.Q!.z1"Z1[!.GlO)|k @ SK657aST[Duv1"7NBLM:<N:/'A
5
C
+
I
V$.TV Y;<7j657H7 LMCA S 657jSQ" u1"7N LM:</ N
:/'

5C:/J<<:!1 10
)J2L=/)1[7NO 657B1"&LM7R)I,sgLtU1[@V$C:]:<,+.Q=1[7N&
oTA+
Exercise 31
Á
# include < iostream >
int main ()
{
& ]av @/A
Ã
unsigned int a;
std :: cin > > a;
Â
Á
Ã
unsigned int b = a ;
b /= 2 + b / 2;
std :: cout < 2;
std :: cout < < c < < "\n" ;
return 0;
}
2.3.7 Challenges
p 57 h>fb>FGaM F^ h)) i P_ vo 1[.IEST:!'A+G S?uv:!1[01[7Ny,sgLM:<<.<./12M.
u10657C=V]L7:<<M657<.Q!.T \_ ;<'I`L=<LMCD:W1" 657XU+W=1[</<M./1Us012<.]uv57< 657
/']L7=V <uP <66{2_J!:/).<2:g,)BC.QP1[v+.`1[&LMChSQ:!'+hST:B,sgLM:/!.<./1qM.]|657/1[:
)J!.<Q<L)?5C)+57<D)I+D<7U+:g.T
# \_ \-Bu` :g./zuv:!1[ 657@<L=<:)+./-?) 657< 657@<L=<:/+: i65C+Q% .Auv5C+G657
Z:fb>F^ .<2)+.vST:o& 4:^,sJ']Lt <-v657^,sgLM:<!././12
Exercise 32
V ZS
(
C
(0,
XCTg
(
V ZS
(0, 1))), 1)
y;<Bu:!1[6<x~1[dK65&10.X1[ P_
001
V &S
XC gzC
V ZS
1
.
p 57U+6<:^.Q *</A SE<L=<:)+.?)<L=/:/+:g.E)J !.EA.qL=//1 <Fd+Z*+01q.Q<{
* </G Sc6573 sgLM:<!.<.12=- .QW8/!01q 7 7 h p ?/Fd+~t+z,sgLM:<<.<./12 1" P_ P- u`
NY 65&:<$N5|6573.Q //S/:/' D0S<?:!1 N5Y auv57<<Fd<:û` )I <L=<:/)-9u`j)J*% )J =V65&1[7N-K;!7cuv57< u` )@ <L7<:/+: i/ S :!100V o-cuÒ<F+~t+E10ST:?657
<L=<:)+.^)1[:<!6ZV| 6573D0S<c SK10P) :<qLtDJG65731[=F&ZFn&)
.Q < ! /'I<M6n.
;!V3657`:<<./7~ S657X<F+~t+012 p 57<Au`NYKB657`,s+.Q n<+B1 !D<'A<M2R# ,+.QK S
C:^,sJ']Lt GY;<F&/-a65&1[.`LM:<Q//,)+.^+.SQ[Duv.Gi/<C:!:/<M6~VGLM:<
!.<.Q,) <L=<:/+:B1[y;/D)!o
XCTgzC
0 0
AND}
| 1 {z
0
C
V ZS
0 0
NOT
1
| {z }
1
V &S
0
1
OR
| {z } 1
1
1
AND}
| 1 {z
1
1
1
V &S
&
4 @0
p E.T/z65C+65&10.X10.X1[)J)Oju`V| SG<F+~t+01[7N?657:!1"N
1[+f,sgLM:/!.<./1q
V ZS
(
C
(0,
XCTg
(
V ZS
(0, 1))), 1),
V$,WSQ:P,sJ'KLtDa'+dPB;<6'E{ LO):uv1"7N3 SO,sgLM:/!.<./1q?0:</Xiq8/!01q 7 7 o
65C+\:!:<!.qL=)+.K?657<Fd+Z*[email protected] *</B1[ P_ ]cOu1[[ )?65C+65&10.]0:</31[.
+~.QjFd+Z1q)PSQ:B657j:!1 N1[+f,s!LM:<!.<./1q
<:</'A!. 657J<0*+P s$<:<<1[.QQ :!1[LM:<N:/'E. and.C - or.C -G) not.C 65C+
:</<1"F&j+.W1[&LMC`j.Q < Sz;<
D@F+~*!.W1[
i q+"<L=<:/)+.]^657BD0S<R S
s
{0, 1}
657 <L=<:/+: od p 57?C LM7R.<577U)|;<^657^.Q *</
65C+ù`N&!];!VI:<2LtUJ<1[7NI657
U+.< <L=<:/)+.K1[ u1[65|657z:<!./Cc S<Fd+Z*+01"7N 657sz:<<.qL=/<01[F&?<L=/:/+:`SQ:3657<'
# ,+n.QI S
s )
-Ru`jC.Q
-K)BSQ:
:O,sJ']Lt /-B
and.C
n=1
1[&LMC
-$LM:</N:/' or.C ./577U)^Cn LM=72657R.Q *</not.C
- u5&1[ .<57CD)XV+12<D)+.
(1, 1, 0)
and
(1, 0)
not

(1, 1, 1)
# Y)Y)10012=-\uv:!1[3LM:</N:/'j.
)
65C+37 LM7W657 .T *</
&;!{
s
57K1"&LM7 one.C
1[+"~V+- u:!10G]LM://N:/'
i2u1065
21[,)I;!VA!LYL=<)1[7N : ^6zero.C
eval.C
j1"&LM7 oj65C+\7 LMC6.X0657j<1']Lt0V .Q *</Ts
p 57`NYQ+* Sc+[765&1[.910.`^/Fd+~t+];/
DzSC!01qM.`1" P_ ;!Vz./1[']Lt~VE,+[Z1[7NB657
/:!:/!.qL=)1"7N.Q *</ S`LM:</N:'E.Ei"LM:///,)J) ;!VO+[
o
-uv57<:<G657j7 LM7
eval
S|^LM://N:/' 10. 7.Q,)y+. 1[&LMCST:I657A,sj1[ 657O.Q <Q # =1Ds)
1[=Js?65&10.H< N&M6~V;<j)JKF
1qA 9E :E,sJ']Lt /-aO/Fd+~t+z657E sJ'KLtD
,sgLM:/!.<./1qES/:<' Y;/F&B1" \_ \- u`z.1['KLtZV 0VgL=z657^/'j')
./ eval |./ zero |./ zero |./ one |./ and |./ not |./ or |./ one |./ and
p 5&1[.,+[~.+"Z10./)zLM:<N:/'E.K1[0h:!=-auv57<:<j?.Q2L7:/+01"7NzLM1L7z./V+'A;/ 5C+.B657
W!W65C+\657A7 LM7X SG657KLM://N:/' O657^ [S<B S31[P10.3C.Q) +.^6571[&LMC| SQ:Ai [1[.
LM1L7)E1"M oE657RLM:</N:' ?657K:!1"N5YR SX102
M.T *<M6~V+-4657auv57DR ST:<<'A<M012,)G/'j')X.<57CD)\./1[']Lt~V\uv:!1[ c.<2M{
)Y:/)I7 LM70-657]:<!.7~P SX657j/Fd+~t+012 P.TE!.<vVhh:RLM://N:/'j.Ku1065A.Q1'I657<:
\_ .Q *</<./-1"AL7:g01q<7U:3657 2&;!F
1q7. ^!.G SK657SQ:!'
./ eval |./ zero |./ one |./ or
i65&10.GK.<577U)IC LM7 oEE'+dK./C:<B65C+a657<V u`:g+.3,s!L=/!,)Y
1
# 10. j/!.<.T:!V 65C+jV$C:LM:</N:/'j.yJ/2Lt?.Q <!.
Sx:
;<100:/:!V
Hint:
s
D<7N65+.91"&LM72 'Qs+1['^C' D<7N65 S Y-YST:K,sY'KLtD<-10../ <1q<M$ST:W+[+LM:J!012,+
LMC:L7.Q!.T
& ]av @R
2.4 Control statements
K:<Ruv5C+ u`\:/2L=+,)h~V )J& s$/!" //<-657<=-10.J!
!; CP?5CY;!1[2
1[[>WC:/M1[yE./C'^'A:!VO S R:!10./6 C% .]1 )J+./p 57]8M:!V SK_v5&10D.Q<Lt5&V p 57 1[F&!.3) efLM1[=1qM.
S]657 :gU)% . R:<,+!.<_v5&10D.Q<Lt57<:g.Gi/k&l Mn/o
p 5&10.3.Q!012 1[M0:/Q) !.PSTC:?/2Lt6.j/M0:/v657 ,sJ/<C012 S^^LM:<{
N:' F^`:) - )>O) - s91`1F -]) uv,8 F p 57!.T/2Lt6.A<Y;gDj7.
O)J<FQ1q+9S:<' 657^)J0STC9Z1[,: /M0:< \u uv5&12!5 ,s$//7!.3.<2+/'I<M
;!VH.<2+<'I/M\S/:<' <L ;/6'A u1[[cD:! 57u(657!.Qx/2Lt6.
:<^1[']Lt <'A<M) 1" `K -K)O57ur !LYLt~V 657/' @/:<+E1"M<:<!./01[7N
LM:<N:/'E.
g F&'7 9$fb FG?FG
P O ./ 9^>)^)o-F^89/Xg79:-`F^FG)TP48F^ 1,`
h
PkFI))>)FD)9)"+:]H`,=)3 .s1 +P^).WFG) )
)^9+N/FI)$+\1,` 8
F"F/3 ) fb_ ~1": /M0:< \u ghF")vE P `>)^9O'tF ,) ^FG)^`>)fb /F
$/` FG)O))Za)!F^,=`!`13!F&:]H`,=)3 )ZFG)Z9`> 3,^ )! ]=`,=)-P )
/-D\1,EFG b, 'h/)D)_89)Z/ /)^9J)9)XE:PNF $# B###8FG)>9FTPN
F^D=9,=)$ g79> b,_'7,93 9fbD) '&)^T!9^/ 'X )
)7$/FI) $# <### F Pr`39/
CXs1fH,9FG)FDFX,93FGs9Xgh>>PN { 3>D)_89>)h )^f1N/ / )F DE('7>PN,{`9/F^F ) `)^^{)^9 9O' P . O">39>3r
2.4.1 Selection: if– and if-else statements
CD9. ^)`,NF^89 'h-) 39:fHN)^+P^$T`)^9O'zFT)
F^>`>)"'&>)> .<2+<'A<M )T+ G)`,NZFG)O))XFZ:]=`>,)^3{Xde 6ZY Yz)^9F&`$/-sE!39!f1N if
g79 FG1)O]F
j
'X>^ F/
if
:]19^F^F^*
f/Ns9 3`NO)P )vE '&F^ f//,F
sE `fb>^)>3 )
E/3 /FX) 9/"F^,FG)F FT FG))>)$
bool
F^_)`>FEFE)4PNO'& TF{>f//,9)^3M/P/3!TP)FEf//, F 0:! F:]H`,=)3 /P)>G'h3Fwde )^9:'73F &
FG))^)8FG9)F )^9-`>)^ 9('
if
)o'7 s/`F gh.f/,9+
P
F^>`>)F 'X`?P4)F^+s9/` F"F :]=`>,)^3{
:]=/89 1) PN('X.F7P `13
*5(
RBI@( R"@
-.*
6M
"@.*
! ( %
`$/
g7
) 27
(7
"@.*O 4)(77
)
=
( "
%*(J
& J @@
int a;
std :: cin > > a;
if ( a % 2 == 0) std :: cout < < " even ";
$3|a,9.sE> P GF )/393 =9,=) )â)8f^Ns9
/3 '&)^ -) FG)939^3
a
even
,=)^9,=)ZP /3a_P Fh>fb{
a
{,;<:/!5 g7&FG1)(]8F
CD) b/
FG)O) )n`/8sE&`>8 )^3 s8/
if
else
j
if
else
/ 3?) F^_/)`FTF"/F!PNO'&F FX ]=`,=)>3?/P)^>^'h3F< )>^'&F^
PN('X9 FZP `13
F!:f/, O)3M Pn)FTf, F
aF&:]=`>,)^3?P)>^'h^39F>
0:! =T:]=/89 =)
int a;
std :: cin > > a;
if ( a % 2 == 0)
std :: cout < < " even ";
else
std :: cout < < " odd ";
$35-1,.sE>.P FG)939^3|=9,=).)-)f/Ns9 g7|P
F >fb F
a
a
even
'&^)^)
)+FG)/393a,=)^9,=)!=)>G'XF^ F7'&)G) )^ FG)/3 O3a,=)^9,=)$
odd
9>?PN_)G) FG))^) )DFX`)9FG>^) 9"s9^$ asE>PN msEPN else /3 if
sEPN 1 >Ofb> /3 ?
3)^3y/3
F /3 'X)
7/F FÔ'& 5):]=/89 sE(fb dePX)^9
'X
else
if
FG)O))hW)Fh- F^T9D))h`$/ /FG+sET)o1E>F^>)DF&.F^T
67`>) fb /3 if-else FI))>)Fh H('XaF .Q!D/<012.<2+/'I<M6. if
2.4.2 Iteration: for statements
VQ+,9` mO'7>^PN,9n'h$ PZ_/9,N)^_)^9`>)^ (' F 9Ôf133 s 10/:/+012
.<2+<'A<M6. kd )>)^_/('XFn).:]=`>,)^" FI))>) _/ )F mF^F^s9 'X)3l >)
/:)>FE$` )^rd )>O) FG))>)Fr^k/F^Z`/3 Q<Lt. /Fr)>G")^^,9 Z
FG)O)9
) E)^)/Y
xFG>fb>/m)^F \1>`>) 93_)^>)8FG))^)^F D`>`>)^fb
PN:^3a) /F /M0:<.<2+<'A<M6. Pn
67F^39:)?s9 P `89,=) ) F^,
P )^9 WFG)
9),=
Sn =
n
i
1,.sE>F PN_?fb
Mx/ @ $/3F8y fONs9i=1
P FG)/3 O3y= ,=) n
n
3>W9F&9)^9:hf/s9
) `)/_) >F^,) =`89,=)F )TF^,)Z/3
W 9/,=)^9,=)F
s
)$ de83>4)!,3>FG)93+'X+)^97/
39>3 sE9fbFn/F4`N/>3 '7&9fb
sum_n.C
.<2+<'I
/M ) :]H9N/
) 3lm>^)h ^)FhPx
for
& ]av @K
Á
Ã
// Program : sum_n .C
// Compute the sum of the first n natural numbers .
# include < iostream >
int main ()
{
// input
std :: cout < < " Compute the sum 1+...+ n for n =? " ;
unsigned int n;
std :: cin > > n;
Á
Ã
Á©Â
Â
// computation of sum_ {i =1}^ n i
unsigned int s = 0;
for ( unsigned int i = 1; i <= n ; ++ i ) s += i ;
// output
std :: cout < < " 1+...+ " < < n < < " = " < < s < < " .\ n ";
return 0;
}
Program 7:
LM:</N. t./C'
gh
FI))>)DFT fb:^ `8 /`>)&PN Pk )^>)-FG))>) /F
for statement.
for
)&`+s9Fh)= FG)))F&X:]HFGF^FD)^8Dde FG)X`/F^F )
FI))>)
for
F^>GfbFD/FZ G`>,9)+1 +/Fh-Mx/ @1nd )FZFIH)O] FZ3>W93s
7
for

'X>^
F8 :]HFGF^ FG)O)) 3`>N)^|FG))) k )a1,
FG)O)) 1F^>!\1>`>) 1A= de/ P{)F^D`$/FGF >93F 'X )
F^`
F^,` )9) )^9: ^ /'hHF{)o'7&F^>`F{ sE>)o'7> )n )F^F /P)> &F^,9/ for
E3>W9F4/3 )^N/ F Xf/NOs )9) F ,F^3 )^"`)^/3 >fb)^, 3
)4)>O)!FI))>!) Fr ]=`,=)r de
9F 73`N) FG))^)
sum_n.C
)9)h3>W9FZ)TfONs9 i
VXFha/
FG)O)) _Fh/:]HFGF^hf/Ns9D3`>N)^
'&F^D)vE
if
`$/ sE.`fb>^)3 )
d )T3>W9FT9(' 8) )^>)-1FT
m9//FT_/F
+d )"F!/O'7>3 )^ O)
-:),=F 0:!*bool
F >89)o- 'X` `$/F^+)FDf//,.F
)>^>)^3|F 0:!* VXF ) 8F^,FG)F |F./ ^s9)G^?:]HFGF^ )^ O)
_/F^ sET8)oE
e'&9` _`$F^T)79/F +:lm`>) : xF7/ ^s9)^^8FI))>)$
d )ZFh^PN>^>3a)8/F7) ;<Q)V P )^9
FG)O))$
for
& J @U
gn19`/ 9/F+)^9
>l `>) P&`9/ af//,_)9) E$F.
{
\ ,`8/ >lm`:)nFxF/3.)!G_ bk>F^Fx)O'hO3F )^>9) ={g77b/HF )9)
1
F ST+~.Q /P)^>
89/F sE8:f/, O)38XW9) ,9.sE>4P9)F de )9) F^>9FG 1>fb:^
>f//,9) P
_ bF4"FG)^>)('h3F4)^9h3P )
FG))>)$ de
for
sum_n.C
_`>^)^F4)&f/s9
'&`FnsE,33PQOsEOfbZ
{xde`/F^F
i
b )FX'&>.).f//,.P F^9.f/s9 F /``>F^F^3 /3 ` 9/3 s1
/3
='7 `$/E)F f/Ns9D) /M0:<4Fd:!1 Y;g P )^9 for FG)O))$
FG))>)$ FG) +O'z$3 )_`F^ 3>W9+)^9+F^_)`>FDP !F+:]=`>,)^3 `5g7>$P)>
F >f//,9)^for
3{|dePX ) :),=F 0:!* /
10/:/+012 P4) 1 FI)^)F> deP >),=F ST+~.Q {)
FG))) <:!'j1"+!. for
)9)hF )^F7`F^FG 3F&>39)b
2n` F^ )^>) P FG))>) `F^FG)FxPW9^FG) :]=>`,=) r/3+)^9>
for
>f//,9)9
{aVXP)^> $/` )>)
F :f/, O)3 /b/r deP7)!^>),=F
0:! =/)> )^>)PN('XF deP >),=F ST+~.Q )
FG))) )>^9)F
g79D ]=`,=)
^39:7F )>PN^
for
{$G,9) :),=F ST+~.Q qr:!) BFhF^ )FZa/`>)^
k6 F^3>7)
FG))>)
for
for ( unsigned int i = 1; i <= n ; ++ i ) s += i ;
/3 F^,=EF^
FG) /) fONs9 F 3>W93+/3+9 )N/ 3T) xg79>
sum_n.C
n == 2
i
1
)hF7)FG)^3'X>)^9:
4VXF
F 0:!* 9)^9TWFG)Z)>)_FG)O^)Fngh FG))^)
i <= n
1 <= 2
F :]=>`,=)
3 F^>)^)^
) 7/3|)>/P)>
F 9`:)3 s5
F^,` )9)
s += i
s
1
i
CD )>) F O'|`89>) VXFx&:]1)xFG): )^9 `>93)^
F >f//,9)^3
i == 2
i <= n
/b/rZVXF
F 0:! {9)^9:D)>)-PNO'&F FG)
FX:]H`,=)3EF^:)^)
2 <= 2
s += i
s
) 4g7>$/P)^> F `>^)^38s1XFG,9`_)9)
4g7DF^`3_)>)Fk9('
3
i
i == 3
`89>)^4gh!FG,s9F^ 1,)&>f//,9) P
)/F
'&9` F S +.T g71,F i <= n
3 <= 2
PN,=^)>h):)8) bF7 /`T/3_)T`F^FG+P{)
FG)))73F4ghDf, P
for
FZ('
=) F^, Pr) WFI)
9),=/{1,.sE>F>
s
3
n == 2
d ) F $/F^ aEF^F^ s .)^ `>^$)+19FX)9) 3)T)>^9) :]=/89 Infinite loops.
`/m)9) sE)^
93
_8sET8)ob(fb> =sE)
/3 n`$sET)",9rFI))>)$4de)F7`$F^"'7 >)h)
FG)))
for
for (;;);
VXF7)">89)o
9/F7f, 0:!* 9:]=`>,)^+)F7FG)))7,9F7)=, )>)^
/P): )^>) '&),=)./`>)^, 3 /1)= g7>PN^ _ sE
^$/35F
(;;)
GPN>fb: = dea>9:/ .FG))^)7'&9` 31Fh)Z):9)Xfor
FZ`$3/ 1[ =1[BD
<L 67$ =W9)D9F&^!:]1)^ ,3F^s9!93 9^/>F )^G93 )fb3
)- Z>fb>G)F^F /F^>)^FmF^,` F{``,= :fb "^$/bPN F^P)o'h/xdeP1b,!>,9
,F^
a`89,=)> b, 9$fbs s9?:]HE:`3 )F 13 P79 +
G9/F =
4, _ /F & 3F^M )T) `89:DF^89 3>)`>) =W9) 19FD/3 'hOFD
sE,=)x) uv,FG)n/F ) `89NF sE,=)xFIH)O]+>^^Fade39>
3 b)^9F '7,93+sE7D)x)
& ]av K#
)" fb&/3 )4'7,3 F^ fbZ_/ s9Fx F^P)o'h&3>fb>9)$ gh79^s zF4)^ O)
=W9) F+^)8'hHF+/F+$/FG )?FIE)8F )smOfb :]/89/ qr F+`$/ sE
>)^)o `89`$O)3H/3mF^F^s9)> a=W9) 'Xa:]H`,=)3 `>^)/
FG)O)Fh9 b
de P `>) )-F^),9)^5FEF^F d )_`$/ sE-FG9('X )9) )s9 PT3>)`:)
=W9)19?
F e` >PN>^3 ) /F8) 5C+01"7NLM:/&;g /' 8`$/)sE-F^ fb3 sy
`89,=)> /F '7X9fbT/3_,93>^FG)/3
)k)39$j
eF^X)^9TSD>)/F g79:PN =FGX`$&F
33'&3F^99 19Fx 9$fbD)+`` Is_ 93 .)9) ) )>O)FG)O))
)>^9)F PN&/EEF^FGs9D9^/cFG))^Fh)^ O)Z`/a1`>`,=$
4,y_ 1O' _ fba/ >3y)^ O),=9^/ sum_n.C F8/`>),9/ 5 s /3
Gauss.
:]=/89hd )DF&s /3 -) FGF^ )9)D)&31FD)X`>fH9`>3FG)G)T) E('7>DP
`)^{FG))^)^F
deaF79^_^FG` 39HF =)! >^_/_)^9>_)`/_67 H3=` /,F^FR(@@@ $Kb0
0 n'h/F )3
).F^,c,=)D,9.sE>^F
g7T)^$/`>&9/3_9N/3_) b>
1, 2, 3, . . . , 100
FFG),3)F8s9,9FIyPN
?'X s9,=) "/,F^F`/a,= '&) )-`^^`>)_F^,)
fb>^
1,` 1b %Z 9/3 _3-'&)^
3('X?)^9+1,.sE>F 9`:$/F^9
3> /3 5050
sEO' `!/b/
a3`>^$/F^ 3>$k67 1)^9 )o'7 ,9.sE>^FZ $/`-`,
F^, ,=
19>9`> H)DOfb:/mFG,9QF 100 101 = 10100 H9/PEPr'&9` _F )^9D,+sE: )9) 'h/F
)
101
/F b>3-PN$
;
$>
UK
UU
$##
$##
UU
UK
$>
;
$#=
$#=
$#=
$> $#=
$#=
$#=
de)F7'h 9
"/,F^Fh3F^`(fb>3a)^9 PN+,N
n
X
i = n(n + 1)/2,
i=1
NP gh
FG))>) `$/?)^9:PN+sE : /`3?s ) +,9`
n
for
sum_n.C

T>b/)Z/3a `)&IF ))>)
s = n * ( n + 1) / 2;
:]H)&>)h)8 $ 9`)_P FG`>)â/3a )>)^ FI))>)F
de )D)^13,`>)G
\1`>)? 1'7T fbD)/b3a )hOsE,=) X,9 Prime numbers.
sE>^F4%&O'y'7,3 Tt .b7)9)4)FG)F4'X>)>n )k fb81,.sE>nF 9^(
VX``>3!) )X,F^,9/ 39:W )^"1,+sm>
FkZP{93_ PE)kF )
n
,n 2
3fHF^s9 s1?/ 1,.sE>
g79 FG)^)^?PN!,=!/F )>PN
d {2, . . . , n − 1}
` ?^)D!1)9)n,9F )^^,9_ )FGD1,+sm>F 1/3_)^FG) $` PE)}PN sm
!3f1F^ P 4deP{!3fHF^ FkPN,9
3 1'7X`/_FG) /3_,)G ,=)7"P /`>) ) P
)!)o'7
n
n
F &2
/
7
(M.*
n(n + 1)
-( %"
%*4J
%*(,7
* %6 M/ I *
#.*
%/
$&
!/ *0,
*
8 N2
n
,
& J K=
1,.sE>F b(fH9 )9)
Fn)4 CX)>^'&F^ b'7h,)G ,=)4)9)
F 9^ Mx}K
n
n
89>)F )FhFG)^) 6hY"Y m,F^ FG))) 9/3-
FG))^) Z for
if
_ /s9 H)
FI))>)Z FD/a8)o
sE13= F^` '7 9$fb" ,=)Z)!3fHF^s9)o )FG)
)a)
for
{ag78E^)/) s9F^>^f/) F )9) ) n % d != 0 3>W ) >)^,^9F S +.T PN
=F^ )9) )D
F ,9/)3_) )>^9)
P4 /3_
d == n
S
+
.
Q
2P
>)^,^9F
$> '779$fb7PN,38D3fHF^xP
+) /
n
{2, . . . , n − 1}
Á
Ã
// Program : prime .C
// Test if a given natural number is prime .
# include < iostream >
Â
Á
Ã
Á©Â
Á
Á©Á
Á
Á
Á©Ã
Á
int main ()
{
// Input
unsigned int n;
std :: cout < < " Test if n >1 is prime for n =? ";
std :: cin > > n;
// Computation : test possible divisors d
unsigned int d;
for ( d = 2; n % d != 0; ++ d );
// Output
if ( d < n)
// d is a divisor of n in {2 ,... ,n -1}
std :: cout < < n < < " = " < < d < < " * " < < n / d < < " .\ n";
else
// no proper divisor found
std :: cout < < n < < " is prime .\ n ";
return 0;
}
Program 8:
LM:</N. YLM:!1['IT
2.4.3 Blocks and scope
ed y6hY"Y )+F.EF^FGs98)?^, F^ 1,`PZ _FG))^)^F+) F^9
FG)O)) )9)xF )+`$/>3. /']L=C)K.<2+<'A<M xF^89" ;gD
! ng7F `9/F^
31F )!_9 ,NO)")+`)G9('z3`>)^b k1`HF"/O' )^ FG)G,`>),=8/ s
,=9 FG)O))F )9) `$ sE9 )>)>$ de ^)`>,9 k)>5- )17)
3F^ EO'7:PN, 93a)Z)^9 F/D)X$/39s9T`)^{FG))^)^F
\H)/`>)`$/ O s91` TFrF^89XhFG 1,`nP >^& FG))>)Fm)^ O) On>9`>F^3
& ]av K
`,=_s/`F>
24/`5PZ)
FG))^)^F+_ ^)`,N sE s9` 4F^?)+F EFGF^s9) fb 9>FG)3
s91`1Fkg7"FG89FG) s9` _Fh) 8)os 1`
{}
4, 9fb $/3=-FG?s9` 1F"2n/`?/ `)/FT8FGE`>N/ s91` m)+FG `$/3
S/h!012 ;<Q)V Pm)
PN,`>){ g79Fns91` +`FGFk)&FG 1,`XPEFI))>)F )9)
main
Fh:]=>`,=)3a'X )
PN,`>) FZ`/3
s1_)!m>)9.FG1FG)-
main
&F^.s91` HF =9 `$/`>$)^"F^>`>)a/3 )>)^ FI))>)F '&9FG sE3=_`)F
F^ ,9>9`> P )v'7
FG)O))F = :]=/89 mF^,=EF^.)9)TPN )FG)^_9,=ÊFGF
'7T'7,93 bD)+'& )T,)&/m G)N/EF^,F73,=+)^9 `89,=))
sum_n.C
for ( unsigned int i = 1; i <= n ; ++ i ) {
s += i;
std :: cerr < ^Z)o'7!FG)))F4Z:]=`>,)^38$` )>)^ P9)& FG) )h:]H)kF^,_/3
&
FZ/333) )^9>)"`>,G)hf//,"P F7'&)G) )^ FG)/3 O3a>^^&,=)^9,=)$
k1` 1FhFG9s,3_
>msETPN_)^)>s3 F FÔ'&smOfbng7 O)7F = 9Xs _E$OF
/P): )DE /3_sEPNX)D`F^!s/` /3 9>F sE>)o'7 D93)3_
>fb>e CD-P ) s 1` -`F^FI)FTPruv,FG) +FG FG))^)D/3 ) W)FT?. )Ts 1` `$/sE PN_)G)3 /F& F^T
ghD)o1EDPE)>FG)h,)G ,=) '7D9fbT`>$O)3
)^9X:fH,F :]=/89XF `/3 )JT;!hN
N
1[7N
7 LMC V ;<hN F7 `+,F^3)^> )+3) !9^/!:^ =`!
G3>s9, FX) `F^FTPxW993_/3 O).F^,` >^^F"d )TFD13 `>)`")'&)^
3>s9,,=)^9,=)") FG)/393 :^ ,=)^9,=)"F^` )!`$ ) +$/F^ sm F^> )3
Pc) I^$/ ./c,=)^9,=)Z)9)h,F^,9/ 1Fh)+FG)/393a,=)^9,=)$
k1` 1F43T)49 FG)^^,9`:),=h&9^/ fHF^,9/"s ,=) )>+/F^D9Ôf137T`$/
Visibility.
sE,39^_^,93
3`N)FeP{fONs9F HPNh:]/89!: VX_39>`NO))9)hE$OF
F^3 s9` F"`$/>3 Q+ ))9)"s 1` mVt1`$n3`)?:]1)3F ,9) )
3 P )^9 s9` a-'X`-)DE$F>XV 9/")9)XFX)^13,`3-s `$/ 3`>N)^
Fh) If1F^s9 ,=)F^39TPr) s9` _'&9:" )ZFZ3`3{ Z:]=/89 H
Á
Ã
int main ()
{
{
int i = 2;
}
std :: cout < 8 > '&) )^9Fh`>13 )hF^F^,F&/a>G&>F^F/
6 )G FG)))FZ/`>)D b"s91`1F&)F^ fbFhgh>^PN
Control statements and blocks.
/ 39>`NO) E$^
?`)^xFG)O))TFD1`/ ))9) `)GxFG))^)!de
^)`,N )^9F 9F!)^ f/s983>W93 )^9
&P& for FG))^)
=& ]/89 1
Á
Ã
int main ()
{
for ( unsigned int i = 0; i < 10; ++ i ) s += i ;
std :: cout < Z) )TfONs9
i
3>W93a ;H
VXP)>_9fH FG )^9>FâWFG):]=/89F '7 'XhO' )G13,`)
Declarative region.
`F^)>^ )9)8/('XF+,F ) 33,à'X`59/>F `$/ sE ,F^3 'X>5)
/- 2n/`
3`N)89/F7 F^F^1`>N)3 )J!D:+01[F&X:< N12 nghFn_Fk)^9X G)
P=)k/[ 'X` ) 3`N) O9m$Fx\1,9` Z+`/+sm Zs91` b&PN,9`:)
3>W9)
1&.`)^^rFI))>)$ de-m)F^ `$/F^>FZ) 39>`NO)
F7F/3
) 9fb D
,+
.Q<L= &V[39>`NO)`/ /F^89fb 'I<.qL7J/G.T/<L= P )XE$F&F^3 + FG /` F^ \1`>) 1B;H 9/ =.3`NO)_)9)hFh,=)F^3 P / ^)^`,N7)>ZFG)^^,9`:),=
9/F N &;+4.Q/<L= V9/ )G13,`3+s1 T3`>N)^
F Fd+Z1q) F
10./1 ;gD 8D O^)4P )F43`NO Scope.
D
)
)M
Fk3`NO)fbh`$/>38) .Q/<L7 PE)D39>`NO){ )8)^9&F^ÈDP
D
9/ )^13,`3 s1
_-sE ,F^>3 /3 /`>)^, ^PN>F!)^
)+3`N)
+de FG)
D
D
`$/FGF )"F^`>E"P +3`) Fh , {)^ )F L=/M01q+4.Q/<L7 gh L=<M01q+J.Q<L= P=h3`>N)^"FI)^)F ){)nE){'X> )k3`>N)^!O9m$F
=Z)^9 9/D) sm"3`3)^9FhFZ`$3 )F L=1[MP Sz)J/!U:/+012 ghTE))N/EF^`>E
:]1)3F&,)m)"3-Pr)"3`) fbT^{
g _>)T) F^È.Pn83`N)
'7.FG)^)TP )^FXE))N/ F^È s9,=)D'7 EF^F^s9
9fbX) (fbXF^& ^)^F P{)$ g79F O9mFk'X)XE)>)N F^`>ED`)/F D
T3`NO)F P )^9 .T'A 9//xVXF&a:]8 H`F^3>&Mx/JUH
Á
Ã
# include < iostream >
int main ()
{
int i = 2;
for ( int i = 0; i < 5; ++ i )
std :: cout < < i ;
// outputs 0 , 1 , 2 , 3 , 4
& ]av KA
D
D
E1
E1
E2
E2
E3
Figure 5:
E3
_`<M01q+.Q<L=!. S )J!U:/+012M.
S657j. 'Ij'A<-X):/u +.
:</!27N !.u10653657\/:!:<!. L=)1[7N3D,
)JE!D1, :E+20,12E3 1[ 6579+LYL=<:D0S/:!<:XiD0S< o
i~1 N5YN:/Vgo
657:!1"N5Y0-u`.Q`657:<!./C01"7N].Q/<L7!.c S i)Y:gPN:V oD
E 1 , E3
)
i2uv5&1[2o&
E2
Â
std :: cout < ^FZ)+)"3`) P R H'X>/FZ)
a"K.PN>^FZ)
gh
i
i
)!3`>N)^P "017gh>^PN 9)"/J,=)^9,=)FhWFI) 93 )
0 1 2 3 4
ndeaF^TF^F^ ) 39>`NO)
9 R.)8EO 3Fh)^9T>f1,FZ3`) P
2
Pt&01 ghF 99>9>9 Fn`/3 'Ic5&1 )1[7N k,=)n'X )Z39>`NO)fb7
i
P ) F^`3-3`) 3F&a"@ =)!F^`>9339>`NO) Ism`F7fHF^ s G '7
F G) ^,9F.,=)+P&F^`m + /3 )WFG)+3`N) ) b>F (fb> /b/{ de| ^)`>,9 TK PN>^FZ).)Df^Ns9T3>W99>3
"0 H'7T>)7)",=)^9,=)
F^`D) 9/
i
2
KH
d )XFZ13/`>)` )^fb3a9/ 9399+F^` )F&,`FGF s9PN,F^`$O)F&) 9^ /-4CD )^9T)>h9/
3 =9/D3./('XF ,`
F e bDaMx/c
U k)+,F^T,=ZP fb )
3)W9>
F!)89/+P7)^9 `)^^nf/Ns9 FG))>) r'X)^9,=) 9fH)
i
for
`` ?'X>)^9:!)> F F^ )^9: 9/
F^>'&9: >F^ ?) - g7F"F!/
i
/``>>)s9"/3>fb-,F^>PN,9 9`$)^P 9/D3=
Z('5'7 `$/+>)x)D) PN_/3>W99 ) P9F^È .) >H`$/F^`
mF^F^s9k>F^`
P +,)9 3`)9FTPk)^9 F/+9/g : gh .T/<L= P7 3`)
F!s)3
D
P)FTE))N/xFGÈ /F!PNO'&F "$/` 3`)
).E))N/xF^`>E P
E
D
F^,`8)9)4sE)
/3
39>`N7)&F^/7 )ZE)^)/=F^ÈZP
F4>Ofb3 P
D
E
E
) F^È"P
,=!0+ fbF&.FG1+sE`&9`>),= P )^9 F^)^, O){
D
de_M ^/~U )^9h3`NO)fb7 P )h3`N) 9Z0"Fn97A $#X Ds 1` J J & Kb0
)FnE))N/9F^ÈTFkD0 $# =93_)^F F^ÈDF D0!9,F &K $#H = )^9D3`>N)^
hR 3`>N)^fb ^ `>)^FG))), E)^)/=F^ÈZ/38F^`>E&OZZR H@1
n$ 1a3O'&?) FGÈF ) F F!?>/4)!EF^FGs9 rP `,=F^ F^` s$ 1F&_
eX__,) 7E$X/FG)h1'&> deP '7 'h/)h) )/
sE,=)ZF^È!-
s1 Xs /F^F H'7!9fb )^ PN_)7)T/J/``>3b
Z>N)3-)8)^9 FGÈ.P4 f^Ns9!FD)F .<:NY?)C:/+01q TghF&)^>
Storage duration.
3)F.)
)^|'&9` )393=>F^F PZ)
f/NOs F.f//3 )9) F 4F^
^
`$)
Fh/F^FG3) )$
=-f^Ns9_'X) 1`/ F^È k)FG)^/
39,=O)|F ,F^,9/ ) )|'X`
)/?
F!`)^ F )fONs9
F!E)>)NkF^`m SD,=a/ ]=`,=) )FX/F&)9)X'&9>9:fb>D) fONs9 3`>N)âFX/`9>3 EFG ^
`$)F
/F^FG393
) /33^F^F sE`>F f/3{4V&93_'X>fb:Z)T:]H`,=)
>)F ).)T3P
) 3`NO)fbn
/)7/FGF^1`)3.^ FxP^3 /3+) f/NOs BF /33^F^FxsE`>F
f/3{ D)>>PNT>)Z GPF^ FG)/` .P{)XfONs9D>fb:^1)X)F 3`)F
:]H`,=)3{
ghF7sE9fHZFZ`$3 7'A+01qB./:/NY^)h:/+012 =X ]/89 1
for ( unsigned int i = 0; i < 10; ++ i ) {
int k = 2;
// do something with k
}
) /33=FGFXPx)!fONs9
_$
`9/ -$` ):)P ) 1r7VXFG ) )/
k
O)_)
) bFh9N/`Ta/à)>){
2
VXFZ+T``>^>)! ]/89 1`F^3>h)"PNO'&+`>13"P/)
Á
Ã
int i = 5;
for ( int j = 0; j < 5; ++ j ) {
std :: cout < < ++ i ; // outputs 6 , 7 , 8 , 9 , 10
int k = 2;
std :: cout < < - - k ; // outputs 1 , 1 , 1 , 1 , 1
}
\ `8;sEF )a)F^È P7)^9 39>`NO) 9 )8>l `>).P78;aF!)
1
`>>)h)"f/NOs "3>W93-a !->fb>^)>O)P )
FG))>)$kqr9 0 for
)X)> 9/
3 smFn) )^9&F^ÈDPE)X3`)8ZAY1)X>l `>)7PEX0!F
)>>PN )3`>>)Z) GPF^ f/Ns9
->fb>G-)>)
9/3-)FX/ 'h$1FZ>F^,)F
k
f/,9 1
de `)^/FG) fONs9 )9)XFZ3>W99>3 - FG /`"F^È Ds rFGÈ 9/F .<2+01q
.<:/N& )h:/+012 g79F $/F!)9).)^F./33^F^F.F 3>)>^3 ) )8sEaP7)
/?
FT:]=`>,)^
/3 )!31>F!)!`9/j
9>9`> GFG)O)` +X"sm`+f//3 ,)
).:]=>`,=) P4) 3F g7.f/s9FX9/3-s
/3
std::cin
std::cout
&2 M'(&
8//S
:/
*
2&*
(7..B
"K.*L7
"
Q7
6#
#R
M
#6%"
%7*4J
& ]av KR
PN FG)/` fbXFI))`hFG)/&3,=)r jkONs9Fx'&)8FG))`ZFG)/Z3,)OX/F^
PN:^3a) /F .<2+012KFd:!1qY;gD!. 2.4.4 Iteration: while statements
.<2+/'I<M
\1P 9'7!9fb F^ )^>)FG)O)) =)
FI))>)$7g7
for
while
F FG89W93
FG)O)) '&>XsE)
/3
D
)^)^3{ d )F
for
FG1)O] F
?
'X>^ !/3 E^
while
/F .
FG))^) VXFrsEPN mFrPN>^^3
for
)+/F )XsE3=PE)
FG))>)$4\1>_/)`$/ b
FG))>) F 1,f//) )
while
while
) `^FIE399
FG)O))
for
Ah
for
v
g79 ]=`,=) ^39:.F!)>>PN
$^,) ?^ ),=F ST+~.Q FG))^)^F._F^ $/FG )-:'&)^_/F
FG))^)^F x'X 39 '7_3
\1`
while
for
)
gh&_ $/F^F ^$/39s9)ob VXFn )F hFG,9>FG)F 1
FG))) F )v9`$/ for
E>^`fb3 /F& `,) aa'&`)!`>^)Pe&39>`>), Px F^9 f/NOs F!^FGEF^s9 PN ) FGF )O'hO3F")>^9){_de )F!`/F^ x) >F^F F FG)
`fb)^ _/3 )^9
FG)O)!) BF
r k,=).) FG),9) `$|sE_
for
`89:] m)4^F^Fr_$X39:E3 ")^9 f//,F{P1F^:fb>/bfONs9F r F^x`>93)^
)9) '7h` ` )^9h*
FxsE13=bxde F^7P )F^h`$/F^>F 1
FG))^)4FxPN:s9
while
g79 :]H)&F^`>)^3F^`:sEF&/a:]=/89
'7!`>9FG3>&)j
"D+ G.Q <
Tfb . O),=/r1,+sm>
The Collatz problem.
n
'X)^
/3
n0 , n1 , n2 , . . .
n0 = n
P
Fh>fb
ni−1 /2,
ni−1
P
Fh133
ni =
i 1.
3ni−1 + 1,
ni−1
=D:]=/89 P n = 5 '7 :)X) F^ ,` 5, 16, 8, 4, 2, 1, 4, 2, 1, . . .&\1`!)^9!F^ 1,`
>)FX^>E>) )fb /FDFG1 /F
E$OF '7 _aFI) O)X)F&E)$DMx $#8$/3FD 1
1,.sE>
/3
,)G ,=)Fn)D)F P{)DF^ ,9>9`>
,)^ )D,9.sE>
E$F>
n
(ni )i 1
1
Á
// Program : collatz . C
// Compute the Collatz sequence of a number n.
# include < iostream >
J & Ã
Á
Ã
Á©Â
Á
Á©Á
Á
Â
Kb@
int main ()
{
// Input
std :: cout < < " Compute the Collatz sequence for n =? ";
unsigned int n;
std :: cin > > n;
// Iteration
while ( n > 1) {
if ( n % 2 == 0)
n = n / 2;
else
n = 3 * n + 1;
std :: cout < < n < < " ";
}
std :: cout < < "\ n";
return 0;
LM://N. 4[U+ +
}
Program 10:
gh-`$/yPT`,=F^-sEa'&^)^)yF_
FG))) '&) 8)o
for
/3
s9,=)7)DF^,)9 f/N)hP )D/cF7>F^Fh^$/39s9DF^` ) )^^F )
/3=fb>^)^F^7)hO)>k`>8 `$)>3+ )>)^ F TF^89 `,)!r VXFkT,7P )^1,+s FG))^)
P )>F aF^89
)9) `$),=>F )1tBF F^F 4,F^_
for
CX)>^'XF^ `F^3>ZPN+,9)!b,=X1/FX
FG))>)$
while
g /1 sE,=) >F^F 8F ) `$O )9) )1,.sE>
/'h1F.E$FcdeP7) x)
1
/
`)F /=W9)h18PN7`>^)8f,FkP 4dePmb, $+'X ) )
collatz.C
n
/ b,-'X s9F^>Gfb )9)
33 EFXPNT ,9.sE>^F&b,-)^^ m),a)F
1
_) b".'&
4, '&mW9
3 =PNX:]=/89 1)9)7)!67N) F^ 1,` PN
F
n = 27
27 82 41 124 62 31 94 47 142 71 214 107 322 161 484 242 121
364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526
263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754
377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079
3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154
3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488
244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4
2 1 d ).F >?sm>fb>3 )9)
:fb),9/ `F!,|PN+/kf//,F P 4s9,=)._) 1
n
_)`N/F79$fb )Xb:) sm s9!)83,9`>. 1PxPx)FX`>buv`:),=DVXFX9`)D/F
)D1 HF m)F&s9 FGFD)sE. fb>^ 93 _)_O)`$/E,=)D)_`>/` eF^> /F^8)
SD>)/F F^`>)^h
Hs9,)hb,-^!`>G)/fH )3 )^ fb )Z.)G^ & ]av KK
2.4.5 Iteration: do statements
<. 2+/'I<M6. +F^X)
FG))>)F r ]=`>)")9)T)+`>93)^ F">f//,9)^3
while
S</: >fb:^):) Pm)X1_9FI)$/3P ;<[SQ:< :fb>^)^>)r gh>>PN H `>)^FG)
Do
) PN /3
FG)))F H)TsE13=
P FG)O))7FZ ]=`,=)>3 )&$/FG)h`
while
do
g79 FG1)O]P FG)O))ZFZFZPNO'&F
do

while
do
'X>^
-F7Px.)oE '&F^ f//,Fh`$/ sE"`>fb>^)^3a)
bool
gh7F^_)`>F Fx3>W938/FxPNO'&F V& )^>)+P=)7 `>9FGFG)FxPWFG) :]H`,=)
/3)a:f/, O)
{ deP
a>),=F 0:! ) /)>Z)> ) PNO'&FnCD)>G'XF^ H)
FG))) )> O)F g7":]H`,=)3>&F7)^9:PN
do {,)^
^>),=F S +.Q VX):9)fb )ZF^>_/)`Fn`>,93 sEh39:W 3 )>F4P )hPN('XT ,9 f/)
for
FG)O))$

for
bool firsttime = true; firsttime ||
* firsttime = false
g FTsE9fbF b ,= GF^+,N) 8Pn)
h
FG))>) {:]H`>)!)^ O)! )+WFG)
while
):)
F )7>f,9)3 e3,D).F^^)7``>,9 ) >f//,9)F^!\1`>) 1B;H<; /3 nFh:]H`,=)3-,`3)9/ b
67F^39:D.F^89T`$/`>,9) )vE 9`$)^ 'X`)",9FG>X)>FX+F^ 1,` P
1,.sE>F E/3?P)> $/`?,+sE:X) 9^/ ,=)^9,=)FX) F^,pPx)+,+sE:FX)>3
F^ P $ n )>9 x),F^>+3`$)^F!)9) )^9iF^,3 FG)r ghF F FI)
0
9),=/ 'Z)^)^ ,F^9 FI))>) 7F^`-)-)>^9)5`>93)^ `$/ 9 5sE
do
`` b3 S<<: ) :]H)Z1,.sE>h9/Fhsm-)>3{
int a ;
// next input value
int s = 0; // sum of values so far
do {
std :: cout < < " next number =? ";
std :: cin > > a;
s += a;
std :: cout < < " sum = " < < s < < " \n" ;
} while ( a != 0);
ed )F7`$F^ )hF EFGF^s9D)+39>`N
'&>T'7 '7,3
,F^,9/3 ) H9/ a
3N)> sEPNh)&=9,=)4FG))>)$ ghZ^$/F^F )9) '7,3 )^9> sE&1`$/=) )ZsE3=
a
P )^9
FG)O))&93 '7,93)7sE f1F^s9T
)
FI))>)! F
do
do
a != 0
& J KU
2.4.6 Jump statements
V )T)FXE) '7 '7,3 b )
:]H)^3?,= F^> 4P4`)^xFG)O))F&'X )?8FIE`N
o) 1E.P4FG))^)^F&)9) >PN>^3 )
/F <C']L .<2+<'A<M6. g7F^.FG))^)^FT.)
`FGF^)!FGF^")9)h)>
'7,3a/O'*b,a)^ 38F^>)^99 '&9` FZ)hEF^F^ s )>^'&F^ deFG)$3 uv,FG)r b
D/3 do FG))>)F 'X`!O4/F^7,9>`F^FO^! )9)
while
F^F^! uv,8aFG)))F79^Of13 /33)9/ 9:]Hs9)oa3F^ )>)^FG)))F
4,8F^,3,F^Z)^9F 9:]= s )o 'X>^>fb> )k/O'&F b,)"8(fbhb,= `>134%&('7>fb> sE
/FGX'h3 )9)muv,8.FG))^)^F F^,3 sE ,F^3.'X)^ `$ F^9`>k)> )>93.)D`89`$)
)h`)^^ 9(' xg7&`89`) P )^9h`)^ ('y9/Fn)^ sEhs /N/`3+s1 FGW9`$/)
b/
DPE)D)> `)F4gh>^PN H) `$>PN,9 sE>PNT)^^13,` &uv,8
FG)O))
9>aXuv,98
FG))) F7:]H`,=)
3 H)D/ 9('*,`39 )9/ uv,89F )+
`>G)/
E)ng7>!T)v'7+3lm:)4uv,8aFG)O))F )9)7'7"'h/)h)^ 3F^`,F^Fh>
.<2+<'I/M )^FZFG1)(]FhO)>&FG89
gh W9^FG)nuv,8 FG))>)hFh`$/3
break
break

FG)))XFD ]=`,=)>3 'X ) / )>O)-FG)O)) ) F^_/FG)& break
`FG|):) FG))>)
)> O)F_3N)^b~gh ]=`,=)*`)1,F-) )
FG)O))&P)>h)")^>) FG)))ReP /Y:=Z:]8 for (;;) break ;
F7)h/W9)&1
s9,)7O)>h `89`$)^38'h_P '&^) 1,EFI))>)$ %&:TF
!Z,F^PN,EE$O/`XP
nde_,= `$`,N)n:]=/89&P M /DKK )k'7,38sE
break
b/)D)^ F^,=F^FT).^>>f//)"/33)-P
)^9.NFG) )>O){ ghFT`$/ sE
0
3 '&) )^9 PN('X9 1r
for (;;) {
std :: cout < < " next number =? " ;
std :: cin > > a;
if ( a == 0) break ;
s += a;
std :: cout < < " sum = " < < s < < " \n" ;
}
% > b'7 FG ) )v9`$/H,F P
&
)7)^>9)!P &1 G F^:'X>
break
)"33 = Z)T)9)7'7!`,3 1,f// )'Z)
do {
std :: cout < < " next number =? " ;
std :: cin > > a;
if ( a == 0) break ;
s += a;
(27
$&
#,S&(*
*
BL%Q.*
switch
( "
%*(,
97
2 J
& ]av U#
std :: cout < < " sum = " < < s < < " \n" ;
} while ( true );
de )^9F`$/F^
F8>PN>s9 h), hF^` )_`>5/3F F GPN^>fb>=zCDP!`>,^F^
for
) F/-PN,`>)9/)o|FEFGF^s9a'X ),)
Zs9,=)_)-F^,)9 `>13- 1,^F
/
break
/33)9/ s 1`
/3a>f//,9)>F
)o'&`
a != 0
do {
std :: cout < < " next number =? " ;
std :: cin > > a;
if ( a != 0) {
s += a ;
std :: cout < < " sum = " < < s < < "\ n";
}
} while ( a != 0);
gh F^`>93 uv,98FG)))_F`$/3
continue
F^89/
continue
.<2+/'I<M !/b/
) FGH)O]yF

D`)^1,7FG))^) F4 ]=`,=)>3 b)7_/3>xP9)7F^_/>FG)x`FG )>)^
FG)O)!) BFsE3 F
F H 9m3T/3 :]=>`,=)*`)1,F)
)?3*P!) sE13=b g7
):)
FG)O))Z )F^PrF )>^9)3{
deP&)
F^,=^,399 )>) FG))^)+F do FG))^) 4)
:]=>`,=)
while
)>>PN `),9>F"s1 >f//,9)9
)F
{+dePk) F^,=^,399
):) FG)O))
F FG))) ) :]H`,=)|`)1,F s >f,9) )F
/3|)^9> )F
for
{Dq{ b ) break FG))) )^9 continue FG))^)&`$/ )>PN^+sE ,F^>3 )
_/9,N)h)!`)^ 9O' G ) 339 .P .1r
de ,=!`$`,N)T ]/89 E)+PN('Xf/N/)!P4) >F"b)^fb 9,=)$
VXb/ O) '7,93 sEkEF^FGs94)^D39Z)F '&),=)
) ) :]HmF^ P/)>xFI)3
continue
s91` for (;;) {
std :: cout < < " next number =? " ;
std :: cin > > a;
if ( a < 0) continue ;
if ( a == 0) break ;
s += a;
std :: cout < < " sum = " < < s < < " \n" ;
}
2.4.7 Equivalence of iteration statements
ed a)>^F&P 9,= PN,`>)9/ )v )
/3 do FG)O))FX^"3,39/) /F&sm)-P
while
)Q`$ ,9 f/) sED:]1F^F^>3a,F^+
FG))^) ghF _8`:$)D)D8>F for
F^ )^ O)
FG)O))F"9fb8+:]19^F^F^fb8mO'7>")9/
93 do FI))>)F
for
while
J & U=
de )F_F^>`>)'7 F^('p)9))F_F
9)_) `$/F^ |/X)= )>O)yFG))^)^F O
PN,`>)9/ 1,f//>)$ T`FG ='7"FG9('wO' )+,F^
FG))>)F /3
while
FG)O))F ) :]1F^F
FG)))F
while
for
deP '7 3) `$/sE ,F^3)^ :]19^F^F +s1
'7 )>PN^!9fb
A
B
A⇒B
FG)))
FG)O))
FG)O))
FI))>)
do
⇒ while
⇒ for
⇒ do
,
'X>^
'7 H('Q) NFG).89`$O) P )>fH,9F+F^`:){ g >)> 4)F.`$
I(fbF ) `N/3_ 1,f//`/
Z) )9)&'7 ,=)Z)"'73 LM:<Fd!. 1,)F /F&,=X^$/F^ `/9)ZsE!`>9FG3>3
PN_{P &de 3>&) $
9^Ofb FG))>)Z b )F '7!WFG)DP /r'7,3-9fb
)_sE+ PN_/x 3>W999)+F^_/)`FDPnFG))^)^F.\1_/)`FTP49^/
N/,9/FZFX F^,=s uv`>)&P )F&('X
/3-)!PN_/{)^))&PxF^>_/)`F&Fh'hsE>b3
'X9) '7
`$/ 3 >/ de|)>.'73F _g7 PNO'&-F./F+ ,` P& I9^1P ?/F b,
'XE>)X>^ 9s9,=)X )&FZF^, 8`)Z) ,3>FG)93a)^9 NO)F sE>)o'7a) )=! )>)^
FG)O))F
do
FG)))F )+:]HFGF
?
do statement ⇒ while statement.
67F^39:Z)
while
while
4,=hWFI)&3$ ('w)^
j
if
F^+,N)Z)FZ,F^9 do
FI))>)
FG))>)7)71 _
bD)F
E,
while
de339)^9F&3,`F7) :]=>`,=)a3> ^,)^ :),=F ST+~.Q /3.) FG))^) )> O)F k,=)x)> F &FG89n)>` `$1s9 P F ZfONs943`N)('7 `$/M ) ,F^ ) /F ) + )
FG))>)$
do
do

%&>^"FZ PN+,N))9)7'7 1
F
do
if
j
else

break

while true
g79Fk3,`F :]=/`>) ) FG))>)! Fk:]H`,=)^39: >^,) _>while
),=F ST+~.Q /3 )"1 F7)>9)^3 , F^9
(
*
2
.*C%27. J
7B.*6
'(Q
7.#*
/K
/
*
8*
U= > 22 *
&(1#%<
%27. JA*
,*
#*-%(B#,S7
Q)#%2%
>
break
#/
22 *
J
/*
& ]av U
\1 ,NO)!)
while statement ⇒ for statement.
7
for

s1 while
FI))>)hF^Fh$/FI

for
FG))>)
E
while
t
ed 334)^9F.'&4'7 CMD!.<. "`>)/9F.
de )
FG))>) continue
for
:]H`,=) '7,93 ) 9^1`>3?'X)^ ).>f//,9) P
rs9,=) )+F^+,N)
FG))^) FTF 1E3r/3 `>F :]1)$.
g79FT^PN+,9)
while
F7)>PN^"'&^9=4%&:"F& fb>F^)9)7'7 HF E
while

bool b false

while b = !b

if b break
t
g79FZ HF7F^>'&9)7T`89`$O)31F^ >)h,F&:]19N/_'X9)hFh {
_ F^,=EF^_)9) )_3)^W9>
3F.9)+O9m$+ )
fb
FI))>)
b
for
e)>G'XF^h'7D`9F^T!3l >) 9/g: Z)^&)9)k)^9X'&&FG))) PNF !F^> O)
s91` /FZ31FD
FG))>)$kV E)>)Nr3`>N)â
k/FZ'7 /FZ)
for
F^`m!P F7)1,F&)38)^ )F s91` b
67F^39: / :]H`,=) P )8,=):
FI))>)$ FG) F">f,9)3 while
/3
P ) >),=F ST+~.Q )DFG))) )> O)F CX)>^'&F^ H)^9Xf/s9
F F^>)7) 0:! b
) 9>
FG))!) BFD`>93)^
m$/ )9) hF ]=`,=)>3 :]H)$ while
deP !31F8) `)/y
)> 1 :f/, O)F )F `>93)^|PN8)^9
break
F^`>93 )"de?3_F^ FTF^:) )^ S +.Q 93 ).`39 )->),=F ST+~.Q gh>>PN b
) 9>"?)>^9)F \1`
F O' ST+~.Q F">f//,9)^3 :]H) rPN('73
b
s1
{ag79F 39,`>F ) for
FG)O)!) BF ]=`,=) 3>
:G,9) _>)^,^9F ST+~.Q /3) ,=)>Z1
)> O)F
(.''(&
8
.6#*"
*:
%72: 7*
R
M*
%$
22
820 %92
%*
/1J
& J U;
de )7`$/FGh'&9: r`)F )Z> )^>9)>F 3N)^ break
/ 3
_/9F 0:! .de?)^9FD`$/F^ {'7 /F^)>^9)!)+,=)> 1 )9)T>>F^)^F",=
b
FG))^)
for
de >)^^FGE`>) T'7 F^,3 O' `` )9)+uv,98 FG))>)F
`$/,F^ 59 ,=
>f1,FT^PN+,9)-P4)
FG)O))T )^>FTP4)^9
FG)))$"? $fb
while
do
)Fh/F&/:]H>`F^
2.4.8 Choosing the “right” iteration statements
!
fb"F^-)9)&P +PN,9`>)^9/EE)ZP fH>' =)
FG))) =)
FI)) for
while
) /3!)^9
FG))^) ^k ,9 f/)$(fb> )
/3
FI))>)F
do
break
continue
T3,39/)$n\) 6hY"Y l >FZ/EP )^9>F^!FG)))F =93 )F7fbF7b,)"P^3
s9,=)h/F^ )^9Xs9,=3h kPr` F^ )TN)&`)^^mFG))^)^F PN7b, G)`,N
/-
? ) /F F7 319/`&`F^Fn2xfb ),_)X9^/Q_83 'X9) b,
'h/)kO) F^ E) )7^ 1,>)Fx`9/ /3 b, 'X b>`9/D)h/ )hPN,)^,^42 fbP9)>^&F `,=^>)+!>3 )!`9/7)ZPN,`>)9/)o P )7/ b,y_ 'h/) ) >9N/à?`89`$O)3| )>)^|FG))) s / ,f) F^89>
PN+,N){ ghT>/m)^9>D9:"F :<0STJ<:!1[7N 4)D9^1`FGFhPr>'Z) ) 8OfbD )Fh^$/39s9)o &FG)G,`>),= ='X b>>9 )F7PN,`>)9/)o,9`9/3{
%&> F!
F^89 ,39.PN"'&)^ G13 19F_6 1F^.) )9)"$/3FT)
) FG) :<Y)YY;gD /3 /<1[.Q PN ,NO){ g7FZ/F
PN>'wFG)))F
PN>'w9>FhP `13
F^8 D`)^^9(' /3
F^8 D:]HFGF^F
V FG)Z>fb>T)>!F 657 ./3 sE>FG)DPN+,N)MO'7:fb> m)> ./'h1F
X
,9s9 s /3 ``>FX'X`-b,?F^,3-)^â)^ fb3{ &F^,9/ 9)^9:+ F^!)G/3lmF b-PN:'7>FP!`3 fb:F^,F ^-`89`$)^3 :]H>F^F^F D93)> F
/FG FG
/,)XP4E:F^9/ )FG).fb fb3{`
4, F^,3 ]HE>^`+/3 W93 ,)D'&9)TF^,)FXb,
sEFI)$
qr:) ,F+1 )+F^:]8 F ) FÔ'Q'&9).'7
/{ \1,=EF^)9) b,|'h/) )
/3
8%XfH9+uv,9FI)!3 sE,=)T)
,=)^9,=)")+133 1,.sE>FTsE>)o'7>
0
100
continue
FG)O)) 1b,-_'& )D)"PNO'& 1r
for ( unsigned int i = 0; i < 100; ++ i ) {
if ( i % 2 == 0) continue ;
std :: cout < PN>`>) `G`>) ms9,=)T) PN('X fb:F^ FXPN>Os F^`.)T9/F
fewer state93
ments
fewer lines of code
for ( unsigned int i = 0; i < 100; ++ i)
if ( i % 2 != 0) std :: cout < )^3
P )
if
for ( unsigned int i = 1; i < 100; i += 2)
std :: cout < 3 '&) 9O'!
FI))>)4/3 1
,9/ F^89 `)^
while
int i = -1;
while (( i += 2) < 100)
std :: cout < ^ =)"`3)
F7T`89`$)3F^`T)h`.s 9>FZF^F^)h/3`8 F^ E:)F> \1,9` :]HFGF^F.`8 )fb 3 `,) ) ,3>FG)93 3,8)-)
>l `>) P9)&F^F^)nm>)r VXF^ )h)N )^!P )
F4`,): ),)fb i −1
fb )9)7'7"3$/{'X ) 9)^,/{1,+sm>F
4, `$/ FGfbn)4N)^)^> s9> 93 ) )^9nF/x)x:) simpler expressions s1 '&)^
unsigned int i = 1;
while ( i < 100) {
std :: cout < X) _F )^ O) b,>)7>F^F7``F^T`39QH)> TO' W9fbT9>F FG)$3 P{)
)o'7 9>F7)^ O)h)^9
FG)O))Z>39F>kd )ZF^F7)9)hPN&)^9 F^89Z9^s cP '& )
for
,=)&331,.sE>F FG)))7'X)
F )!P `
`/
for
i += 2
2.4.9 Details
Nested if-else statements.
6 F^3>h)"FI))>)
if ( true ) if ( false ); else std :: cout < < " Where do I belong ? ";
d ) Fn) D9`n'&9)k )Fn:lm`>) F xP9)
s9/` 8smF4)^!)&,=)>
)>
else
if
'X sE+ ,=)^9,=)XeF^9`>.)^9.`3) 9/FDf//, 0:! Es ,=) P )
s9` sEF
else
)+)">
'7 >)h)",=)^9,=)
if
Where do I belong?
gh.),)fb , FX)9)X)^9
s9` sEF&)^_)^9
3N)^8`>3
else
if
) r ,=!`$/F^+) ) 9:
g79:PN ) ,=)^9,=)"F
93 '7
if
Where do I belong?
F^,3a`>),9/8PN_)7)"FI))>)h bD)F if ( true )
if ( false )
& J Ub0
; // null statement
else
std :: cout < < " Where do I belong ?";
>fb> b,OZ,F^,=&sE,=) ,F4b )F b,8`$8_ b 7)ZFG)G,`>),=h`$4)=,
:]1 `) s9` 1F if ( true ) {
if ( false ) {
; // null statement
}
else {
std :: cout < < " Where do I belong ?";
}
}
kFG3F if else )>-:]HFG)F FG`3yF^`:)yFG))^) The switch statement.
.<2+<'A<M 4d )kF , F^PN,)!FG`>)4sE>)o'7>8_/ /)>^ O)fb7FG))^)^F
6hY"Y )
switch
,F^.)"PNO'&+FIH)O]E
j
switch
g79hf,&
P
+,9FI)nsEZ`fb:^)s97) )>/=)o1E4g7FnFk8`>)^FG)k) )
)>Z`)G{FG))>)F '&9:
9/Fh).sE"`>fb>^)^s9 )^ bool nF7,9FG, s 1`)9)Z`)FhFG>fb>/ UY;<!~. P )"PN
case
'X>^ FT8)>/ P4)^/ )oE =")o'7_NsEFXF^9/ )^9>F^.)>/F&9fb )
F/Df//,ngh>^"`$-/F^+sE"+NOsE
default:
ghDF^>_/)`FkPr
FG)O)) Fk)^9&PN('X=
8F >f//,9)^3
/3_)^9
switch
FG,9 ) Fk`>8 O38) $` _PE)X )>/F4'&9` _O9m$ _"NsE9 GxdeP{PN7/
P) ) f/,9>Fn ) :]H`,=)+`)^1,F4)x)7FG)O))x3N)^TPN('X9
) NsE ndePr)>"FZ />)7s9,)&
NsE 1)" ]=`,=)a`)^1,FZ)h)
default:
FG)O)) 3NO)-PNO'& )
NsmeCD)>G'XF^ DF 9^3
default:
/3) :]=>`,=)a`),9>F&/P)^>Z)^9
FI))>)$
switch
Z)D)9)
_F^>`>)Fh/>)^^_E)7PN7) `F^FG+P H)731F
switch
) :]H) 'X)X ]=`,=)_^$/`Fh/)> NsmexdeP{&'h/)Fk) F^: O)X)X3l >)
/)^>9)fb>F n
9/F+)^ ,F^
e/3|)F.F.) >b/ ,F^P break ,=)FG3 P
break
/ )>) FG)O),) : 67F^39:.PN ]/89 )^9PNO'&9`_Ph`13 n/3 >).,F
F^,=EF^T)9)
FZ.f/s9TP )o1E
x
int
switch ( x ) {
case 0: std :: cout < < "0 ";
case 1: std :: cout < < "1 " ; break ;
& ]av UR
default : std :: cout < < " whatever ";
}
=
)n,)G ,=) F
PN x==1 )n,=)^9,=) F 1 )>G'XF^ '7k:) )4,)G ,=) whatever x==0
01
g h
FG)))4FxEO'7:PN, +)ZFGF^Z)9)4)n/O'&F )Z3l >) )>9)^fbF
switch
)"F^9h`134%&('7>fb> b)FxE('7>k/F^"_ bF
FG)))F 93 ) $/38/38>^
switch
V P 1,)k9^s tFn)9)k9ZPN>)Fn)!9,=) 'X>h)>XFG9,3 sE&
break
g79:PN '7 )^
> PN `89>)>9>F^F"b8>fb> )^9: 9 ?
switch
PN>' /)^>9)fb>F )+sm"3FG)^,9F^
3 9N_)hF/PN"/3,F^
>
)>h)9/
if else
switch
gh9/ )-9^s F P&)
The Halting Problem, Decidability, and Computability.
PN,3 )rs9Fh-)!)G-Px`89,=)){hdePN_/ _FGE$ 1 )!s9>
F
)53`3 e,F^5/ )+'&>)>- fb 9/)^FA )^>9)>F 'X
:]H`,=)3 ?8fb> =9,=)W FG))^g: gh ): I/ 8_ PN>T)
_6hY"Y / s ,=)Z/FG ) /c-/ )^9:&` N/,9/
g )^)/` )Ts9QPN_/ )^9 n)^F^
_)_)`N/ VXN/ g ,=G
$U= $Ub0/A+
3>W93? F^ r E:. G_/ N9,9/Q { 9^/ )FDN ,9/ F HO'&-/FZ p C:!1[7N^'AJ<5&1[ g ,=.9Ôfb3a)^ O)h)^9 9/)9 s9cFh,39>`39s9TPNXg ,^ _/`F s ,=)h)
F/ ^,9>)F7`$/a/F^.sE ,9FG3a) (fbT)!F/TFG)))7PNT6hY"Y[F
O) 3F G,3`39s9 $/ 4 fb F^>.7FG89 !PNr'X` )E'h/FE /PN,
>f13)T)9) ) F / W9)!1 E9fbM
) '7(y
4F {33? 39>`3+)^9.9/)^
s9 PN _/ ``>:) /F> Z93`>39s9 /F")9B
) e ^)^`,N 13
P4`89,))h
7)^9:.`$/)DsE+/ /) )^ O)T39>`3FT)+9/ ) s9 PN +[
L=.<.1q;gD 9^/F
SDFG9) ) F^8 `)o mg ,^
_` F ^ 'X39> /``>>)3 39>xPk`89,=) )M
9-P /`>) uv,FG)D b"_/`"N9,9/ g ,^ _/`Fh`$/ 3>fb>^) )9)T6hY Y
/Fk`$/a3 H:]H`>)h)9)7)>_,F^,9/>3a 1, 1,+sm>7Prfb:^_9^)fbZE> )F7PN&)^ O)$
V )8)aF/
) /F_g ,= k)^9VX>`/5_)_O)`N V& 6 1,=`w
$U#; $UUb
0 X3>fbE>3 `89,=))9/r13 `$/>3 e`$/`,,F VXF )D),=3 ,=) EFT13
λ
FZ 1,f//)X)g ,^8_/`FZa)>FZPx`89,))9/mE('7>$Zg79 67,^` g ,=
)F^FZFG))F7)9) G>fb>^ PN,`>)
)9)hFZ9)^,/ b33a/FZ`>8 ,=)Os X`$/sm"` 9,=)3 s1 -g ,=-_/` =-VXF )>^F.9-,F 3>W99 ) P7'X9).F G9),=
bO33-/FZ`89,)s9 )^9FZFG)))7Fh9)& )cs9,)&.1m)F^F7)9)Z`$/)
sE!(fb _)_O)`$/bxVXFXP )39$ ) 1E)FGF&9/FX)ZsE 3FG(fb3{hde )^9 >)^`$/4`89,=)>"F^`` ) )>^ /']LM72Y;gD ,F^>3 '&),=) PN,G)> 1,9/W9`$) F!
FG19HJPN G`89,=)s9Zs1+g ,^ _/` Ge ,f)^ = 6ZY Y[9^/
:
CD,= O9b` Pk3>W9E))N/ FGÈ /3?F^`m 9 s F
Point of declaration.
F^89 W `)>fb>PE)T`13DF FG,9 )s98PN^_)^)3_93_'7T89fbT9D3`>N)^
E>Z4g7 )^^,)âF )9)h) E)hP 3`) P i
int i = 5;
& J Ub@
F7_) '^1q)Y)hD P )T3`NO)1/P)>7) 9/FhE$O3{4ghTE))N/mFGÈ
i
)>>PN"3F79)h`,93X)TPN,E s ,=)h8)D ^)hFG)O^).P
kg7F :]1 /F
=
'X9)k9EFk8)^9&PN('X `13XP) bs9,=) PN^),9) )FnFk`>9FGFG)) 'X )8,=
s1 Tb/`{nde
Á
int i = 5;
{
int i = i;
}
) 9/
/P)^> )
;
PN>F")) 3`>N)^?? ;H
6 F^ ,9>) F
i
=
i
)/ >38'X) )F^>P )F7 =$99 )^ O)Z )Fhf//,T'X sE ,93>W939/3a) 5
de )> F^),9)9F ) _ 9E4),94)9) )^9 E$O/`
PD 9/_ )
3`NO) P )8F/ 9/+^PN>F )a LM:<<FQ127. 3`NO)?P )F"9// 9(' '7D`$/
$/FG $fb3
FG,9` F^,=s9)^>)>Fks18)^9DPNO'&", /3`>N)^8F^,3_`)/
) T)^ sE 3`N^3a_`
g76 N) FG 1,`F"s /` ))T>_/
The Collatz problem and the ?-operator.
_)_O)`N q{)9D67) ^$U=$# $UU# '&9FI),393 )X-)$U;#& FD\1:fb>/ F
9fb.sm?l >3 )
/b 'X8(fbF T39FGOfb>FT)^9.ùv`>)^,^.)^ O)X).1,+sm>
1
E$OFZ) 6 NO) F^ 1,`!P >fb>^1,.sE>
g7!P /,Fh%&,bN_) _)`N/M /,{2 3 R
F $U=$; $UU
R 4l >
3 1'&9n` 1F +,` _s1_F FI)/393`
F e ,F^>3
500
).lm:&+,` ('7>h/,)^F PN7fb>^
3 8`,)k9^s F :x2x
3 FhF/3_)9) a)_O)`F
Fh)hb:)&$3
PN&F^,` as9>F =xde3
3 =)"ùv`>)^,^!FhFG)E,9F^ fb3{
" fb 9^F^)3-)"`>8 ,=)O)_P ) 6 NO) F^ ,9>9`> /FX-9`$)^_P )
FG))^) m)9,).)9).)_`3)9/n` 9
P
F )1 `89`$)3 )^
while
n
FI))>!) F
{ ? x)9<) BF ).:]`>) )^^,9 )_39>F^>F
9,=).).)^ for
PT6 ) `,=F^T) 6hY"Y r 3? '7$ HF^F PNTfb>^ `8 /`>)X`13+/3 `/ ,=?'X )
) )1001q+<L7<:/+: )9)Z/('XF7,FZ)^ F^+,N)
FI))>)F s1 :]19^F^F^F gh
if
FG1)O] P )^9F /:!:!V m>)R
O)o
; 7F
*
%&>^ . F4/ ]HF^FG P X)vE7'XF^ f,F `$/ smh`>fb>^)^3 )
/3 * 8/3 - :]H>F^F^Fngh F^_/)`F7Fh/FhPN('XF> bool
FG) F >f//,9)^3{ndeP{) ^>),=F 0:! *
F >f//,9)31/3 )^F f/,9DF >),=3/F )
f//,DPE)D`8EF^)^&:]19^F^F^{nCX)>^'XF^WeP 8 >),=F ST+~.Q _F
>f//,9)>3 /3 )Fxf,ZFx:),=3{ gh7) b
F4 F^ 1,`Zm)eFGD\1`:) 1 1$# $/8)9) >l `>)F"P O 9^1`FG?F^3?sE>PN >)> * a^">f//,9)^3{
&F^ ) `3) E>)^ n
) 1|PDM ^/
'&^)^)/FhPN('XF
$# `,3
,)
`8
/`>)?sE
for ( ; n > 1; std :: cout < < ( n % 2 == 0 ? n= n /2 : n =3* n +1) < < " " );
$fb")h,) b,)^ 3`3T'X>)>hb,abT)^9Fhf/N)7sE>)^)^>$
& ]av UK
gh 39F^`,F^F^5sE,=)+FG)/
3,=)|sE(fb31>F ) )^ )
'&
Static variables.
FG)G -) F8/F^?EF^FGs9
)^ 3>W99 f^Ns9F+'X)|1`$ZF^`>E)9) 9$ fb-FI))`
FG)/
3,=){
ghF F39 s15>m399 )^9!
b>1'73
) )f/NOs a
3 `N){ static
:]=/89 H
for ( int i = 0; i < 5; ++ i ) {
static int k = i ;
k += i;
std :: cout < < k < < "\ n";
}
/
) /33^F^F7P _Fk)DF/D3,=./ )>O)F H93
F )N/ 3 )^
k
k
i
) WFG) )>){Tg79+smOfb 9`.P4`13 'X )>^PN.,=)^9,)D)^9.F^ 1,`
PEf,F
>+sE: /,F^F :x),=) ) static b>'73H)XF^,) '7,3
0, 1, 3, 6, 10
F^89 sE )^9 F^ ,`!P :fb-1,.sE>F
0, 2, 4, 6, 8
\))`7f/s9F 9$fbhsE 1,)h,F^PN, 6 PNn:]=/89 )!`>,9)k('yP)_DFGE>`W9`
9` P `3"Fh:]H`,=)M
3 6hY"Y =)> F^Fh8E^)/)$
=XfONs9FhPxPN,3 )m)vE!)")N/mf//,"_$_sE",93>W93 /F&a)^9"3>W )
%&('7>fb> )&f,&F4,93>W93 P 9/Fk/,=)_)` FI)/Z39,=O){
int x;
x
de `)^/FG) xf/Ns9>F 'X) FG))`8FI)/_3,=) ^_/'hHF /:<{q1"=1001q+Z1 +) x)9).F W93_'X )a > P )"9^9^N)X)o1m
MZV {
g7>_)v'7 uv,85FG)O))F. 6hY"Y )9).'7
9$fb>t ) 3F Jump statements.
`,F^F^>3 5)^9F F^`:){ CT-PX) F )
FI))>) )^ O) b,//3= 1O'
return
o\1`:) 1A+ )k_.``,= .8"PN,`>)/38)F4:]H`,=)8>)Fn)^97/ 9O'
uv,8 )^8)+>93 P ) `^FIE399PN,`>) sm13=bTgh.)> uv,8 FG))>)&FX)
FI))>) s9,)XF^`!)FZ F&O 3
3 /3-F^>'& O)&3 `,)&),F^! '7
goto
)7)
2.4.10 Goals
V7)7)F7E) 1b,aFG9,3?
Dispositional.
g H('w)^9 FG1)O]/3aFG_/)^`F7P if else for while 9/3 do FG))>)F/
,93>^FG)/3 )^9_``:9)^F.s9` F^>`>) )>)x3`NO)fb F^È k/3
FG)/ 3,=)M

;,93>^FG)/3 )Z``>) PE=W )^ 1_93 sEX'hhP )Z398`,)o+Pm3>)`:)
F^,` -19F<
A+,93>^FG)/3) ``>),9/{ ,9 f/`"P PN ='X
0 H
('w)^9 FG1)O]/3aFG_/)^`F7P
R
H/33
FG)))F<
93 break FG)))F<
continue
H('[O)&/FG)hPN,=&`>^)> )^"uv,39T) `13 1,9/)o_P )>O)
FG))>)F>
& J UU
de ^)`,N b,-F^,93 E
s "s9D)>$
Operational.
o+g_`` ?_ fb?F^8 !/ F 3>W93 sE('`&PN"FGH)/`>)^`$/x`G`>)FGF!/3
E)h,=)hmF^F^s9T>^^F<
o ^$/3
/ 3 ,3>FG)93 fb?F^89 //3 :]1 / '&9)!9E>9F 3,=
)^Fh ]=`,=)t
o ;W93I E))N/DkW9)D19F7a.fbaFG89X/?
oTA+W93 )^9"_)` 3`N)
PND.fb 3) W :!
o"0_
3>)^>D3`NO)fbD>a/3aFGÈ!P + fba3`NO)M
o R
^PN+,9) 8 fb for while E do FG)))X ,f)a,F^9_-Px)
)>h)o'7 FG)))F<
o"@_
`>8 OZ)D`3 1,9/)o Pr)o'7 fb> )^>)8FG))^)^F /3 ` )D9X)^ O)
F7PN>s9K P /J
o K_
3FGF^8 X9^/F PNX fb)/F HF
g79X)^> ./1"'KLtD9LM:</N:' PN>^Fh).!/t)9) `F^FG)F Pr _/8PN,9`:)_'X`
,=8)"PN, EFGF^s9.FG)>3Cx
):)+FG)O))F4E$ 9,F4F^7F^`:)8FG)))F
&)^,/ 9 !)7PN,939/>)/)o1EF4/3+E>O)F 3F^`,F^FG38 ) `3TF^`:)F
,F^3{
2.4.11 Exercises
:!://!\+"./V+M2Qs <:!:<:g.X1[ 657PLM:<N:/' ;<!Du 5C+P)J
!.]657]:<!./C01"7N
LM:</N
:/' C LM7tSQ:z657SQ" u1[7N1"&LM76.
i!o A i;2o # i2o
i,)!o ;
o+go
Exercise 33
Á
# include < iostraem >
int main ()
{
unsinged int x = +1;
{ std :: cin > > x ; }
for ( int y = 0 u ; y < x ) {
std : cout < < ++ y;
return 0;
}
Ã
5C+\10.^657KLM:<&;gD<' uv1[65@657OQ)JI;<< u
:<!.7~01[7NOQ)Jj'KLMC!.T
Exercise 34
Ã
Á
unsigned int s = 0;
do {
int i = 1;
if ( i % 2 == 1) s *= i;
} while (++ i < 10);
1Ds1[K),sgLtU1[u5C+W657
o o ";
& ]av $##
:cJ!5zF:!1qY;gDP)J!D:+012 1[?657*SQ" u1[7NLM:</N:' N1[Fda106.`)J/!U:/+01"F&
:< N12 )|1[6.j.Q<L=E1" 657\ST:!' 0Z1[GsdV 5C+P10.657O7 LMCK S^657KLM:</N
:/'
o o!0
Exercise 35
Á
Ã
# include < iostream >
int main ()
{
int s = 0;
{
int i = 0;
while ( i < 4)
{
++ i;
int f = i + 1;
s += f;
int s = 3;
i += s;
}
unsigned int t = 2;
std :: cout < < s + t < < "\n ";
}
int k = 1;
return 0;
}
Â
Á
Ã
Á©Â
M.1q)J<:G657WLM:</N
:/' N1[Fd<};/!DuSQ: ,J!5 S3657G~1[.<)A1[&LMC=C'A;/<:g.T
>3!<:!'^1[^657F+~*!.E S - -X) +X;/qN1" S3657 :g.< F&G10<:+012M. S3657`SQ:g{
D
<LM-^;/0SQ:<I657 )10012 x 10.s <Fd+Z*+i,)Y 5C+)JQ!.A657GLM:</N
:/' C LM7vSQ:O657<.Q
i,!o i;o
i/2o
i)!o ;
1[&LMC6.
o o";
Exercise 36
Á
Ã
Â
Á
# include < iostream >
int main ()
{
int x ;
std :: cin > > x ;
int s = 0;
for ( int i = 0; i < x ; ++ i ) {
s += i;
x += s / 2;
}
std :: cout < < s < < "\ n";
return 0;
}
Exercise 37
1[)A+9D+.< STC:`LM:<&;gD<'E.]1[ 657^Q)JBN1"F&<;/! u
o";,o +A !0
& Á
Ã
Â
J $#=
# include < iostream >
int main ()
{
{ unsigned int x ; }
std :: cin < < x ;
unsigned int y = x;
for ( unsigned int s = 0; y >= 0; - - y)
s += y;
std :: cout < < "s= " < < s < < " \n";
return 0;
}
:uv5&12<5]1"&LM7*=h'A;/<:g.v1[. 657PC LM74 S657LM://N:/' N1"F&<A;/!DuuR<[C)J!{
Exercise
) 3810.<t657.Q1[&LMC 47 LMCYLC1[:g.9)K:qN uv5&VWVhh:v~1[.<t1[./'KLtD!Q o"; +A o"0
Á
Ã
Â
# include < iostream >
int main ()
{
unsigned int x;
std :: cin > > x ;
int s = 0;
for ( unsigned int y = 1 + x ; y > 0; y -= x )
s += y;
std :: cout < < "s= " < < s < < " \n";
return 0;
}
R[SQ:!'jCD+\657z/T)JB;<!Du h1[Fd+D<M6ZV^1"@:/)J<:X31"'KLM:<F&P106.R:<Y)YY;!1[{
100VJ\>G<.Q<:!1q;/z657`LM:</N:'y% .7 LMCR+.3cS/h!012y SX106.X1[&LM7
o Ro"@
Exercise 39
Á
Ã
Á
Â
n
unsigned int n;
std :: cin > > n;
int x = 1;
if ( n > 0) {
int k = 0;
bool e = true ;
do {
if (++ k == n ) e = false ;
x *= 2;
} while ( e );
}
std :: cout < < x;
R[SQ:!'jCD+P657LM:</N:/' ;<3!.Q<:!1 ;<^657XLM://N:/' % .?7 LMCX+.jKS/h!012} Sz106.z1"&LM7
x
o o "R,o!@
Exercise 40
& ]av $#
Á
Ã
Â
Á
# include < iostream >
int main ()
{
int x ;
std :: cin > > x ;
int s = 0;
int i = -10;
do
for ( int j = 1;;)
if ( j ++ < i ) s += j - 1; else break ;
while (++ i <= x );
std :: cout < < s < < "\ n";
return 0;
}
Exercise 41
=C';<<:
n
:!10| LM:</N:/'
fak-1.C
/']LM7 657XSTJ!:!1 +
n!
SA N1"F&<}1[&LM7
o K
:!10GBLM:</N
:/' dec2bin.C 65C+v1[&LMC6.3j+0C:/+=h'A;</: n )I7 LM76.
657^;<1[:!VO)1"N
106.G S 1[ :<<F&<:g.TE:/)J<:
4:^,sJ']Lt <-MST:
657jC LM7v10.
)
n==2
01
SQ:
657^7 LM7vn 10.
i.Q/G+~.Q s$<:</10.Q o&
o K
Exercise 42
n==11
1101
:!10PaLM:</N:/' cross_sum.C 65C+ 1[&LMC6.`\+0C:/+J=C';<<: n )37 LMC6.
^
S 657Ii)J//1['A+ o )1 N106. S :O,sJ']Lt <-SQ:
657A7 LM7P10. )
n
n==10
1
657^7 LM710.
o"K
Exercise 43
6 57E.C'
SQ:
n==112
4
:!10GKLM:<N:/' perfect.C j!.<uv57!657/:^N1[F&/ +0C:+ =C';<<: n 10.
L=<:6ST/!2 r=C';<<:
10.z,+[D) E>PN>`>) 1 SB)MZVj1 S\10 10.B *+ ^657]./h' S\1[6.
P
LM:<<L=/:)1"F
10.Q:g.-f65C+n10./ :G,sJ']Lt <n = k , s.t. k<n ∧ k|n k
28 = 1 + 2 + 4 + 7 + 14
10.L=<:6ST/!0-vuv5&10D
1[.]2
s+<)x6573LM:<12/N<:'1 +2 + 3 ) + +4[+L=6/:6SQ/<K=C';<<:g.|;<<0uR< 1 ) n u '=V
L=<:6ST/!9=h'A;/<:g.^,s10.<v1[@657K:/7NY
o"K
Exercise 44
[1, 50000]
!: 10KPLM:</N:' dec2bin2.C 65C+1"&LM76.Wz+0h:/+==C'A;/<: n )j7 LM76.
657E;!1":!V )1"N
106.j S 1" 657?/:!:<!K:/)J/:
4: ,sY'KLtD<-SQ:
657?7 LMC910.
n
n==2
)cST:
657j7 LMC1[.
i.Q+~.Q s$/:<<10.T o&
o 10
K
Exercise 45
n==11
1011
_R<x) ~1" LtDV}})1q/ON&'AN&1[M.<?,J!5 657<:
_`!5C+.O65&:<
QSExercise
h:g{2./1 )J46
) i"LMV+:/'^1q)Y+ o@)12<-GJ!5xu1065?STJ!.E=C';<<:<,)Hk- Y- d- Z1[}5C+.E0u`
./1Us{2./1 )J)Ai/<t;!12,+ o?)1q/<-,J!5 uv1[65]S J/!.c=h'A;/<:<)Ok- Y- d- - -ndP_`!<:) Z1[
:<" 657<1[:)12/)@'KL7:<j2+. E657^5&1"N57<.<\2+ u1[M.T p 57^:/!./7~\1[.? ):/u 1 S
657z2+~.3:<j *+U
5C+B10.657LM:/&;/Y;!1[~1[0V65C+K_vV+:/'j1 )Y+`_R!;<,+6. t;!12 ~1"
5C+B10.657
LM:<&;Y;!10Z100V|65C+ *;!1q ~1"w;/+6.G_V:/'^1q)Y+_R< R)uv5C+`1[.z657cLM:<&;Y;!10Z100V@ S
):/u X.BA/M.Q //<-v10.c1[`RST1":WN&'I<-9)1 Sc0- uv57û`7U)VhI:/+657<:G;<
& J :!10\LM:<N:/'
=C';<<:g.A Sj657cST:!'
MxuI>`>)X2 ,> -v.Q
$#;
65C+f7 LM76.657\ SQ:/<'I<M01q)`LM:<&;Y;!10Z1001q!.P+.:+012+
,)|F&<:g.12 S_v:/&;g /' S/:/' 657
o "K
dice.C
yi p 5&1[.1[. .1['KLtZ1
p/q
o
http://projecteuler.net/
V$:g.B
Exercise 47
3Qu S/:<' 8/!01q k&[k?65C+910Q =:/M B!~.Q j:/C) 65&:<
)?657STJ!:!1 ++012
761838257287 193707721
Sc657 /:g.Q<=X=C';<<: 67
;!VE5C) ,+ /7U+012M.T :!10zWLM:</N
:/'
2 −1
65C+ L=/:6SQ:!'E.]657K.T'IK2+.<Ii257<L=0S/C[ZVE1[ <.<.K65C@65&:</zV$:g. o&
|u1[[/,) 657z0VgL=
-.Q/K8/!01q 70k&[k$
Hint:
mersenne.C
o"K
ifm::integer
2.4.12 Challenges
!57!./.T;<Q:) ./ <5
p 57 1,FZs9> 1[.K]LtDJ //M.jy
65C+G0u` n<M. 65&:<+< J!5 657<:
4n:!'+[ZV-^65&10.O'AnM. n 65C+657<:<10.O
57:!1 +M2+~-vF&<:g01q+Z-\:j)1 NY+uv1[65I'I:/365CH */<1"@102 :!1[jzLM:</N
:/'
65C+c7 LMC6.K657B=C'A;/<:? S3)1 W<:/<M`.Q~C012M.] 657 { //M.\LM:<&;gD<' ST:ÊN1"F&<
1[&LMC X.<.C'j1"7NA ;!1[./Vd./<'j-657PLM:<N:/' ./577nU)Eu`:gjL@
+ ,+.<2
n
n=9
57/!E65&:/hN5 ?u`Q;3.Q:<<5Ou57<657<:G657z=C';<<:g.B65C+9Vhh:PLM:</N
:/' /']LM7!.G:<
/:!://!2
Exercise 48
Exercise 49
m?1[.
p 57KU:2NY!.< <:g.Q<=\LM:!1"'IKQu+.G S]8qLt<'A;/<:
243,112,609 − 1.
# s$<:<<1[.Q|k Q-cu` 5CF&I+./d) V$ ) 657?=h'A;/<: S)J<1['+R)1"N
106.j65C+W65&1[.
= C';<<:O5C+.T}# 65&10.|!5C+" <7N&<-zu` :!1 N1[+"~VuM)+.< VhwZ10./+[657<.Q
)1"N
106./-B;<7W1[w657E1"M<:<!./ Sj657 p 65C+X5C+.j ':gIVhh: .QZ7012M.-\uÒ)J/<1 )J)
./u10!5w 657GSQ" u1[7N@Fd:!1qM :!1[|LM:</N:/'
65C+
famous_last_digits.C
7 LMC6.X657zU+.<Pk )J/<1"'A+f)1"N1[6.3 SK657Y;<F& <:g.Q<=RLM:!1['A
& ]av $#A
2.5 Floating point numbers
=C:g657<:!'A:<<-105C+.c:</F&+D) 657W:+012 SW657G!57:/) )A:/ S
=1[<0V )JqN://!./-ùv5&12<5O10.^+.z.T<F&< <1"N5Y0-\) +.Q 6573:+012
S?657O)1 NY+`) ./1 )JO SA . *:<?uv5&12<5x1[.A+.E< .Q<Fd<=-B)10.T! .1[7NI657\SQh:g65 1[']L=:g2MS J!0-c65C+c657j:/+012@ S
657^)1 'I!/:j)O<1":<<h'\SQ/:<</B10.3+. Fd!{STC:g65Y.]cSQh:

# )1q C.Q]b910[ K&
7 -
2 2
10 -) 1
π
=
8
2=
7
π
n
-)J =1"7N
i/km$lo
= 5/16
)
p 5&10.W.Q!012x)10.T<7.<.T!.c657 \Q+01"7N]L71[M=C'A;/<:K0V L=<.
float
double
ST:w!LYLM:/Ts+1['+01[7N}:<+z=h'A;/<:g.T uv1[[K ,:! Y;<C PQ+01[7Nx
L=1"M
=h'A;</: ./Vd./<'E.1[NY/<:/+Z-B)@Y;<Cc657E# v .<2)Y:) 65C+K)J!{
.Q/:!1q;<<.O0u`x.qL=//1 \T+01[7NOL=1"Mz=C';<<:./V.<<'j.T |u1[[fL=1[MEC
657 .<0:<<7N65Y.)HuR,+Q!.<.T!. S \Q+01"7N L=1"ME=C'A;/<:g. )N1[F& Vh
65&:<Nh1q)J!Z1[<.]F&1q)K/'^'IKLM1[ ST+". 1[/']LM701"7Ncu1065 \Q+01"7NL=1"M
=h'A;</:g.T
9> `fb>G)?3>F_6 F^,F.) /=).'&) )^9_/
fahrenheit.C
\1`>) 1 7'7 _ bFG) b>F = ]/89 3^F67F^,F
3>F
28
82.4
/=) =s9,=)X) 82 /FZ,)G ,=)hs1 fahrenheit.C g7"$F^-PNX)FhFG) b FZ)9)
)7)^> 3fHF^.89(b3.+)^9 F^89 G`,=)F l T)^97P`>)9/1 G)$ 9)
'7!>3 F&.)vE")^ O)D/O'&Fh,Fh) >>F^)D/3a`89,=) '&) P/`>) ,9.sE>^FZb
82.4
=k)F 6hY"Y (fH3F )o'7 PQ+01[7NRL=1[M =C'A;/<: )o1E>F float /3 double de33 P '7 FG89: /` ).39>`NO)
s1
int celsius
fahrenheit.C
float celsius
) >F^,) ,=)^9,=)F
PN+/ = ,=) f,8P
b)
E)!1,.sE>F
82.4
28
/FGF^fb /)> 9^s )^ O)"'7+9/3 'X)^ ) )oEF
/3
int
unsigned int float
/3
9fb +,9` >
f, 9?/3* )>PN^?F^,)Os PN GF^>,F double
`89,=))9F> de!P /`>) `>8 ,=)O)F 'X ) 9b)97E)r1,+sm>F nfb:^!P FG) 39:
9N)PN^ 13,")+FGE`>N/ 3 `F^F^^F
dePb, ) .OsE,=) 9(' )T>FG)n3`_/11,+sm>F b
Fixed versus floating point.
82.4
,F^- W=]H3 1,.sE> P 3`>_/ 3)F]$#3)^F FY x 9)^,/nFG,=) F")F !b,
^) ) )^98$#8f/Ns9 39)Fh)@ 39)F7sEPN!)^9!39>`_/EE) 9F m/3 ;83)F
/P):Z)"39>`_/ m)$ g7ab,a`$>FG)X/{3`_/ 1,+sm>F7P ) PN
6
X
βi 10i ,
i=−3
'X )
βi
{0, . . . , 9}
PNX/
i
nghF7Fh`$/3a
s$,)zL=1[Mv:<2LM:<<.Q<M2+012
& + v z
$#b0
gh>^ O'7>fb: )o'7sfH,F!3F^/3=f/)/F P _W]H3 E)"^>F^))rCT
)&Z 93b)Zf//,7/&F4fb>^ )^3{r?&9fbD//3= FG
\1`:) 1 1L0T)^ O)
) N^FG)
f//, FDFG F^_r)^ O)T)D93 ('XFX )>FI)`89,=))FR /F
int
/ ]/89 E)^G ,=)!M ^/
F^.NO>"= ,=), +VzW=]=3 m)T>>F^)) F
>fb '7F^T_)F FGm`>) =F^9`> ) F^:^fbF&FGDP ,=7`,F 3)^F PN7) P/`>)9/
^)+/P)^> )3`_/4E) x>fb P )F^_3)F!q) .) PN,
9>3y
3 e/F :

82.4
/3
gh F^`3 3F3f//)/.F `F^ -N)^
3 :fb ), ).)v'7
1,.sE>F
82.4
9fb.)^9.F 1,.sE>TP4F^W9`$/)X3)\
F e9/
; m).N)^)^>D1,.sE>DF )
0.0824
>>F^)s9 'X) 39)Fx/P)^>x)739>`_/bE)$x%&>^ '7h 'h/FG)^X) 3)^F
3
7
sEPN^") 39>`_/ m) Hs9,=)7'7 ^"N` 1 3 )F7/P)>h)!3`>_/ E)$
V PQ+01[7NL=1[M*:<2LM:<!.T<M2+012 >F^fbFrsE) FGF^,F s":9^F^)X71,.sE> F^89
/F4)FxF^ 1,`&P 3`_/139)F /+)>4`$/3 )^9 ./1 N=1 ,) Lt~C. )hPN_)
I'X>^ )+39>`_/ E)"F =8g `9`/ {+mF^F^s9 )v)$ .)F F )^FG)^8/
,sgL7<M F^,à)9)7) :9^F^)3 1,.sE>hFhP ) PN
./1 N=1 ,)
10
.
=& ]/89 82.4 = 824 10−1 ,
0.0824 = 824 10−4 .
2.5.1 The types float and double
g79 )o1mF
93
O.PN,939/)/ )oEFT9Ôf133 s 6hY Y /3 )> FI)
float
double
1,.sE>F7! bO) E) ^>F^))r
9 ).PN,939/)/r)vEF
/3
O.$/)&)O9$]=_)")^9
int
unsigned int
G_)_O)`$/7)o1E>F /3
XFGm`>)fb> &)?b P sE) float /3 double F
) O9$]=_)8)FG>
) P&$&1,.sE>F>y\1` )^9:a^+,9` _$Z,9.sE>^F
)9/ )>>F )F b/ F^>F :fb5_/.s9),K
F /3|>F^F $/FG)g` T)9/ )^GH-)
$]=_)^
F 'X)^ W99 ) f//, 9/ Z>fb:^)FGF ) )o'7a)o1EF float /3
Ohfb:^ ,F^>PN,99+9/`>)`/9`$)^F g7
9b)9DE) >FG)O)8('XF
double
f//,F&)9)X" ,`-N:&)9/ /f//,!P )oE
/3
&de-P `>) )^9
int
unsigned int
f//,D/F P )^9 9b)^ E)7,+sE: )o1EF
/3
^ FG, `)h FI)
float
double
9`$O)F
jn/,FTPk)F^.)o'7 )o1mF!O+^PN>^>3 )^/F \Q+01"7N^L71[M\=h'A;</:g. r'&>
double
,F^,9/
('XF >R
e9/ )J*;gD `Fâ$]=_)^ $/{1,.sE>F>
CT )X)vEF
/3
'7T9$fbD)TF/TO)>)` N)^9/ /3a/F^FG float
double
) E>O)Fh/F )/ )o1E>F H'X)^_)^9DF^/T/F^F^`N)^fH)^F 93_`>39>9`>F g7
:]=`:9)^ F")^ O)")83,9,9FTE:)F
/3
f//NOs +PN )>/ )vEF
%
%=
bxg7Fn_ bFnFGF^ 1FG`&3f1F^+Ofb:
93
F4$/)4)^!13=)7)^,
float
double
3fHF^
Ofb>
['Xà9/Fh _39:$
& ]av $#R
q b+)/ )o1EF r) 9b)E)T1,.sE>D)o1E>F!^
r
`89>)^F )"FG)7P PN,3 ){)>)^`&)o1E>FZa6ZY Y :!1[65&'I!01qj0VgL=!. /3 )^9F
q{)>F P )o1E>F
/3
h7`89`$O)3
Literals of type float and double.
float
double
)9/ )>/F PH)o1m
=x:]=/89 FxZf//3
)>/ int unsigned int
1.23e-7
double
>>F^)^!)&f//,
4q{)>FnP )vE
1 .)&F^/X/Fn):/F Pm)o1E
1.23 10−7
float
=PN('73
s1
)":)^)>
double
f
F
de?)^FDFG)T>/xPN^ E
)>/ `F^FG)^F Pn/ 1[MqN&<:KL7:g {PNO'73 s1 double
S/:J!012+CL7:g eFG)O^) 'X)^
) )J/<1"'A+CL=1"M /3/ ,s!L=<M01q+7L7:g eFG)G)
.
'X ) ) >)^)^>
:ng79 )^>/
F&/EP )F^T ^)F
e
E
1.23e-7
k) )^9-): O^)
/F'7DF_)^9 P`>)9/& G) /P)>)^9 3`_/ZE,) O
)^ '&> Px) _
sE 8)o b $/
F^ ,`FTP 39)F&P^
0
9
.1
!/3
e$/

: g79 ]HE)N/k G3

) /P):+)^9>)^)>
!F./FG 0.1
1.
1.0
e
E
F^ ,`_Ph3 )F `>33 s / 9)^9/
|
10657<: )P/`>) n ^) : )
+
:]1E)/m ^)Z_$sm")G)3{ gh,F /3
f3
)^>/F s9,=)
123e-9
1.23
double
Fh) =^39:&) fb3a`PN,F^'&)
)>/F
123
int
ghXf//,DP{)X):/ F s)/3s18F^`$/9 )^9DP/`>)9/ 3`_/f//,D3>W99>3
s
) )>D ^) 93-).P/`:)9/ G)Ds1
'&> e FX)3
eF^9>C
3 h39>`_/ )>>
10e
:
)":]1E)^N/E ^R
) e3>W93-/F P )^9 :]1E)/m ^)ZFZF^F^+
0
g FG9(' 9b) E)71,.sE>F7-`>)
H:)Z,F7'& ) /Q)9) G`89,)^F +
PN, 933 ^$/r1,+sm> H9/ 8)!2 ,>Z`FG)/)
∞
X
1
= 2.71828 . . .
i!
i=0
4,8_.`$/9)9)4)F4F^,}`fb>F1,`H F^!'7ZF^,3/^$/3=+>)
11389^(] _)^ PNk)X2 ,9> `FG)/)n'X8'7XF^,t,_)hW9^FG)D$#")^>F Fb4M / &3F
:]=/`>)_)F>
Á
Ã
# include < iostream >
Á
Â
// Program : euler .C
// Approximate Euler ’s constant e.
int main ()
{
// values for term i , initialized for i = 0
float t = 1.0 f ;
// 1/ i!
float e = 1.0 f ;
// i - th approximation of e
std :: cout < < " Approximating the Euler constant ...\ n ";
// steps 1 ,... , n
for ( unsigned int i = 1; i < 10; ++ i ) {
& + v z
Ã
Á©Â
$#b@
e += t /= i ;
// compact form of t = t / i ; e = e + t
std :: cout < < " Value after term " < < i < < " : " < < e < < "\n ";
}
return 0;
}
LM:<N. 4<C <:

Program 11:
9>ab,,)T9^/ H)^FZ,=)^9,=)h_$_1 _bD)F>
Approximating the Euler constant ...
Value after term 1: 2
Value after term 2: 2.5
Value after term 3: 2.66667
Value after term 4: 2.70833
Value after term 5: 2.71667
Value after term 6: 2.71806
Value after term 7: 2.71825
Value after term 8: 2.71828
Value after term 9: 2.71828
d )F^>F )9) '7a3 >) 11359^(]H_) PX)a2 ,>`>9FI)/) 5)F+'h$b
9) _F")asE :]H9N/9>3 F O't) ]=>3 :]H>F^F^
a(0F
e += t /= i
3$/)"'X)^ )9) `)/F!m>/3F P )o1EF
/3
Z) )9) F^9`>
unsigned int
float
) O)>)`-/F^F^9>)
E>O)F ^) /F^F^`N)^fb g s9 ?*M /?AbK X)^9F
:]19^F^F^ F 89`>) ^)FG 3/F
x:f/, O)3
)>)^ e += (t /= i)
i
)X)^9:PN.WFG)T3fH39>F
s

`^FIE3998)^_)^9 FI)> P^
)

E/3
t
i
1/(i − 1)!
1/i!
)a )&393F7) ^F^,).f,
) )^9"$]=_)^ 1/i!
e
2.5.2 Mixed expressions, conversions, and promotions
g79 9b) E)Z,+sE:h)oEFXO"3>W93a)+sE T>/{)9/a )>/m)o1E/
g71,F H_ ]H3_`8EF^ )Z ]HF^FGF =)^/ E>/3F >) `>fb>^)^3 ) )X>FGE`>)^fb
9b)aE).1,.sE> )o1m F^ /F^ \1`>)^ 1 1L@'X>'7WFG)+F'Q)F `9/F^
PNh ]=>3
]HF^FGFh(fb>7)X)o1E>F
/3
ng7DF^,)9 f//,DF )^9
int
unsigned int
>>F^)s9 f//, :<<.< )^a)89/ f//,ade ^)^`,N P )89/n):
f//,DF _)Df/,9X/DPr)D>f) 9b) E) ,9.sE>k)vE H)Df//,X_/9F
,` 9/3{ deP )>
8)o'7 $>FG) f,F ).F 8 )) 3>W93 'X` F
`F^{
ghF7_ ^)`,N ]H9N/F '&8)T`9/ P
)
)
int celsius
float celsius
/
$/3F7) )^9 sE9fH7'7 'h/) 43,= :f/, O)Pr)!:]1F fahrenheit.C
F^
=/m)/ E:/3F7D>fb),9/ `fb:^)3
) float HF^
9 * celsius / 5 + 32
)9)7) `89,))) b>FZ9N/`T:]H`,F^fb>
Ofb>Z) )oE
float
de!) 9^/
'749fb4)4F/ H93"P1`fb>F^ r )n ]H3 :]HFGF^
euler.C
Z)
E>/3
>)^F
`>fb>^)^3 ) )-)vE
PT) )>
t /= i
unsigned int
i
float
& ]av $#K
E>/3 t
gh )o1E
F 3>W93")^ZsE >b)^ ")x)o1E
g71,F / `8EF^)^
double
float
:]19^F^F^ fb fH E>/39F P{)o1E>F
/3
F Pr)o1m
xFG,9`
float
double
double
/ :]HFGF^:)F7>f,9)3/_E>/3
P )o1m
F LM:/'I) )
Z`$
float
double
Pt\1`>)^ 1B;H &)9)x) FnX):z,9FG3 ) 3)7`>^)/.9^fH3+`fb>^F^F
'X`_.PN_) >)F FG) de
^)^`,N b)Xf//,X/DP
,FG) `>)/
double
)Tf/,9T/ P
float
de F^,_^ x):` PDO)>)`8)vEF P )
$/FI) >/h)^?)FG)
>{)oE!F
bool int unsigned int float double
/^$/3= 1O' )^ O).a`fb:F^ _ /F^- Pi) >9:/ ) a)>F^F
>{)oE F^ \1`>) 1 1L@14g7FZ9E>9F7PN&:]/89D)"3`>N)^ FI) )>)
int i = -1.6 f;
9b)XE) 1,.sE>4F `fb:^)3 )^!/ )^> )7P/`>)^9/H ^)4F 3F^`$^39>3r
deP )hF^,)9Tf//,&F4 )^9hf,h9ZP )Z)>)4)o1E b'7&:)k)F f, )^9:^'XFG
)&`>fb>FGF4,93>W93{ de8)Z>f1,Fn ]/89 )F ,h)N F 'X)
93
i
−1
'&)_)!$>FG)Z^>F^)s9Tf,

−2
9>
f//,FDO `fb>^)3 )
'7 /b/a>)X) $^FG)X:9^F^)s9
double
float
f//,W
'X ))F s b / 8 )) 3>m39>) 'hJ
hM <.<. )T9/f//,
F4N:nnF^_/>x)9/ /
f/,9/ de )F N)^)^>4`$/F^ b)&`>fb>FG8F4,93>W93{
float
2.5.3 Explicit conversions
6 fb>FGF&sE>)o'7 )/ /3 9b) m)&1,+sm>&)o1mFD!`aa/`:)`
= :]/89 ).`fb>^F^?Pn9bO)fb float f//, x )_).)oE unsigned int
`^^FGE3Fk) )&'7> 1O'& \
: SC!01q
)9)n,93F 3('X8) )^9&9 ]H) )>
x
6 fb>FG m)X`$/ _ b F^>9FG )E>PN / )>/ `>8 ,=)O)(fb>" 9b)9+E)
1,.sE>7)o1E 9P )F7N)G)>7)o1E 9/FX N^>Zf//,T/
2x]19`)8`fb>^F^ /('XF )^ `fb>G)?f//, P / )>)^` )o1m-39 `>) 5)
/ )> )>)^` )oE 'X ),) ) 39:),= P+3>W9 :]1)^5f/Ns9 b g s9)/ )^9
f, FG,9 ) P^)
f//,
{'7 `$
int i = -1.6f;
int
float
−1.6
F^89 '&)^T)^9 :]19^F^F^
int(-1.6f)
gh">/{FG1)O] Pxa:]H9`)7`fb>F^
9/FG `$3 ,+.<\,sgLM:/!.<./1q 9F

`
'X>^ F& )o1E 93
.FZ/a:]1F^F^rng79 `$/FI)X:]19^F^F^Fhf//3P /3a_P
) `^FIE399+`fb>F^-P
)+) )oE /Fh
F73>W99>3r
T x =
= `>^)/ G`89`$)3 D)o1EZ9/F b)kFn`>F^FG8)" ^)FG bZ )
`$/FI)&:]1F^F^
(unsigned int)(1.6f)
& + v z
$#U
2.5.4 Value range
=.)^/k)oEF )_)^9:)`+E>)^F _ P /n) `89,)^8`^>`>).F^,)F 3,
) (fb> !
,3> 9O'! g79F F sE>`$/,F^_)
f//,/
P&$/`5)/ )o1E
F+
/M01 N C. F^,=s FG>)&P = 'X)^ G>F sE>)o'7{
9 b)9 E)r1,.sE>{)o1EF )FrF{) )^^,9 r '&)TW )^ve/3 >fb '&) `,)s9!
=
f//, / =
)ZF78EF^F^ s &) ^>F^)D.F^,=s9F^>)ZP ['X)^ D)9/9 >) s9,=)
>F de `)^/FG) {
(fb> Z ,93> 9 ('XFT^+F^F Pk/ F^F^, X
)^9 :9^F^)s9 f,F
,F^,9/+FG /_ 1,&)>Gf/ +,` N>n)9/8PN )/=)vEF4dePmb,8)k)ZNFI)
double
f//, b,=Z9NO)PNQfHN ) :]H>F^F^
std :: numeric_limits < double >:: max ()
b,?)DPN":]8 >)T) ,=)^9,=)
Z`$ )9)T)FD$/F
1.79769e+308
1.79769
308 9.>)G)v
NO",+sE:$
10
qr:)D,FDb/à)!FGF^, PxFZ'&)-+fb>âFG89D9^/ )^ O)D/F 1F&)!,F^:X)
=9,=) )o'7 9b)Zm) 1,+sm>F ) ) 3lm:` g7 9^/ )^9>+``HFx'&9:)>
)F7Fh33) `^>`>)X3l >`nMx/ .m>PNF )F7)/F Á
Ã
// Program : diff . C
// Check subtraction of two floating point numbers
# include < iostream >
Â
Á
Ã
Á©Â
Á
Á©Á
Á
Á
Á©Ã
int main ()
{
// Input
float n1 ;
std :: cout < < " First number
std :: cin > > n1 ;
float n2 ;
std :: cout < < " Second number
std :: cin > > n2 ;
=? ";
=? ";
float d ;
std :: cout < < " Their difference =? ";
std :: cin > > d;
// Computation and output
std :: cout < < " Computed difference - input difference = "
< < n1 - n2 - d < < " .\ n" ;
return 0;
}
Program 12:
LM:</N. )1 c
& ]av $#
P
%&> F /5:]=/89,
1.5
First number
=?
Second number
=?
Their difference =?
Computed difference
k,=)Z)^9"/,=)Fh`$/-O9
FÔ'& )9)+) /,=)F Oas9)?`^`:) F^,=s)^/`>)
1.5
1.0
0.5
- input difference = 0.
^) First number
=?
Second number
=?
Their difference =?
Computed difference
`^^`>)
F^,=s)^`>)
1
P
1.1
1
1.1
1.0
0.1
- input difference = 2.23517e -08.
9)hF7 >(-VXP):Z3,=s9D``H ,=h)/mO)>)` '7T+,FG)7``,3
)9)D !) BF&)^9 'KLM7/: /3 9)X,9FD'&9`/9)T`>^`>)^-F^,=s)^/`:)$Tg _
, 93>^FG)/3 'X '7 fb ) ) b!.F^>'& O)7`F^>Z )9b)^.m)h1,.sE>F7 >/ 2.5.5 Floating point number systems
=10 \Q+01"7NKL71[M=h'A;/<:3./V.<<'
V
Fh W99 )DF^,=s9F^>)ZP
93>W93
s1
PN,=&,9.sE>^F
)^9 ;/+.Q 1 p
) LM:</<1[./12 e
)^9 ./'+[D!.<K sgL=<M
2
β

) U:2NY<.<c,sgL=/M :
/3
e
gh FG>) P ^$/ ,+sE:FZ:9^F^)3-s1 )FZFIHFG)> `F^FG)^F&P /
(β, p, e , e )
9b) E)h1,+sm>F7P ) PN
p−1
X
s
di β−i βe ,
i=0
'X>^
PN&/ 9/3
s {−1, 1} di {0, . . . , β − 1}
i
e {e , . . . , e }
gh 1,.sE>
Fh) ./1 N H)"FG 1,`
F7`$3_) .1"N=1
s
d0 d1 . . . dp−1
) 1,+sm>
F7) ,sgL=<M e
)v9`$/ 8'& ) 9b)9 E)h1,.sE>h )^9 PN
d0 .d1 . . . dp−1 βe .
=+:]/89 ,F^-s F^
/3a
0.1 100 0.01 101
) /3
x)_1,+sm> 0.1 `$ sE'Z)^)> F 1.0 10−1 n/3 /F
β = 10
_/_)>h'hHF
gh>>F^)) Pha1,+sm> sE`>F ,91,8'X '78FG)G`>).,=F^>fbF )-)
4ghT3O'&F^3
F^>) P :!'+~1 ) ,9.sE>^F He 1)TF 'X)
(β, p, e , e )
d0 = 0
P )F!F!)9)!'7F^8F^ 1,+sm>z
F e O^)`,NO")8,+sE: s9,=) ><) BF!) '7G^
0
sE,=) )^9F 9('` `>F^ x_ )^?FGF.:]/`:) )_1,.sE>F!P&s9F^,)^
f//,"FG_/>k)9/
eF^"/FG82 ]=>`>F^ 0/+A βe *
/
%
I %*: %7"
& + v z
=D
W=]=3-:]1E)
e
H)"F^_/>FG)
EF^ )fbT_/ 3_1,+sm>hF
1.0 . . . 0 βe = βe ,
'XX)"NOFG)Z"F

(β − 1).(β − 1) . . . (β − 1) βe =
p−1
X
(β − 1)β−i βe = 1 −
i=0
1
β
p
βe+1 < βe+1 .
g79FZ$9F )9)7) _/ 3_1,+sm>FZGF^^)3 s1
:]Hm9>)=
FG) 9b)9+E)&,9.sE>&FGHFI)F&,F^>3 a`>)`! ;<1[:!V $/.)9)&)^9:
9fb s /F^
8de s99^ FG1FG) {) 3`_/x1,.sE>F
/3
) > β=2
1.1
0.1
F^)s9 bF '7k'X1F^ :]1!) b`F^ ,9>) :^F4k_/3 .`fb:^)X) ) 9b)^
E)h1,+sm>F /3)^9Fh ]H9N/F ) FG)^/TsE9fHhPxMx
1
de ^39:h)^ `fb>G)& fb
mF^)fb 39>`_/
Computing the floating point representation.
1,.sE>
) _/ 3 s99G 9b) E
) 1,+sm> FG1FG) 1'7
x
(2, p, e , e )
WFG)Z`89,=)T)F ;!1[:!V| sgL7M./1q
∞
X
x=
bi 2 i ,
{0, 1}
bi
PNX/
i.
i=−∞
g79FnF4F^N ) )hs99G ]H /F^ PE 9)^,/1,+sm>n/Fk3F^`>,9FGF^38\1`>) 1 1<KH
g79 3lm>^`F.)9).'79fb )^ /O'c b)fb_E('7>F+P FG`
`$|sE
2
x
^s9)G 8`FG ) nghTs O^_:]H /F^aP
PN&:]8 DF
0
1.25
1.25 = 1 2−2 + 1 20 .
)+3>)>^4) F^_FG) /3.N>FG) f//,F P /3 PNx'&`
F >
i i
i
bi
_ sm
1
s9,=) F4W99 )&F^9`>
FnW9)g ghX,9.sE>
i
−∞
i
x
i−i+1
{∞}
F7)",+sE:ZP ./1"N
=1 MR)1"N1[6. P )
='7 >) d = 0 /3 x
di := bi−i
0
e)h)9)
x=
i
X
i
bi 2 =
i−i
X
bi−i 2
i=0
i=i
i−i
=
i−i
X
di 2−i 2i .
i=0
g Fh89>Fn)9)
h
P /3_P
/3
x
(2, p, e , e )
i−i3_^>F^))aF 'X)^9
)"/('hs9&/
dea`89,=) )Ts O^_:]H /F^P
=>)Z,F&F^F^,TPN&F^8 `)o+)9) x < 2 x>0
g79FnF F^, `)k) :]H9N )hF^F^,7'X)^ )h39>`_/H,+sE:F
/3
938/=)>
1.1
0.1
%BL7
A 2"O
%6(
A*S
+
"
Pn
i=0 xi = (xn+1 − 1)/(x − 1)
% ( M/
$&
B@7
-.6>*
xJ = 1
s
& ]av `$/FGF8`$/5sm 3,`3 ) )F `/F^s|F^: O) 3 $ '&) )^9 N >FG)8>fb>):
F^_/> . 1,9/ ) '&^)
'&)
/3
x'7>).)8s99^
x
x = y + 2k
k
N
y < 2
:]1 9FGP
s1 `+s9 ) :]H /F^9FhP
/
3
y
2k
= x < 2 Hx'7!9fb
x =
0
X
bi 2 i = b 0 +
−1
X
0
X
bi 2 i = b 0 +
i=−∞
i=−∞
i=−∞
0
1 X
bi−1 2i .
2 i=−∞
| {z }
bi−1 2i−1 = b0 +
=:x
ghFD39>))oa(fH39>FT_FG89 /)c) `>8 ,=)").s99Ga:]1 9FG P deP
x
m) FG)DFGW9`$/)&3)
F E)>^'XF^ )TF g7+)^9:T39)F
x 1
b
1
0
bi i −1
`$/F^,=s FG 1,)
sE" ]H)^`>)3s 0 H )!FT)^`9 , )
x = 2(x − b0 )
SD+)F7PN
13F7)"PNO'& F^ ,9>9`> P 3)^F
x = 1.1
2(1.1 − 1)
2(0.2 − 0)
2(0.4 − 0)
2(0.8 − 0)
2(1.6 − 1)
2(1.2 − 1)
=
=
=
=
=
=
2
2
2
2
2
2
0.1
0.2
0.4
0.8
0.6
0.2
=
=
=
=
=
=
1.1
0.2
0.4
0.8
1.6
1.2
0.4
→
→
→
→
→
→
→
..
.
b0
b−1
b−2
b−3
b−4
b−5
b−6
=1
=0
=0
=0
=1
=1
=0
4O' F^ )9){) s O^T:]H /F^!P)^943`_1,.sE>
F{E>^13` m)^9 `G 1.1
FGE3Xs99^!1,+sm> F
b/3 )x9/Fx=W99 ) _/ F^W9`$) 3 )Fx\19`>7/
1.00011
1,.sE>F ) 9b) E)71,.sE>7FG1FG)F
/3
(2, p, e , e )
(2, p, e , e )
9fb )ZFG)
FGW9`$/)h3)^F =)hPN('XF )9)
Fh)Z^>F^)s9"- s O^
9b)7E) p1,.sE> FGHFI) ^b3FGF P p, ex /=3 1.1e xgh F^/nF )^,kPN x = 0.1 ?-9$fb F^('Xy)a9^>fH,F8 9 )9))_F 8EF^F^ s The Excel 2007 bug.
) `fb>G) F^_` 3`_ 1,.sE>FGeb
) s99^ 9b)9 E)
1.1
0.1
1,.sE>F 1'X)^9,=)k_ H FG_/ :^Fng79F F7)^9D>+s ^/F^F^ `F^ 1,`T)9) )
)o1E>F
/3
O",9s9D)+>F^>)&)^9Tf//,FhP F^TP ) & )>/F
float
double
SDFG9)T)F79^s = ,",9.sE>ZP 3`_/ e)^ es99^ `fb:F^F&) b 9N/`>"
`89,=)>F.'73='&3 4)^9
,)^_b,5^$/35)F =8:]=/89 'X>fb>8b, >)>8
1,.sE> ) FG$/3F^>) Zb,y3 )8y3`_ZPN_O)$ k,=)`9/`F
9y)^ O)
)>9/ ).1,.sE> FD`fb>^)3 )93 >FG)>3 s99G 9b) em)&PN_)$
g79 F^_/&>^F )F^fb>F O?,F^,9/ y9)
) s9>?
Ts9,=))^9 FG,9 ) I'7>3 9b) E)Z1,.sE>F&:]H)^^ `F^ )^F^ G` _3`_/rf//, _:]HEFG )>
s9F ) -
V `>) F^,` F^F^,8)9).9/F ^`fb>35a).Ph)^)^) F HO'&|F.)^98)^9?
s$!
@;!hN &FG>F 9fb_>E^)^3|)9).)
,) `$) P
'X)^
`>F^P)
77.1
850
2 ]=`r31FZ)h13

) _)_)^`$/ +`^`:)Z>F^,,) s ,=)
65, 535
100, 000
& + v z
$;
`>FGP)!$`>)3 ) )F s 39 )^)8).s , rs ,=)!)!)^9 F/ ) E)
,=)
)9) )^9 /'KLMC) Fd+Z F `^>`>) 4/3 )9) )8>G. 9EF!'& )F!f//,
F )1[.qLtDV$) ) F^>) k,=)TO' `/?)X O9m )9)&)+`> )>>Df,
F
65, 535
`^^`>) 3FG9Nb3 w? )!3F^M )! O9m 'X b, +,9 )9
'&)
PN
65, 535
1
:]=/89 1)TF^,)ZFh`^>`>)
3FG9Nb3/F
65, 535
gh4E)rF{)9)r)n`>8 ,=)>3"f//,kF $s ,=) F^4)> ,+sE: ]H)^>
65, 535
`FG ) )$ gh $F^?FX)^ O) 8F^_/rs ,=)T,9fb39s9!>G FD_/3 `fb:^))
3`_{f//,
)+) 9b) E)7,+sE:&FG1FG) ):9/_,F^3-s-2 ]=` b
77.1
/3
H) 1,+sm>
9/FZ+W )^Ts99G>F^>))^{
1.1
0.1
77.1
ghF{>G `/ P1`>,^F^n){sE I> /^3 7sT)n+,)9`)7'X)^
F^&2x]H`>)F
850
f//, fb>^-`>F^!)
hghFh'7,3asE /`>`>)s9 9s9,)X:]/`:) PN 65&10. f/,9
535
/3 )>^F m/`>`3 )65,
`:F^P,) ) 3FG9N PN,9`:)9/)o
9/FD.s ,= &),= Pr
I'7 3 +,+sE:F7,)hP / 9b) E)k1,.sE>Fh"l `>)3 s1_)F s9,9 = )
Fh$/FG
)h) 3>)>`>)Z)^9Ts9, 3,=+^,N7)FG)^F
9 `>F^P) $^9>3 1,) F^-3`>,9 PN
) 2x]H` ##b@ s9,
ePN
'X` )
1,` 1 lm>>3 _W=$] ) F^,3 / P /FGF sE /3)^)>3 )9) F^,` s9,9F `,3 /F^
9fb!1`>`,=^3aaF^P)o'h P )>hfb3FXFh'7e
deP9'7h^h)ns9 )T>>F^)kX$/=,+sE: :]/`:)./F4Xs99^ 9b)^
Relative error.
x
E) ,9.sE>! )8FG1FG) )!F 9),=/ ) $]=_O) )!s1 )
e , e )
9b) E)h1,+sm> :<!./ ) (2,x p,
9)ZF7)":^h'7"_ bT )Fh9^(]H_) \1,=9mF^")9)
F EFG)fb /3a9/F7s99^_:]1 9FG
x
i
X
x=
'X>^
bi 2i = bi .bi−1 . . . 2i ,
bi = 1
i=−∞
'X)^
ZF^F7F^W9`$) 3)^F4CT
g >^!^ )o'7+ O),=/ 'hHFhP 9^(]H_)
h
x
p
'h
Fh)^ ,33('X=F^,)9.
)!,9.sE>
i
x = bi .bi−1 . . . bi−p+1 2 =
x−x=
bi 2
i=−∞
i−p
bi , i
i−p
X
i
bi 2 i .
i=i−p+1
g79Fh)^^,9`$O)FZ/m) 39)F
i−p
X
i
X
/3)
>^h'7!_ b
TF
2i = 2i−p+1 .
i=−∞
VX):9)fb '7!`>,93 ,3a,=a).)!,9.sE>
x = x + 2i−p+1
N'
%*?7
K2/
bi−p+1 = 1
M7@
7PL7
(9 *S
/#*3
,I
M2/
2//
"
2i−p+1
/#*L2
2i−p+1
O(7BK/.6#Q(
N2
%*
/1J
&' 9(Q"L
2%.B3#*
%"L
7
p
@/6>M
A ( -
-.6#*
L(
6>*
*&/.6> JR -
"L
(*
p−1
.6#*
L *:
M
*
2i−p+1
bi−p+1 = 0
/.6> N2
*
/
>;
x
J
,
.*
Q*
/?
( 9*
& ]av A
'X>^",=h9^>fH,Fh>^ZFG)^_)DF^('XFh)9)h39>3
939F>
x x
ghFn$/Fn)^ O)
Fnsm>)o'7)o'7 1,.sE>F4)9) ^ i−p+1 G) FG )&$: Pm)
x
2
)o'7-1,.sE>F"F ) FG) i−p $'h?P
aCD )8)> 9/3
F!F^ +W ,+.^
FD)
x
^
'AJ<5&1[X2Lt./1[ 2−p ,
bO)8E)T1,+sm>D$>FG)")^ g7.1,+sm> −p EPN>^^3?) / FD)
F )^9 :/!D+01"F&]<:!:<: _/3 O9x$]=_) x 'X2)^+ )F $>FG) bO )
E)h1,+sm> x
^
gh8>f1,F , )o /F^-3F PN.b)^fb
/3 ) `GFGE3asEFI).
x
$]=_)^ F^+)9)7'7!>)h) >/ :<!U+01[FdE<:!:<: PN+,N
x
^
|x − x
^|
2−p , x = 0.
|x|
ghF
$/F8)9)_) 3FG)/` P
)|)^F
$OFG) 9b) E) ,+sE:_F )
x
'7FG) `$/F^am^)9/ )?)^9F^ aP ghF F sE>`$/,F^)9b)^ E)8,+sE:F
x
) ,9/ FG /`>3y/ )^9
$/h567F^
)^
k) 3F^)o F 9 4s9,)+)
0
'7+'h P
{)+FI OF^>")>-sE`/ VXF F^8 :]8 E`F^3>T)
0
_/ 3 bO)DE)4,9.sE>nFG1FG)
g7&F^_FG)xmF^)fb7,9.sE>4F
F (2, 3, −2, 2)
b
/
3
^
)
9
7
N
O
G
F
)
7
F

$
`
/
1)9)x)73)FxO s O^J
: gh
1.00 2−2 = 1/4
1.11 22 = 7
3FG)^^s9,)^?P7/4EFG)fb 1,+sm>F!(fb> )^9 ):^f
F!F^('X )8PNO'&
[1/4, 7]
9`>),=/
0
1
2
3
4
5
6
7
8
H~)^9F49`>)^,^ )nFn`$4)9)n)^97)fbh>^nPN+,ND`$/)n3 PNkfb>^ N^
k,=)h/F^.P F fb>^
`FGX) > =)X)fbD>GhPN ,N"_$ P /e de P /`>) )>TF
x
x
_F^,=s FI)/)^N/ b?sE>)o'7
/3?)^9+F^_/FG)XmF^)fb _ 3 ,9.sE> Z,+sE:F
0
x
)9) bOOX)7`F^F^8(]H_s9 s189^_/ >3 9b)!E) ,+sE:Fk'X )
)fbT>^^X)ZFG) −p 2
9:DFk)XFG) bh,= `$/`,N)F )^9> -g7>XFk9!FG) b s9,=) )&`/`,N 1[. P /`:)D 9b)
)Fh 9`$Os XP )^9 9b) E)71,.sE>
$>FG)Z)^
x
^
x
E)&,+sE:& )!FG1FG) '7"`F^3> 9e P )&9/FZ)Fh:]HE)D)^9!/('73
/
4g79FZP /F7P
F ) NOTZ) FG_/e
{e , . . . , e }
x
^
M >PN9h/33) >F^,=s9)G/`>) (+,)9`) /3"3f1F^D'&)
Arithmetic operations.
9b)+E)&1,.sE>FZF&$FG-a)G 7/FX)FG !^$/ ,+sE:F '7!F^89E>^PN
)7 )>)`4E>O)F (fb> ) F^>) P=$=1,+sm>F<P=) >F^,) F )x>>F^)s9
,= 9b)9 E)k,9.sE>kFGHFI) '7X F^h,399 ,ceFG,9`
/F `1FG!)
$>FG)Z^>F^)s9 bO) E)h,9.sE> :
& + v z
(0
de
`>)` 9b)!E) ,9.sE> )>)^`ZF 9)7^X38`,)k)9/_)>>h)^ >)`&q{>)D,FD,FG)^)^T)^9F&'X)^ / :]/89/X\1,=EF^!)^ O)
m/3-)9)X'7 9fb.
p=4
s99^_FG1FG)?H'7 'h/)7)+E>PN^J)"/33)
1.111 2−2
+ 1.011 2−1 .
g79XWFI)7FG)> Fk) +Z1"N ^) 9X)o'7.1,.sE>FkF^,` _)9) )^9:9$fbD)DF&:]1E)$ng79F
$/F ) G39>9^_/ 9
P )T)o'7 ,9.sE>^F =B= ) F^`3- 1.111 2−2
+ 10.110 2−2 .
Z(' '7-`$ F^8 |393,=y)a)v'7 F^9 W `/39F
>>F^)){4gh F^,)hF
uI,FG)_b'7 /33y)^>F
s99^
100.101 2−2 .
9/ '7 _/ D/3as)/
1.00101 20 .
hO'y/ )9)4)F :]=/`>)4F^,)nF49) :9^F^)s9Z'&)
FGW9`$/) 3 )F F^
p=4
'7!9fb!)+,3r dea)FZ`/F^ )!$^FG)&^>F^)s9 ,+sE:ZF&s)/3
s1F^89
3=9+) N/FG)7)o'7 3 )F 1.001 20 .
2.5.6 The IEEE standard 754
gh 6ZY Y~FG)/3 O3a31FZ)&F^`:sE )"f//,"9"P )")o1mF
Value range.
float
/3
d )! FI)9,N)FD)9)T)+f/,9./+P
F `)3 )+f//,
double
float
/ P
F^,` a)^ O)&
f/,9 `$/ sE )^3
)^8
f//,
double
float
double
de /`>)` FI)"9N)^PNFTF^,=9m^)z f//)F P2T) # v .<2)Y:)
wPN ^> FG)9a/3 `89,)^_'&) 9b)^
E) 1,.sE>F &39:!)F"FG)/393{) f//,
/ P ) )vE
Fh) F^>)
float
F (2, 24, −126, 127)
P ./1[7N LM:<<10./1q 9
^_/ >3 9b)!E) ,9.sE>^F 9,F F^XFIE`Nm1,.sE>F9 `
fb) Fh9 P )^9>F^!FGm`N/{,9.sE>^F :4ghTf, 9 P
F7)"F^:)
0
double
F (2, 53, −1022, 1023)
P )J*;gDfLM:<<10./1q 9^_/ >3 9
b)&E) ,+sE:F b /b/ 'X) FG FGE`>N/H,9.sE>^F
/333`,3 0
& ]av $R
ghFG& />)^>F _.F^tF^>'X9)4O^s9)^G+)nWFI) s ,=)4)> h)f/)3+s1 `+^ $b,=)4 '&`
s9)^F4PN^ T^ `>e de33
s9)F4Pm^
32
32
!,9FG3 )^ >FG) FG"`>F^-,9.sE>&g7 F^W9`$/3^ 1,>F ; s9)F<9`$/
)9) Z_ 3 s99^ 9b)Zm) 1,.sE> FGHFG)^ )4W9^FG)x3)rPH)kFGW9`$/3
F7/ 'h$1F H`T) 9>3) :]19`)^ smDFG)3{4g79T:]1E) ^ 1,>F7/)>
s )F
1
8
PNh>F^>)8 )F
mF^F^s9Df//,F 9/3-/)>hs9)hF733-PN&)^9 F^{
= 28 − 2
="3,=s9 `254
F^ 1,+sm>F m).F^W9`$93- 1,F"0 8s9)F m) :]1E) 9/F+
s9)FDPN")F
mF^F^s9 f//,F {93 .s9)TF >39>3?PNT).F^{ de ))/ 2046 = 211 − 2
)F7fbF
s )F
64
Z)T)9)h sm)a`$/FGF =)o'7 T:]1E)hf,FZ`>,93sE"/``3 O)38'X ),)
`>/F^.) ))/{1,+sm>hP s9)Fng79>F^!:]1)^ f,FX^" P /`>)&,9FG3-PNZ^>F^)
) FGE`/{1,+sm>F7)^3OsEOfb 99`>,399 0
gh!6ZY Y FG)/393
31Fh)79^F^`> sE )^9"/` Requirements for the arithmetic operations.
`,=/`:.P ^)>)`4E>)F (fb> ) )o1EF
/3
s9,)x)^9hd 242n2 FG)/393
float
double
@0/A 31F>ng7& ,)F OD/F FG)G`>) /FkEF^FGs9 )&>F^,) Pr//33)bF^,=s)^/` )
m+,)9`$)
HT3f1F^-FX) >F^>)s9 f//, :/!.< )^_)^9 )^^,9 f//,"deP
)> !)o'78$^FG)Xf//,c
F $/+)^ O)Z)!)G,!f,!FX9/P'h$
asm>)o'7-) Fn` F^>r g79&F/7,&O99F4)^!)
)&h)9) 9/Fn$/FG)kF^W9`$/)439)
dp−1 = 0
`fb>F^P 3`_/ f/,9>FZ b
)^ )7s99^ 9b)^.m)7>>F^)){
(fb> =`8 ^F^FnPmf//,1.1
Fk fbX) smX:]`>) ,39:7/9N)^9/E>)â
F o\1>` ) 1<;H :
2.5.7 Computing with floating point numbers
k
fb FG )9) PNx/ 9b)Zm) 1,.sE> FGHFG)^ )>^ ,+sE:Fr)^ O) ) `$/)
>>F^) k/3 )F^
_).`FGF fb>^ :]H)` 4/F.,=.:]=/89 '&) )_39>`_/
1,.sE>
F^('XFTCD ) )>T9/3 9).d 2n242 FI)/393 @0/A,9/)FX)9)&'7 'X
1.1
>) ) $>FG) >F^>)s97,9.sE> /3 ) F 939FxPN ) >F^,)4P /. )>)`
E>)
O,.)\
! r(fb> r/3 ,3> 9O'&F Tfb )F /9n)rsE4)8)3!)^XsE:fb
)9)Z)"FG,9 )FXP FG)Z`89,=))9F7fbfH9
9b)+E)Z1,.sE>F&"`F^!)^ )
_)_O)`$/.`^>`>)&>F^,)F ='X )
^FGE`:)&)+N)^fbT>^$
de39>
3 )F{FE)^,4!_D`$F^F
= :]8 (,r)^N//
`89,=)F
euler.C
9^>)^)o 113?(]H_)-Pn) 24,>"`FG))$ Z>fb:^)FGF F^.`$O 9/FT)
sE
) b>8>e ghZb/ P )FnFG`>)F4)"E)n,=)k`>+9)P F b/38) Of13
FG,9 ) ,3F PN GF^/PN 8`89,=))^Fk'X)^!
9b) E)71,.sE>F
DFG)^) '&)8)&WFG) /3
FI) 8EG)/)4,397)9)k_$ /^$/3= smDs1fH,9F4)
b,-)h)F E)$
Z:fb>X`8 ^T)o'79b) E)Z,+sE:F7PN
Floating Point Arithmetic Guideline 1:
,9/)o 9P )h$FG)& P )cFG,9 )FZP^ :]=/`>)9b) E)Z`>8 ,=)O)F
&(. /
<>=
-< S27
%&#*
/.*6
"L/
*C
%*
/
0*
(B#J
& + v z
(@
2 fb> fb>^ F^89 ]HF^FGF fbfH9 9b)^ E){,+sE:F{_$&sE _)_)`$/
,9 f/) s ,=)hFG) :),=3l >)hf//,F =F^`T)^>3N)^ZFG,9 )F7D,33{ng '7
F^,` ]HF^FGF O
93
g79:PN 7)FI) )
x * x - y * y
(x + y) * (x - y)
FG,9 )F&P )o'7 bO)+m)h`89,=))9F PN& 1,9/)o_,9FG )"^N) mE>O)F
==
_ bF")^) F^>9FG \1`8 1,9/)o F F^>9FG)fb )â) )FI) >^^F '7 '7M )!>)
!=
, )o_FI)Z`/F^F 9>fb-P _)^9>_)`$ '7T'7,93r
fb> ).PN+,9)-P )+sE(fb+,3 b,?_a'73> O'[)
)^ '&9:)>
^)^`,N 9b) E) `89,=)) F :]/`:)--)$J2x]=/`>)FGF-,9FG, 3>E3F
|)_>FG)O)593|F n)^9:PN k O3 ) `N/ |>/e %&('7>fb> k)^9:O
`>G)/ E>)F"'&` $/FG FG )asE:]=/`>)$ = 9FI)/` x+,)9`) /3
3fHF^s-+mO'7>XP )!s /F^z
e,F^,9/ Z38)X`9/!) FGW9`$/
3 s9,=)X
)
2
:]1E)$4g71,F H)FGDE>)^F D:]=/`>) =,FGF )>$3_)^ (fb> n7,3> 9O' ) :]HE)$
(fb> x)"F"F/PN )a/F^F^, )^ O)") NFI) :]1E)!B
F e+,`C
D> )9/ )
`F^
p /3-a)F&`/F^ '7 `$ F^8:]=/`>) :9^F^) /r)>>FXP s9F^,)^"f//,
F^_/>)^ 67F^ 1,)^ !)> 393)^F TF^,=s9)G/`>)9F !/3w ,)^9`$)^F
βp
'X )
)^9F7/!^":]=/`>)$
gh 9 ]H) )v'7 ,3F +F^>'&9)TF^F"s1f1,F r/3?'7 ) f)^ )s W9^FG)
F^('X&) ,3>1Zs9- g7^,9,=) O'77/F^F^, hs O^ 9b)^hE) ,+sE:
FG1FG) P `F^
p
\1,=EF^7'7 'h/) )"/33 )^9 )o'7
Adding numbers of different sizes.
/
3
4
9
h
)
X
'
E
s
^
)
9
T
G
F
9
,
) a)_)`$/ b)ZF
2p
1
2p + 1 =
p
X
bO)DE) ,+sE:F
bi 2 i ,
i=0
'X )
\19`>Z)F s99^+:]H /F^89/F
FGW9`$/)
(bp , bp−1 , . . . , b0) = (1, 0, . . . , 0, 1)
p+1
3)F p
FD)D>>F^)s9 'X)^ >`F^
&3>D).d 242n2 FG)/393?@0/A )
2 +1
p
FG,9 )"P )+/33) F p ` F^> P ) )v'7
$>FG) `$/339)F p 93 p
{F^
2
2
2 +2
)Fh/33)
9/Fh9 >lm>`>)$
ghX>9:/ 9 >^XFn)^ O) /33 9b)9TE)k,+sE:FkPm3l >) F^ F
HF 8).FGF"F^9 W `/)D3 )FXPn)^9.F^_>D,+sE:X
e , :]=/89 +" )FD3 )F :
g79 N^>Z)^9 F^ "3l >` =) T3=/FG)`"F )">lm`:)$
g `f1` b, )9)x)FxFx9)48^)^W9`N/ :) ,9F `F^39:x)7s9>
PE`89,=)"%X`71,.sE>a
F eF^$^` )&'7:s PNk) /+L=/" !:7% .aLM:<&;! <' )
W93 / ):FG)9
1`>`,=^`+P %XO` 1,.sE>F : =
r)^9 ne)^ :!'A=12
n
=C';<<:
F73>W93a/F ) F^,cPr)T` 9^1`$/F Pr)TWFG)
9),=E1,+sm>F 1)^ O)
Hn
n
F Hn =
n
X
1
i=1
i
.
& ]av $K
d )TF^,3-O' sE./ $/FGa:]=>^`F^.PNDb,-)'& ) / )9)D`>8 ,=)>F
PN
H
fb
z4, 9>3 F^+?, )=, ) ,+sE:F
,= )n
n
1
n
/33!)n` 9^1`$/F,9FI)7/Fk'79b,
`$/__ bhb, 1,_P
3('X)
93
n
1
F^, ,= )^9+` 9^1`$/F ?) r)^ O)!F^,3 )!_ b+/ 3lm>^` ) qr:) ,F
)^^.sE)^ fON/)Fk93 F^Z'&9)n'7h>)$4g7h
F^('X sE>('y`89,)^F
harmonic.C
)T)v'7 F^,Fh/3a,=)^9,)^F7)^9>-
Á
Ã
// Program : harmonic .C
// Compute the n - th harmonic number in two ways .
# include < iostream >
Â
Á
Ã
Á©Â
Á
Á©Á
Á
Á
Á©Ã
Á
Á
int main ()
{
// Input
std :: cout < < " Compute H_n for n =? ";
unsigned int n;
std :: cin > > n;
// Forward sum
float fs = 0;
for ( unsigned int i = 1; i <= n ; ++ i)
fs += 1.0 f / i;
// Backward sum
float bs = 0;
for ( unsigned int i = n ; i >= 1; - - i)
bs += 1.0 f / i;
// Output
std :: cout < < " Forward sum = " < < fs < < "\n"
< < " Backward sum = " < < bs < < "\n" ;
return 0;
}
LM:<N. t5C:!'I=1qQ
Program 13:
gh PND+F^`3 /3a`$E'X )&FZ8E^)/) ) 'Z)
- $R /3
1 / i
H Z(' >)h,9FZ9$fb".1 )&/a ]=`,=)P )T/-
Compute H_n for n =? 10000000
Forward sum = 15.4037
Backward sum = 16.686
g 9X^F^,)F 3l > F^9 W `/) b g79D3l >`Xsm`F :fb ^X ) 'X'7X)^G
7
N^>h= ,=)F>
Compute H_n for n =? 100000000
& + v z
$U
Forward sum = 15.4037
Backward sum = 18.8079
33 9) `9/ n
'&9` |`$/)+sm `G`>)$
Z)`>
)9).)
PN^'h3|F^,
$]=_)^
&F^ )
1
1
< Hn − n − γ <
,
2(n + 1)
2n
'X>^
F )^9 2 ,> a/F^`>X`>9FI)/) h'7->)
PN
γ = 0.57721666 . . .
Hn
18.998
kg79)ZF H)Ts ` 'h3aFG,9c(fH3FZ++,`sE>)^)^>Z9^(]H_)_P
n = 108
Hn
31F )PN^'hO3 FG,9ism9$fb_F^ s /3 g78$/FG F FG89 .VXF )N:
F^,_/3Fh 3933-,=-WFG) 9) )>^3N)Xf//,!P )^9"F^, ) sE `89,=)3 O'&F
e`8 )fb J P /FG)4V )kF^ZE) b)&F^ X3lm:`&sE:)v'7>)Z ^)/F^,Q/38)
F^,_/3 1 )_sE /339>3 FTF^ )9)T) 393)^ 39F 9)"`9/+). ^)N F^,
/H (uv,i FG) b& p
4gh,9F 1b^39F^F PE(' _/ ZF^,_/3FnO
2 + 1 = 2p
/333) ) H) F^, FI)HF7)"F
de_`)^/FG) b)&s /` 1'h^3F^,}FG)O^)Fn)^ /33,=)XF^_/=FG,9_93FxWFG)$ g7>PN^ )&f//,XPm)& G)N/9F^, ('X9
F `8 ) fbJ
FGO'& /('XT)XF^_/9FG,9_93F
) `)Gs9,=) gh_F^,_/3F")^)3 )^9_3 Ph)_F^,_)?9fb FG)^ -13
`9/` ) 9,9>9`>_)^9
F^W9`$93 P&)_ ^)N F^, nF^9`>
)> Î
e`8 )fb J

N^
gh .uv,9FI)&s9F^>Gfb3-$/3Fh,Fh) ,=&F^>`3-,939>
Floating Point Arithmetic Guideline 2:
3l >&^F V&fb3a/33+)o'7+1,.sE>F7)9)h`F^3>Os Cancellation.
67F^39:Z) 1
,9/3=)^`T 1 ,9)
ax2 + bx + c = 0,
a = 0.
d )ZFh'7> H
('X)9)h)^Fh)o'7.1)FZ fb s1
b2 − 4ac
.
2a
de- /c)9)h`89,=)F )F^ 1)F '7 ) )>PN 'h/)h) `89,)^T)^9Tf//,
Ph) )1[.Q<:!1"'j1[M deP 2 /3
O>F^>)s9 /F9b)aE)
d = b2 − 4ac
1,.sE>F"'X) >`F^
p x,= :fH,F b>G.FG4ac
)_)F", O/)^ )9) )^98^F^,) ^ P
d
)!W99/ FG,s)^/`>)-9/FDFG_/ENO)fb!:^ Dg79FX$/F >fb P
F
−p
^
|d − d| 2 |d|
d
`FG ) > ^ '& sm!'h
P
s1 'j*!5ID!.<. )9/
)"3FG)/`"P
) >H
d
d
d
gh s9>iF^F Pn)81,.sE>F 2 /3 /
O8)!:9^F^)s9/F bO)
b
4ac
E)&,+sE:F a'X`-`$/F^!>^FX _/3"-`89,=)+)^9>- VXF^F^,
b = 2p , a =

/
)
^
F
&
,
+
E
s
:
4
F
h
:
]
/
:
`
)
>
>
^
F
)
9
s
g
:

7
g
)
h
:]/`:)nf//,
2p−1 − 1, c = 2p−1 + 1
r1,2 =
−b
& ]av #
P
F 4g79Xf//, 2
F7":9^F^)s9 9b) E) 1,+sm> s9,=)
d 4
b = 22p
4ac = 22p − 4
F ) bF^`7)F 1,.sE> 9/F
FGW9`$/) 3 )F /=P9)z 1,9/=) )Fxs O^
2p − 2
1
:]1 9FG{ g79n$FI)9b)7E) 1,.sE> Frs9)/3!s",3&,=G /33 93
4
/P): )^93
>^ ePgDF^,=s)^/`:) '7 >) ^
!g7 N)^fb >^ P )FX`89,=))^
d=0
F7)>PN^
FG)$/3 P −p 1
2
gh7$/FGFx)9)4 F^,=s)^`>)T)v'7"1,.sE>Fx)9)nZ/FI)x 1,9/ b)^9h7F^P `$) 3)^Fk`$/`>m$/`)^9:$ deP =)D)>79/3 )X_/99 FGF F^W9`$/) 3)F
/^$/3=_`$^G FG :^FhPc9^>fH,F7`89,=))9F )TF^,=s)^/`>)^,9>
/89W9F
)F^-:^F )-`$/`N) 9^)F )^9a:fH,F^|F^FFGW9`$/) 3)F+) ,`
TF^W9`$)73)F P ) >F^,)$
VXb/
O)k:]=/894'7nbfb > F ^) W `>N/ $s9,=) sE F^F^,=3 )9) `$/`N)"9EF
`>)`/ 2 fb )
,9/3=)`7 ,9)8 ]/89 )k)xsE7)9)n)^9h , O)F4)^ O)
`.,= ?/ 9`)-9fb ) E>G)v )9)T) "3FG`>9/) 2
F"+,` b − 4ac
u
0
1
"
F^_/> )9/
/3
)^9>F^fbF>nde )FZ`/F^ `$/`>N)
9E{
a, b
c
gh"39F^`,F^F^ `$/sm"F^,_^ 3_aPN^ P .)^3,3
V&fb3F^,=s)^/`:)+)o'7 1,+sm>F7P /FG)7 1
,9/
Floating Point Arithmetic Guideline 3:
F^ P )^9>F^!,9.sE>^FZO ^F^,)FhP )>9b) E)h`89,))F
2.5.8 Details
g79_d 24242 FG)939^3 @0/A-39:W F )o'7- 9b)
Other floating point number systems.
E)D1,.sE>TFG1FG)F ./1[7ND!{,,s<)J,)ELM:<//10./1q
{/3 )J*;gD!{ s<)J)^LM:<!{
p = 32
<1[./12
93 FG
9N)^PNF.lm>8 ))F!PZ)F^)o1mFgh>^F
p = 64
/FG
) d 2n242*FG)/393?Kb0/A )9)"/O'&FXs /F^
rPN"s1fH,9FT$F^F )^9+3`_
β = 10
PN_)ZF&)^9" a'Xà'7!)^9 sE,=)&1,+sm>F /3-a'Xà'7 ,F^,9/_>F^>)
1,.sE>F> de O^)`,NO k-s /F^ FIHFG)> 9/F 9 F.|)_f,
9 ) 3`_/
10
P/`:)9/E1,.sE>Fh b
93
1.1
0.1
FG)&9N)^PNF )!)o1EF
/3
`GFGE3-)
IEEE compliance.
float
double
)+F^ /3 3,s9 `>F^ bO)
m) 1,.sE>FTPk)^9 d 242n2*FG)939^3 @0/A r)^9F
`^^FGE3`8F",F^,9/ ) ) :]/89 rPkb, +)^^H9
):9^13,`
) `$/`)
:]8 X'7"bfb =b,-) '& )
float b = 16777216.0f ;
float a = 8388607.0f ;
float c = 8388609.0f ;
// 2^24
// 2^23 - 1
// 2^23 + 1
std :: cout < )X):)X)!3`>)^3-'& >F^,) k,=)D)X_ $/F^9E> )^ O)&b, >)
0
)D`^^`>)7F^,) =>fb )^9,_b,= )PN `>N/F4).PN(' )Td 242n2|FI)/393@0/A=
4
g79 FI)X b>/F^ F&)9)X)^9 9NO)PN )>^ ,9FGFT FI)>D'X )-"s9)^F&)
& + v z
H
E>^PN ) `89,)){& )FDFGFDb.813 3$8 >/ m)T`$/ sE P )
PN&.c'&F^"PN,`>) )o`> )`$/ >Fh)"d 2n242 FG)/393a@0/A=
4, u10" FG)7 b _F^")^9 `$/`>N)
:lm`>)& )^9 PN('X9 FG 1,f//)
f/N/)hP )!OsEOfb `39/
float b = 16777216.0f ;
float a = 8388607.0f ;
float c = 8388609.0f ;
// 2^24
// 2^23 - 1
// 2^23 + 1
float bb = b * b;
float ac4 = 4.0 f * a * c;
std :: cout < < bb - ac4 < < "\n" ;
%&>^ ) F^,)F&P )!)>3NO)T`89,))Fh '&)G)s /`)^
f/NOs F float
s s9^F^,)8 ) :]1E`>)^3 ^,93P
Tg79> ) W99/ F^,=s)^`>)->fb/F
4ac
)!`/`NO):lm`>) M !./. Px`>,^F^ )^9"`89>h3`39>F&) b: )^9 fONs9
ac4
^FG):X'X)â">`F^{=X)FZ^$/F^ b, `$ )o `$/_(fH3 `89>
)
) _ b"F^,=T)9
) 9b) E)71,+sm>FZ 9
) b>)Z >FG)>^F
O).F!)8 P )F 4, ,F^,9/ `$/) PN, )^,FG) ) # v /']Lt~1 /
P4 9N)PN^ 93 )&FX)^9:D$/FGD'7^)'&T) 3`>)X9(' 9b)^ E)X,9 sE>^F :]`>) sE9fb8 FGE` W `+9N)PN- d ) F"+8E^))TPN!b, )!
HO't/3
,39:FG)/3 bO)!E)71,.sE>7FG1FG)F > =/ 'X ))7 ))F g79F
H('X>39D'&{/O' b, ) 3)P /3'7 ^,93
9^s F )9)h)7`T,=-
FGE>`W9` 9NO)PNF
g7!6hY Y FG)/393FG`>sE>F7/)>7PN,939/>)/ 9b)!m)
The type long double.
1,.sE>X)oE `$/3
Dd )FD)>FZ>93 'X )a) >)G)>
E/3 )DFX,9 long double
l
L
/)3?)^ O)D)+f//, /.P
FT`)/3 ) f//, / P
double
long double
SDFG9) )F O)k`fb>F^ P
)^
F ) 39:W 3 )^&sEnh)
double long double
s1_) 6ZY Y FG)/393{
/3
,9FG, `^^FGE3|) F^
/353,=s9
>`F^ P&)
9
float
double
d 242n2 FG)939^3z@0/A )>|F F^,` 3P /,) `9`|PN
de `>)` long double
) F^89 sE_aFG1H PN
xs9,=).) ) /FâsEF^>)^99
long double
double
FG4CT8)^9h9N)PNz,F^38s )^9Z/,=)F PNn:]/89 `GFGE3Fn)^")
long double
_/ 3 9b)^ E) ,9.sE>!FGHFI)

)^9F!F":]/`:)?)^9
F (2, 64, −16382, 16384)
3,=s :]1)33`>F^ P ) d 242425FG)/393 @0/A=
deP b, 'h/)!) 19('~) />)^>F!P )^9 bO) E)"1,+sm> FG1F Numeric limits.
)F+sE3
93
yb,=89NO)PN nb,y`$>8 O )
float double
long double
'789fb,F^3 sEPN^8 ) \1`:) 1 1<01
%&>
numeric_limits
limits.C
T)T>f//)X:]19^F^F^F7)>)>h'X ) ) Z/F HF^('XPNZ) )oE
float
& ]av :]1F^F^|eP )o1m

int
std::numeric_limits<float>::radix
std::numeric_limits<float>::digits
std::numeric_limits<float>::min_exponent
std::numeric_limits<float>::max_exponent
$/
β
p
e + 1
e + 1
_ )^ O)
3 1F fb_)
F^_/>FG)
std::numeric_limits<float>::min()
f//,? sE>`$/,F^P )F^ s9) )^9F F^_/FG)!f//,8F ^F 89 )b)fb P7)^9
float
N^FG)7f,g =s9,)7)"F^_/>FG) _/ 3 L=./1[01[F& f//,
9fb.)^3)9)X)^9 9b)^ E)XFG1FG)F&>F^`>sm3 sa)
Special numbers.
d 242n2 FI)/393 @0/A`)/ F^ FGm`N/ 1,.sE>F< )T`399 ,9FGFT:]HE)Df,F
)9)h3 )Z1``,=Za_/ 3_1,.sE>F>
CTa)^9 "9/
3 =)>"T) )J/:!'A+Z1 ,) 1,.sE>FhP )^9 PN
d0 .d1 . . . dp−1 βe ,
'X )
V~39>9^_/ >3 ,9.sE> 9/F"F^_/: s9F^,=) f,+)9/ / _/ 3
d0 = 0
1,.sE>$4de G)`,N F&.3_/ 3_1,.sE>$
0
gh)>!FGE`>N/k,+sE:F!`$9) ^$/ sE `$/>3 1,+sm>F gh>^ f//,F ^> FG)9
/3
/3 )> >),=3 s1 Ofb> ('X9E>)^F
gh )>
+∞
−∞
DFG>fb>/mf//,Fk`$/>3
F ePN G)7"1,+sm>+ )^ O)7X:),=3
s8E:)Fk'&)
NaN
,39:W 35^F^,) n b_) H9 )F ,9 ^1) PD b)^fb ,+sE:$5g73$-sE3
)F^Xf//,F Fk) (fH3D
]=s9 )v+3$!'X ):]=`>>)9/ FG),9)F de9FI)$/3
P F^89 sEG) ) 'X F^"E>)^-P /F )&_ bFXF^>9FG.)^8>),= /
:]H`>)9/7f, g79`$/: P&)^9 E>O)5`$5) 3`3('c) 3$/7'X)^|)
F^)^, O){
2.5.9 Goals
V7)7)F7E) 1b,aFG9,3?
Dispositional.
g H('t) 9b)^
E) ,+sE:")o1mF float 93 double /3 )9)!)> >/{)9/ )^9 )/E)o1mF<
,93>^FG)/3y)-``>)_P! 9b)^?E)81,.sE>FIHFG)> &/3 ^)`>,98)^F
/3=f/)/Fh(fb>D W=]=3m)h1,.sE>hFG1FG)?
; H('[)9)Z).d 2n242 FG)/3 O3 @0/A3F^`> sEFXFGm`W9` 9b)9+E)X1,.sE>&FGHFG)^F
,9FG3 /Fh13F PN
93
a_/89N)PN^F<
float
double
A+ H('w)^9T)= bO) M )hV& )>)`TT,3F<
08
sE.'h )9)T`>8 ,=)O)FZfbf1 ) )o1EF float 93 double _a3fb:T :]/`:)&FG,9 )F =
FG) 3,9 )+9FZ )"f//,T/
& + v z
;
de ^)`,N b,-F^,93 sE"s9D)>$
Operational.
o+g_:f/, O)-:]H>F^F^F_fb fH?) )>)`_)o1EF int unsigned int float
/3

double
o _`>8 ,=)X)^9 s99^>>F^)) Px+ fb / ,+sE:!
,=) ) 9b)+E)&1,.sE>X$OFG)X)^_ fb-$ 1,.sE> 9
'X)^FGm`>)
) .W9) 9b)9 E)71,.sE>hFG1FG)?
o ; _`>8
oTA+'7 '&) + fb9b) E)h1,+sm>hFGHFG)^?
o"0
^` ",9F^/"P bO)+m)h1,.sE>F7)9)hfH)Fh/P )^9T)= M )ZV&)>)^`DT,939>F<
o R
'Z)DF )9)7E>PNJ`89,=))9F '&) 9b) E)Z,+sE:F
b)^
2.5.10 Exercises
F +~t+ 657GST[Duv1"7N},sgLM:<!././12M.|.<qLt{,;!V{2.<qLM-IJ/:/)1[7NH657x/M{
F&<:g.12 :!C <. S 'j1Us$) ,sgLM:<!././12M.T y+.<.C'I PQ+01[7NL=1"Mj:<2LM:<<.Q<M2+012
J/:/)1[7N|I# v -R65C+\1[./:!:<!.qL7)+.j
) )JC{
float
F (2, 24, −126, 127)
;gD|/:!:<<.qL=)+.?
I+~.Q@+.<./h'Ij65C+
;!106.A:/?7.Q,)
32
:<2LM:/!.Q<M
Fd+Z <F.T (2, 53, −1022, 1023)
i ]k<o
Exercise 50

sC
`!
3h
int
6 / 4 * 2.0f - 3
2 + 15.0e7f - 3 / 2.0 * 1.0e8
392593 * 2735.0f - 8192 * 131072 + 1.0
16 * (0.2f + 262144 - 262144.0)
']LM7K657;<1[:!V , sgL7M.12M.G SK657vSQ" u1"7NA)J<1['+4=h'A;/<:g.T
;o
2o
!) o
i o
Exercise 51
!o
0.25
1.52
1.3
11.1
:X657c=C';<<:g.\1" sJ<:<<1[.Q hk-'KLMCW:<!./ PQ+01[7N\L=1[Mf=h'A;</:g.
1[ 657z./V.<<'j. )
i To
Exercise 52
(2, 5, −1, 2)
(2, 5, −1, 2)
5C+R:<K657BU:2NY<.<R)E./'A+" !./ EFG)fb :!'+~1 )E./1[7NDG)A)Jt;g LM:<<10./1q \T+01[7N]L=1[Ma=h'A;/<:g./-J/:/)1[7N E657K# v .<2)Y:) i o
Exercise 53
uw'A=V \Q+01"7NWL=1[M =h'A;</:g.B)JG657W./V.<<'E. M21[
Exercise 54
(β, p, e , e )
Exercise 55
']LM7K657KF+~*j SX657BF:!1qY;gD
float d = 0.1;
X.<.C'I]657K# v .<2)Y:/)

d
(β, p, e , e )
i
)
o
S<<:B657^)J/<D:/+01q.<2+<'A<M
i To
& ]av /A
5C+10.]657Ei L=<M01q+ o]LM:<&;gD<' u1[65O657ST[Du1[7Nj <L
Exercise 56
o!0
for ( float i = 0.1; i != 1.0; i += 0.1)
std :: cout < < i < < "\ n";
5C+10.]657Ei L=<M01q+ o]LM:<&;gD<' u1[65O657ST[Du1[7Nj <L
Exercise 57
o!0
for ( float i = 0.0 f ; i < 100000000.0 f ; ++ i)
std :: cout < < i < < "\n ";
:!10 LM://N:/' 6 5C+]C LM76.9ST:N1[Fd< )J<1['+1[&LMCR=h'A;</: x - 0 <
Fd+Z 3Vhh:9LtD+ ST:!'A p 5737 LM7.<57CD) M21[ 657
i;!1":!V oj)1"N1[6.B Sc657X./1 N=1 ,)- .<2:g01"7NGuv1[65 -a)657Gi)J<1['+ oE,sgL=<M2?
5C+.
'AV@+././C'A^65C+\657 PQ+01[7NL=1[MP=C'A;/<:?./V.<1<' h)J<:g~V+1[7NO657j0VgL=
float
;/+.T

i ToJi ]n/o
Exercise 58
-f106.c:!'+~1 )
x<2
float
β=2
:!10 ^LM:<N:/' 65C+\!.<6.Guv57!657<:AN1"F&<yFd+~*A SG0VgL= double 10.EJ!{
0*+"~V 1[M NY<:!-9)!.<v6579LM:</N
:/' u1065?Fd:!1q7.\1"&LM76.cZ10d
- ]8=1"'KLtZV=F&<:g01"7NIA Fd+~*j SK0V L7
)|<570.5
/!Q1"71N uv1234567890
57!657<:G65&10.
1234567890.2
!5C7NY<.W657WFd+~*B)J
!.Pu`:gz1"ONY<<:/+Z-v./1[X657\int
N1"F&<Fd+~*c'j1 N5Ya;<3|1"M!{
NY<:3C6./1q)JP657cFd+Z P:/7NY3 S
E 'AV?+.<.C'IP65C+ 657 PQ+01[7NPL=1[Mf=h'A;</:
./V.<<' C)J/:g~V+1[7NE657z0VgL= int 5C+.3;+.Q

i ToJi Kn/o
Exercise 59
p 57?=h'A;/<:
0u`O SX657<'
Exercise 60
π
= 1−
4
π
= 1+
2
double
π
β=2
, ;/O)J ) 65&:<hN5xF:!127.^1[ = 10 .C'j.T <:/O:<
1 1 1
+ − + 3 5 7
1 1 2 1 2 3
+
+
+ 3 3 5 3 5 7
:!10IjLM:</N:/' SQ:A/'KLMC01[7N@ !LYLM:<Ts+1['+012 S
5&12!5]SQ:!'^7Uj1[.G;/!6<:SQ:B65C+ LMC:L7.Q
π
-K;+.Q)xH657!.QcST:!'j7U+.T
i ]n/o
p 57/:< 10.Iu`![~{2Qu 1[<:/+01"F&3LM:<
/,)C:< i657 sHN/ :)13 ozST:
/']LM701"7N657O. *:<A:<
E S|OL=.1001[FdI:<,+R=h'A;</: 8M2:g01[7NES/:<' =VxFd+~*
-vu`^/'KLMCE.Q <
SXFd+Z <.GJs/:/)1"7N E657SQ:!'jCD
Exercise 61
x0 , x1 ‘, x2 , . . .
x0 > 0
s
1
).
xn = (xn−1 +
2
xn−1
# P,H;<B.<57u@65C+
xn =
s.
65C+4:<Y)+.`1[A657`=C';<<: )^/']LM7!.c ! LYLM:<Ts{
:!10X`LM:</N:/'
1['+012 S 7.babylonian.C
1[7NI657Eb\Y;!V =1 'A!657Q)Y p ;<I/s<:<<<-X657KLM:<N:/' <. 577U)
7 LMCa657 :gs.<a=h'A;</:
./ <5I65C+
i ]n/o
n→∞
xi
|x2i − s| < 0.001.
& + v z
0
:!10W`LM:</N:/' fpsys.C ]F
10.*+Z1\]:!'+~1 ) \T+01[7NL=1"M=h'A;</:
-)
./V.<<' p 57 LM://N:/' .<577U)c:<,Y)]657 L7:'I!<:g. (2, p, e , e )
p e
+.G1[&LM76. )3SQ:AJ!5L=./1001"F&j=h'A;/<: S/:<' ):u @<1[:<< eO S
(2, p, e , e )
:/Y)1"7. :<h)z657]:!1"N
1[ 4.QR657P~1 ;!:/:!xV
65C+ 10.cFd1[DY;! W+657]/h:g.Q
x
libwindow
57'IqL7NYE <:<+^N:/!Lt5&12,+7 LM72 4.Q^657KLM:<N:/' IF&<:!1 SV 657j=h'A;</:g.jVh
/']LM7,)?1[ s$/:<<10.T
i oJi Kn/o
Exercise 62
2.5.11 Challenges
p 57 a/3s)nF^>) 10.X./*;!.Q!9 SP6573/'KLtD,s]LtUW65C+a;<'ARL=<LMCD:
65&:<$N5 1[6. P/`>) ./5C!L=?)O657;<C01 S7):/u1"7N. SG1[2bc! u Vhx.Qj657^.Q!Q% .
_/
`$^393 )AI)J!21[f SX10R+'j <5A5&1 N57<: +
' .Q+DQ
Exercise 63
p 57 )J!U;!:<P.Q!10.j)J )+.9SQ[Duv.T :
S3'KLtD,s =h'A;/<:g.]N1"F&<;<V
c
z0 (c) = 0
z0 (c), z1(c), . . .
zn (c) = zn−1 (c)2 + c,
- u`/M./1 )J<:657.Q * </
)
n > 0.
SQ:K+" i65&1[.]&;!FQ127./~VG5C!LYL=/M.ST:z,sJ']Lt p 75 <:<K:/c0u`?+.Q<. W<1[657<:
|zn (c)| 2
1S
o-\:
SQ:G.Q'A i65&10.j&;<F
12nC.<~VA5C!LYL=<M.9ST:?,sJ']Lt G1 S
o&
(c)| > 2
.Q!X/M./1[.<6. Sj+[n ST:Gu5&1q!5 uR :<1[y657 :g./],+.QQ?# |c|SQ>[D2uv.
p 57c = 0)J<D;!|z:/nc
65C+9657 )J!D;<:<9.Q!P/M21[M. c)E10.3/M21") 1[yI)10./ SW:/Y)1"7. :<h)
0
2
0
1[ 657j'KLtD,s3LtUQ
:!10\LM:</N:/' 65C+):/uv.\i!LYLM:<Ts+1"'A+012I STo\657 )J!U;!:<.Q<0-:<!.<0:!1q!)
zW:<!27NCD:P./*;!.Q!f S657c/']Lt ,sRLtUQ# ./577U)B;<fL7.<./1q;!
' 1"=-'I,M{
1[7N65C+657W:<!27NCD:K./*;!.Q!9;</'I<.].'A+[D<:!-a)G'A:<z)J!21[.B;///'AKFQ10./1 ;g \1[
657W):/u1[7NKu1[)JuBeW;!FQ127./~V+-Vh|, % MLM:<
!.<.W+[h1" =10!ZV3'A=Vj/'KLtD,sz=C'j{
;</:g. 1" 657X:<!27ND<-P)PSQ:WN1"F&< -fV$@=v:<+"~V|!57!Gu57<657<:
|zn (c)| 2
SQ: /c -9.Q?1[10.K/!./.T:!V|O)10.T<:<!0c1 G6573:</<27N 1[MzLM1Us$!~./-\)|
!./2Y;g~1[.<5
n
.Q'ABLYL7<:E;<h)
x657B=C'A;/<:? S]10<:/+01qM.T]#US
ST:j+[
-V$
N
|zn (c)| 2
n N
'AVA./1"'KLtZV+.<.C'IB
65C+ 1[.B1[x657 )J!U;!:<`.Q<2G_R
/:^.T<-9657<:/310.BE
Nt:/M/
c
65C+W657E:<<./7~01[7N@):uv1"7N|
10.<F&< !D.Qj 657 )J!U;!:<W.Q!3i!.qL=<1q+"~Vx+ <:
D<F&!f SB)J<210 o-a;!CtST:B657].T+dG Sz&;!21[=1[7N=1q/`LM12!0h:<!./- u`GNY<<:<C.<ZVjND.<.
F&<:z65&10.W10.<. Q
& ]av R
'AVA7.QG657
~1 ;!:/:!VOGLM:<Q)*/657E):/u1[7Nd p 57 ,sJ']Lt libwindow
LM:</N
:/' 1"@106.3)J
<h'I<M2+01q.<57CD)N1[FdKV$ 1q)J,?57u 65&1[.G,;<^)JQ
Hint:
Z1q;!:/:!V
Exercise 64
p 57SQ" u1[7NO<'A1[4u+.z.T<MO?'A1[~1"7N Z10.< ST:z7.T<:g.^ SK6573.Q S/0u`:/

Hi all,
This should be a very easy question.
When I check if the points (0.14, 0.22), (0.15, 0.21) and (0.19,0.17) are
collinear, using CGAL::orientation, it returns CGAL::LEFT_TURN, which is
false, because those points are in fact collinear.
However, if I do the same with the points (14, 22), (15, 21) and (19, 17) I
get the correct answer: CGAL::COLLINEAR.
!o 1") Cu5C+R65&10.?<'1010.jY;<C P1"|L7:g012<CD:!-`uv5C+`10. -auv5C+`10.G657
:!1q<M2+012| S`9L=1"M0:!1LtD<-uv5C+10.
-uv5C+)J
!. 2/[Z1[,:
CGAL::orientation
'A=-)Euv5&V?10.]657]uv:!1[<:E SX657^<'A1[./C:LM:!1[.Q)AY;<Ca657&;g.Q<:!F&,)O;<<5CF{
1q:
;o>W:/ S/^ M./u`<: 65&1[. <'10R65C+j,sgLtU1[M.O657 &;g.Q<:!Fd+01qM. SA657
C.Q<:z65C+vu:<z102

& av G v @
2.6 Arrays and pointers
`Y)1[7N 1"M :!:/V u10657Cc'A+Q1"7N@ ./1[[ZV /:!:<: 10.A;/!{
V$)G657KY;!1[~1[0V? SX/'KLtD!cFQ12/<.c{9;!V657c01['A\V$?NY! 65C+
:!1"N5Y0-vV$:<Bj 7N&<:jO'KLtD!zF
1q/Q
b Q:!]8M0:<C.<0:!LM- `K 8M0V ) p !5&=1 *
p 5&10.3.Q!012 1[M0:/Q) !. :!:Vd.E+.E/M21"<:g.`SQ:^.Q *</<. SE&; /<6. S
6 57.T'I0VgL=<- u1[65K:/)J' J/!.<.\1[)1"F
1q)t+'A<'A;/<:g.P S657a.Q *</T
R :!:V1[.P657P'I.<MLM:!1['^1001[Fd];!C+657\.T'IP01['AKBF&<:!V <1q<MaM{
21"<:9SQ:K.<:!1"7N-LM:/
/!././1[7N-`)E1[<:/+01"7NOF&<:zD:2N&^'IhM6. S3)Y+2Y
|u10" +.TED:!Y;/7L71[M<:g.G+.,s!Lt~1q<10R&; !PY)Y):<!./.Q!.3)IY;<C
657<1":!D.Q|:<!U+012M.<5&1~L u1065 :!:/V.T 5&10D 657 `K .<2)Y:/)y~1 ;!:/:!V
M21[M.\D!.<.LM:!1['^1001[Fd])zN&<<:/+"~V ;<!6/:z+<:!+01"F&!./-f657 ``:9)^F ;<<{
5&1")I:!:/V.G)KL=1[M/:g.:<^ S S/C)Y'A<M2+1[']L=:g2/T
dea\1>`>) 1BA.
`)^ÊFG))>)F '7D9fb $3 sE,=) )D``:9)hPr)>){
= :]/89 '7.`$/ (' )>) (fb>")^9+F^ ,` Pn1,.sE>F 1, 2, . . . , n /3 m>PN
F^ E:)F b7/33 ,=8/1)71,.sE>F 3)PHX) ,9.sE>^F /
)-4\1N b'7 `$/a)^>)T(fb>&)^9 133a,+sE:F H)TEO'7:FZP )v'7 =:)`
de /9 `$)9F /O'7:fb> '77P)89fb ) `F^Fv /3+ G)`,N )>)7Ofb:,
F^ ,`F&P )Y+2 h:]=/89 Prb, 'h/) )+3)P8)T(fHX)$O)>F7_)('X )^ O)
F^(' b,=Z3FG3
Of1&)) bb,9fbT)^+ )>)^XOfb:h)^9DFG 1,` P (fHZ)$)^>
>m>^)^F ghF^+>E:^)>F"+,FG)TsE FI)3 FG>'X> /3 )> +,FG)TsE _'h )
FGE>`>)Z)^9>J),={4de 6ZY Y 9'7 `$/a3$r'&)
F^,` a)/F HF7s1 ,F^ :!:/Vd. 2.6.1 Array types
VX ^+P >9)^
b)>F
s uv`>)^F4P9) . 'I )o1E |)"DF^ ,9>9`>kg !/`>`F^F
n
n
Px)./b)^3 s uv`>)^FW ) !D<'I/M6. m'7 ,F^ )^F 1[)J,s ./*;!.Q<:!1Lt mF^)h
|)^9
F^ ,`5VX7)^9>F^>9)^ F^ ,`F8PN / G )oE '&9FG
f//,_/
n
`^^FGE3F )a)8_)^9>_)`$ )o1E n -de )`89,=)><BF _/ G / G
1`>`,= F&.`),,F7 G) H'X)^ ) )F7FG)>3aF^3 s1 eF^3B F^ ,="R:
qr:) ,F FG)^)ks18FG9('X9 /
^8 `>)
f :/+.<657<<.h%Y8=12/F& F "P /FG) >)^93PN
`89,=) / &,+sE:F F^_/>k)9/ fb
1,.sE> Hs /F^3
`:F^F^.,=) )
n
1,.sE>F4)9)kX)k d ) '7 HFk bZ)F b,'&)^Z3('X8)&FG 1,`DPE,9.sE>^F
sE>)o'7>
/3
4\)^)^!P
b,
/'hHF4!)")X:]H)k1,+sm>k)k`>F^F^>3 ,=)
2
n−1
2
b>) =:E^)Z)Z/Fh =/3)`>F^FZ,)&/E )F 9Ê>Z ,) F
qr:!) BFn9)n3='78)&`>^`>)^9>F^F P )F4>)3 s9,=)k!^)4)")^9Z89))r
dePXb, ) sE,=) )8PN_ 1,=) 4)_/uv 1,FG)5F )F O' 3 '7`>^F^F,=)
1,.sE>F
& ]av K
gh PN('X/,F^F / ^ )vE f/s9
PN!)^9 FG) {'&>
crossed_out
/ f//,
F.P&)oE
/3|^>F^)F )^9e`9/+ PN_)
crossed_out[i]
bool
'X>)^9: )^9+1,.sE>
9/F"/^$/3=-sE?`>F^F^>3 ,=)! )$ V&^$ 3`>F"'hHFDFG)O^)
P
{F^ ?3>T)
i >) )
39 ] n − 1 {'7+3 ^$ Pn) n .gh.9^/
0
,F&2x)^FG)<F \1>fb"PN
n = 1, 000
Á
Ã
Á
Ã
Á©Â
Á
Á©Á
Á
Á
Á©Ã
Á
Á
Á
Â
// Program : eratosthenes.C
// Calculate prime numbers in {2 ,... ,999} using
// Eratosthenes ’ sieve .
# include < iostream >
int main ()
{
// definition and initialization : provides us with
// Booleans crossed_out [0] ,... , crossed_out [999]
bool crossed_out [1000];
for ( unsigned int i = 0; i < 1000; ++ i)
crossed_out[i ] = false ;
// computation and output
std :: cout < < " Prime numbers in {2 ,... ,999}:\ n";
for ( unsigned int i = 2; i < 1000; ++ i)
if (! crossed_out[i ]) {
// i is prime
std :: cout < < i < < " ";
// cross out all proper multiples of i
for ( unsigned int m = 2* i ; m < 1000; m += i)
crossed_out[ m ] = true ;
}
std :: cout < < "\ n";
return 0;
}
Program 14:
LM://N. 4<:+.<657<!.T
VX G f^Ns9 e+F^89 ^J 'X)^
)^F P
Definition.
n > 0
0VgL= Fh3>W99>3)=,) PN ('X 3`NO){
C)J<:gZV+1[7N
% >^ &
+,9FI)!sE /M.<2M3,s!LM:<!.<./1q P )^/ o) 1E '&F^ f//, F
!= :] /89 {
)>/F b 1000 )>)^` :]H>F^F^F (fb> )>/F] b 1+1 n `FG)/)
& av G v U
:]19^F^F^F<)>D^X)> `FG)/)k:]HFGF^F 1s9,=) / PE)t fb&)^9Z9Ê>^)o )^ O)
)Zf, F1O'& O)X`89D)7ghFX/('XF7) `89:h)^ W9,^ ,=)XO'[ ,`
^ )!G
f/s9T9>3F
gh.)o1E.P !
F ms9,=) '7.9,=) )^9FD?3,=s9 1,)F >ea)
)X)^9>
[n]
N)^>, :gh8$F^ F!)9
) F ) )8 `N/k '7`$/M
)!'&)^
PN
[n]
int[5] a
:]=/89 1) 3`N^!/^
Pr)vE
a
int[5]
ghhf/,9h/hP F n b)&FG>) PEF^ 1,`>F
'X )8/
sm
[n]
(t1 , t2 , . . . , tn)
ti
Pm)o1E ghX,3>1!)o1E y)4PN ]/897sED/8PN,39/)/=)vEc
e b
int

/3. )F `$F^ ) f//,F P=)
^$!>)F ^_/ ,)/ >3
bool
double
n
s1_)"39:W )^{
gh-P /`:)8)9)8)-O^5>9)^ ,FG) sE H('X)8`>8 )^ `>$ |)F )
,F^PN,F^F Pn^-f/Ns9>F =":]/89 )FT ))
31F ) /(' ,FT)'& ).
fb>FGPm2x)FI)<F F^>fbZ 'X`+)&1,.sE>
F $/3P^~)Z= ,=)$
k,)4'7Z'&
n
F^^) ?F^>9('~)F">FG)^`:) `$/ sm8Ofb:`, PN )^9 ) sE >!) BF"F^89 fb
'X ))
2.6.2 Initializing arrays
g 9k39:W )^!P/.O^"'&) ,3>1&PN,39/)/)vE 31>F ) )/ )kf,F
7
P )"^
>)F>4?!`$/a/F^F^ f/,9>Fh)+)!)^FZP)>^'h^39FReb 7
' 3 )Z
Mx/pA+ s ,=) '7D`$/F^ 9^Of13X)Xf/,9>Fh3^`>) 1/F _)DPN('X 3 `>N)^
FG)O))$
int a [5] = {4 ,3 ,5 ,2 ,1};
\1` )^9 1,.sE> P7O^?>)F!`$ sE833,`3 P
Z10.< ='7 `$/-/F^+'Z)
) ) P )
1"=1001q+Z1+<:
int a [] = {4 ,3 ,5 ,2 ,1};
g79 3`>N)^
'&),=)X-)^N/ )
FXf39)^9,mF^9`> )D31FD)
int a[]
PN,_3>):D) )o1m"P F
)9)
9/F 1"/'KLtD!B0V L= )^9Fh`/F^
a
a
2.6.3 Random access to elements
g 9!FG)X`/3 ,9FGPN, 'haP4/``>F^F^_/3 13P1+) >)F&P / G
7
F7s1 :/)J' J/!.<. kdeP
.FZP )>/E)o1E"/3a9/F7f/,9 =)"f//,
i
F7P )^9D)o1m",3>1.)^9 G
/3 ^PN>Fh)+)^9 e) >)`e`,).P
P
a
i
0
nghT1,.sE> F `$/3) 1")J,s ./t;g.Q<:!1~Lt P{)T>)$xdeP
F7)^9D) P a
i
n
a
) 39 ]
,FG)hF)^F^P
ngh"E>)^
Fh`/3
) .*;g.Q/:!1LtP<L7<:/+: i
0 i<n
[]
gh
F^>'& O) FG)^9_3`) PN^_) PX5^$ x'&) 9 :]1 `) )oE 9/
E$O HFh)f/)3_s1_) F^,=s9F^`>)ZE>O)$4de39>3)^9 3`NO)
& ]av $;#
=de
)F4F^F^ )4Fk889` ) 3>W9)+P
F4)o1m
a
g786hY Y}N/,9/ lm>^F"
PN,9`>)^9/)oaPN!/``>F^F^_) )
Watch out!
Pr/O^?eF^> \1`>) 1<RH A!sEO' PN7X_)F :xVXF )^9Z9^/> V$
,FG)!.sE> )) b,=F^P /3 V$ O8FGm9FGs98PN._ 1F^,=
)9)4Xfb ^ 93:] 33+F)FGW9F
'X> n F )7)+P9)^9
i
0 i<n
^b"de39`FD)9) )T )FX9+^ `$/>3 CX S^;<C) &F^FDb,=
`89>!lm>^F FGE` W `3>s9,aP /`>)F ),F/P7,=) eP esE,3 93`>F
5)aF^,=s9F^`>^)E>O)8F 39:)`>)>3 ) ,)/3y$39F+) ,39:W 3
sE9fHhP )^9T/-
`$/ sEh^$/3_/F
FnP9)vE
79fb&/ $/3= 3F^`>,9FGF^3 )^9 ): /3z/``>F^Fn ``>)^+'&).)7`89,)^>!BF
_/ G@o\1`:) 1<;,k/3 /``>F^F /F.)^ O) <F&<:!V G ).`$/ sE
/``>F^F^3 ?) F^/.,9PN^p'h /3 '&) e/FG),7) F ``F^F ) E _)^)>
'X9) )F893:]|F 2 f//,9)^ )a:]1F^F^
F/F8P FG)
/F8>f//,9)^
a[0]
a[10000]
de `)^/FG) m) )` -9!P m399_fb`F s / -)^/F^PN>^F"93 f/,FD)>D E:F
b,= 39>F 3F ) F^,=E^)_/3 /``FGF ) )a) W993 / )> F
^,9
E^)^9/ ) )F73>)'X )
)^9T9
de.P /`>) 93[/``FGFx./+^!39 `>) ">39,`>F )&/3 ``F^Fx.)^9k`89,)^>! BF
_/^ 7F^9`> / ^ /'hHF
1`>`,= F `),,F FG>)P!^5`>F &F^
,="RH
a
{
a[0]
a[1]
a[2]
a[n-1]
s Figure 6:
Ry:!:VQ/<+LM12!.3|/M01 N C.`LC:gP S]657B'1[ 'A<'I:!VJ F&/:!V< <'A<M
1[x0C:!w
<LM1q!. 'I/'I:!V/!"./-au57/:< 10.z657B'I<'A:!VI:< C 1":<)I .<:<
?./1"7N KFd+Z SXs657BC)J/:g~V+1[7NE0V L= s
g +/``>F^F ) >) Pr39 ]
_)^9TO^ = F^89Z`89,=))+'X)^ 393=>F^F^F
i
a
)>>PNXF^, 8`>F deP
Fn)^9X/33^F^F9 mF^)h 'X>^Z)&WFG)k>)nP
GFG)^)^F 193
p
a
FX) ,9.sE>XP4>^
`FZ)9)T8F^ f, P )^9 ,3>1 )o1Ew``,=9F s
) ) >) P993:] FG)^)^F ) ) G"`>1'&9FG7/33^F^FxF
F^ ,=h@1
i
p+si
2.6.4 Arrays are not self-describing
V&^$_)o1mF&O!:]H`>)9/E 6ZY Y g79 PN('X9 `>13!P)h,9FI)^)>Fk)F
int a [5] = {4 ,3 ,5 ,2 ,1}; // array of type int [5]
int b [5];
b = a;
// error : we cannot assign to an array
a
& av G v p+s
p
$;=
p + 2s
p + (n − 1)s
a[0]
a[1]
a[2]
a[n-1]
0
Figure 7:
1
n−1
2
p 57:!:/V|!D<'I/Mc S]1")J,s
i
.<2:g6.3+9657^Y)Y):<<.<.
p + si

+
/F^`$/)T)N "/ ^ Pp/)> O^b FX)F5V&^$1FT+83$/3
9/3aPJ) N/,9/"6 /3)"39>FâPxO^HFZ 6 FRePJ)1397 F
E)DPxfH:`
' ,9 )!)^fbZgh _/a'7 HF^FDFX)9) 657G=C';<<: Sj!D<'A<M6. S
:!:/V@1[.EX:/2LM:<!.Q/M)1" 657O:!:/V4% .EF+~* 6 'h/F 3F^3 ) `8E:) 'X )
_/`DN/,9/T 8`>9`: 93 )F73939M
)7$fb ^1JPN&,H]H,Gb4V&a^$_f/NOs F4> >F^>)3s )^Fk/33=FGF 13>W93/F4)&393=>F^FkP )^9Z^ `>='X>h)
WFG)4>
) )^9 hP 93:] FG)^)^F Z/F^TF )9)n/ ^$+Fx) .Q! S<{ )J!.Q<:!1 ;!1[7N 0
)h31Fh) 1O' )F7('Xa){
Z('Q'7 >).) ) /F^F^) F^FG,9 _)^9: F+ 'h PZ/,=)_)^`$/ `>1H9 /
) 'X ),
) H('X(' _/)^FZ)F^.O k,=)X'7
>)FZPx).^
a
b
P `,=F^ 3 )Fh_/, +fHN..F^89D 1F^`T'7 19(' )!O^
)F7P
).3`N)<F {$^?6 9^/>Fh'7>+) b:)!FGm3 , )
`89N/ OsE,=)
F^,`-7`fb>9`Fv
oCD)")^9:&9/
3 H)Fh$fbFZ/89X1cPNZs ,F
9>6hY Y*'h/F43>fbm3 ,`8NO)> &3F^ b/9'hF4)! fbX6yF "FG,s9F^:)$ VXF
.`F^ 1,` GHFZ!FI){^,93 -6hY"Y 2.6.5 Iteration over a container
q{>)!BF ) bT!FG):s /` HPN^>)hsE,=) )X)>` `$)>FnPr^$1F7PNh!) 1/3_.PN
s >7 `>),=
-9fb /$3|3`$)>3 )a)G13,`>)5)^ )F8F^`:)y)9) )a`F^F_P
):)_(fb> _F^ ,`8Pk39)_F",=s9 1,)^,9F> gn19`/ m)+39)_+FG)>3 ?FG
/M21"<: 9/3
'7D>3 ) E:PNJ `>^)/_E>O)PNZ/ )F )D`)/>$
dea>/ `)/>ZFh/as9uI>`>)7)9)h`$/aFG)^")^9:&s9uv`>)R
F e)F7)^F =/3)^ O)
lm:F&F^T'hHFZP /``FGF^ )F^">)Fkgh"9 G9^3 + ,9 )&>"FZ)^ O)
8`)/>D+,FG)Xl >T)!EF^FGs9)o
Pn )>)^ (fb>"/ )FX>)F>Dde )^9FXPN_/
F^F^ /a^ FZ93>3-+`>)/9: =F^` ) 93J/``>F^FXPN,`>)9/)o`$/sm!,F^3
) )>O)TOfb>Z)">)F
& ]av $;
qr><) BFX>)&s ` )_^$1FXd )^>)9 Ofb: / ^ P4)
Iteration by random access.
`$/
sE 39Ds1_93J/``>F^FhbT
F
$;+P Mx/A=x 9$fb F^>)9)7)n
/3 /``FGFDPN,9`>)^9/)o_P ^$1FXFZ):9/s /F^>3 393=>F^FD )>)`kSD,^
) ):)
=) PN('X9 F^ ,9>9`>+P4/33=FGF^FTFX`89,=)
3 p, p + s, p + 2s, . . . , p +
H'X>
93
9fb") ,9FG, r/F
(n − 1)s
p
s
ghF ^ 1,>F 9+,)9`$) /3 _/33) PN+/ /33=F^F ]=`>).)8WFG)$
k,=) P b, ) sE,=)!) )8+,)9`) `>F sE`/,F^8'7`89,=)8$/`
/33=F^F+P F^`>)`
k3>E3) P )_>fH,9F.F de P /`>) )
F/F^>) P
/33=F^FGF `,3h 8`>)^893^&9),=/ sEX`89,=)3 s18FI)^)9 'X)^
/3
p
>m$)3
393
G )+)":]1)&) +:
s
&F^!/39}/``>F^F H'7X`$ .1['jCD+ ^$ )>)
s9,)k'7DXF^F^T)XE> )
P G9") )X:]H) ) Y9 )FkE>)^8_ bFn)>)8(fb>h!`>)/9:
9),=/9/38 8`>)$4g79hPN('X9 /9/.,FG)^)F )hE) b, Pm`,=F^h$/3
+sm1 _s1 FG)G)+'&)
=)-`F^ )TsE1 E )&/b/ /F
1
2−3
`FG-) E-). /F
>)` k,=).,9>F^F bt, F^>'&9) ``>)^^` b,
4−5
s s9PN>Z)"uv,FG)h),=)T /Fh sE:)v'7>{
V&GHF{lm> 9),=//3! `) )>)^D)=, L=1"M<:g. xM ):
Iteration by pointers.
f//,F&`/asE!),)ZP /F&/`>),9/ /33=F^F^>F m/3a)>/O' E:)Fhb^/33 s
3>h) .).)T9 ]H)h>)7
)"O^b4%&> F7O' '7 `,3
1,f//>) '&)^
) )>O)
9>F" $;+P Mx/ A.'X)
E)>F
bool * begin = crossed_out;
// pointer to first element
bool * end = crossed_out + 1000; // past - the - end pointer
// in the loop , pointer p successively points to all elements
for ( bool * p = begin ; p != end ; ++ p)
* p = false ; // * p is the element pointed to by p
VX3)^)>39 ()^9F HF ^n`89`$)3!) WFI)xF^) )9/ )k/39 / ``>F^F fb>FG
s9,=) '7
B :]H9N8'X9)! F _3>)/ )D:]H)7F^>`>)F4de_)>F P ,^ @ 1'7
9fb">9N`3 )>)^
s93:]_'&) )>O)_s1 /33=F^F
2.6.6 Pointer types and functionality
=X_)o1E ) `^>FGE3
L=1"M<:30VgL=
F
` / ) ,93> H )o1m P $XV& ]HF^FG P )oE FD`$/3 L=1"M<: )
:
gh+f/,9+P E)>T) FT) /33=FGF!P /?s uv`>)"Pn)oE kE `/ )FT)
s9uI>`>) L71[M)? s1
) m):$
+`$/?f1F^,9/ .8E)^>
F" ^Ô' E)8) _`x ) `>8 ,=):!BFD_/
^
)X`{'&>")^9 s9uI>`>)7E)^3 )+s1
FI)^)F =F^ ,=!KH
& av G v Figure 8:
$;;
rL=1"M<:? :<2LM:<!.T<M6.E657 Y)Y):<!.<.? S? &; /)^ n93 )_`8 FG E>)^F
93
=
==
!=
_3>W93|PN / E)> )o1m $ g7_N)^):.F^89 )FG).'X>)^9: ) 393=>F^F^F+
1,FI)-^ ) F/TZ9)
de)N/ ) 93 /FGF^)Zà)!f//,z /F&,9FG, D 'X`-a)FX`/F^ $/F7)
` / /33^F^F< ),F P
E)FT)F^.s9uv`>) r) /FGF^)
9/F ) :lm`>)
j
i = j
)9)
9('~/FG
E)FX)_)FTs9uI>`>)$ g7+s9uv`>) )FGP4FT9) `9>3r" ._ )9)
i
E)>")N )^ /3 /F^FG)T , )^9 )vEF"P sE) E:/3F")
sE :]=/`>) )!F^/, 89`>)k`>fb>FGF&3M
)h'7 m deP b,a)^9 sm,)h) )^9FZFZ`$O$ de_/
)9)7)Tf^Ns9
FhP )o1m
93 )9)7b,-`,3 'Z)
i
int*
double * j = i
\1`
s9uI>`>)Fh,9FG, ,!^ >^`Fh)^ s uv`:)F '7,3a('
double
int
j
sE&TE)>4) s9uv`>)4)9)k9`>,39>FnG `Fk )4sET) double
i
g79FX`$/ 93
sE `$3 G`>fb>FG =Xde P /`>) F^` '7 `1-/ /33^F^F )^9:
`$/)7sE /9HF^`/{`fb>^F^Pr)TFG)3 f/,9 =>fbaP{)T^ 1,^)^F
P )^9 )o'7+)o1mF&9E
) sm )"F/
h`$/ s) DE)>x)"/ fb s9uv`>) s1 9H9D)7,9^
The address operator.
Y)Y):<!./.<L=/:/+: )8
f//,D)9)7PN>FZ) ) s uv`:)$4deP )" f/,9TF7P )oE =)
T
)TF^,)ZFZ/ ^f//, P )o1E
kg7"FG1)(]P /a/33=FGF&E>)h`$/{F
T*
dea)!PNO'& `>13!P)Z'7!,F^!)!/33^F^FXE:)h)8)^N/ T+f/Ns9
iptr
P )oE
'X ) )"/33=F^FhPx/as9uv`>)hP )o1m
9/3 int*
int
i
int i = 5;
int * iptr = & i ; // iptr initialized with the address of i
=^ E)^> X'7?`/*>)s /`)|) s uv`>) E)3 )^
The dereference operator.
)=, )J<:<[SQ<:</<1[7N 1")1[:<!012 ng79Z, O^ )J<:/0SQ<:/</G<L=<:+:
9>3 ) /
*
^f//,TPrm): )vEDH>3F7/ f/,9X^PN>^9.).)Ds uv`:)hm)>3
)=4deP{)D^f,DF
P )oE H)) F^,)hFhP )o1m k g7"FIH)O] Px+3>^PN>`!E>)^Z`/{F
& ]av $;A
=('X
,=-a,=h>fH,9Fh`39 P/>) H
'7 `$/)^9:PN '&)^
int i = 5;
int * iptr = & i ; // iptr initialized with the address of i
int j = * iptr ; // j == 5
g .9/ FG` PxE)>&)vEFDFD)^fO)3as1) 39:PN>>9`> E:)$Tg7
h
3`NO)
`$/a/F^.sE"^$/3O 93 P `>)X>b/8sm 'Z)^)>t'7"3M )Z3+)F )^9,h /F
g79"F^`3-fb>FGa89`>) 3>W99>F&) )oE!P
sF^$1 )9) F&P )o1m k g7F
p
F7)"F^/ H3P 89`)n3>W9)_)9)h'7!/ $/3= H('wP O^
39>`NO)F
,="U ,FG)^)^F7/33=F^F&/33>PN:` E>)$
m):P fb-/F7^f//,g
s9uv`>)Pefb-FZf//,g
Figure 9:
p 57\Y)Y):<!./.P<L=<:+:Ki2 0S/ oB)X1[6.1[=Fd<:g.Q<-4657\)J/:<0SQ/:<</]<L=<:+:Xi2:!1"N5Y o
=./ E)>!)oE8)>8F f//,83FG)9,FG Os +P^ /?)>
The null pointer.
E)>Tf//,.g79F f//,.F `$/>3 ) =7"*L=1[M/:EFd+Z +g79+)>Tf//,
`$?sE
0
`fb>^)3 )
/-E)>&)vE!gh f,./P)> `fb>F^ FX).1, m):Df/,9/!de
) 3`)
PND:]=/89 =)!f/NOs iptr >)FX)^N/ 3
'&))^9
int* iptr = 0
1, E)^> f//,x T/F^ F)9)
F =7[JL=1[M/: kg7 , E)>kf/,9D+,FG)
iptr
)Zsm"3>>PN>`
3 F^` )h39Fh9)Z`^^FGE3) /
:]HFG)9 /33=F^F>
&F^!)^9&1, m):kf//,&Fn)XF/PNX'h8PE93`)T)9)k)>^XF s9uI>`>9
) b:,)
)8E)&)^=&g7./)>^ O)fb P4$fH )^9"E)^>D,)N/ 3
FZs /
3 )>!FX8'h
Px)FG)^ 'X>)>D E)>&)^ O)DFX)T , E)>&3FZ)./33=F^F&Pn )_)
& av G v $;b0
s9uI>`>) &
' >)> ) 3F!F^ I/3 a/33=FGF!F^,)âPi$f1 ) m):
,)N/ {
3 de?) ^) )> `$F^ r39:PN>>9`>)+m):!,F^,9/ -`>/FG9>F") /-+67F^39:
)F7`39 int * iptr ;
// uninitialized pointer
int j = * iptr ; // trouble !
VXP): )F 3`) )^9hE)>
Fk,39:W 3 f/,9 '&9` /`:)`h$/Fx)9)
iptr
)D_ `^>FGE3 )^_/ ^s9)Ôâ393=>F^FD ^C 3>>PN>`
)X/F&)_/``>F^F
)T^ `)>)&)7)Fh/33^F^Fnde>/ H)F7/33=FGFh'&m)7sE ).) G)
Pr^+) 'X`
)^9X/Q9/F7/``>F^F<=)TE>)^ FGHFG)^Q'X) 39>
/``>F^F
) )Z/3)> O)&)T9^/c'X) .QqN'A<M2+012?S 7~ 2.6.7 Array-to-pointer conversion
VX ^ Pn)o1m `$/?8 `) sm+`>fb>^)^3 )_)vE $.g7+^F^,)f//,.F
[n]
)"/33^F^F&Pr) W FI)&)hPr) ^$b=Z:]/89 1'7"`/a'& )
int a [5];
int * begin = a ; // begin points to a [0]
gh"39>`NO)
int * begin = & a [0]; // address of the first element
F ,f) /F P /F )8FG,9 )af,P
F ``>3 s9,) )^9:F.aF^,=s)
begin
3lm:` )")^)>h3`) >f//,9)^F
='XD) PN>h31Fh)$
a[0]
gh&m): FG)o1h^>9N/`>)n`13XPNn)X188F& P{Mx/ A!)9)n'7
9fb F^)3 )") 3 Ph\1`>) 1<RHL0__ bF",9FG P7^$ e) E)>"`fb>^F^ )TW9^FG)& bool * begin = crossed_out;
// pointer to first element
g ^ ) em):"`>fb>FG FT ,=> ``>),9/0 ?) _/` F^3 )
h
9EF =")F r'7.`$ P ,="$> 3F^`,F^F^ \1>`>) 1BRHBA_)9)" 6 /3
)>>PN /F^ 6ZY Y -^ GF F^89 )!/33=FGF&P )FhWFI)&)
V&^ e) E)^>!`>fb>FG /,=)_)^`$/ ) bF"9N/`>+'&9> / ^$ EF! /
:]19^F^F^{ uI^9\)^,FG)G,= )_3F^>+P+6hY"Y k,FG)^)^F!)F s1 F^$1 )^ O)
6 57'I S :!:/V /=F&<:g6. L=1[M/:@1[6. :g.<< <'A<MA+^657 .<Z1"N5Y!.<
LM:<F&Q+012 kdeFG)^ )^9:h'7^39F 1)>"D+E>)^F aGHF >fb>G1).)9) '7
``>),9/ 3!'&)_/
G8F 8$/ )v 3&'X)^"E)>!
)Fk_ G)`,Nn9>F
)_) /3 ``F^F"E>O)mF^.). O/9 `$/3
)^,=)asE,=)7/3 ``F^F 8 )^9 :]H)&F^`>)^{
/
*
2(
%7.#*E M7
5
R*
2-.6#6 %B>TU7T 1#* R*
$&
% @*
(7B
-*5
9*
(
(( 8 97
7:"
6
^M
5 B.*6
):"FG,s9F^`:) E
T)
%2*
/322*
'(9(%B R(*
.6#*"
R
*J
N
-.6>*
/@7J
& ]av $;R
2.6.8 Pointer arithmetic
dea3>h) ,93>FI)/3'X_)"`3"P)
bool * begin = crossed_out;
// pointer to first element
bool * end = crossed_out + 1000; // past - the - end pointer
// in the loop , pointer p successively points to all elements
for ( bool * p = begin ; p != end ; ++ p)
* p = false ; // * p is the element pointed to by p
33+F^>)F 1)^F P9)^97O^
) '7 9fb )T,93>FI)/3 L=1[M<:
crossed_out
:!1065&'A!012 )a^) PD`89,=)-'X)^5/33=F^FGF 39>sE>) `/ )F8 ^^) F^`.E)>" )>)` `>FD'X)?
)TPk9)P /F ms9,)T'&),=) FPN>)o?>)$ CD?)
)> 9/3 ) /,=)F.PN7)9).)>F F^ `:^)/ sE$/,=)o ) _F^pP
E)>.O)>)`d <) BF b83=fH9a/ 3)^> !) BF ,93x)<BF!3 `,) )-FI)> 4F^$O)F
",9`>PN^)s9 H/3)>
F& $O)= k,=)h)!` '&) ) ngh!3)^> HFhF^
+,9àsE:)^)>X)^ 13>a`$ h>fb>^)F^F EP)>T3=f1 ) 3=)>hPN +'X )
'XHs s9.),=,=)k)9)4sE$/,=)o Fk) >9,
/38)9)kF^/PN>)o/3,Fs9)o+&^
8EG)/) P /`>)Fh)^9 .,r
gh s O^T/33)Dm>)^F O 39:W 3!PN P)rE>/3F
Adding integers to pointers.
+ Ph/?E)^>!)o1m -93 ) m>/3F Ph/?)>/n)oE h`$/n)9) Ph/ F.9Ôf1335/F+) P)+E>/3k )+'& 8 `) sm `fb>^)>35) -E)>+,F^
^$ e) E)>h`fb:F^{
= )^9 sE> f1TP + )sE.3>W99>3 m)^9: +,FG)XsE+/?^ P4F^ ) n EF^,`
)9)h)!>P)XE>/3
.FX.E)^>Z)^ )!)ZP F^"3:] k, 0 k n )
^$b4g7D`$/F^
F /O'73_93
`^^FGE3F ) )DF^)^, O)8'X>^ !F !E)^>
k=n
/FG)")8N/FG)">)!Pk)^$ '7 `/ )F L7+.<6{2657<{<)L=1[M/: ) )^ O)
F^,` .m):Z+,FG)h)hsm!3>PN:`h
3 :
deP ) F^`3-E>/3
.9/FhF^Tf, FG,9`)^ O)
=)
i
0 k+i n
XY F.E)>!)a)
e)^ )!Ph)^9F^/^$b dePN_ '7:)+E)^>
(k + i)
)9)T9/FDsE> (fb3 >)FZ)
)^9 )? '&`?/`:),9/ $/FX)).P)TP F
i
i
b) fbg : g79:PN P
F )^9Df//,DP

/ /33=F^F H)_)Xf/,9TP
p
ptr
ptr + expr
F4)D/33=F^F
/F^F^,T)9) / f//,XPm)X,3>1")o1mX1``>,9>F ^
p + si
s
`F4ghX9$F^ P /`>)hF )9) '7D3M
)79fbD).`$ sm,) H) m>)
s
ptr + expr
'&9` HO'&F s P})^9h)o1EXP ptr 43Fk)FnPN ,F /3lm:F7 )o1m 3>E>93)k'h
P (fH9+ E)> >)F )+) ^)$
i
VXF&sE>PN P
'7!>)T+ /FI) e) e3-E)^>$ jn/,FhP i FG,9` )9) k + i F
k+i = n
)Zsm>)o'7
/3
/3 ) ,93>W93sm9$f1$
0
n
qr:)&,FZ^>E$)h)TE)7)9)7'7" fb"_/3XsEPN"a``>) 'X ) 93J/``>F^F
a\1`>) 1<RH<; xs3P /,) H)> O!Os FG,=)_ `` HFZ)9)7) sE(fbT 1,^)^F
& av G v $;b@
33 3 /3 ) F ) b,=8FGm9FGs9)o )?_ bF^,= )9) )F+F+)a`$/F^/
/,= ) 39 F^ 'XZ>F^,)_ / `>/F^F ZFG)^/asE> f1_PD) / s s9y) '7^FG)aF^`>9+_F^y_Tsm9$f1 Xs9,=)'&) )^9 E))N/TP
),=+)+FG)^/ sE> f1Z)X_) 1&-
)>Z_/`99/
gh>^PN 9>)Z,FZF^,_^ &)T ,9 )F79`>! /3
+,FG)EE){)h)^9 )rP3:]
.
,FG)h9fb!F^Df//,
k9^
F^,=s)^/`:)
FZFGN
− k
F^,` a)^ O)
i
.
deP
F^4^ P)
0
k+i
Fhf//,
i
n
n
O'X>
0
k
n
F^,à)9)
0
k−i
n
9)
H3Fh.E)>7)+)!O^
)hP 3:]
k−i
/3
P1)4)o'7ZE>)^F `$!sEk,F^>3 'X)"P) E>93F
ghk/F^F^){fb>F^F
+=
-=
Pm):r)oEkF{'7> ('X)T) ,F^,9/b$/= \1N^ :)4, O^T`>>) /3!3`>^ )xE>O)F
/3
7f//NOs nPN E)>F>x\1` `>3`Fn/3 F^F^1`>N)f1)>F
++
-T)>3).)"E:)ZFGH.sEF 1)> "/F&ag s9+ /"AbKH
Z('w'7 `$/,93>FI)/3) F^`3-DP )^9"sE(fb `13!P/) bool * end = crossed_out + 1000; // pointer after last element
FI) ) ^ crossed_out F"`fb>^)>3 )â
m): )â)FTWFG)")B ) P 3:] : \1` )^9!^ 9/F
)^F /33+)")>>
13FZ. /GF ) 0
1, 000
1, 000
) e3E)>
PNZ)"^bngh F^,=s9F^ 1,)X
end
for ( bool * p = begin ; p != end ; ++ p)
* p = false ; // * p is the element pointed to by p
F8`>$ 9(' F '7 FG)^)^?'X)^y E
)>
)^ )WFG))^
)
p
p = begin
>)ZE)^3a)FXFG>)D)
:Xgh-'7 `>>)

FG8) 9)D)ZE)^F
*p = false
p
) )D:]H)7>
)
:x D>m$) )F7F
./F
F 39 lm>>)7P )X /FG) e) e3
++p
p
E)>h9/3
end
9$fb /^$/3= 3F^`,F^F^>3?)+^N) E>O)F
/3
)9)
Pointer comparison.
==
!=
F^89)^FG)X'&9:)>&) )v'78m):FZ 1,FG)aE)h) )^9!F/ s uv`>) k,=)Z'7 `$/
/FG `8 ^T)o'7+E)^>F7,F^ ) E>O)F 9/3 >= nVXbH9^`3`F&/3
< <= >
/F^FG1`NO)fH )F P /m)9/mm>)^F&^!/Fhg s9 + /"RUH
= )&FG,9 ) )!sE&FGE` W >3 )>X+,FG)ksmD/
^ PEF^&>9)^ n F^,`
)9)n)
P)&E>93
g_F& E)>h) ) >)&PxF^ 3:] k , 0 k n ) ^$ 1
/3)DF^`3 E>/3 F !E)> ) )^9D>) P{F^
X3:1]
k2 , 0 k 2 n
)kF^/kO^b VXb/ /3
('73 /3 `GFGE3 )Z)n /FG) ) >93
k1 = n
k2 = n
`$/FG
& ]av $;K
fb> )F )^9+F^,)"P ) m):!`8 ^F^?F!3>)>^3 s ) )>> ` FGaP
/3
7de-)>&'7^39FP /3 1,)^!),)fb J 1)^9 E)>h) )!>)
k1
k2
)9)h`F7WFG)Z )^9"^$
F7) F^_/:79/
de|,=.`39_P) )^9`>8 OF^
`>,93 ,f)^?sE: /`3 s1
p != end
)D:]19^F^F^
'X`H39
F "/F ./F
E)Fk)+/
/`>)^, E^>) p < end
p
,9 f/) /FZ9 /F
F7)X /FG) ) >93 E)^>$
p
678 ^Z)o'7XE)>F )^ O)x3D9) >) ) OsEOfb ^ 1,>)F $/3F )^ ,FGE>`W93
9/3 >= FG,9 )FZ ) PN,&E>O)F < <= >
g79:8F 8^ ^)>)`.E:)? E)^>F_VXF^FG,9+)9)
)+) P F^.3:]
F^+G?Pn)
k1 , 0 k 1
n
E/3-) F^`3 E>93 FT E)>&)8) >)XP F^"3:] k , 0 k
n
|)aF^/O^H
/FI) e) e35E)>F /('7h
3 : g75)FG,9 )8PX)2 L71[M2<:
n
Pointer
aF"subtraction.
_m): )^
./t;g0:/J!01q
− F&)^9!)>>
Xg71,F E)>&F^,=s)^/`>)^)F&,F GO' P TO O^) ) )o'7G
k −k
>)FZ/ g719 sE29fHZP E)>ZF^,=s)^/`:)F&,39:W 3P /3
")
E)>F ) >)F7O h /FI) e) e3aE)>F7P2 ) F/ ^$b
M )>DF^,=s)^`>)|
'&`-89O1FZ)!s99GF^,=s)^`>)-E:) 9F^.g s98 /TAbK+PNZ)F FIE`W9`F 3FZ)7``,=&
)T`39 P/>)7PQ)DsE P )^9F
F^`:){&V[)o `$/r,9FG F&)3>): )^9!1,+sm>XP >)FZ / ^$ )9)DFZfb
s1 E)>7) )FhWFI)&)h93a. /FG) e) e3E)>$
de_$/)o )DF^,=s9F^`>^) E>O)
Pointer subscripting, or the truth about random access.
[]
/F )^3,9`>3
\1`>)^ 1<RHB;!31F )kE>)^X_^1F 1s9,=) E)>F defb H!)^9F
E>)7 /aG`FG)^,`>)^F&/
:]19^F^F^/3
)^9:PNT)^:F7/ ^ ) em):
`fb>F^{
fb>-.m): +93/a:]1F^F^
.P )/E)vE =) :]HFGF^
Fh 1
,f//>)R /FG )F ,9 )F7
+ &Y
,
./3
)
ed P
9/F f//, H)TN)^): :]H>F^F^
H39Fn)"O^>) 9N/`F )^+)X)7P
i
i
)49 E)3T)7s1 $ de O^)`,NO (P kF^,)FrPw/ O^ ) eE)>{`>fb>FG
)Fn/>F4'&) )^9hF^_)`>F4P /3t``F^FnPN GHFn/Fk)^39,`>3 _\1>`>) 1BRH<;H
g s9TA FG,9_O F )^9 >' E)> eFGE` W `Ds99^_m>)^F
& av G v Description
FG,s9F^`:)
3>^PN>`
393=>F^F
Table 4:
$;U
Operator
[]
*
&
Arity
Prec. Assoc.
(@
P)
$R
)
$R
)
_v:</,)J</!.W)3+.<.QQ<1q+01"F
1001q!.R S4L=1"M<:X<L=<:/+:g.T p 57.*;g.Q/:!1Lt<L=/:/+:
,sgL7/!6.9:!Fd+Z <.R+.`<L=<:/)+.P)c:<!0C:!M.R ~Fd+Z T p 57R)J<:<0ST<:<<X<L=<:/+:
,sgL7/!6.3 :!F+~*G)^:<!0h:!M.G ZFd+~*<-fuv5&10D]657Y)Y):/!.<.z<L=<:+:j,sgL7/!6.
@ZFd+Z G)E:<<0C:!M. :!Fd+~*Q
\1 P ).F^F )9) )^9_ ,F^
P7E)^>F.F )
What have we gained with pointers?
_ ba )>)^5)=,/ O^y? )^) a `))9/y)>O)5s1539 ]E k,=)
,F^F4'7XZ )h$/~PE:]1)^ ) e`>)`$/H9F )&FfH9FkZ_9/e =
)&F^ bXPm$3 Os )o '7&)>>PNXP)FI)9,F^X):)+s1 3:]E4\1 '& O)kF4)&$
uv,FG)W9`$)^ PNh)TE)^>h``>) gh>^ /`>),9/ -)v'7+uv,9FI)W9`$)9F 93 Pk) '& sE 3F^`,F^F^>3 )!'h
)D:]H) F^`>) xE)>^F D3FGmFs9hPN7>)^) I/`:)`$/ ^$1F 'X)^)
) 1O'&a)Z`89X)
gh F^`3-uv,9FI)W9`$) F")"b>) ,3?) `^9: Fâ'7+'Xx9 s )^,9`
)Z>7V&^$1FD s1P X)&)^9"`>)/9:FZPNXF^>)FXPx39)1k-'7!89>)
39).9^1`FGF^/^)F 1'7 FG9,3a)^9:PN _ b!F^,=!)^ O)&)> '7 PN
^$1F
=T:]/89 W93 `)/>&>)&'&) fb-9Ê>^)oO
Of1 )^9)>X)^ O)
9NHF b,= P $fb^)(fHg F^,93 sEEFGF^s9PN =V `)/>^F.)^ O).lm>F )_PN,` )9/)o P )^>)9 Ofb>T )FD>)F&gh.a,PN )v_'7 3 FX )^9 )>)^
1`>F^F& )F^P VX53 O) e`F^FG / ) PD) 6hY"YFG)/393ysO^
'7'XZF^>-FâP
) N)> .'7 1F 5)F+'h -) :]Hm`>)F8) ,93> H?`>)/9: ) l > 1[<:/+:g.
`PN
) F^ '7 3>W93 ):) `>9`>>)$-g79FIE`W9`F P )^9`>)/9: )F^>P
^^>f//)ZPNh)!/)-
%&>.FX'&> E)>F&` h)^9: ).):)F&lm>>3 s ^HF>"g7>PN^ >fb Pr'7!3M
)h,F^!E)>^FZ ,=X('Xa`3 9'7 9fb") 1O' sE,=)h)Ja3>h)
sE"s9D)8O99_FI)/393a s9^_/)Fk) ^$1F
2.6.9 Dynamic memory allocation
q >)4,9F4Ts ` .)"Mx/JA 9(' d )Fn_ 3='&s /`+Fx)9)x)Z,+sE:
{
F 93='X^3
n
/F
)F / uv,FG)4sE`$/,F^Z)Z>9)^8PE/O^ 9/F )!sm19('X_O)k`89
1, 000
)/
V )!/FG)T )FDFGE>`>) ^HF".)_FGE`>N/ m),{ VX )oEFT)9)T'7 9fb
>) :`

=/3 bool k9$fbD)XE:^)o)9)h F^Xs9uv`>)7P{)
int unsigned int
)o1EZ1``>,9>F W=]H3_/,)4PmG H('X8)!)^9Z`89>
PN :]=/89 ; Ts9)FnPN
& ]av Ab#
/
s uv`>)"?_-9N)PN^F :!)?^1F ?s1f1,FT9>3 FGFD)` `,+fb>)
int
)F FI)^`>)^{
de*6hY"Y T^$1F'XF^ ) F3>)>9>3 )_,)a`$ sE s)/3y)=,
)V+'j12'I/'I:!VH+[D
,+012 gh=, F^,` 5/ /1`$)^ '7 `:$) / s uv`>)+'X )
)V+'j12 FG)"3,=)r
CDs9uv`>)F )9) '79fb_F^|F^ P .'7>
/4)3 )-fONs9F x '&` `$F^_^
>)Fa/F^F^3 )^|) e/3*FP>3w/b/h
_)3>)^>3E)^F3,= :]H`,=)y
/,=)_)` /3 FG))`FG)/_3,=) 4\1>`>) 1BA=<
; CXs uv`:)F Ph3=H9/`
FG)/_39,=O) 9).)3 ) f/Ns9>F k93 )> _ GFG)G)+)^ fb e>) ^
/F^FG3 ) )
/3 G3 e>))^9>^ P>h
3 _O) =V E)83,= :]H`,=){ g79Z9^/> `$39:)>7)F^hE)^FnfH
/3
:]1F^F^9F>
new
delete
ghX}9/F F^P
)o19`$
1,)^XN! Pm)X`89,=)>! Fk_/ ^
fNs9 )^ FI)_3=H9/`$ /`$)>3 s uv`>)^F g7F F `$3 ) 57!L ?d ).F
)/_,1,F^>3 ms9,=)X'&9>?/ s9uv`>)TFD3=H9/`$/
1`)
3 9)DF&smFG)>3 )
$ F^ )^ O)h)^9 ^/`>),9/ ,F^3as1_)T/c('XF
%&>.FT(' )FX'7 1F PN"2x)FI)<F r\1>fb/ Z>+sE:&)9)D'7 'h/) )^9 FG)TP
.1,.sE>F sE>)o'7
/3
_g78PN('Xf/N/)!/3F") ,+sE:
P
2
n−1
n
FG)939^3 = ,=)!/3 3=H9/`$ /1`$O)F"/ ^ P )
gh ^_/3> P )
n
/ F /F!smPN :]H`>) )9) '7:]19`)^ fb8)-P^)319/`$/ /1`$)^3
FG)/" ) 3r
Á
Ã
Â
Á
Ã
Á©Â
Á
Á©Á
Á
// Program : eratosthenes2.C
// Calculate prime numbers in {2 ,... ,n -1} using
// Eratosthenes ’ sieve .
# include < iostream >
int main ()
{
// input
std :: cout < < " Compute prime numbers in {2 ,... ,n -1} for n =? ";
unsigned int n;
std :: cin > > n;
// definition and initialization : provides us with
// Booleans crossed_out [0] ,... , crossed_out[n -1]
bool * crossed_out = new bool [ n ];
// dynamic allocation
for ( unsigned int i = 0; i < n ; ++ i)
crossed_out[i ] = false ;
// computation and output
std :: cout < < " Prime numbers in {2 ,... ," < < n -1 < < " }:\ n ";
for ( unsigned int i = 2; i < n ; ++ i)
if (! crossed_out[i ]) {
& av G v Á
©Á Ã
Á
Á
Á
Á
A
// i is prime
std :: cout < < i < < " ";
// cross out all proper multiples of i
for ( unsigned int m = 2* i ; m < n ; m += i )
crossed_out[ m ] = true ;
Â
}
std :: cout < < "\ n";
Á
Ã
delete [] crossed_out ;
// free dynamic memory
return 0;
}
Program 15:
LM:</N. 4<:/+.<657<<. 7
Z) )9)Z)^9"f^Ns9
FhO'[+m):ZO)>&)9/ -^Cm/P):&)
crossed_out
3`NO)H )hm)F )+) WFI)&
)7P +3=H9/`$/ /1`$)^3O^
P )
new
n
The new expression.
)=Tf/N/)F
new
new
new
=
/ )vEm
new
:]H>F^F^ `$/ `" /P ) PN('X
^
de /7`$F^F k)^9
:]19^F^F^5^>),=F |^f//, Ph)o1E $ d )F.f/,9
F )a393=>F^F
PZ/ s9uv`>)+P )o1E ~)9).9/F sE>|3=H9/`/ /1`$)^3 )$r?ghs9uI>`>)
)FGPkF!/H,F Es9,)"'7 ,F^,9/ FG)+) >F^,) /33^F^F!,3> f/Ns9+9/
de-Mx/(0 ='7 `$/{)
crossed_out
de5) WFG)8/35F^`>93 f/N) k)^9 >lm>`>)8PX)
]HF^FG F.) 3=H9/`/
new
/1`$)^7 ./1"7ND s9uv`>)nP )o1E
)h$r jnN/)ZZ$fbF4)Zs9uv`>)n,99 )N/ 3 P
F &PN,39/)/)oE 'Xnf/TN/) &)N F{) >' s9uv`>) '&) 'X9):fb> E$OF
- )F^F> =D ]/89 9) PN('X9 3`N)F&9 )N/ D) f^Ns9F
/3 i
j
sE) P )o1E
='&) )^9"/33=F^FGF&Pr)v'7 >'ws uv`>)^FZP )oE
int*
int
int * i = new int ;
int * j = new int (6);
// * i is undefined
// * j is 6
Z) O' 1Pm'7X'h)3)o'7 ^F ,`s9uv`>)F PE)o1m
'7
B3_)> ,9FGXf^Ns9Fk'&)
int
/,=)_)^`XFG)/"3,=)^93 '& )^
int i ;
int j = 6;
// i is undefined
// j is 6
X)>>FG) PN ,F F )Z)^3 fON/)$xdeP "9/Fn)^>kf//, n 0 b)&>l `>)
P )
:]1F^F^ F )^-319/`$/ /`$) / ^ P7>9)^
'X ) ,9
3> H
new
n
& ]av A
)o1Ew ) $rTg7 >),= f, F&)./33=F^FDPx) W FG)T>)$Xg7FDF&'& O)
'7 F^!a.$R P M /(01
VXFh,9FG, H)
O^
)F _/ ,)N 3P &
F .PN,3 )m)vE
n
SXH9/`$ /1`$)^3 ^ )9).F >.33 F^,93
The delete expression.
sE"P3{ de 6hY"Y )^9T/:739>`3FXO)Z'&9` E)7)FhFh)"`$F^ SXH9/`
FG)/.39,=O) 89Fh)9)T3=H9/`$/`$)3 s9uv`>)FTfb.,)^ ) / )> 9)>F b,9>F^Fk)>8X:]19`)^.P3{nSXH9/`$ 1`)38>^.F 9:]= s )9/ FG))`"G s9,=)Z>),=)Z/F^+fb fbFZFG"399FG)^)^fb&>l ^)$
gh
:]1F^F^9Fh) b `$"P P>8^b gh>
`T)o'7.fON/)F
delete
delete
delete
de m
s ) f/N)F __$sE+
,9 E)^> E 'X` `/F^ )^9
:]H>F^F^ 9/F
delete
> l `>)$
CD)^9:^'XFG )ZWFI)nf/N) T+,FG) sEZTE)> )! FG7s uv`>)4)9)49/Fn fH,9FG sE>83=H9/`$"/`$)>3.'&).)hWFI)n4F^>`3 fON/)nP)
:]H>F^F^{
new
g79&>lm`:)7Fk) _ bh)X`^^FGE3 ^+f//NOs &/bPN F^,=s9F^ ,9>)73=H9/`
/1`$)^F ) 9r
=X:]=/89 =)Z E)Z )T9^/Q'X> ) )o'7 int s uv`>)^FZ3=H9/`$/ / `$)^3)=,
int * i = new int ;
int * j = new int (6);
// * i is undefined
// * j is 6
>h33'7 '7,3
'& )
delete j;
delete i;
gh73>4P939>>).31>F4)4_)^)^> > s9,=) _/!9^/>Fr`F^3>4) `$/
)
E)>^F&-) fb>F^ ^39:TPx319/`"/`$)kdeP b, 3 ),3 )o'7
delete
FG): F b,aWFI)&,3+) F^`3-FG)^>r
de?)+F^`3 f/N/) Pk)
:]19^F^F^
_+,9FI) sm E)>T)_)+WFG)
delete
>) P/+^!)^ O) 9/F >f1,F^ sE.3=H9/`/T1`)3!'&) )k)^9 3 fON/)
P )
]HF^FG{ng7Z'&9ZG.1``,=93 s1 )^9&^ F ,=)ns ` 8)&$O
new
PNh,F^ ghFh9EF7a ; PxM ^/(01
deP )"9N
FX9>3 ) + ,9rE)>7)9)Z31FZ)ZE)h)8+319/ delete
`$ /`$)>3 F^ s9uv`>) )^98sm9$f1 F ,93>W93{-g7F/8F")^,8P7 )^F
"
%
%"L7B
*
,SB>#
((
-..F,S$
0J
% 7.#*K
A75."5
"
:
9
82*
CD ,& 5"K(7@
8"K*S
2
/
A
( Q."5
#6%"@"@.*6E2*6#'6
7
/B57
*
*SBK(
#*
A2
N %*
.*
%"5
GO7
(
5(. /
O, :
I
%*R6(7'6 9#. 7.#*C 7
6
7 %*
R@/B*'"@..BK.I(( /
6
"
7
A6#2"
J
"K(7..B
%< < 1
J
.*
"L2
72
OJ
(7B
6 *
*
/
-72 /
#"
*S
%..BE"L K
7( /L."5
"
@'..B
/ % %
+@
%$&
/
7T
%*(*0J
% 8$&7
*
& av G v Ab;
)
/?O^a'X>^ )> FDa8F^ s9uv`>)$DVXF / 'h$1F&'X )aE)>F
delete[]
6hY"Y N/,9/T31Fh)&l >&/
$/F7P 3>)>`>)+F^,` -:^F
)
VX),+/=^ /`$)>3+s X~Fn,)^_)`$ P3 'X
Memory leaks.
)! )> O)Fh9^_/ =)XF&fb>^s 3 `>)`")8a )^9FXP /`>)DPNTP^
3=H9/`/ 1`)3 ^b5dePT / 39F8)8 ]H9` ) Pa/h319/`$/
/1`$)^3 ^ )FF^/3y) fb 'I/'I:!V D+ [\1,` $ 1F_-P)> F^P
s /3`>13=4gh>
,F^,9/8 fb"9+3NO)Z`>9FG 1,`F s9,=)h'&),=)7P9 ,1,F^3
FG)/ 9^/Q,99+PN&+.)W
)^9 _P E>O).FGHFG)^ ,=)9>F _$
O)
F^DE)hF^89:]H9/,FG)Z)^9"fNs9D$aFG)//
gh>^PN '7"9fb")^9 PN('X9.,39/
Dynamic Storage Guideline:
>' /33:)" ]HF^FGF&F^,3a'hHFh`T_)^` /F:
2.6.10 Arrays of characters
\1 ,9>9`>FXP `9`>)>FZ`F^>3aa3,=s9 1 ,)FhbT
std :: cout < < " Prime numbers in {2 ,... ,999}:\ n";
$` /3 ./0:!1[7N?Z10/:/+. \1_P D'7 9$fb ,F^>3 FG)G):/Fh'&)a,=)^9,=)D:]19^F^F^F ms9,=)X'7 `$ '7 'X )!) )> `):]1)Fx/F '7>eFI)x)s9 &FG)^9Z):/`$/ sE ,F^>3.)^D)N/ / ^ P !5C:J!<:g. ?679/`>)>^F )8s9,3 s9` 1F P ):]H)+F '7 HO' )$-de
6hY"Y 9)^9:O"33_s1
)"PN,39/)/m)o1E
)9)7'7"s 9_3FG`,F^FZ:]1)$
char
g7 PN,39/)/k)o1m
^>F^)F `9`>)>F>|6 9/`:)>F.`,3
The type char.
char
) <6<:g.
)=,
e/a'X)^ )^9>.`$9)nfb>FGF

)=,
) )1 N106. 0
a
z
A
Z
)=, Fh'7 FZ,9:,Fh)^9: .qL7/<1 +<5C:/J!<:g. b 1ng79 9
char c = ’a ’;
3>W9FkXf/NOs P )o1m
/3+f/,9
:9^F^)")7>)G)> a ghh:]H>F^F^
c
char
’a’
F h):/bP1)o1E
xg71,)>Fx,3!) /`:),9/`9`>)> FGH.sE `>F^FG
’a’
char
3>&)+3FG)9,FG
) ):/
Pc)"39>)W9> ’a’
=_/ 9)+)oE char F / )^/ )o1m T) F )a.F/.E:)F /FT).)vEF
=/3) 6ZY YwFI)/393_>fb_EFG),NO)F ")^ P char )
int
unsigned int
d ) F FGE` W >3 ),9
r)a'X` )>")8` O/`>):
int
unsigned int
’a’
F 'X sE")>3r &3>Z) '&3_,F^3
`13K
>^`$/ )939^3
13"PN
ASCII
A
S
C
PN_O) )^>`9/g )hF7)")>
I
I
97
*.P Q2
.
7PS..B#,$
8$*
2
&*
H. % ,
/
7
7.*
6A. % ( 22
-.* %
7.*6 N 2 J
,2
'-BI.*
6A8"L/+BN
%"
*
/ *
/
& ]av AA
ghF FG>)^) _8)7F^>Qfb>^_,F^PN, =93
33
) _ b>F )^)^&F^>9FGT)^+3f13X
`9/`>)^> s1 /)>$ CD )^97)>49/3'7h`/ PN4:]=/89k)4)^9h/ 9sm>) )=,
"FG89D1I /F^FG,99
`399+ 2x]H`,=)aP )
ASCII
for
for ( char c = ’a ’; c <= ’z ’ ; ++ c)
std :: cout < < c;
'&^)F ) ` 9/`>)>&F^ ,9>9`>
abcdefghijklmnopqrstuvwxyz
) FG)/393 ,)G ,=)$nTfb)F H
b,a_) )9)Z)^9 std :: cout < < ’a ’ + 1;
)F
s9,=) ).39F^M ) \1`)_E>93F Ph)
`>8EF^) :]H>F^F^ ’a’+1 ^
’b’
P439 lm>>)D)o1E>F E) P) E>93 P )vE
'X /,=)_O)`$/sE )3-))
char
Z>/ )oE
PE)X) m>/3{4gh>>PN 1)X)o1mDPm)D:]HFGF^
int
’a’+1
F
E/3 )F&f//, F

F^F^,
`399+,E/3 )^ O)!BF&'&9)D>)FD)3{TdeP
int
98
ASCII
b,'h/)
)^ sET)^
3 Hb,a ,FG)h,F^ )":]19`)7`>fb>FG
’b’
char(’a’+1)
ghn`$)G!PHFIE`N`9`>)>F /F^Z`,3F /M0:<M!5C:/J!/:g. )9) )J F^:)
'X)^3{[g7F^ -'& )^) 'X ) $/3 s /` HF^NF^
X/3y) FI)8E^)/)
`)^{` O/`>):XPNZ,9FZF
='X` `$/,F^F&+9Ds$ m
’\n’
CT
FG)49N)PNF 1
f,X1``>,9>F
s9)FkPE^7
b'X>)^9:k)&f//,&9
char
8
`^^FGE3 )^ )TF^>)ZP )>^F
) F^3
`$/F^! 7h)TF^>) {0, . . . , 255}
. . . , 127}
) ,F^3 `$/F^! DFX89){−128,
) 3>W93{ \19`>./ VT\=67dGd `9/`:)>FT9fb+):
f//,Fh
H)^9: `/sET>>F^)^3 sE)^`$/FGF
{0, . . . , 127}
V ) ]H) FnF^89+!FG 1,`DPE`9/`:)>F 93_`$8sEX1338
From characters to text.
6hY"Y[)=,a-^_'X ),3>1.)oE
=Z:]/89 1) "3`>N)^
char
char text [] = { ’b ’ , ’o ’ , ’o ’ , ’l ’}
3>W9F&-^
P >9)^
) 9)7>F^>)F&)^9T):]1)
4
VX):9)fb ? /3^ >` fb)Y ='7"`$'& )
char text [] = " bool "
g 9F HO'7:fb> =F 1,f//>) ) )TPN> 3`>N)^{ 9>a/ ^_Pr` O/`>):F
7
F7)/ >3 'X)^ FG)G.)>/ )^9D):9) <:<A!5C:/J!<:
P )> f//,
’\0’
TF!,)^_)`$E33 ) )^9 ^bg7F!`9`>)>!39F!)"`^>FGE3 )-/
0
)s9!`9`>)>$&VXP)>X)^9 )^)>&3`N)) ^
)>PN^+9/F&) text
5
g79
WFG) PN, >)F /3
/3|)
W P)^5>) F+)^9 >^
’b’ ’o’ ’o’
’l’
`9/`>)^>
’\0’
7`$/=F^,` / ^ <:/{2<:!'^1[+) & bk9^_/HGHF >^ e)>^9)3+^$1F
H(' )"){8g >) )F") r'7 F^89-9fb ))>O) (fb>")8^$ /3
`,)h) 1,+sm>hP >)F sEPN^ ) )>9)^
’\0’
%&> FD/ `$)
P ^$1FDP h` 9/`>)>^F 8M0:!1[7N'A+<5&1[7N FX)^9"9^s P
W9399")XW9^FG) h/ 1``>,G`F Pr fb_F^`FG)Gz
e,F^,9/ F^^) n
fb):]1)
e,F^,9/ + :
& av G v A0
gh s1f1,F F^,=)_F )TPN('X xF^F^, )9) )TF^$^` FG)^.9/F7)
m
'7+`8 .) ` O/`>):^'XFG.'&) ) )^F
Pn).):]1)$ dePk
F^_)`-F
1, 2, ..., m
PN,3 PNXF^ >) ='7 FI) /3 :]1)D`8 ) F^$O` FG)G 'X )) )^F
P)7):]H) /3 F^ {4\1>)F4P
`>9FG`,=)fbh>)F
2, 3, . . . , m + 1
m
i, i + 1, . . . , i + m − 1
) ) ]H)DO"`/3 u1")Ju g FZ/^) FhP /FG)&/Fh /F7) F^$`-FG)G FZF^^) Hs =
h
, )Z)h_sE`T
`>)TPNTF^$O` FG)GFWeF^.2 ]=:`F^ @ :Xgh>.FX8 ^F 9FG)`$)3 )
) W=765Y{ :!:!1[.<{2_:/+6\+ NY:!1065&' )9)&FZ'hHFhP /FG)$
g 7PN('X9&M ^/Q$RD89>)F{) s1f1,Fx/ )- d ) _)F{)o'7 ^$1F
h
Pk`9`>)>F +PN")+F^$^` ?FG)^ r/3?+PN")^9+`,=^>) 'X3O'!! +8EFG `>1``.3> )^9 '&39(' )^9 WFG) )T3^`>)aPN('XFX).N/FG)Tg,r)FT_ b>F
).$FG ) F^P).)8'X3(' /` xs1 FG89 ^>9N/`-)
P7)_)^:]H).'X )
i
>)
/3 )&)^9"F/T) /3=f//` ) `$/mWFG)ZEF^)^P ) 'X3O'
i+m
s1
g :
Á
Ã
Â
Á
Ã
Á©Â
Á
Á©Á
Á
Á
Á©Ã
Á
Á
Á
Á
// Program : string_matching. C
// find the first occurrence of a fixed string within the
// input text , and output the text so far
# include < iostream >
int main ()
{
// search string
char s [] = " bool ";
// determine search string length m
unsigned int m = 0;
for ( char * p = s ; * p != ’\0 ’ ; ++ p ) ++ m;
// cyclic text window of size m
char * t = new char [m ];
unsigned int w = 0; // number of characters read so far
unsigned int i = 0; // index where t logically starts
// find pattern in the text being read from std :: cin
std :: cin > > std :: noskipws ; // don ’t skip whitespaces!
for ( unsigned int j = 0; j < m ;)
// compare search string with window at j - th element
if ( w < m || s[j ] != t [( i+j )% m ])
// input text still too short , or mismatch :
// advance window by replacing first character
& ]av AbR
Â
Á
Ã
Â
if ( std :: cin > > t[ i ]) {
std :: cout < < t[i ];
++ w ;
// one more character read
j = 0;
// restart with first characters
i = ( i +1)% m ; // of string and window
} else break ;
// no more characters in the input
else ++ j ; // match : go to next character
std :: cout < < "\ n";
delete [] t;
return 0;
}
LM://N. t.<0:!1"7N 'A+!5&1"7N&
Program 16:
9>5'79 )^9_/ ) )):]H) P&)W9
n)_/
eratosthenes.C
,=)^9,=)F&Mx A ,=a) )^9TWFG)Z1``>,G`!Pr)"FI)^
"bool"
// Program : eratosthenes. C
// Calculate prime numbers in {2 ,... ,999} using
// Eratosthenes ’ sieve .
# include < iostream >
int main ()
{
// definition and initialization : provides us with
// Booleans crossed_out [0] ,... , crossed_out [999]
bool
VcPN>'c`)F 3|)^ sm
_398'&) FGm`>).) )
9/3-PhFG)/393 =9,=)
>tg79 / /3F
) ):]1)`9/`:)>s` 9/`>)> P
D,9)X)F
std::cin
FG)^^$/}sE>`FG8)o = g )FG) )F '7X,F^X)DP /`:) )9) FG)^^$/}f//,F `$
8 `) sE
`fb:^)3 )
k'&) )^9_FG,9 ) sE> -/F 9 /F+)> 'hF O)^)8) )
bool
$399+ /FG)&)^9"3 P )!FG)G$/-7\19`>")"f//,"P
FZ)!FI)^$/
std::cin >> t[i]
S</: (f/=P Z`9/`:)> b)Z`>fb>FG )
]/`>)^ )F4,F 'X>)>4)>hFG)
bool
'h/FZ+`9`>)>Z) FG)^/ HZ9)
FG) `>fb) )79^/ F , s1.>39 `>)9 FG)939^38=9,)x)^ DW h`>)/ )D):]H)nde
)F `$/FG H)TFG)^$
'X sE>`D8)o8:]=/`>)
O) ) >93 P
std::cin
)TW 4gh!
std :: cin > > std :: noskipws ; // don ’t skip whitespaces!
F`>F^F^y) )>&)-FG)^^$/ )9) uv5&10<.qL7J/<. s9N/ HF h>'XF 7>)` F^,3)
sE 3} s13P /,) Z)> !:zghF
/O'&F,F
) F^$O` PNFI)^F)9)
`)/
'X)^FG /`F =/3_)h/('XFn,F ).,)G ,=) )^9X)^:]H)è,=
) )XWFG)71`>`,=^`TPE)DF^$^` FG)^^+
)Fh mNb,=)$
& av G v A@
2.6.11 Multidimensional arrays
de 6hY"Y '7"`$a9$fb!^1FZPxO^HF = & ]/89 )"3`)
int a [2][3]
3 `NOF )ZsEk/.^$ PH)
'&9FGn)^F ^ ^$1F PH)
'X)!,93> H
a
2
3
)o1E
! /FG
F^$ )9)
FT 'j7~01q)1"'I<M.12+:!:/V e )FD`$/F^+Pk3>9FGF
int
a
/3 :ngh")oE"P F 93 )!,39:H9 )o1m !F int[3] de>/ 2
3
a
int[2][3]
) 39>`NO)
3>W9F/yG
PD>9)^
f/,9 P
.'&9FGa>)F -O^HF PD)
a
n1
f//,
P
C 'XF^
>)F+/$Gb,5>)+)_ `>),= g79_f//,F n , . . . , n
n2
`$/3 )^9 )1['A<M./12M. PZ)G /3|)
:]H>F^F^F
>$ 1+,FG).sEk
`FG))Z:]HFGF^FZP )>/m)o1E!/3 EFG)fbDf//,
&/3i``F^F ,)^3F^9/ ^1F!'7 HF /F!:]1E`>)^3
F")^9 >)
a[i]
P 3:] E/3 )FX>)DFT/ ^-)^F^P 6 F^ 1,) FX)^9 >)DP
i
a[i][j]
3:]
)"^
9/3aFG {
j
a[i]
VX), '7 ,F^,9/ ) P+,)3F^ OGHF /F )Os F _)^^`F O) ^ Nb,=).F ) - bPN e39F^9/n^HF> = ]/89 )^9)o'713>9FG9/
^$ 3`3|)^^,9
1`>`,= F `),,F+ ^)+P&)^9 ^ x'&)
int a[2][3]
FG /`"PN
s uv`:)FZP )vE
F^ ,= $#H
6=2 3
int
a[0][0]
a[0][1]
a[0][2]
a[1][0]
a[0]
Figure 10:
,)3F^
)Tf/,9 PN&)^9
a[1][1]
a[1][2]
a[1]
<'A:!VDV$7R Sz?0u`Q)1['A<M./1q+f:!:/V
H^$1F `$/sE&)N/ 3._"'h F^x) 3F^ ^HF/
:g.< /3 MZV ) WFI),73>9FG__$sm")G)3
int a [][3] = { {2 ,4 ,6} , {1 ,3 ,5} };
g79Fh3>W9FZ/aO^_P )o1m
'X>
Fh,F^>3)+)N/ Z)^9T)
int[2][3]
{2,4,6}
9/3 {1,3,5} F7,F^3-PN a[1] a[0]
g7 ,3 3F^9FDP +,9 )3 Dynamic allocation of multidimensional arrays.
F^9/1^ _!9)xsm HO'& O)x`894)k.'&`+`/F^73=H9/`k/`$)!F `$/3
PN$Dq{>)D,FXFI)^)X'&)a) `/F^ 'X>^ / 39F^Fhs9,=)&)^9 WFI)T 1O'& O)D`89
)/8deP
9/F f,
E)> )-_3=H9/`/ 1`)3 ^ P ) n
n
0
'X ),3>1.)oE e rFhs9)/3
P :]19^F^F^
n2 nk
new
& ]av AbK
new
& F!f, n i = 1, . . . , k 'X>^
VXn3F^9FTs9,=) ) WFG)! ,FG)"sE8`FG))
:]19^F^F^F!deP4b, ) i sE,=)D )TPN!,=) )^9FDFD)TF^,=^F^= =T:]/89 m
3>{)h>)^n_/`xN/,9/ `>13nPNr/3 /``>F^F m>)9F !) 319/`$/
/1`$)^3_^ )D`89>n ,FG
) HO' O' _/ ^+`F F^X>) PE)
,39:H9 )o1m e r1``>,9>R
F F^!\1`>) 1<RHB;: k,)7)F7F78mF^F^s9TP{)
n2 nk
f//,F
$ n H('X-O)&`89&)
n2
k
dePT'7'h/)8) ,Fâ) sE(fb
]HF^FG)^ 9 )N/ E)>
Pointers to arrays.
new
f/Ns93 'X)-)+/33=F^F P )+WFG)T>)TPn). ,)^3F^9/EO^J m'7+3
) )oE IE)>+) e =yVXF+b,y_$ F^,FGm`>) '7a9PN^_/?`$/7)F )oE
n2 nk
n n Es9,=) '7.`$t )T'&)^ )Tb )9)T 6ZY Y F^9`> n e n FT)")oE
9/2hVXb/k
'7 9fb )8F^G)D)_/ 89` )739:W )âP ) 3F^2^3-Ek )^>&f/NOs 9/Fh) PN('X `39 P/>)$
p
int n = 2;
int (* p )[3] = new int [ n ][3];
// type of * p : int [3] <= > p : int [3]*
g )F^F 49>`F^FO^!>^ /F^9`>
h
'&`"F{) F/4/F int* p[3]
int *p[3]
3`NOF
)_sE / O ^ P4E)>^FT)^
eF^> /F^_9 ]H) O/9h: 6ZY Y FGH

)O] F
p
int
s9)^):FG'7>)
deP bM
, F H PN.+,9 )3F^9O G?'X ) `FG)) 3
Arrays of pointers.
H)D `N/E9FI'7>7F )>^ F 9 k,=)7,39: )D`,)> F^F 9
n2 , . . . , n k
b,a`$s9, fb>^ 13a)O){
CT_WFG)+F^,=) )9).FG,9>FG)F+)F^P7'& b, `F^3>.) ).^?Nb,=)
Pk+,)3>9FG9/{^1F"FD)F Xb, 3=H9/`$/`$).3F^9/ ^ P
)
/3 ^)^W9`N/ ^) ) )&) FG,s GHF&s3 FG
n=n
n
nk
uv, 'X)
1 3`2F>
qr:) ,F 39F^`,F^F )&)o'713>9FG9/=`$/F^D+).fb3_) PN^ ,N//xV )o'713 F^9/n^$ 'X ) 3F^F
/3
`$/ sE F^+,9)3 s1?3F^9/xO^ P
n
m
)
nghX>)k'X )`$993`>F
/3
nm
i {0, 1, . . . , n − 1}
j {0, 1, . . . , m − 1}
E$OF&O)&3:]
) 3F^9/EO^b4jX`Dfb>F 9) >)hP 3:]
mi + j
`
) 3>9FG9/mG_ Fh`$/m93`>F
3f
/3
3
kg7F '7 HF
i=`
m
j=`
m
sE`/,F^") PN,9`:)
(i, j) →
mi + j
s9Buv`>) fb _9FZ) F^:)DP `$/{3`>F
) ) FG>)TP 1,+sm>F
(i, j)
{0, 1, . . . , nm − 1}
de)^,9 )fb ()F _ 9 )^)>9F ) _/9GX)s9nP Ô'&Fx/3
`,Frs F^89
n
m
9,=)^) (' P)>_/)> VXF b,`$/yF^>-P ,= $# 7)^9F F8 ]/`>)^|'& O)
)+`89>DF"89`)
3_PN +,)3F^ O{^1F"'X ) M.<2M 3F^F
n1 , . . . , n k
& av G v AbU
SD)D:]1 `) PN `FG)/)D3>9FGF&FX- '7 /,3 ),9mFG`
'7+F^ ) ),) fb ))
r(fb> E)FD'7 /,3 sE`>FT>fb a[i][j]
`,.sE>F^X'&) 99:3F^9/ ^1F
V sE>)^)^>_F^,=) )9) b>9F)-))
93y)9)FG1) :]1)3F8)
a[i][j]
>X3>9FGFhF&)!PNO'&^
b/
='7 3FG`,F^F&) `$F^ P )o'713>9FG9/
^$ '&)?3>9FGF
/3
Tb, W9^FG)!3=1 `$/ -/1`$)^ 9 ^ P n L71[M6{
n
m
<:g. n/3 ) b, >).>fb>G|F^9 E)> E) )-)W9^FG)+) PZ/|3f13, 3=H9/`/ 1`)3 O^5PD>9)^
g79PN('X?`>13aP/>) 39>9FI)^)>F
m
)F
// a points to the first element of an array of n pointers to int
int ** a = new int *[ n ];
for ( int i = 0; i < n ; ++ i)
// a[ i ] points to the first element of an array of m int ’s
a [i ] = new int [m ];
g &)oE
h
F Im):n)"E)^>n)^
=
F4)>PNX E)>4)
eF^>
int**
int a[i]
int
)Z /98 m):kF^,=s9F^`>^)!\1`>) 1BRH<K /3
F / f/,9hP )o1m
a[i][j]
buv,FG)Z bT I>,9.)v'73F^9/ ^b
int
ghG $b,=) F 3lm>^) ),9
,=a$# F!>9N/`>3 s1 ,^--g79F
$/F ).)o'739F^9/ ^ FX )`3 ,?PF^>) P
9>39F^9/ ^HF n
s9,=)!)F^
GHF ^8) `FGF `F^`>,)^fb?^9>3 ^bde P /`>) )
n
^1FX_>fb> 9fb 3lm:)X)FZghFZFX,F^PN, PND ]/89T'X-b,-'h/)Z)
n
9/FT)
FG)^ ('7> )^N/,N&_)^ =
] E )FT`$/F^ r )"FG, `F"P4)^9 ('zPk93:]
i
i+1
a[0]
Figure 11:
a[1]
a[1][0] a[1][1] a[1][2]
a[0][0] a[0][1] a[0][2]
<'I:!VzDV$7 S\K0u`Q)1['A<M./1q+*:!:/Vz:/+Z1,)j;!V:!:/Vj S L=1"M<:g.
qr:)7,9F `>9`>,39X)^9F F^`>)
'&)
/
):FG)9.O99`$) P
Computing shortest paths.
e+,)3F^ OD ^$1F de_/ `>)9,NOTP /`>)^ 91 rFG,sm39 fH33 )
F , O
`Fn\1DPr)^9T`>F7Ds9` b>3 'X )_s FI)/`>F )F^T`,3 PNZ ]/89&sm _` F
.`,=sEb3F s9,=) :).,9F.Os FI)^/`>)^ `$n) I'h/F + V sE) F )/ `$)>3 O)
F^8`>
)F^,=`! 4/3 )^9_b/nF ) (fb )8sE) )^ FG8)>.` T )
S
)^>,) :_V ) /?) r)^9+sE)!`$/ _ b FG)> Pi )F!`,=^^) `>n)^-/?P )
PN,=X3buI/`>)Z`F =s ,=)hPNXsfH,F $F^F&)h___,F^!`>F )9)& 8)vb
fb> )F F^:),= H'7D'h/) ).W 3 F^^)FG)7EFGF^s9&sm) O) P^
)
e7W 3
S
T
,=)T)9)T F^,` )^?:]HFG)F %&>^ E)+) Pn8sE)D ) FX).1,.sE>TP4FG)>9F
& ]av (0#
) b> s1 ).sE)T3,=_)FT) P ^)`,N )X) b>F
FG)^>9FT)^_^$/`
P 0
S
`F>
8 12
S "!"! "!
!" !" !"
"!"! "!
Figure 12:
)
).)/ ` > 31>F") `,)!r
S
S: T ,=_ EeP) &^F OS'&F" :]=/89!'&)
A B
A BA
B
A B
A BA
B
AD
BA
B
C BADC
DC
C
DC C
DC CDC
DD
D
T
- .
- &
% .- &
% &
% &% $
# $
# $#
.
-
.- -
.- %
&% -.- %
&% %
&% %&% #
$# #
$# #$#
..
&. &
&& '
$('
$('$(
'
'
')(
(
)
)(
*
'
'(
S (
) *
) *'(*)
*
*
),
)+) *),+)
*
*
+*
),
*
+ ,
+ *,+
++,
+ ,++
,
,
/, +
/, 0/,
0
// 0
// 0//
0
0
0
10
10
2
11 2
11 21211
2
2
222
0S< 5C+a10.zE.<57:g!.<a:/&;< LC+65BS/:<'
S
T
; <
; >
= <; >
= >
= >= @
? @
? @?
<
; <
; >
= <; >
= >
= >= @
? @
? @?
<
;;:
== <;;:9 >
== >
== >== @
?? @
?? @??
<;
<
9 >
<;:9
<
>
<
>
>
>
@
@@
9 :
9 :9
:
9
9
:9
:
:
9
:8
:7 98:7
7 9
8
77 877
87
8
8765
8
55 8655
55 6
6
6
T
53466534
63
6
43
3
343
4
4
3
3
444
91"N5Y p 5&10.G
dea)F& ]/89 +)^)^X) H.>fb$F&)9)h)>! FGF^)^N/)o'783l >)&EF F^s9)FkPN&)"^sE)Z)^ $` )h`$/a /F^FhsE('*)!`>8E)hP 'h/Fh/3uG`)h)
T
m sE(fbTd )X)^,^9FD,=)D)9)& F^F^_sE(fb FXP FG)> {93 >F^,)F^^)>FG)D ) eP
S
)
XF!3>9`>)^3 ,= ),: Z).)9)"?>/4)^9: F ) ,91,
21
F^^)>FG)n )^r de ,=4:]/89 )7W99/H)x),= P ) ) `>,938/FG 9fb7sE8_39
"h)o'7 `>F7PN,=^)>X3('X{
'h/)D)'&)^ / )9)&W 3F 8F^^)FI)DsE)D )
mfb> ).3F^F
1,+sm>.PhÔ'&F /3 m e1,.sE>.Ph`,F Ph)P /`>)G 9 )_`139)^F
n
P F^,=`+/3 )>) E/3 )^9 'hFD%ZO'[`$/ )FXsE 3( kPN $399PN,=^)^9: m'7
`,=/Xb,_) ) 8sE,=) )Fks9>}PNh!'XxM $/F^D)Z)9) ) ;!:!C!{ST:</
!LYLM:<QJ<5 P7)^^1 /kmF^F^s9 )^9F.93 F^`>)a)_F^^)>FG).
F 9)+ ) F^` ) ,9.sE> P9F^,` )F FxF^8 ")TN7/ $/3= PN439:)
9 39F^F
kFG3F O'w3 b,a:fb->)^!/m)F^" )^9FT
%&>-F/ 9^b/`ys /FG3 )V+'j1qELM:</N:/'^'j1"7N wghF8>9:/&)>` ,9F
9`$Os )-s9F 'XF^F^,=)9F!`$/ 1,` H?sEs)/3 Pi)_F^,=)9F )
F^_/>7F^,=ss9F7P ) F/ FG)^,`>),=/ g79!^)Za3=H9/`X9^/ F7) W93
) I) F^,=ss9F /3)F _+ 1,^T!X F^F P $/`.>/ )
P )^9"9/ 9^s -
CTà'7 fb-3)W935FG,9 )s9
F^,=s9s9>F k'7FGfb-/hPD) 5),=
P
)+F^_/:X) )^9+N^> 9>F 93? ") F^,)^F.gh9)T'h E'7 9fb8 )
PN_)^!)9)x'773 ^39:4)^ ,` H `>8 ,=)n)hFG,=) ) Dfb+F^,=s9s9>
Pc)"FG,=)F7P )]
//3=_F^fbh
3 F^_/:7F^,=s9s9>F
de-,=&`$F^ '7!> T) s9> F&PN('XF 4PN +[ 8)o
`F
a)^9 91 C
`89,=)7) D<7N65 PE F^^)FG)n )P
)
'&>Z)^9hf,&F ∞ Pm FG,9` )
S C
:]HFG)F &`>N/ )9)4)Fn/F^"F^fbF ,=n9/Hs9>tPm`>8 ,=)9 F^^)FG)n )
P
)^
VXFGF^,T)^ O) ) ) P F^^)FG)7 ) P S ) T F ` < ∞ e)>^'&F^
S
T
& av G v (0H
'7 19('w)&'h)9)Z)>!F&8 ) )X/D:n /F^ F )9)
F :<,J!5CY;gD P
T
FI)>9F
S
`
r)>8+,FG) sE8
`k/3uG`)")
Z('}P
)9)!F"$` 9s9 P
T =S
T
S
`−1
FG): F m/3 /3uI/`)X) )^9FX `> ^$/`9s9!
FG)>9FX>)^` =O'&+FG,9` `9/-P
`−2
`FZ,)m'7 >)Z)^
fb>FZ,FZ. )P )
'&`FhF^^)FI)ZEFGF^s9
S
`
qr:) ,F >9=/FG_)^9_>/ >3 s9 .'7'h/) )^ sE 8)o `
'X )
C
9bO)fb )>X
mF^F^s9
X)^ O)T39`$)FX)^9 ) Pk8F^^)>FG)T O)?P

∞
S
)
%&:
)_F^,=s9s9>F!)-'X` '7_9N )^ >39,`)F 8PN afb|):
C
{NOsE / )^9.`>FD)9)".$/`9s9.P ?) FG) FG)^>9F i 0
−1
/`:),9/ EPN!F^ FG_/>Df//,g r)F NsmF / S`F )^ O) ^ i$` 9s9+Pi = Snm
)!/ F^`".F^^)FG)h )'&m>fb:X)>D
`E)o'X`
%&>4FE) 3,`>)"P N>{)hF^_/>EF^,=ss9F {F^F^, )9){'7n9fbn/$3
F^fb>3 )^9!F^,=s9s9> PN
e '7 9$fb!NsE3-/ `FZ)9)X!/` Os P S
i−1
'X )
T>F^FDFG)>9F de 3>D)F^ fb ) F^,=s9s9> PN '7 FG) 3 )_NsE>
i−1
i
) `F )9)Z^ /` Os "
FG): R
F s9,=)&)hFGF :
k,)7)FhF7F^89 1F^`T)F^"`F
i
:]=/`>)_) ,9sE>3 F&/3uG`)h) `F '&)sE
i−1
,=$;a,FG)^)FTO'~) P^): P Nsm3 `>F ('XF! )F"`F^F PN i =
0, 1, 2, 3
67),99
?)^9F"P /FG9 r'7 W99/-^fb ) ) F^)^, O)?3>9`:)3 ,=aA / 8)v_`>F 9fb smsE>A
3 93 P /`>)h$/`9s9DP
)F ]/89g g S
W93 FG9G)FG)" O)?P
)
r'7 FG)G)!P

'X` 9/F NsE
D/3 PNO'~
S
T
T
21
) P 3`>$F^ NOsE
F
,9)m'7 W 9/8$` S 20, 19, . . .
qr><) BFX>)D)^8)+6ZY Yt89>))^_P4) OsEOfb :)13{
The shortest path program.
>>F^) ) 9
s 3=H9/`$/|/1`$)^3 )v'73F^9/&^
'X)
floor
3F^F
/3
/3)^>F P )o1E
=_/ Fk E)>n)!)
n+2
m+2
int
floor
WFG)h>) P / ^_P
E)^>F )
Hs ,=) '7TFG)m`$/ )F7 )o'713F^9/
n+2
int
^$Y
: ghFG+3>9FGF&$fb.FG /`.PN :]1)^ 'h/FXF^,=^,38) $"\1,` :]H)G
'h/FZ(' ,9FZ) >)Z3-P FIE`N `$F^F 9X`FZ9fH9 F^Fh)9/-PN,=D/3uI/`>)X`>F
de|>9:/ n/ ^)W9`/k39)-)^i)9).,93F )
/`>),9/ 39) /bFG) FGE>`N/ `$F^F+F
`$/3a .Q<M01"! gh $O^)XP ) 9^/(
'X`-E$OFX/F&M ^/i(@.sE(`
' FX 1a)^ O)&` 9,=)F )hFG,=).)"FG,ss9
P~)^9hF^,=).)^"F^,=ss9
PN i = 1, 2, . . .
i
i−1
g79TF^,)^8).F^,=s9^s F $39f//Ns9 '7TF^>)7)
)^G_`^FIE399
0
floor
)
) /3 ) )^F `^FIE399Z)&) 8)o"`Fr)
)^9F F $/)r)D3`$O)
S 0
−1
)9)7) `r9/FZ)7sE>-NsE3
b:,) : /FZO!'hHF7NsE>3 '&)
)")>
−2
de-)>)
P )! ='7 FG89 )=, /{)^9 b>)D,NsE>3`>F&/3-NsE
i
:]=/`>)_) Fh'&)
)9)h9fb /-3buI/`>)h`{'&) NsE
ng7! )^>9)>F
i
i−1
/FTF^?/FTF^FTFT_/3 H^ m$/ )^ O)T9_>'z`> `,3 sE.NsE>3 ) `,=^)X):){x%Z>!F7)^9"`13
// main loop : find and label cells reachable in i =1 ,2 ,... steps
for ( int i =1;; ++ i ) {
& ]av (0
0 S
OPO OPO GGOPO GHG GHG GHG E
EE EFFEE EFFEE
PPOPPO GPPO HHG HHG HHG FF
FJ
I
IFJIIFJI
I
J
JJ
I
2 1 0 J
KK JILKK JILKK
L
L
LL
2 1 L
KN
MM LKNMM LKNMM
N
NN
2 N
MR
QQ NMRQQ NMRQQ
R
RR
QT
R
SS RQTSS RQTSS
T
S TTS TTS
T
!"! "! "!
!!"!! "!! "!!
!"!" "!" "!"
T
.-.- .-.- .-.- &%&% &%&% $#$# &%&% $#$# $#$# $#$#
.--. .--. .--. &%%& &%%& $#('#$ &%%& $#('#$ $#('#$ $#('#$
('(' ('(' ('(' ('('
)(') *)(') *)(') *)(')
1 0*
**) **) **) **)
+*)+ ,+*)+ ,+*)+ ,+*)+
1,
,,+ ,,+ ,,+ ,,+
0/0/,+ 0/0/,+ 0/0/,+ 0/0/,+
0/210/ 0/210/ 0/210/ 0/210/
2112 2112 2112 2112
21 21 21 21
AABAA BAA BAA
ACABBADCA BBADCA BBADCA
CCBDCC BDCC BDCC
CDCD DCD DCD
<;<; <;<; <;<; >=>= >=>= ??>=>= @?@? @?@? @?@?
<;:9<;<;:9<; <;:9<; >=>= >=>= ??>=>= @?@? @?@? @?@?
:9:9:9:9 :9:9
87:98787:987 87:987
876587876587 876587
6556 6556 6556
465344653446534
T
343343343
434343
i=0
i=1
ccdcc dcc dcc
ddd
eceedcfe edcfe edcfe
effe ffe ffe

^]^] ^]^] __^]^] `_`_ `_`_ `_`_ baba baba baba
^]\[^]\[ _^]\[ `_ `_ `_ ba ba ba
\[\[\[\[ \[\[
ZYZY ZYZY ZYZY
ZYXWZYXW ZYXW
XWXWUXWXWU XWXWU
T
VVUVVUVVU
UVUVUV
i=2
qrq qrq qrq iji iji ghg iji ghg ghg ghg
rrq rrq jji jji hhg jji hhg hhg hhg
3 rqr
lklk lklk lklk lklk
2 1 0l
knm lknm lknm lknm
nmm nmnm nmnm nmnm
3 2 1n
popo popo popo popo
3 2p
ots pots pots pots
tss tsts tsts tsts
3t
vuvu vuvu vuvu vuvu
vu vu vu vu

~}~} ~}
~}~}~}~} ~}~}
|{|{ |{|{ |{|{
|{zy|{zy |{zy
zyzywzyzyw zyzyw
T
xxwxxwxxw
wxwxwx
i=3
Figure 13: ¡ ¢¤£¥ ¦§¨ i ª©« ¬®©¯°±¬°² C £³´©µ±¬©« ¦ª·¶£¨ S ¸¹º¥»¼©¯
¨½³ i ³¢¾¿¸¹º½ºÀª³ÁºÂ¶¿º³£¦¬³Ã¢©¯º¶Ä£³¨ S C Å
bool progress = false ;
for ( int r =1; r <n +1; ++ r )
for ( int c =1; c <m +1; ++ c ) {
if ( floor [r ][ c ] != -1) continue ; // wall , or labeled before
// is any neighbor reachable in i -1 steps ?
if ( floor [r -1][ c ] == i -1 || floor [ r +1][ c ] == i -1 ||
floor [r ][c -1] == i -1 || floor [ r ][ c +1] == i -1 ) {
floor [ r ][ c ] = i ; // label cell with i
progress = true ;
}
}
if (! progress ) break ;
}
ÆÈÇµÉ¿ÊÌËÂÇµÉÍÏÎÐÑÍËÂÒÓÊÔÕËÂÇµÉ
Ô×ÖµØÚÙÄËÂÛÜÊØÝÐÑÍÂÉÞßÊÌÍÉàÊÌÍâáÜÉÒÒãÒËÍäÐÛÜåÇ¯ËÂÔ×ÊÌÍæÈÐçÍè¾éëêØÚÛ²ËÂÛ×ÐáÜá²ìíÚæ¹É
ÍÂÉ¬ÐçèËÂÇµÉãèµÛÜÞßÉØÚÒîÛÜÊØµÒïÔÜÍÂÊÞðÒmain
ËäÐØµèÚÐçÍÂèÛñØµÎÚÖ«ËòÐØµèóèµÊËÂÇµÉÏè«ì¥ØÚÐÞßÛÜÙÓÐáñáÜÊôÙ¬ÐÑËÂÛÜÊØ¤é
@ A"
@ A"
@ A@
A"
AC @"
AC @"
AC @BAC
,- 5,"
B"
B"
B"
- $",- $"
% 6$"
% !"$% !"
# 7!"
# !# 8 9@"
4 "
@C"
@C"
@C"
A"
,- -"
, $"-, %"
$ %"
$ !"%$ #"
! #"
! #!
B A"
B A"
B A@CB 15 16 17 18 19
"
, -"
, $"-, %"
$ %"
$ !"
!'"
!
!
B
B
B CB
-"
%&"$ #"
"
#
#
"
C
"
C
C"
&! '"
&! '&#! 9 10C"
3 -"
, -"
, $"-, %"
$ %"
$ !"
B C"
B C"
B CB 14 15 16 17 18
&"%$ #"
& #"
& '&
'"
'"
&"'"
& '"
& '&
0&"
&)"
&)"
("'"
( '"
( '&)( 10 11 12 "
:; 13
: ;: ="
< 14
< =< ?"
> 15
> ?> 16 17
;"
="
?"
("
(
(
(
:
:
:
<
<
<
>
> ?>
"
)
"
)
)
"
;
"
;
;
"
=
"
=
=
"
?
?"
*) +"
*) +*() 11 12 13 :"
8; :"
8 :98; <"
+"
9"
9"
1*"
("
= <= >"
? >"
? >? 17 18
*"("
* ("
* +*
8 ;9"
8 98 = <"
+"
+"
9"
2
1
3
2
4
3
5
4
6
5
4 0"1"
0 1"
0 10 8
7
6
5
*"+"
* +"
* +*
8 9"
8 98
9"
2*"
*/"
*/"
8"
87"
."+"
. +"
. +*/. 10 11 12 9"
67 9"
6 9876 20 19 18 19
."/"
. /"
. /.
6 7"
6 76
7"
3."
.1"
.1"
65"
65"
0"/"
0 /"
0 /.10 9 10 11 7"
4 7"
4 7654 21 20 19 20
0"1"
0 1"
0 10
6
7
4 5"
4 54
5"
3"
9 10 3"
425"
42 35423 22 21 20 21
"
5"
3"
2 3"
2"32
3"
3"
2"23 23 22 21 22
8 9 2"
3"
2"3"
2"2
i = 23
Figure 14:
¾ ¢¤£³¥ ¦ª³¨ i = 23 EDô¬ º «Á ³ £©¯ GF¢¤£³¥ ¦ª³¨ © F
º £¬°ÁÌ» ©¯¾¢¤£¥ ¦§¨ HÄ©½³£¦¬³ï¢©¯º ¥ ¦î©° IF K J ³ î©£¦°Á¿¶£³¨ T © F
¶ç»ª¸È°Á½©¢©¯ºÂL
¶ FµÄ±³£³Ä©¯Ä»Á ×©« ³¬NM¥Å
// read floor dimensions
int n ; std :: cin > > n ; // number of rows
int m ; std :: cin > > m ; // number of columns
// dynamically allocate twodimensional array of dimensions
// ( n +2) x ( m +2) to hold the floor plus extra walls around
int ** floor = new int *[ n +2];
for ( int r =0; r <n +2; ++ r)
floor [r ] = new int [m +2];
O IÉ PôË¬í¯æ¹ÉïÍÂÉ³Ðè ËÂÇµÉRQÚÊôÊÌÍ ÎÚá×ÐØ¿ÔÜÍÂÊÞ ÒËäÐØµèÚÐçÍÂè ÛñØµÎÚÖ«Ë¬éTS É·ÐçÒÒÖµÞßÉ ËÂÇÚÐçËÃÛñËÃÛÜÒ åÛVUÉ³Ø¿ÍÂÊÑæòæ ÛÜÒîÉ
Ð ÒÐ ÒÉXWôÖµÉ³ØµÙ³É ÊÔ
ÙäÇÚÐçÍäÐÙÄËÂÉÍÂÒ³íÃæ ÇµÉÍÂÉ
ÐØµè
ÒËäÐØµèÔ×ÊÌÍÒÊÖ«ÍÂÙ³ÉÐØµè ËäÐçÍåÉË¬í
nm
’S’
’T’
ÍÂÉÎµÍÂÉÒÉ³Ø ËÒ ÐæÈÐáñáí ÐØµè
ÐØ ÉÞÀÎÚË ì Ù³É³áÜáé½Æ¹ÇÚÉ®ÛÜØ«ÎÚÖ«ËZYÚáÜÉßÔ×ÊÌÍ¿ÊÖ«Í ÛÜØµÛñËÂÛ×ÐáÃÉIP«ÐÞÀÎÚáÜÉÀÔÜÍÂ’X’
ÊÞ
’-’
[ ÛÜåÖ«ÍÂÉ \ æ¹ÊÖµáÜè ËÂÇµÉ³ØóáÜÊ Ê ] ÐÒÈÛÜØ [ ÛñåÖµÍÉ êÔôÊÌËÂÇµÉÍ¾ÙäÇÚÐçÍäÐÙÄËÂÉÍÂÒ¡ÐçÍÂÉÃÔ×ÊÖµØÚèãÛÜØãËÇÚÉ ÛÜØ«ÎÚÖ«Ë^ÊÌÍ¾ÛÜÔËÂÇµÉÕÛÜØ«ÎÚÖ«Ë ÎµÍÂÉ³Þ®ÐçËÂÖ«ÍÂÉáñì_ ÉÙ³ÊÞßÉ³Ò¤É³ÞÀÎµË ìa`Äí
æ¹ÉàåÉ³ØµÉÍäÐÑËÂÉàÉ³ÞÀÎµË ìóÙ³É³áñáÜÒ³
é S¼ÇµÛÜáÜÉâÍÉ¬ÐèµÛÜØµåÀËÂÇµZ
É QÚÊôÊÌÍ ÎÚá×ÐØ¤íÚæ¹ÉëÎ Ö«Ë ËÇÚÉ ÐÑÎÚÎµÍÂÊÌÎµÍÂÛÜÐçËÂÉâÛñØ ËÂÉ³åÉÄÍÂÒ
ÛÜØ¯ËÂÊ¿ËÇÚÉÏÉ³Ø¯ËÍÂÛÜÉ³ÒÈÊÔ
í ÐçØÚè æ¹ÉÏÍÂÉÞßÉ³b
Þ _ÉÍïËÂÇµÉÏËÂÐçÍÂåÉËÈÎ ÊÒÛñËÛÜÊØ Ô×ÊÌÍòá×ÐçËÉÍ¬é
floor
// target coordinates , set upon reading ’T ’
int tr = 0;
int tc = 0;
//
//
//
//
//
assign initial floor values from input :
source :
’S ’ - >
0 ( source reached in 0 steps )
target :
’T ’ - >
-1 ( number of steps still unknown )
wall :
’X ’ - >
-2
empty cell : ’ - ’ - >
-1 ( number of steps still unknown )
8 12
------X - - - - -XXX - -X - - - - --SX - - - - - - - ---X - - -XXX - ---X - - -X - - - ---X - - -X - - - ---X - - -X -T - -------X - - - -
Figure 15: Â¥¢¤¾¶£È£Á£©¨
±£¬£¬¢ F°Á óº µ©¨¿¢¾§ Â¶
°Á £³
for ( int r =1; r <n +1; ++ r)
for ( int c =1; c < m +1; ++ c ) {
char entry = ’- ’;
std :: cin > > entry ;
if
( entry == ’S ’) floor [ r ][ c ] = 0;
else if ( entry == ’T ’) floor [ tr = r ][ tc = c ] = -1;
else if ( entry == ’X ’) floor [ r ][ c ] = -2;
else if ( entry == ’ -’) floor [ r ][ c ] = -1;
}
O Ê¦æ æ¹ÉãÐèµèËÂÇµÉÏÒÖ«ÍÍÂÊÖµØµèµÛÜØµå¿æÈÐáñáÜÒ¹ÐÒÈÒÉ³Ø ËÛÜØµÉ³áÜÒ³é
// add surrounding walls
for ( int r =0; r <n +2; ++ r)
floor [r ][0] = floor [ r ][ m +1] = -2;
for ( int c =0; c <m +2; ++ c)
floor [0][ c ] = floor [ n +1][ c ] = -2;
O ÉIPôË Ù³ÊÞßÉÒ¾ËÂÇµÉÃÞ®ÐÛÜØÏáÜÊôÊÌÎãËÂÇÚÐçË¤æ¹ÉÕÇÚÐXUÉ ÐçáñÍÂÉ¬Ðè«ìÓèµÛÜÒÙÖÚÒîÒÉ³è Ð _ ÊUÉÌé ê»Ë á×Ð _ É³áÜÒ¾ÐáÜáÌÍÉ¬ÐÙäÇÚÐ _ÚáÜÉ
Ù³É³áñáÜÒ³íÑÒÊÈËÇ ÐÑË¾æ¹ÉÃÊ µ_ ËäÐÛÜØãÐïá×Ð _ ÉáÜÛÜØµå¹ÐÒ¤ÛÜØ [ ÛÜåÖ«ÍÂÉ é [ ÍÂÊÞ ËÂÇµÛÜÒ¤á×Ð _ É³áñÛÜØµå«íæ¹ÉÕÞ¿ÖµÒîË¤ØµÊÑæ ÉGP¥ËÍäÐçÙË
ËÂÇµÉ·ÒîÇÚÊÌÍîËÂÉ³ÒîËÕÎÐÑËÂÇÀÔÜÍÂÊÞ
ËÂÊ é ÒÕÉI¥P ÎÚá×ÐçÛÜØµÉ³èÀÐ _Ê UÉÌí¯ËÂÇµÛÜÒÃÙ¬ÐØL_ ÉòèÚÊØµÉ_ôìÔ×ÊáÜáÜÊ¦æ ÛÜØµåÏÐãÙäÇÚÐÛÜØ
S T
ÊÔ ÐçèÐÙÉ³Ø ËÏÙ³É³áñáÜÒ æ ÛñËÇ½èµÉ³ÙÍÉ¬ÐÒÛÜØµå á×Ð _ ÉáÜÒ³é [ ÊÌÍãÉ U ÉÍì Ù³ÉáÜá ÊØ½ËÂÇµÛÜÒ Î ÐçËÂÇ^ÉIP¥Ù³ÉÎµË `Äíæ¹ÉàÎÚÖ«Ë
ËÂÇµÉ ÛñØ ÂË É³åÉÄÍ
ÛÜØ ËÂÊ ËÂÇµÉ Ù³ÊÌÍîÍÂÉ³ÒîÎ ÊØµèµÛÜØµå
É³Ø¯ËÍì ËÇÚÛñÒâÐáñáÜÊÑæ·ÒÓÖÚÒÏËÂÊóè«ÍäÐ³æ ÂË SÇµÉÎ ÐçËÂÇ ÜÛ Ø
−3
floor
ËÂÇµÉàÒÖ Ú_ ÒÉXW ÖÚÉØ ËÓÊÖ«ËÎÚÖ«Ë¬é êÔ ØµÊÀÎ ÐçËÂÇ æÈÐÒ Ô×ÊÖµØµè ^»ÊÌÍ ÛÜÔ ËÂÇµÉÍÂÉëÛñÒ ØÚÊÀËÂÐçÍÂåÉËG`´í ËÂÇµÉ _ Ê è«ìóÊÔ ËÂÇµÉ
ÒîËäÐçËÂÉÞßÉ³Ø Ë¹ÛÜØ ËÂÇµÉãÔ×ÊáÜáñÊÑæ·ÛÜØµåàÙ³ÊôèµÉÏÔÜÍäÐåÞßÉ³Ø¯ËòÛñÒ ^ÙÊÌÍÍÂÉ³ÙËáñìa`ïØµÊÌËòÉGP«É³Ù³Ö«ËÂÉè ÐçË çÐ áÜáé
while
// mark shortest path from source to target ( if there is one )
int r = tr ; int c = tc ; // start from target
while ( floor [r ][ c ] > 0) {
int d = floor [r ][ c ] - 1; // distance one less
floor [r ][ c ] = -3; // mark cell as being on shortest path
// go to some neighbor with distance d
if
( floor [r -1][ c ] == d ) - - r;
else if ( floor [r +1][ c ] == d ) ++ r;
else if ( floor [r ][ c -1] == d ) - - c;
else
++ c ; // ( floor [r ][ c +1] == d )
}
[ ÜÛ ØÚÐáÜá²ìíËÂÇµÉ ÊÖ«ËÎÚÖ«Ë æ¹ÉÞ®ÐçÎ½ËÂÇµÉÛñØ ËÂÉ³åÉÄÍâÉ³Ø¯ËÍÂÛÜÉÒãÊÔ
_ ÐÙ ] ËÂÊÙäÇÚÐçÍäÐÙÄËÂÉÍÂÒ³í¾æ ÇµÉÍÂÉ
floor
_ É³Ù³ÊÞßÉ³Ò

í ÊÖ«ÍÈÎÐÑËÂÇóÒîìôÞb_ Êá»éêØµÒÉÍîËÂÛÜØµå
ÐçËÈËÂÇµÉãÍÂÛñåÇ Ë¹ÎÚá×ÐÙ³É³ÒíÚæ¹ÉÏÊ _µËäÐÛÜØÐÙ³ÊÌÎ ì
−3
ÊÔËÂÇµÉòÛÜØ«Î Ö«ËÚQ ’o’
ÊôÊÌÍ¬íæ ÛñËÇËÂÇµÉ·ÒîÇÚÊÌÍîËÂÉ³ÒîËÕÎÐÑËÂÇ®ÐçÎµÎ É¬’\n’
ÐçÍÛÜØµåãÛÜØÀÐèµèÚÛ²ËÂÛÜÊØ¤é"S É·Þ¿ÖÚÒË ÐáñÒÊâØµÊÌË Ô×ÊÌÍåÉË
ËÂÊèÚÉáÜÉËÂÉÓËÂÇµÉãè«ìôØ ÐçÞßÛÜÙ¬ÐáÜá²ìÀÐáñáÜÊôÙ¬ÐÑËÂÉ³è ÐçÍÍäÐ³ì¥ÒÈÛÜØËÇÚÉÏÉ³Øµè¤é
// print floor with shortest path
for ( int r =1; r <n +1; ++ r ) {
for ( int c =1; c < m +1; ++ c)
if
( floor [r ][ c ] == 0)
std :: cout
else if ( r == tr && c == tc ) std :: cout
else if ( floor [r ][ c ] == -3) std :: cout
else if ( floor [r ][ c ] == -2) std :: cout
else
std :: cout
std :: cout < < "\n ";
}
<<
<<
<<
<<
<<
’S ’;
’T ’;
’o ’;
’X ’;
’-’;
// delete dynamically allocated arrays
for ( int r =0; r <n +2; ++ r)
delete [] floor [r ];
delete [] floor ;
return 0;
ê ØóÙ¬ÐÒîÉâÊÔ Ê Ö«ÍòÛÜØµÛñËÂÛ×ÐáÉIPµÐÞÀÎÚáÜÉíôËÂÇµÉãÊÖ«ËÎÚÖ«ËòáÜÊôÊ ]ôÒÈáñÛV]ÉãÛñØ [ ÛÜåÖ«ÍÂÉ é ÍÂÊåÌÍÂÐÞ
ËÂÇµÉÏÙ³ÊÞÀÎÚáÜÉËÂÉ Ò ÊÖ«ÍÂÙÉâÙ³Ê èÚÉé
ooooooX - - - - oXXX -oX - - - - ooSX - oooooo ---X - - -XXXo ---X - - -X -oo ---X - - -X -o - ---X - - -X -T - -------X - - - -
Figure 16: Óñ¢¤ÏÂ¶¹£³ÁÌ£©¨
# include < iostream >
º¿»¥¢¤ãÂ¶
°ÁÌ£³ ÒÇµÊÑæ·Ò
int main ()
{
// read floor dimensions
int n ; std :: cin > > n ; // number of rows
int m ; std :: cin > > m ; // number of columns
// dynamically allocate twodimensional array of dimensions
// ( n +2) x ( m +2) to hold the floor plus extra walls around
int ** floor = new int *[ n +2];
for ( int r =0; r <n +2; ++ r)
floor [r ] = new int [m +2];
// target coordinates , set upon reading ’T ’
int tr = 0;
int tc = 0;
// assign initial floor values from input :
// source :
’S ’ - >
0 ( source reached in 0 steps )
// target :
’T ’ - >
-1 ( number of steps still unknown )
// wall :
’X ’ - >
-2
// empty cell : ’ - ’ - >
-1 ( number of steps still unknown )
for ( int r =1; r <n +1; ++ r)
for ( int c =1; c < m +1; ++ c ) {
char entry = ’-’;
std :: cin > > entry ;
if
( entry == ’S ’) floor [ r ][ c ] = 0;
else if ( entry == ’T ’) floor [ tr = r ][ tc = c ] = -1;
else if ( entry == ’X ’) floor [ r ][ c ] = -2;
else if ( entry == ’ -’) floor [ r ][ c ] = -1;
}
// add surrounding walls
for ( int r =0; r <n +2; ++ r)
floor [r ][0] = floor [ r ][ m +1] = -2;
for ( int c =0; c <m +2; ++ c)
floor [0][ c ] = floor [ n +1][ c ] = -2;
// main loop : find and label cells reachable in i =1 ,2 ,... steps
for ( int i =1;; ++ i ) {
bool progress = false ;
for ( int r =1; r < n +1; ++ r )
for ( int c =1; c <m +1; ++ c ) {
if ( floor [r ][ c ] != -1) continue ; // wall , or labeled before
// is any neighbor reachable in i -1 steps ?
if ( floor [r -1][ c ] == i -1 || floor [ r +1][ c ] == i -1 ||
floor [r ][c -1] == i -1 || floor [ r ][ c +1] == i -1 ) {
floor [r ][ c ] = i ; // label cell with i
progress = true ;
}
}
if (! progress ) break ;
}
// mark shortest path from source to target ( if there is one )
int r = tr ; int c = tc ; // start from target
while ( floor [r ][ c ] > 0) {
int d = floor [r ][ c ] - 1; // distance one less
floor [r ][ c ] = -3; // mark cell as being on shortest path
// go to some neighbor with distance d
if
( floor [r -1][ c ] == d ) - - r;
else if ( floor [r +1][ c ] == d ) ++ r;
else if ( floor [r ][ c -1] == d ) - - c;
else
++ c ; // ( floor [r ][ c +1] == d)
}
// print floor with shortest path
for ( int r =1; r <n +1; ++ r ) {
for ( int c =1; c < m +1; ++ c )
if
( floor [r ][ c ] == 0)
std :: cout
else if ( r == tr && c == tc ) std :: cout
else if ( floor [r ][ c ] == -3) std :: cout
else if ( floor [r ][ c ] == -2) std :: cout
else
std :: cout
std :: cout < < "\n ";
}
<<
<<
<<
<<
<<
’S ’;
’T ’;
’o ’;
’X ’;
’-’;
// delete dynamically allocated arrays
for ( int r =0; r <n +2; ++ r)
delete [] floor [r ];
delete [] floor ;
return 0;
}
Program 17: ¢¤£³Á ³£¦¬³ ¢©¯ºÅ
2.6.12 Beyond arrays and pointers
·ÍÍÂÐ¬ìôÒßÐçÍÂÉ UÉÄÍì ÖµÒÉ³Ô×Öµá Ô×ÊÌÍ®Þ®ÐØ¯ì ËäÐÒ ]ôÒ®ÐØµè ÐçáÜáÜÊ¦æðÖµÒ Ë Ê ÒÊáVUÉóØµÊØ ËîÍÂÛVU¥ÛÜÐáÈÎµÍÂÊ _ÚáÜÉÞßÒáÜÛ ]É
YÚØµèÚÛñØÚå ÒÇµÊÌÍËÂÉ³ÒîËÓÎ ÐçËÂÇµÒ ÛñØ ËÇÚÉëÎµÍÂÉIU¥ÛÜÊÖµÒ ÒÉ³ÙÄËÂÛÜÊØ¤é [ ÍÂÊÞ Ð ÂË ÇµÉ³ÊÌÍÂÉÄËÂÛÜÙ¬Ðá¾Î ÊÛÜØ¯Ë ÊÔ U¥ÛÜÉæâí ÐçÍÍÂÐ¬ìôÒ
ÐçÍÂÉÏÛÜØóÔªÐçÙËòËÇÚÉÏÊØµáñì®Ù³ÊØ¯ËäÐÛÜØµÉÍÂÒ¹ËÂÇÚÐçË¹æ¹ÉÏØÚÉÉ³è¤é
Ø ËÂÇµÉÏÊÌËÇÚÉÄÍòÇÚÐØµè¤í¥ËÂÇµÉÍÂÉÏÐçÍÉÓË æ¹Ê¿Þ®ÐÛÜØ®è«ÍäÐ³æ_ ÐÙ ]¥Ò¹ÊÔ¡ÐçÍÍäÐ³ìôÒÈËÂÇÚÐçËïæ¹É æÈÐØ¯Ë¹ËÂÊ¿ÍÉ³Ù¬ÐçÎÚÛñË
Öµá×ÐçËÂÉÓÇµÉÍÂÉé
Ø¯ì¿ÐçÍÍäÐ³ìíÉ UÉ³Ø ÛÜÔ Û²Ë ÛÜÒ èµìôØÚÐÞßÛÜÙ¬ÐáñáñìãÐáÜáÜÊ Ù¬ÐçËÂÉ³è¤íÌÇÚÐÒÃÐY P«É³èáÜÉ³ØµåÌËÂÇ¤é
ê Ø®ÊÌËÂÇµÉÍïæ¹ÊÌÍÂèµÒ³í¯æ¹É Ç ÐXUÉ·ËÊ]ôØµÊÑæ
èµÉ YÚØµÛÜØµåëÊÌÍ¹è«ì¥ØÚÐÞßÛÜÙ¬ÐçáÜáñì ÐáñáÜÊôÙ¬ÐÑËÂÛÜØµåâÐçØ ÐÑÍÍäÐ³ìßÇµÊÑæ
³

ç
¶

³
£

Þ®ÐØ¯ì É³áÜÉ³ÞßÉØ ËÂÒ æ¹É¿ØµÉ³É³è ËÂÊ ÒîËÂÊÌÍÉ¿ÛñØ ÛñË¬é ÔÜËÉ³Ø¤í¡ËÂÇµÛÜÒ ÛÜÒÓÖµØ«ÍÂÉ¬ÐáñÛÜÒîËÂÛÜÙé [ ÊÌÍÏÉIPµÐÞÀÎÚáÜÉíÛÜØ ÒîÊÞßÉ
ÐçÎµÎÚáÜÛÜÙ¬ÐÑËÂÛÜÊØ æ¹ÉâÞßÛÜåÇ ËÈØµÉ³É³è ËÂÊÀÒîËÊÌÍÂÉàÐ ÒÉ WôÖµÉ³ØµÙ³ÉàÊÔ ÛÜØ«ÎÚÖµËòØôÖµÞ_ ÉÍÂÒí _ÚÖ«Ë·æ¹ÉëèµÊØ Ë]¥ØµÊÑæ ÛÜØ
Ðè UÐØµÙ³ÉãÇµÊ¦æ Þ®ÐØ¯ìßØôÖµ
Þ _ ÉÍÂÒïæ¹É æ ÛÜáñáåÉË¬é Ë ìôÎÚÛÜÙ¬ÐáîÒÊáÜÖ«ËÂÛÜÊØ
àÛÜÒïËÂÊ¿è«ì¥ØÚÐÞßÛÜÙ¬ÐçáÜáñìÐáÜáÜÊ Ù¬ÐçËÂÉ
Ð UÉÍîì á×ÐçÍÂåÉâÐçÍÍäÐ³ì ÐØµè ÖµÒîË ÇµÊÌÎ ÉëËÂÇÚÐçËòËÂÇµÉàÒÉXWôÖµÉ³ØµÙ³É YµËÂÒ ÛñØ¾é·ÆÈÇµÉëÎµÍÂÊ _ÚáÜÉ³ÞßÒÈæ ÛñËÇóËÂÇµÛÜÒ ÐØµè
Ð _ ÉËËÂÉÄÍ ^ _ÚÖ«ËÈÒîËÂÛÜáñá Ù³Öµb
Þ _ ÉÄÍÂÒÊÞßÉ `ÒÊáñÖµËÛÜÊØÐÑÍÂÉâÊÖ«ËÂáÜÛñØÚÉè ÛÜ
Ø P¥ÉÍÂÙ³ÛñÒÉ é
ÍÂÉ³Ðá ÒîÊáÜÖ«ËÂÛÜÊØ ÛÜÒ Î ÊÒÒîVÛ _ÚáÜÉ Ûñ
Ø ËÇµÍÊÖÚåÇ ËÂÇµÉóÖµÒÉÊÔ
é¼Æ¹ÇÚÉÒÉ ÐçÍÂÉÙ³ÊØ ¥
D
Ä

¬
±

¦
£

ËäÐÛñØÚÉÄÍÂÒòÔÜÍÊÞ ËÂÇµÉÏÒîËäÐØµèÚÐçÍÂèáÜVÛ _µÍäÐçÍì ËÂÇÚÐçË·ÙÊb
Þ _ÚÛÜØµÉ ËÂÇµÉÏÙ³á×ÐÒÒîÛÜÙ¬Ðá¾ÐçÍÍÂÐ¬ì Ô×ÖµØµÙËÂÛÜÊØÚÐáñÛñË ì ^ºÐØµèóÛñËÒ
É Ù³ÛÜÉ³ØµÙì `ïæ·ÛñËÂÇ ËÇÚÉÓÎ ÊÒÒîVÛ _ÚÛÜáÜÛñË ìÀÊÔ¡åÌÍÂÊ¦æ ÛÜØµå ^ºÐØµèÒÇ«ÍÂÛñaØ ]ôÛÜØµå `ÛÜØáñÉ³ØµåÌËÂÇ¤
é ÃÉ³ÙÄËÂÊÌÍÂÒÈÙ¬ÐØ _ ÉÏÛÜ
Þ ÎÚáÜÉ³ÞßÉ³Ø¯ËÂÉ³èßÊØßËÂÊÌÎ ÊÔ¾ÐçÍÍäÐ³ìôÒ³í«ÐØµèßËÂÇµÉìßÇÚXÐ UÉ ÒÊÞßÉËÂÇµÛÜØµåãÒÛÜÞßÛÜá×ÐÑÍÃËÊëËÂÇµÉ ÞßÉ³ÙäÇÚÐØµÛÜÒÞ ÊÖ«ËÂáñÛÜØµÉ³è
ÛÜ
Ø P«ÉÍÂÙÛÜÒÉ N_ÚÖÚÛñáñË ÛñØ «
é ÕÉÙËÂÊÌÍÂÒ®ÐáñÒÊÝá×ÐçÍåÉ³áñìÍÉ³ÞßÊ UÉ ËÂÇµÉ ØÚÉÙ³É³ÒÒÛ²Ë ì ÊÔÓæ¹ÊÌÍ ]¥ÛÜØµåÝæ Û²ËÂÇ
Î ÊÛÜØ¯ËÂÉÍÂÒ
é S Éãæ·ÛÜáÜáåÉËÈËÂb
Ê UÉ³ÙËÂÊÌÍÂÒ ^ºÐØµè ËÇÚÉÛñÍòÍÉ¬ÐáÜÛ ³ÐçËÂÛÜÊØ `Õá×ÐÑËÂÉÍÈÛÜØ ËÇÚÛñ
Ò _ Ê Ê ]é
Arrays have fixed length.
Æ¹ÇÚÉÈÖµÒÂÐåÉÈÊÔÊÖ«ËÊÔ _ ÊÖµØµèÀÐçÍîÍäÐ³ìÛñØÚèµÛÜÙÉ³Ò ÛÜÒ ØÚÊÌËÃèµÉËÂÉÙËÂÉ³èÀÛñØàíôÐçØÚè
ËÂÇµÉ·ÒÐÞßÉ¹ÇÚÊáñèÚÒÃÔ×ÊÌÍÎÊÛÜØ ËÂÉÄÍÂÒÃËÊëÐèµè«ÍÂÉ³ÒÒîÉ³Òæ ÇµÉÍÉòØµÊâÎµÍÂÊåÌÍäÐçÞ Ê _ É³ÙËáÜÛVUÉ³Ò³éS¼ÛñËÇ ÒÊÞßÉÈÙ¬ÐçÍÂÉí
ìÊÖ Ù¬ÐØ æ·ÍÂÛñËÉëÒÞ®ÐáÜá¾ÎµÍÂÊåÌÍäÐÞßÒÈËÂÇÚÐçË ÖµÒÉ¿ÐçÍÍäÐ³ìôÒÓÐØµè Î ÊÛñØ ËÂÉÍÒ ÛÜØ½Ð Ù³ÊÌÍÍÂÉÙËÏÞ®ÐØµØÚÉÄÍ¬í _ÚÖ«ËÓÛÜØ
Ù³ÊÞÀÎÚáÜIÉ P®ÎµÍÂÊåÌÍÂÐÞßÒ³í¥ËÂÇµÛÜÒòÛÜÒòØµÊÌË·É¬ÐÒîìÐçË ÐçáÜá
é É _ÚÖµååÛÜØµå¿ÔªÐÙ³ÛÜáÜÛ²ËÂÛÜÉ³Ò¹ÊÔ ÞßÊ èÚÉÄÍÂØÙ³ÊÞÀÎÚÛÜáñÉÍÂÒ¹Ù¬ÐØ
ÇµÉ³áñÎ¾í _ÚÖ«ËÏÉ UÉ³Ø½æ¹ÉáÜá ËÉ³ÒîËÂÉ³è½ÐØµè ÔÜÍÂXÉ WôÖµÉ³Ø¯ËÂáñì ÖµÒÉ³è½á×ÐçÍÂåÉãÎÚÍÊåÌÍäÐÞßÒ èµÊ®ØµÊÌËÓØÚÉÙ³É³ÒÒÂÐÑÍÂÛÜáñì åÉËÓÛ²Ë
ÍÂÛÜåÇ¯Ë¬é êØÀÔªÐÙÄË¬íôÒÊÞßÉ¹Î É³ÊÌÎÚáÜÉ ^áÜÉÄ!Ë ÒÕÙ¬ÐáÜá«ËÂÇµÉ³Þ ÐçËËäÐÙ ]ÉÍÒ `ÕÐçÍÉ·Þ®Ð ]ôÛÜØµåã
Ð _ÚÖÚÒîÛÜØµÉ³ÒÒÊÔ GÉ P¥ÎÚáÜÊÛñËÂÛñØÚå
ÎµÍÂÊåÌÍäÐÞßÞßÛñØÚåÏÉÍÍÊÌÍÂÒ ÍÂÉ³á×ÐÑËÂÉ³è®ËÂÊ¿ÐçÍÍäÐ³ì¥Ò¹ÐçØÚè®Î ÊÛÜØ¯ËÂÉÍÂÒÛÜØ ÊÌÍèÚÉÄÍïËÂÊ ÙÍÂÉ¬ÐçËÉÏÞ®ÐçáÜÛÜÙ³ÛÜÊÖµÒ ÒÊÔÜË æÈÐçÍÂÉé
" Ö«ÎÚÎÊÒÉëËÂÇÚÐçË·ËÇÚÉëÐçËËäÐÙ ]É
Í ]ôØµÊÑæ·Ò·ËÂÇÚÐçË ÒîÊÞßÉãÎµÍÂÊåÌÍäÐ$
Þ #¼ËÇÚÛñaØ ]®ÊÔ ÐçØ ÊÌÎÉÍäÐçËÂÛñØÚå Òîì¥ÒîËÉ³Þ
ÍÂÊÖ«ËÂÛÜØµÉ¿ÊÌÍàÐ®æ¹É _ÚÒÉÍ UÉ%Í # Þ®Ð¬ì ^ÖµØµÛÜØ¯ËÂÉ³Ø ËÛÜÊØÚÐáÜáñì `Èæ·ÍÂÛ²ËÂÉÛñØµÎÚÖ«Ëãè ÐÑËä
Ð _ÉìÊØµèÝËÂÇµÉ _ ÊÖµØµèµÒâÊÔ
ÐØ ÐçÍÍäÐ³ì&
é ÖÚÉÀËÊ ËÂÇµL
É UÊØ O É³ÖµÞ®ÐØµØ ÐÑÍÂÙäÇÚÛ²ËÂÉ³ÙËÂÖ«ÍÂÉí ËÇÚÉßÎ ÐçÍË ÊÔïËÂÇµÉ®Þ®ÐÛÜØ ÞßÉÞßÊÌÍì _É³ÛÜØµå
ÐÙ³ÙÛÜèµÉ³Ø ËäÐçáÜáñì ÞßÊôèµVÛ YÚÉ³è ÛñØ ËÂÇµÛÜÒ æÈÐ³ì Þ®Ð³ì Ù³ÊØ ËäÐçÛÜØ ËÂÇµÉ ÐÙËÖ ÐçáòÎµÍÂÊåÌÍäÐÞ ÛÜØµÒîËÍÂÖµÙËÛÜÊØµÒ³é Æ¹ÇµÉ
ÐçËËÂÐÙ ]ÉÍëÞ®Ð³ì ËÂÇµÉ³Ø _É Ð _ÚáÜÉ ËÂÊ ÎÚÍÉÎ ÐçÍÂÉÀÐçØÝÛÜØ«ÎÚÖµËÓËÂÊ ËÂÇµÉ¿ÎµÍÂÊåÌÍÂÐÞ ÛÜØÝÒÖµÙäÇ Ð®æÈÐ³ì ËÂÇÚÐçËÓËÂÇµÉ
ÎµÍÂÊåÌÍäÐÞ ÞßÊôèµVÛ YÚÉ³ÒÓÛñËÒÉ³áÜÔËÊóèµÊóæ ÇÚÐçËÂÉ UÉÄÍâËÂÇµÉÀÐÑËËäÐÙ ]ÉÍëæÈÐçØ ËÂÒãÛñËâËÂÊèÚÊ«é®Æ¹ÇµÛÜÒâÞßÊ èµVÛ YÚÙ¬ÐçËÂÛÜÊØ
ÍÂÖµØµÒ æ ÛñËÂÇóËÇÚÉ ÒÂÐçÞßÉàÐÙ³Ù³ÉÒÒ ÍÂÛÜåÇ ËÒÓÐÒ ËÂÇµÉ ÊÌÍÂÛÜåÛÜØÚÐá¾ÊØµÉÌí¤ÐØµè ËÇÚÉÒÉ ÞßÛÜåÇ Ë _ É ÐèµÞßÛÜØµÛÜÒîËÍÂÐçËÂÊÌÍ
ÍÂÛÜåÇ¯ËÂÒ¹ÛÜØËÂÇµÉÓæ¹ÊÌÍÂÒîËòÙ¬ÐÒÉé
êØ ËÂÇµÛÜÒ æÈÐ¬ìí ÐØ ÐçËËÂÐÙ ]ÉÍÀÙ³ÊÖµáÜ'
è îÇÚÛ îÐçÙ ](½ËÂÇµÉ Ù³ÊÞÀÎÚÖµËÉÍ ËÂÇÚÐçËÍÖÚØµÒ¿ËÂÇµÉ ÎµÍÂÊåÌÍäÐçÞ íÐØµè
ÒÖ _ÚÒXÉ W ÖÚÉØ ËÂáñì ÞßÛÜÒîÖÚÒîÉÏÛ²ËòÔ×ÊÌÍ·ÛÜáÜáÜÉåÐáÐÙÄËÂVÛ U¥Û²ËÂÛÜÉ³ÒïáÜÛ ]ÉãÒÉØÚèµÛÜØµå¿ÒîÎ ÐÞ í¥ÊÌÍÈÎ ÐçÍäÐáñ(ì ³ÛÜØµåàæ¹É _ ÒÉÄÍ UÉÍÂÒ
_ô
ì QÚÊôÊ èµÛÜØµå ËÇÚÉÞ æ·ÛñËÂÇ ÍXÉ WôÖµÉ³ÒîËÒ³é
[ ÊÌÍïÖµÒËÂÇÚÐçËæ¹É·ÐçÍÂÉ ØµÊÌ
Ë ^»ìÉGË `ÃÎµÍÂÊÔ×ÉÒÒÛÜÊØÚÐáµÎµÍÂÊåÌÍäÐÞßÞßÉÍÒ³íÌËÂÇµÉ ÒÉ³ÙÖµÍÛñË ìßÐÒîÎÉ³ÙË ÛñÒ áÜÉ³ÒÒÕÊÔ¤Ð
Ù³ÊØµÙ³ÉÍØ ÇµÉÍÂÉ*
é )ÊÌÍÉ ÛÜÞÀÎ ÊÌÍËäÐçØ ËÃÛÜÒËÂÇÚÐçËÕÎµÍÂÊåÌÍäÐÞßÞßÛÜØµåÓÉÍÍÊÌÍÂÒ èµÖµÉ ËÂÊàÛÜÞÀÎµÍÂÊÌÎ ÉÍÕÖµÒÉ ÊÔ¾ÐçÍÍÂÐ¬ìôÒ
ÐØµèóÎ ÊÛÜØ¯ËÂÉÍÂÒÈÙ¬ÐØ _É UÉÍìÇÚÐçÍÂèËÂÊ Y Øµè ÐçØÚè ÊÔÜËÉ³ØóÍÂÉ³Þ®ÐÛñØÖµØµèÚÉÄËÂÉ³ÙËÂÉè ÖÚØ¯ËÂÛÜá ËÂÇµÉì ÒÖµèµèÚÉØÚá²ì
ÍÂÉ³ÒîÖÚá²ËÓÛÜØ½ÒËÍäÐØµåÉ ÐØµè ÒÉ³É³ÞßÛÜØµåáñìÛÜØµIÉ P¥ÎÚáÜÛñÙ¬Ð _ÚáÜÉ _É³ÇÚKÐ UôÛÜÊÌÍ ÊÔ ËÂÇµÉàÎÚÍÊåÌÍäÐÞ é áÜÒîÊßÇÚÉÄÍÂÉÌí¤ÖµÒÛÜØµå
UÉ³ÙËÊÌÍÂÒëÛÜØµÒîËÂÉ³Ðè ÊÔ¹ÐçÍÍäÐ³ìôÒâÇµÉ³áñÎÚÒ³í ÒîÛÜØµÙ³ÉËÂÇµÉÍÉßÐçÍÂÉÞ®ÐØ ì Î ÊÌËÂÉØ ËÂÛ×ÐçáÃÉÄÍÍÂÊÌÍÂÒâÍÉ³á×ÐçËÂÉèÝËÂÊ ÐçÍÍÂÐ¬ìôÒ
ÐØµèÎ ÊÛñØ ËÂÉÍÒ¹ËÇ ÐÑËÈìÊÖ ÒîÛÜÞÀÎÚáñìÀÙ¬ÐçØÚØµÊÌËòÞ®Ð ]É æ ÛñËÇ UÉ³ÙËÊÌÍÂÒ³é
Arrays are insecure.
O ¦Ê æ ìÊÖ Þ®Ð³ì Ð Ò ] æ ¯Ç ì æ¹ÉëÇÚÐXUÉëÛÜØ¯ËÍÂÊ èÚÖµÙ³Éè ÐçÍÍäÐ³ìôÒ ÐØµèóÎ ÊÛÜØ¯ËÂÉÍÂÒ·ÐçË
ÐáÜáæ·ÇµÉ³ØËÂÇµÉÍÂÉâÐçÍÂÉÏÞßÊÌÍÂÉQÚÉIP«ÛV_ÚáÜÉÏÐØµè ÂÒ ÐÔ×ÉÍ Ð áñËÂÉÍØ ÐÑËÂÛVUÉ³Ò³é·ÉÄÍÂÉâÐÑÍÂÉÏËÇÚÉÓËÂÇ«ÍÂÉ³ÉÏÍÂÉ¬ÐÒîÊØÚÒé
Why arrays, after all?
é ·ÍîÍäÐ³ì¥Ò ÐØµè Î ÊÛÜØ¯ËÂÉÍÂÒ ÐçÍÂÉ ËÂÇµÉ ÒÛñÞÀÎ áñÉ³ÒîË¡ÞßÊôèµÉ³áÜÒ¡ÊÔÚÛñÞÀÎ ÊÌÍËäÐØ¯Ë¾ÒîËäÐØµèÚÐçÍÂè áÜÛV_µÍäÐÑÍìâÙÊØÚÙÉÎµËÂÒ
^ÙÊØ ËäÐÛñØÚÉÄÍ·ÐçØÚèÛñËÂÉÍäÐÑËÂÊÌÍG`Äé
Ø ÉÉ³è Ë Ê èÚÛñÒÙ³ÖµÒÒàÒîìôØ ËäÐÙÄËÂÛÜÙ¬Ðá
\ é·ØµáÜÛ ]É UÉ³ÙËÊÌÍÂÒ³íÃÐçÍÍäÐ³ìôÒëÙ¬ÐØ _ ÉÀÛÜØ¯ËÍÂÊ èÚÖµÙ³ÉèÝæ ÛñËÂÇµÊÖ«ËâËÇÚÉ Ú
ÐØµè½ÒÉ³Þ®ÐØ¯ËÂÛÜÙ¬Ðá ÐÒîÎ ÉÙËÂÒÓÊÔ Ô×ÖµØµÙËÂÛÜÊØµÒÓÐØµè Ùá×ÐÒÒÉÒ »^ ËÇ ÐÑË æ¹¿
É ÒîÛÜÞÀÎÚáñìèµÊØ ËÏÇÚÐXUÉ
ÐçËòÊÖ«ÍÈèÚÛñÒîÎ ÊÒÂÐá¾ÐçË¹ËÂÇµÛÜÒ¹Î ÊÛÜØ¯ËG` éÈêØÝÊÌÍèÚÉÄÍâËÂÊ ÍÂÉ¬ÐáÜá²ì½ÖµØµèµÉÍÂÒîËÂÐØµèÝá×ÐçËÂÉÄÍâÇµÊ¦æ ÒËäÐØµèÚÐçÍÂèÝáÜÛ _ÚÍÂÐçÍì Ù³ÊØ¯ËäÐÛÜØµÉÍÂÒãÐØµèÝÛñËÉÍäÐçËÂÊÌÍÒ
ÐçÍÉÏÍÂÉ³ÐáÜÛ³Éè¾í«Û²ËòÛñÒòØµÉ³Ù³ÉÒÒÂÐçÍì ËÂÊb¥] ØµÊÑæ Ð
_ ÊÖµËòÐçÍîÍäÐ³ì¥ÒòÐØµèÎ ÊÛÜØ¯ËÂÉÍÂÒ³é
ÆÈÇµÉÈËäÐ ] É ÇÚÊÞßÉÈÞßÉ³ÒÒÐåÉòÇµÉÍÉ ÛÜÒ ËÂÇµÛÜÒ ÛñËÕÛÜÒÃÛÜÞÀÎ ÊÌÍËÂÐØ Ë ËÂÊãåÉË ÔªÐçÞßÛÜáÜÛ×ÐçÍ æ ÛñËÂÇ¿ËÂÇµÉ
_ É³ÇµÛÜØµè ÐçÍÍäÐ³ìôÒòÐçØÚè Î ÊÛÜØ¯ËÂÉÍÂÒ³í _ Ö«ËÈÛñË¹ÛÜÒ¹áÜÉÒÒ¹ÛÜÞÀÎ ÊÌÍËäÐçØ ËËÊ_ÉâÐ _ÚáÜÉ ËÊ ÐÙÄËÂÖÚÐáÜáñìÎÚÍÊåÌÍä±ÄÐÞ ± æ î¢¾Û²ËÂºÇ
ÐçÍÍÂÐ¬ìôÒ·ÐçØÚè Î ÊÛÜØ ËÉÍÂÒ¹ÊØ Ð á×ÐçÍÂåÉÏÒÙ¬ÐáñÉÌé
2.6.13 Details
êÔ¾ìÊÖæÈÐØ ËÈËÂÊÖÚÒîÉ ÍäÐçËÂÊÒËÂÇµÉ³ØµÉ³Ò " ÛÜÉ U ÉÓËÂÊ Ù³ÊÞÀÎÚÖ«ËÂÉÏÐáÜáÎµÍÂÛñÞßÉ ØôÖµÞ
Constant expressions.
_ ÉÍÒÒÞ®ÐáñáÜÉÍ ËÂÇÚÐØ
íÌìÊÖ ÇÚÐXUÉòËÊëÙäÇ ÐçØÚåÉ ÍÂÊåÌÍÂÐÞ ÛÜÀ
Ø ÒÉ UÉÄÍäÐáÎÚá×ÐÙÉ³ÒôØµÊÌËÕÙ³ÊÖµØ ËÂÛñØÚå
10, 000
ËÂÇµÉ Ù³ÊÞßÞßÉØ ËÂÒ¤ËÂÇÚÐçË ÒÇµÊÖµáÜèZ_ ÉÖµÎè ÐÑËÂÉ³è¿ÐÒ æ¹É³áñáíçìÊÖ ØµÉ³É³è ËÂÊ ÍÂÉÎÚá×ÐçÙ³É Ô×ÊÖ«Í
ÒT_ôì
Òé
1000
10000
Æ¹ÇÚÛñÒÛÜÒ ÙÖÚÞ_ ÉÍÒÊÞßÉÈÐØµè ÉÄÍÍÂÊÌÍ» ÎÚÍÊØÚÉéS¼ÇÚÐçËÕìÊÖ®æÈÐçØ ËïÛÜÒÐãÎµÍÂÊåÌÍäÐÞ ËÂÇÚÐçËÒîÎ É³Ù³Û Y ÉÒËÂÇµÉ Ö«Î Î ÉÍb_ ÊÖµØµè UÐáÜÖµÉ ÛñØ ÖµÒîË ÊØµÉ®ÎÚá×ÐÙ³ÉÌé [ ÊÌÍËÇÚÛñÒ³íÃæ¹É ØµÉ³É³è Ð ÞßÉ³ÙäÇÚÐØµÛÜÒÞ ËÂÇÚÐçË ÐáÜáÜÊ¦æ ÒàÖµÒ ËÂÊ
åÛVUÉàÐØ ÐçÞßÉãáÜÛ ]É ËÂÊßÐ ÙÊØÚÒËäÐØ Ë ÉIP¥ÎµÍÂÉ³ÒîÒÛÜÊØ áÜÛV]É
éò
ÉÍÂÉëÛÜÒ·ÇÚÊ¦æ ËÂÇµÛÜÒ Ù¬ÐØ _ ÉâèÚÊØµÉëÔ×ÊÌÍ
n
1000
ÍäÐçËÂÊÒîËÇÚÉØÚÉÒ " ÛÜÉ UÉÌé
// Program : eratosthenes.C
// Calculate prime numbers in {2 ,... ,n -1} using
// Eratosthenes ’ sieve .
# include < iostream >
int main ()
{
// define a constant n
const unsigned int n = 1000;
// definition and initialization : provides us with
// Booleans crossed_out [0] ,... , crossed_out[n -1]
bool crossed_out[n ];
for ( unsigned int i = 0; i < n ; ++ i)
crossed_out[i ] = false ;
// computation and output
std :: cout < < " Prime numbers in {2 ,... ," < < n -1 < < " }:\ n ";
for ( unsigned int i = 2; i < n ; ++ i)
if (! crossed_out[i ]) {
// i is prime
std :: cout < < i < < " ";
// cross out all proper multiples of i
for ( unsigned int m = 2* i ; m < n ; m += i )
crossed_out[ m ] = true ;
}
std :: cout < < "\ n";
return 0;
}
Program 18: ¢¤£³ÄÁ ³£©¯º³ ¬ Å
ÆÈÇµÉ ]ÉÄìôæ¹ÊÌÍÂè
ÛÜØëÔÜÍÂÊØ¯Ë ÊÔ Ò¡èÚÉIY ØµÛñËÛÜÊØâÞ®Ð ]ÉÒ ÐÈØÚÊØÞßÊôèµÛVY Ð _ÚáÜÉ
æ ÇµÛÜÙäÇ
const
n
n
±
Ä

ÒÉÍ UÉ³Ò ÐÒòÐ Ô×ÖÚáñáñì Q ÉèÚåÉèÙ³ÊØµÒîËäÐçØ ËÈÉIP¥ÎµÍÂÉÒÒÛÜÊØ¤éÕÆÈÇµÉÏÙ³ÊÞÀÎÚÛÜáÜÉÍ æ·ÛÜáÜá ØµÊÌËòÐáÜáÜÊ¦æ ì ÊÖ ËÂÊ ÙäÇÚÐØµåÉ
ËÂÇµÉ UÌÐçáÜÖµÉ ÊÔòÐ Ù³ÊØµÒîËäÐØ¯Ë¬é [ ÊÌÍ¿ÉIPµÐçÞÀÎ áñÉÌí ËÂÇµÉ®Ô×ÊáÜáÜÊ¦æ ÛÜØµåóáÜÉ³ÐèµÒ ËÊ ÐØ ÉÍÍÂÊÌÍ¿ÞßÉ³ÒîÒÂÐåÉßèµÖ«ÍÂÛÜØµå
Ù³ÊÞÀÎÚÛÜá×ÐÑËÂÛÜÊØ¤é
const unsigned int n = 1000;
n = 10000; // error : can ’t assign to a constant
»ê Ë ÛñÒ ÐáÜÒÊØµÊÌËÓÐáÜáÜÊ¦æ¹É³èËÊÀáÜÉ¬ÐXUÉàÐ¿Ù³ÊØµÒîËäÐØ¯Ë ÖµØµÛÜØµÛñËÂÛ×ÐáñÛ³É³è¤í«ÐÒòËÂÇµÉÍÂÉëÛÜÒòØµÊßÙ´ÇÚÐØµÙ³ÉãËÂÊßÐÒÒîÛÜåØ Ð
UÐáÜÖµÉÏËÊ ÛñËÈá×ÐçËÉÍ
const unsigned int n ; // error : uninitialized constant
ÆÈÇµÛÜÒòÞßÉ³ÙäÇÚÐØµÛÜÒÞ É³ØµÒÖ«ÍÂÉ³Ò·ËÇ ÑÐ Ë·ËÂÇµÉ UÐáÜÖµÉãÊÔ ÐÙ³ÊØµÒîËÂÐØ Ë·ÛÜÒ]ôØÚÊ¦æ Ø ÐÑË·Ù³ÊÞÀÎÚÛÜáñÉÓËÛÜÞßÉÌí ÖµÒîË
áÜÛV]É ËÂÇµ
É UÌÐçáÜÖµÉãÊÔ ËÇÚÉÏáÜÛñËÉÍäÐá
é
1000
ÍÊåÌÍäÐÞ Ô×ÊÌÍëÒËÍÂÛÜØµåÞ®ÐÑËÂÙäÇÚÛñØÚå«í Û²ËâÛÜÒÏØµÊÌË UÉÄÍìÝÙ³ÊØ UÉ³ØµÛÜÉ³Ø¯Ë
é S É¿ËÇÚÉØ Ç ÐXUÉ¿ËÂÊ ÍÂÉÙ³ÊÞÀÎÚÛÜáÜÉàËÂÇµÉ¿ÎµÍÂÊåÌÍÂÐÞ ÉIUÉÍì ËÂÛÜÞßÉàæ¹É
æÈÐØ¯ËòËÂÊ¿ÒÉ¬ÐÑÍÂÙäÇ Ô×ÊÌÍ ÐØµÊÌËÂÇµÉÍÈÚÒîIËÍF ÛÜØµå«é
ÞßÊÌÍÂÉQ ÉGP«ÛV_ÚáÜÉòÐáñËÂÉÄÍÂØÚÐçËÂÛVUÉòÛÜÒÕËÂÊëÎ ÐÒîÒËÂÇµÉ ÒÉ¬ÐÑÍÂÙäÇ ÒîËîÍÂÛÜØµåàÐÒïÐ
±¨¨ ©F ° À©£îÁ ¨ ³¤
ËÂÇÚÐçË¹æ¹ÉãÎµÍÂÊU¥ÛñèÚÉÓÖ«Î ÊØóÙ³ÐáÜáÜÛÜØµåëËÂÇµÉÓÎµÍÂÊåÌÍäÐÞ é
ÆÈÇµÉÞ®ÐÛÜØ Ô×ÖÚØµÙËÛÜÊØÝÙ¬ÐØÝÐÙ³Ù³É³ÒîÒëÒÖµÙäÇ Ù³ÊÞßÞ®ÐØµè½áñÛÜØµÉÐÑÍÂåÖµÞßÉ³Ø ËÂÒÓÛÜÔÃæ¹ÉÎµÍÂÊU¥ÛÜèµÉ¿ÒÖµÛñËäÐ _ÚáÜÉ
Î ÐçÍäÐÞßÉÄËÂÉÍÂÒ³é ·ÉÍÂÉ ÛÜÒÓÇµÊ¦æ ËÂÇµ
É YµÍÂÒËÏËÉ³Ø áÜÛñØÚÉÒ ÊÔ ÍÂÊåÌÍäÐÞ ÇÚKÐ UÉ ËÂ
Ê _ É ÙäÇ ÐçØÚåÉèÝÛÜØ ÊÌÍÂèµÉÍ
ËÂÊÞ®Ð ]É ËÂÇµÛÜÒïæ¹ÊÌÍ ]é
Command line arguments.
ËÂÇÚÐçËÏËÂÇµÉÒîÉ¬ÐçÍÂÙäÇ ÒîËÍÂÛñØÚå ÛÜÒ
êØ
// Program : string_matching2.C
// find the first occurrence of a string ( provided as command
// line argument ) within the input text , and output text so far
# include < iostream >
int main ( int argc , char * argv [])
{
if ( argc < 2) {
// no command line arguments ( except program name )
std :: cout < < " Usage : string_matching2 < string >\ n" ;
return 1;
}
// search string : second command line argument
char * s = argv [1];
ÆÈÇµ
É UÌÐáñÖÚÉÒ ÊÔ
ÐØµèàÐØµè
^»æ·ÇµÛÜÙäÇëÛÜÒ ÐçØàÐçÍÍäÐ³ìãÊÔôÎ ÊÛÜØ¯ËÂÉÍÂÒ¡É¬ÐÙäÇàÊÔôæ·ÇµÛÜÙäÇëÛÜØâËÂÖ«ÍÂØ
Î Ê ÛÜØ¯ËÂÒ ËÂÊ ËÂÇµÉ YµÍÂÒîËÃargc
É³áÜÉÞßÉ³Ø ËÂÒ ÊÔÐ argv[]
³ÉÍÊ ËÂÉÍÞßÛÜØÚÐçËÂÉ³è ÐçÍîÍäÐ³ì ÊÔ ÙäÇÚÐçÍäÐÙÄËÂÉÍÂÒ ` ÐçÍÉïÛÜØµÛñËÂÛ×ÐáñÛ³É³è _ ìàËÂÇµÉ
ÊÌÎ ÉÍÂÐçËÂÛÜØµå ÒîìôÒîËÂÉ³Þ æ ÇµÉ³ØÛñË Ù¬ÐáñáÜÒ ËÂÇµÉ
Ô×ÖµØÚÙÄËÂÛÜÊØ¤éTSÝÉïæ·ÛÜáÜáôÉGP¥ÎÚá×ÐÛÜØ Ô×ÖµØµÙËÂÛÜÊØ Î ÐçÍäÐÞßÉÄËÂÉÍÂÒ ÛÜØ
main
èµÉËäÐÛñáÚáÜÐçËÂÉÍ¬í¯ÇµÉÍÂÉÈæ¹Éòæ ÛñáÜá _ É·ÒÂÐÑËÂÛÜÒ YÚÉ³è æ·ÛñËÂÇßÐçØÀÉIPµÐçÞÀÎ áñÉÌé " Ö«ÎµÎ ÊÒÉòËÇ ÐÑËÕæ¹É Ù³ÐáÜá«ËÂÇµÉòÎµÍÂÊåÌÍäÐçÞ
áÜVÛ ]É ËÂÇµÛÜÒ ^ºÐÒîÒÖµÞßÛÜØµå Ð ·ØµÛ P Ë ìôÎÉãÒîìôÒîËÂÉ³Þ `
./ string_matching2 bool
ÆÈÇµÉ³Ø
^»æ·ÇµÛÜÙäÇ ÙÊÖÚØ¯ËÂÒëËÂÇµÉßØôÖµÞ_ ÉÍâÊÔòÙ³ÊÞßÞ®ÐçØÚè½áÜÛÜØµÉßÐçÍåÖÚÞßÉØ ËÂÒ `ÓåÉËÂÒàÛÜØµÛñËÛ×ÐáÜÛÉ³è
argc
æ Û²ËÂÇLUÌÐçáÜÖµÉ éÃÆÈÇµÛÜÒ ÙÊÖÚØ¯Ë¹ÛÜØµÙ³áÜÖµèµÉ³ÒËÂÇµÉ ÎµÍÂÊåÌÍäÐçÞ ØÚÐÞßÉ·ÛñËÒÉ³áÜÔ ^
ÛñØ®ËÇÚÛñÒ
Ù¬ÐÒîÉ `Äí¡ÐØµè 2ÐØ ì½ÐèµèµÛñËÂÛñÊØ Ðçá ÒîËÍÛÜØµåÒÏÎµÍÂÊU¥ÛÜèµÉ³è½ÊØ ËÂÇµÉ¿Ù³ÊÞßÞ®ÐØµè "string_matching2"
áÜÛÜØµÉ ^ ÖµÒîËãËÇÚÉ¿ÒÛÜØµåáÜÉ ÒîËîÍÂÛÜØµå
ÛñØ ËÂÇµÛÜÒÈÙ¬ÐÒKÉ `ÄéÆ¹ÇµÉ ÐçÍÍÂÐ¬ìôÒ
ÐØµè
åÉËòÛÜØµÛñËÂÛ×ÐçáÜÛ³É³èÀæ·ÛñËÂÇ ËÂÇµÉãÒîËîÍÂÛÜØµåÒ
"bool"
2
argv[0]
argv[1]
ÐØµè
ÐÒàèµÉ³ÒÙÍVÛ _ É³è ÛÜØ " É³ÙËÂÛÜÊØ \ é é Ð _ ÊUÉÌé&ïÊØÚÒîÉXWôÖµÉ³Ø¯ËÂáñìí
"string_matching2"
ÐÔÜËÂÉÄÍâÛñËÒ èÚIÉ Y ØµÛñËÛÜÊØ¤íËÂÇµÉàÎ ÊÛÜ"bool"
Ø ËÉ
Í UÌÐçÍÛ×Ð _ÚáÜÉ ÛñØ ËÂÇµÉÐ _ Ê UÉàÎ ÛñÉ³Ù³É ÊÔÙ³Ê èµÉ¿Î ÊÛñØ ËÂÒ ËÂÊ®ËÂÇµÉ YµÍÂÒîË
É³áÜÉÞßÉ³Ø ËòÊÔ $
Ð ³ÉÄÍÂÊ ËÂÉÄÍÂÞßÛÜØÚÐçËÂÉ³èóÐçÍÍäÐ³ì ÊÔ Ù´ÇÚsÐçÍÂÐÙËÂÉÍÒ·ËÂÇÚÐçË ÙÊÌÍÍÂÉ³ÒîÎÊØÚèµÒ·ËÊ ËÂÇµÉâÒîËîÍÂÛÜØµå
é
"bool"
Æ¹ÇÚÛñÒëåÉËÂÒëÖµZ
Ò _ ÐÙ ]ÝËÂÊ ËÂÇµÉßÒÛñËÂÖÚÐçËÂÛñÊØÝÛÜØ ÍÂÊåÌÍäÐÞ ÐçÔÜËÂÉÍàáÜÛñØÚÉ í ÐØµè ËÂÇµÉ ÍÂÉ³Þ®ÐÛñØÚèµÉÍÒâÊÔ
_ ÊÌËÂÇ ÎµÍÂÊåÌÍäÐÞßÒÈÐçÍÂÉÏÛÜèµÉ³Ø¯ËÂÛÜÙ¬Ðá»é
2.6.14 Goals
Ë¹ËÂÇµÛÜÒ¹Î ÊÛÜØ¯Ë¬íôìÊÖóÒîÇÚÊÖµáÜè é³é³é
¹
Dispositional.
`L]¥ØµÊ¦æ
æ ÇÚÐçË·ÐØ ÐçÍÍÂÐ¬ìÛÜÒ³í ÐØµè æ ÇÚÐçË·ÍÂÐØµèÚÊÞðÐÙ³ÙÉ³ÒÒÓÐØµè ÛñËÂÉÍÂÐçËÂÛÜÊØóÞßÉ³ÐØóÛÜØóËÂÇµÉëÙÊØ ËÂÉIPôË
ÊÔ ÐçÍÍäÐ³ìôÒ
\ `ßÖµØÚèµÉÍÒîËäÐØµèËÂÇµÉÓÎ ÊÛÜØ¯ËÂÉÍÈÙ³ÊØµÙ³ÉÎµË¬íÚÐØµèóÇµÊ¦æ ËÊÙ³ÊÞÀÎÚÖ«ËÂÉ æ Û²ËÂÇóÐèµè«ÍÂÉ³ÒîÒÉ³Ò
`L_ ÉãÐ¬æÈÐÑÍÂÉÏËÇ ÐÑË ^ÐØµèóÖµØÚèµÉÍÒîËäÐØµèæ·Ç ìa`ïÐçÍîÍäÐ³ì¥Ò ÐçØÚè Î ÊÛÜØ¯ËÂÉÍÂÒÈÞÖµÒîË _ ÉÏÖµÒÉ³èóæ·ÛñËÂÇ Ù¬ÐçÍÂÉ `L]¥ØµÊ¦æðËÂÇÚÐçËÙäÇÚÐçÍäÐÙËÉÍÂÒÀÐçØÚè ÐÑÍÍäÐ³ì¥ÒÊÔ ÙäÇ ÐÑÍäÐÙËÂÉÄÍÂÒ Ù¬ÐçØ _ ÉÖµÒÉ³è ËÊ Î ÉÍÂÔ×ÊÌÍÞ _ ÐÒÛÜÙ®ËÂÉIPôË
ÎÚÍÊôÙ³É³ÒîÒÛÜØµåËäÐçÒ ]¥Ò Ë ^»ÞÖµáñËÂÛñèÚÛñÞßÉ³ØµÒÛÜÊØÚÐçá `¾ÐçÍÍäÐ³ìôÒïÊÔ UÐçÍÂÛÜÐ _ÚáÜÉ·áÜÉØÚåÌËÇßÙ¬Ð
Ø _ É Ê _µËäÐÛÜØµÉ³L
è _ ìßè«ì¥ØÚÐÞßÛÜÙ
`L]¥ØµÊ¦æ ËÂÇÚÐç
ÞßÉ³ÞßÊÌÍì®ÐáñáÜÊôÙ¬ÐÑËÂÛÜÊØ Operational.
^
êØÎ ÐçÍËÂÛñÙ³Öµá×ÐçÍ¬í¯ìÊÖ ÒÇµÊÖÚáñè _ ÉãÐ _ÚáÜÉ ËÂÊéé¬é
`ßÍÉ¬Ðè¤íÌÖµØµèÚÉÄÍÂÒîËäÐØµè¤íÐØµè ÐçÍåÖÚÉ Ð _ ÊÖ«Ë ÒÛñÞÀÎ áñÉ ÎµÍÂÊåÌÍÂÐÞßÒ ÛÜØ UÊáVU¥ÛñØÚåòÐçÍÍÂÐ¬ìôÒ ÐØµèëÎ ÊÛÜØ¯ËÂÉÍÂÒ
\
\ `ßæòÍÂÛñËÂÉãÎÚÍÊåÌÍäÐÞßÒ·ËÂÇÚÐçË èÚÉIY ØµÉ¿ÐçÍÍÂÐ¬ì UÌÐçÍÛ×Ð _ÚáÜÉ³Ò ÊÌÍÏè«ì¥ØÚÐÞßÛñÙ¬ÐáÜáñì ÐçáÜáÜÊôÙ³ÐçËÂÉ ^»ÞÖµáñËÂÛñèÚÛñÞßÉ³Ø
ÒÛñÊØ Ðçá ` ÐçÍÍäÐ³ìôÒ
^
`ßæòÍÂÛñËÂÉ ÎµÍÂÊåÌÍäÐÞßÒ®ËÂÇÚÐçË ÍÂÉ¬Ðè Ð ÒîÉXWôÖµÉ³ØµÙ³É ÊÔàèÚÐçËäÐ ÛÜØ ËÊ Ð ^è«ìôØ ÐçÞßÛÜÙ¬ÐáÜá²ì ÐáÜáÜÊ Ù¬ÐçËÂÉè
Þ¿ÖµáñËÂÛÜèµÛÜÞßÉ³ØµÒÛñÊØ Ðçá ` ÐçÍÍäÐ³ì ^
`ßæòÍÂÛñËÂÉ ÎµÍÂÊåÌÍÂÐÞßÒ ËÂÇÚÐçË Î ÄÉ ÍÂÔ×ÊÌÍÂÞ ÒÛÜÞÀÎÚáÜÉèÚÐçËÂÐ ÎµÍÂÊ Ù³É³ÒÒîÛÜØµåÓËÂÐÒ ]¥Ò_ôì ÖµÒÛñØÚåÓÍäÐØµèµÊÞ
ÛÜØ ^ Þ¿ÖÚá²ËÂÛÜèµÛÜÞßÉ³ØµÒÛÜÊØÚÐá ` ÐçÍ ÍäÐ³ì¥ÒòÐÒÈËÂÇµÉãÞ®Ð ÊÌÍïËÂÊ Êá ^
`ßæ·ÛñËÂÇµÛÜØ ÎµÍÂÊåÌÍÂÐÞßÒ³í¯ÛñËÂÉÍäÐÑËÂÉ ÊUÉÍòÐ ^è«ì¥ØÚÐÞßÛÜÙ³ÐáÜáñì¿ÐáñáÜÊôÙ¬ÐÑËÂÉ³è
_ ì ÖµÒÛñØÚå¿Î ÊÛÜØ¯ËÂÉÍòÐçÍÂÛñËÇÚÞßÉÄËÂÛÜÙ ^
^
`ßæòÍÂÛñËÂÉÏÎµÍÂÊåÌÍäÐÞßÒÈËÂÇÚÐçË Î ÉÍÔ×ÊÌÍÂÞ
ËÂÉÄÍÂÒ
ÐÙ³ÙÉ³ÒÒ
Þ¿ÖÚá²ËÂÛÜèµÛÜÞßÉ³ØµÒÛÜÊØÚÐÑá `¡ÐçÍîÍäÐ³ì
ÒÛÜÞÀÎÚáÜÉÓËÂÉIPôË ÎµÍÂÊôÙ³ÉÒÒÛÜØµåßËäÐÒ ]ôÒ æ ÛñËÇ ÐçÍÍäÐ³ì¥Ò ÊÔ ÙäÇÚÐçÍäÐÙ 2.6.15 Exercises
Exercise 65
©KM Ý©¯ Fµ¬ ºÈ¶°§¸¹»Á¿¢¤£³ÁÌ£©¨
ñ¢¤ ©Fó¸ÈJ
# include < iostream >
int main ()
{
int a [] = {5 , 6 , 2 , 3 , 1 , 4 , 0};
int * p = a;
do {
std :: cout < < * p < < " " ;
p = a + * p;
} while ( p != a );
return 0;
}
M
£³ Á« ³£©¯» J ó¯¢«èµ¢ Û ÉÄ ÍÂÉ³Ø ºË ©¯ß» º ¢¤£³XDî ¢¤£³Á£©¨ a ° ©¯ IF¼¸¹º
¨½à ³ ±ÄßÂ¶ n
¨½ ³£¦ë» {0, . . . , n − 1} Hî¸Ó àº¥»¶£ n = 7 °
ºÏ¢¤£³X DÌ ¢¤£³Á£Ä©¨ M¥Å È£³ Dôßº©¯·ºÏ¢¤£³Á£©¨ ³£¬¨ » ©¯¬¿» º¥»®±´©¯Å
^
Exercise 66
ª³Áº n Å
©KM
a
³ ¨ àº©¯È» ¨½Ó¢¤£Á£©¨
`
© ©£¬£©JÝÂ¶âFµ³£¦ J °Áó J¦¢ int ©F
D¥³ © ôD ©£¬ ©« ¦ª i Â¶ J¦¢ int ¸¹º D ©¯¡ 0 i
n ¸ ±Ä© JÚ ¥ ¦î©° ©
¢ °¤³£ p ºó³§³¨½³¤àÂ¶°Fµ i ° a H ¶ i = n º¥¿®©¯ »Áâ¶£©
¢©¯ îº ´³ß
F ¢ »¤£ÌÅ M
M
D¥³ ©¢ °¤³£ p ç ¨ ¬§¨ ³¤ë° a â¸ ±´© J ¥ ¦î©° º » Fµ i Â¶
º¥»¬ª³¨ ¤ H ¶ p »¿
©ë¢©¯³ îº ´³F ¢ »¤£ ïºë»Fµ ®Fµ IF ©¯ n Å M
£¬ ± Fµò¶£©Á¨½³¤º º©¯ã±¨¢¤ p ¶£³¨
i »¼©KM½©F i ¶Ä£³¨
p »
^
M¥Å
`
º Ù³ÊÞÀÎÊÒÛñËÂÉ
¶ß©F ¤ J½ ¶ F D ¦§ KJ
Exercise 67 ¬ ®±Ä©¯°¹© ©¯ £©¯ ¨½ ³£ k
µ ©µ±³ºEJ k F ³ £³¤ã¢¤£¬»¨½ ¨½ ³£¦çÅ £«©¨¢¾ª ã¢¤£¬°¨ ß¢ ¸Ó³£¦ ©£³ ´±¨¿¢
©F 6 = 2 3 ©¯ÓÏ
¸ ³» ©¯ 20 = 2 2 5 ©£³ Ä±Ä¨¢ »Å
£¬ ©â¢¤£³ÁÌ£©¨ k_composite.C
0 ©F k
0 ¶Ä£³¨ º »¥¢¤Ý©F º ñ¢¤º ©¯» k
º©¯ £³Ä© F¯ ¨½ ³£¦ n
±¨¿¢ » ¨½ ³£¦ ° {2, . . . , n − 1} Å
¸ ¨½© J ´±¨¿¢ ¨½ ³£¦ ©^ £³ \ Ìº^ ³£³ `
¶£ n = 1, 000, 000
£¬ ©Ï¢¤£Á£©¨ invert.C º©¯ » Dô³£¦º © 3 3 ¨ ©¯£¬ A ¸¹º®£³´©¯ ¤£¬î¬çÅ
Õâ¢¤£³Á£Ä©¨ ³ F £³´© FÝºß ° ß¨½©¯£¬ ¤£¬î¬ ¶Ä£³¨ ºß°¥¢¤ â©F º³ ñ¢¤
ºà» Dô³£¦à¨½©¯£¬ A−1 H£ ºà»¶ç£¬¨½©¯ º©¯Èºà¨ ©¯£¬ A â ï° D¥³£¦ ¬§ MôÅ Â
© F F º ¢¤£³ÄÁ£©¨ ³ F ñ¢¤ïºà¨ ©¯£¬ AA−1 » £ Fµ³£¿ ª¬¹ºà³£®±¬Ä±
¸È¬º³£½º ±¨¢¤î©¯î Â¶óº » Dô³£¦ ¸ ©¯ ©µ±±³ £©¯ H» º ¶°EJ ©µ±±£©¯ ±Ä©¯
ºÀ×©¯º³£â
¢¤£ F ±¬ò º Fµ³¤ J ¨½©¯£¬ M¥Å
ïÍäÐçÞßÉÍ! Ò ÍÂÖµáÜÉ Å ¢«¢¾I F
Hint: £ëº±¨¿¢¤î©¯î Â¶Óºâ° D¥³£¦ JÚ ±Ä© ³¨¿¢¾§ J
−1
−1
ºß±¨¿¢¤î©¯î¼Â¶ëºà° D¥³£¦ ïR
J ¬ F¯ëº©¯ Aij Hîºß³¤K£ JÝÂ¶ A ° £³¸ i ©
F
±¨ ¤ jM» ÁÌ D¥³ K J
èµÉË ji
(−1)i+j
(A )
−1
Aij =
,
èµÉË
Exercise 68
(A)
èµÉË
(M) àº Fµ¬³£¬¨ » ©¤ÏÂ¶© ¾©£³ ½
¸È³£³
¨ ©¯£¬ M ·©F Aij àº 2 2 ¨ ¯© £¬ ¥ ¦î©° I Fà¶Ä£³¨ A KJ Fµ¬ª¬»Á£³¸ j ©FÝ±¨ i Å
± " Ðç¨ÍîÍÂ¢¤ÖµÒ « ÍÂÖµºáÜÉ Fµ¬³£¬¨ » ©¤âÂ¶© 3 3 ¨ ©¯£¬ RJÚ ¨§Á«ò¸ ©¤· ^ \ ¿Ì^ º Ì¸Ó
^ ¬ » `
¸¹
Å
¬£ ©½¢¤£³ÄÁ£©¨ read_array º©¯£³´© F¯ © ³ ±ÄÂ¶ n °¤Á«³£¦¿¶£³¨
³î©F«©£ F °¥¢¤ °¤ ©Ý©£¬£©JµÅßÕ ¨ ³³£ n Èº £¦³ °¥¢¤ Ã©F¿º³ º¢¤£³ÁÌ£©¨
¦¢±¬º JÚ »¥¢¤ © º³£ n Dô©¯¡¬çÅ
¶³£ £³´© F»Á º n Dô©¯ ¬ ßº ¢¤£³ÄÁ£©¨
³ F ñ¢¤ãº¨ ° º ç©¨½ I£ Fµ³£ÌÅ H ×¶ JÚ ±Ä© Fµ º¥» J ©D¥à¢¤£³ Dô³ º©¯
JÚ ©£³ó ªÁ«³£ © ±¨¿¢¾§³ DÌî±Ä ß©µ±±£IF»Á ¤£³£¬ ¢ Å M
£ µ©¨
^ ¢¾\ ªÌ ^ ß `
»¥¢¤ 5 4 3 6 1 2 ºÏ¢¤£³Á£Ä©¨ ³ F½ñ¢¤ 4 3 6 1 2 Å
Exercise 69
±³³¤
Exercise 70 È¤© ±Äßºã¢¤£Á£©¨ read_array.C ¶Ä£³¨
Ú ³£±³
º©¯ º ³£ ³
»Á ¢¤£³Á£Ä©¨ sort_array.C £¦º º¼©£¬£©J ¬ª³ ¨ ³¤º » ¤ ©¯±Ä³F»Á £ µF ³£ ¶£³
ñ¢¤º»Áº³¨½Å £¿£¦»Á ©¯ Á«£¬º¥¨ FµÌ¬ ©D¥½ ã¢©£¦±³×©£¦ J
º ¨½©» º¥»Á ³£³ ®º©¯Ïã¸Ó£ Ì ±Ä£¬£³±¬º JµÅ ¬ JÚ£ ¢¤£³Á£Ä©¨ ¨ ×©£îÁ«³£
»¥¢¤º H§¢¤£³¶£©« ¦ J½£³´© Fà¶£¨ © § Ó©ä¶³£À£³IF°£³±³»Á½³î©F«©£ F»¥¢¤VMôÅ ^ \ £óÌ^ µ© ¨¿
Ì^ ¢¾§ `
°¥¢¤ 5 4 3 6 1 2 ºã¢¤£³ÄÁ£©¨ F½ñ¢¤ 1 2 3 4 6 Å
Exercise 71 È¤© ±Ä º ¢¤£³Á£©¨ read_array.C ¶£³¨
µ³£³±³» ëëº©¯¡º·£³¬Ä»Á
ÎÉÍÂÞ¿ÖµËÂÐçËÂÛÜÊØ
¢¤£³ÁÌ£©¨ cycles.C »¤³£¢¤£¬º ºë»¥¢¤ïÙì¥Ù áÜÉâ
¡èµ³ É³Ù³±Ê®ÞÀÂÎ¶ ÊnÒÛñËÂ»¤ÛÜÊØ Á«³£¦À©¯¿©
π Â¶
{0, . . . , n − 1} ©Fóº©¯·Ïñ¢¤º¿º
Â¶ π Å
¨ Ý ¦ ¢¾ª© ©¯¤ ©£³ » £IFµ³£ ©ó¢ £¬¨î©¯ π © ³ Ä±¬ Dôó¨½©¬¢«¢¤°ÁÀ¶£¨
º à³ {0, . . . , n − 1} ßº³ ¶ º³£³»¶£³ ¹ºâ»¥¢¤ïç ¡³ ±ß±´© ë»¤£¢¤£³¬IF ©¯ëº
³ ±ÄÝÂ¶ Dô©¯ ¬ π(0), . . . , π(n − 1) Â¶ó© ¢ ³£¬¨ î©¯î π ¶ó©F¤ J ¶ß»±¤î©°¤
XD¥£KJ ¨½ ³£ ¶£¨ {0, . . . , n − 1} µ ©µ±³ºEJ ±çÅ
¢¤£³Á£Ä©¨ cycles.C ³ F £¦³±¬Ä± ¸È¬º³£ ºÝ°¥¢¤ ¡³ ± Ñ©¯ ¬
º¥½±F ß©F ¶ ¿³£¬¨ » ©¯ ¸¹º © ±£¬£¬¢ F°Á¨½¬³Ùçìô©Ù³Á«áÜÉ çÅ èµÉ³× Ù¶ó
ºÎ ÊÒ°¥ÛñËÂ¢¤ÛñÊØ
ÊÞÀ
»FµÄIF ³ ±Ä Fµ¬À©ë¢ ³£¬¨ î©¯ π º ¢¤£³Á£©¨ ³ F ñ¢¤ïº
Â¶ π Å
±XJÚ±¬ª » π ß© J ³ ±ÄóÂ¶¿ºò¶ç£¬¨ ( n1 n2 nk ) ±³½º©¯
n2 = π(n1 ),
n3 = π(n2 ),
nk = π(nk−1 ) ©F n1 = π(nk ) Ó©F
...,
n1 »º ¨½©¯°§³³Ï³§³¨½³¤â©¨ Á n1 , . . . , nk Å
J ±XJÚ±¬§ ¬ J Fµ¬³£¬¨ » ¬º π ND ©¯¡¬ Â¶À©¯° º®¬ª³¨ ³¤º ºº³£ß©F
XD¥£KJó¬ª³¨ ³¤ï©¬¢«¢ ´©£¦ » ¨ à± J ±¬ªbHî¸È¥î±¬À¨ °Á« ¿Â¶ ºÓ£¬ DÌ ©¯ç¶£¬¨ (n1 ) ¨ ´©
»Á º©¯ π(n1 ) = n1 MôÅ½Õ¥¿»¨¿¢¾¬ º©¯ ºâ¢³£¬¨î©¯î Fµ±Ä¨¢ ³ »¤Ý© ¡
¬ãÂ¶ßX± J ±³§¬ÑÅ £ «©¨¢¾ª Èºã¢ ³£¬¨ î©¯ π Á D¥³ KJ
π(0) = 4,
π(1) = 2,
π(2) = 3,
π(3) = 1,
π(4) = 0
Fµ±Ä¨¢ ç¬°¤óºÀ¸ÓÝ± J ±¬ª¬ ( 0 4 ) ©F ( 1 2 3 ) Å
^
\ Ì^
`G^
`
Exercise 72 Ó¤ÄNFµ³£ º ³ £¬»Á½¨½©¯±¬¥°Á ©¯ Á«£¬»º¥¨ Â¶ È£Á£©¨ ôÅ ¹£³ D¥ º©¯Ã¶ç£
©¯» m > 1, n m ïº³£ ³º © ´©£³±¬ ³£¬»Á s Â¶ëª³Áº m ©F © t Â¶ëª³Áº n
¸È¥î±¬ º©¯Á«£¬»º¥¨ ° È
£³ÄÁ£©¨ ã¢ £º¶£¬¨ó m(n − m + 1) ±Ä¨¢©£¬»¤ ¬¸Ó³
°Á§ ±¬©£©µ±¬£¦çÅ
Exercise 73 Ó¤ÄNFµ³£Óº ¶ç»ª¸È°Á ¢¤£³ÁÌ£©¨
©¯©£¬£©JÚÅ
^
º©¯ Fµ ¬â©F¿» »©¯ ¬Ó©º¥£³GF»¨
# include < iostream >
int main ()
{
int a [4][2][3] =
{ // the 4 elements of a:
{ // the 2 elements of a [0]:
{2 , 4 , 5} , // the three elements
{4 , 6 , 7} // the three elements
},
{ // the 2 elements of a [1]:
{1 , 5 , 9} , // the three elements
{4 , 6 , 1} // the three elements
of a [0][0]
of a [0][1]
of a [1][0]
of a [1][1]
`
},
{ // the
{5 , 9 ,
{1 , 5 ,
},
{ // the
{6 , 7 ,
{7 , 8 ,
}
2 elements of a [2]:
0} , // the three elements of a [2][0]
3} // the three elements of a [2][1]
2 elements of a [3]:
7} , // the three elements of a [3][0]
5} // the three elements of a [3][1]
};
return 0;
}
¬£ ©Ý¢¤£Á£©¨ threedim_array.C º©¯ó³¤© ±¬ º¥ ¢¤£Á£©¨ K J © Hî ¬³GFKM
int Dô©¯ ¬
ªÌ³¢ º©¯È»³£©¯¬® D¥³£¿ºß©£¬£©J a ©F ºà¾ ©£¬£©J à½ñ¢¤ ©¯»¾º
º©¯À©£³ ³£³G F ° a ©
F »º ¡ Ä©£¬£©JôÑÅ
¿ó£©Fµ¨ ©µ±Ä±¬ Fµ º¥» ^ ¬ `
¢ °¤³£©£¬º¥¨½¬î±çÅ
Exercise 74
£¬ © ¤
¢ £Á£©¨ frequencies.C
º ©¯£³´© F¯ © ò¶£³¨
H ô » ¹
£³ÁÌ£©¨ M ©F ñ¢¤º ºã¶£³ ³ ±î¬ÝÂ¶ º ª ¬º³£¦°
³ î©F«©£IF°¥¢¤
ºó à¸È³£³
¸Ó Fµó F»³»Á ³ ³³¸ÏÄ³ ª¸Ó³£ó©F ¯¢«¢ ³£ó±´©¯ßª¬º³£¦çÅ £ßº¥ Ú³£±³ J
¨½©J ©¯³¨½ º©¯âº JÑ¢ char »¨¢¾ª³¨½³¤º
³ ± F»Á¥Å ¥»ß¨ Ä©¤ º©¯¿©¯°
±¬©£©µ±³³£¦À©D¥À°¤Á«£ D ©¯¡¬° {0, 1, . . . , 127} Å
£³ Dô³£ »
òºL
Dô©¯ ³ Â¶
º 26 ¯¢«¢ ³£ ±Ä©¯»³£©¯ ’A’ ¢ ’Z’ ©£³ ±¤Ä±³ D¥ ¨½ ³£¦¿° {65, . . . , 90} ^ ¶ £ `
ºÀ ªÖµ¸Ó

£Ä©É³ÐçÁ«ËÂáÜ É³Ò »`È Ô×{97,
ØµØµ³£ó
ÛÜØµå±´©¯ËÂÇµ ÛÜÒ·ÊØó£©¯ËÇÚ² Éâ’a’
áñìôÍÂÛñÙ³ Ò·¢ ÊÔ ’z’ ÈºbDô^º©¯ÆÈ ÇµÉ
ÊÌÍ ÉI.P«.Ð.ÞÀ, 122}
ÎÚáÜÉÏÒÅ ÇµÊÖÚáñèóì¥ÛÜÉ³áñèËÂÇµÉ
¬

£
«
F
©
J
Ô×ÊáÜáÜÊ¦æ ÛñØÚåàÊÖ«ËÎÚÖ«Ë¬é
Frequencies:
a:
45 of 520
b:
5 of 520
c:
5 of 520
d:
28 of 520
e:
65 of 520
f:
4 of 520
g:
13 of 520
h:
27 of 520
i:
j:
k:
l:
m:
n:
o:
p:
q:
27 of 520
0 of 520
3 of 520
20 of 520
10 of 520
30 of 520
43 of 520
4 of 520
0 of 520
r:
s:
t:
u:
v:
w:
x:
y:
z:
Other :
19 of 520
36 of 520
31 of 520
9 of 520
6 of 520
19 of 520
0 of 520
34 of 520
0 of 520
37 of 520
2.6.16 Challenges
Exercise 75 Õ¶ç©µ±¬ïº©¯·©
© £¬£Ä©J©¯
¢¾ª ¹» Ú ³£±³ ½©F» Ú ³£³±
ÚIF
ª³Áº ¿Â¶³³ ° ± ¥D î³¤îÅ £ µ©¨
¹º ¨ ³³£Â¶ ³§³¨½³¤º ½ ³ £Ä© F ° ¤ º
©£¬£©J © F ¢¤£³ DNFµIF ©¯ º £¦³Ï»¥¢¤â» £IFµ³£ë¶£óº¢¤£³ÁÌ£©¨ ½©« ¦ª
FJ¯ ©¨î±´©¯» J ©¯»ªÌ±Ä©¯À© ©£¬£©J Â¶ãº ©¬¢«¢¤£³³¢¤£¬ ©¯à§ÁºÅ » ¢¤£©µ±¬î±Ä ¹ºàª³Áº
Â¶¿º°¥¢¤ò ¡³ ± ®Â¶³³ ¸¹ © ¢¤£¬£¬Å
¸Ó F º³£¶£³ ôÝ ¸¹£¬ ©½¢¤£³Á£Ä©¨ º©¯ £Ä© F¯ © ³ ±Ä Â¶ »¤ Á«³£¦
¶£¨ ³î©
F«©£ F °¥¢¤ »¤ © ©£¬£©J ¸È³£³®º®ª³Áº Â¶Àº® ±® ¸¹
³»¶£³¬©
F H´©
F ¢©£¦ Â¶·º·»¥¢¤V M ºÕ¢¤£³ÁÌ£©¨ ³ Fë»¨¿¢¾EJ £³´© F® ¨ ³³£
©ä¶³£ © º³£À ¤ º ³£³Ä©¨ ³Ä±¨ ³³¨¢¾ JµÅ
Ó ¢ ¦ª ³£Ä©¯Á J »Ý FJ ©¨ î±´©¯» J ©¯°§Ì±´©¯ © ©£¬£
© J Â¶ ª©£Á« §³Áº ¬§Á
³ ÚÁ £³ © Jß¢ ³Ä ¦ª°¥¢¤ò ¡³ ±ç
Å ò ¶¿ºÀ ³ ±ÄÀ¿³£¦ Èº¥ ß©
¥ Á¥ ¸ ©¯³ Â¶À¨½³¨ K£ J ë©
F½ ¶ßº ³ ±Ä »L
D¥³K£ J ªÁ Óº ©£¬£© JÝ¨ °Á«Ï³°Õ
³ ª©£Á«³ ÚÁÅ
©KM
¬£ ß© ¢¤£³ÁÌ£©¨ read_array2.C º©¯È£³´© F¯À©ç ¡
³ ± Â¶ë»¤ Á«³£¦ßÂ¶ë ¸¹
ª³Áº »¤ © ©£¬£©J ©F º³ ñ¢¤ºÝº ¡
³ ± çÅ Õó¢¤£³Á£Ä©¨ ³ F
ç©¯»»¶ Jóº·¶°§¸¹°Áó¸Ó¢¤£³¢ ³£¦î³çÅ
HîEM Õ ©¨ ¤ Â¶ F J ©¨ ÎÚÍîÊÌ±´Î ©¯»ÊÌ ÍJ ËÂÛÜÊ©¯ØÚ»ªÐÌá«±´ËÂ©¯Êë
IËF ÇÚÉ·¨½Øô³Öµ¨ Þb_£KÉJ ÍïÊ»Ô ÒÉXW çÖµ É³Øµ KÙ³J ÉÓÉ³ºáÜÉ³ÞßÉ³¢¤Ø¯£³ËÂÒÕÁÌË£Ç ©ÐÑ¨ Ë
_ F É³É³©¯Ø ·©ÍÂÉ¬ ÐJ
çÐ Í ³
ÇÚÐXUÉ
è Ò°Ê ¨ ÔªÀ
Å ±Ä ±³£³¬ º³£³ ¨³ ³ß© ¢ » ¥D ®±¤î©¤ a
Ä ±¬Ýº©¯ò ó¨ £³ º© ak ±³»Â¶ FJ ©¨ î±Ä©¯°EÝ
J ©¯°§±Ä©¯IF ¨½³¨ £KJ ©£³»
¸ÈÉ ³ Ù³ÛÜkÉ³ØµÙ¬ìª³¨ ³¤º ©¥D ³ £³´© F k 1 Å £ ¶³£®óº¥ ¢¤£³³¢ ³£¦ J ©¯
ÒÎÐç Ù³Éâ
Å äã¤£³³ º©¯âXôD ³ ¥D ³£KJ ªÁ ³ ±Ä¬±´© ³À£Ä© F îH ¯¢
óº ©¬¢«¢¾î±Ä©« ¬§¨ ³¨½£K½
J »¨ ºNM Ó ³ º©¯ £¦· ³ ±Ä¬ó±¤Ä¨ ¤EJ
ººªÀ¨½³¨ £KµJ Å
Hî»EM Õ ¨½ ³£ Â¶ ©¯³°ÁÌ ¨ ³¤º H Â¶ ôD ©¯ ¬ ©£¬£©J ¬ª³¨ ¤ºNó
M ¢ ³£º¶ç£¬¨ GF
¶Ñ©£Ó³ À
F ©¯© J »¨ ã ³ ¢¤£³³¢ £¦ ©¯ º ¨½ ³£¿Â¶· ³ ±Ä¿³§³¨½³¤º
º©¯Ï©¥D ³Ä³ £Ä© F ¶ç©£ ¸¹º ºËÂ®ÛñÞß
ç©ÉÃ¨½É Ù³¨½
ÛÜÉÄØÚ©ÙÄ ì °Á Â¶ã¢¤£³³¢ £¦î ©¯ J ©¯
©« ¥D Å £³¶ç³£Ó º¥Ã¢¤£³³¢ ³£¦ J ©¯
Å ä ³¤ £³¬ º©¯ º ¢¤£
ÁÌ£©¨ » ¤ J KÝ
J ©Ý±Ä¤³î©¤ ¶ç©µ±³£À³ª¸Ó³£ßº© ºÓ¢¤£³Á£Ä©¨ read_array.C
º©¯ ¸È º ¡³ ±®ª³Áº½» © FD © ±Å
M
¬ ³£¬¨ » Óºë±Ä¤³î©¤ºàÂ¶Ã¢¤£³¢ £¦î ©¯ J a ¶ç£ ¢¤£³³¢ ³£¦¬ HîEM ©F Hî° MÀÂ¶J £
¢¤£³ÄÁ£©¨½Å
Ì£¦ È
£³ÄÁ£©¨ ±Ä© ± ¨½ » ° ±³³¤ à° ±½XD¥³£KJ
Exercise 76 £ó×©£îÁ«³£
ÐáÜá
³î¢ i µ©¨ » ¬
±Ä¬»½Â¶®º Ì£ » £ Fµ³ £ F º H§¢ ³ ¦EJ Dô³£KJ®¶³¸ M ¬
º©¯â©Dô ³ ª©« ¬ªIF ¸¹º i ° º©¯Ï³î¢ Å
³¬ º£ î ¸Ó F ó µ©¨ »
¤ J º °Á £¦ Â¶®º ±Ä¬» º©¯à©£³ ©¯£Ä© FJ ×©« ³¬ªIF ¸È»º i − 1 Ï° ± ¤ J º¬
©£³ ±Ä©FNF«©¯³È¶ç£Á«¬º°Á ×©« ³¬ i Å
£¬®©¿¢¤£Á£©¨ shortest_path_fast.C º©¯È£³´©¯ ¬ º¥à FµÄ© ·©F ¨ ´©¯£³º
¢ ³£º¶ç£¬¨½© ±ÄÁ¥©° ¨½ ×©£îÁ«£
Exercise 77 Â
n2 + n + 41
Ì£¦®Â¶J £ ±³î±çÅ
¤©£IF È§£ F±Ä D¥³£³GF º
¡© F£©¯±Ï¢ J¯ ¨©¯
¸¹º ºã¶ç»ª¸¹»Á £³³¨ ©£ ©« ¦ª¢¤£³³¢³£¦ J ¶ J XDô©¯¡©¯® ¶£ n = 0, 1, . . . , 39 JÚ
©¯¸Ó©J Á«¬ë©®¢¤£¬»¨½ß ¨ ³³£ ë©F ¨ £ D¥³£ ë©¯°º¬çë¢¤£¬»¨ ß ¨½ ³£¦©£³ F ³£³¤îÅ
³£ º »³ãÂ¶ß©¯» º 40 ¢¤£¬»¨ ¿ ¨ ³³£¦ Á«³ ³£Ä©¯IF KJ ¹ª³£ ¢ J ¨ ©¯
41 43 47 53 61 71 83 97 113 131 151 173 197 313 347 383 421 461 503 547 593 641 691 743 797 1033 1097 1163 1231 1301 1373 1447 1523 1601 Å
³£³ ¸Óß©£³®± ±Ä³£¬ IF ¸¹»º½º
¢ J ¨ ©¯ ° º½³¤½º©¯ÀºXJ
º©¯ã©
223 251 281
853 911 971
¬³ ¸È¬º³£º³£³®©£³¿³»» ³¬º³£ ¡© F£©¯±
¡
Á¥³ ³£©¯ D¥Öµ³áÜ ÉÍ¨½
WôÖÚÐ£³áÜ Û²Ë ¢¤
ì £¬°¨ ¨½ ³£¦çÅ ç©J
2
© F£©¯±Ï¢EJ¯ ¨ ©¯ an + bn + c ©¯
¡
p ¶º p ¨½ ³£¦
|an2 + bn + c|,
n = 0, ..., p − 1
© £³ F ³£³¤â¢¤£¬»¨ ¨½ ³£¦çÅ J î© °Á ©« ¦ç Dô©¯ ³ ¸ÓÝº£³¶£©¯²ç¼©¯° §¸
Áô©¯ D¥Ï¢¤£¬»¨ ³ Å ®©¼µ©¨¿¢¾§ òª¬· §Ì ©¯·ºÏ¢ J ¨ ©¯ n2 − 10n + 2 Å £
n = 0 ï¸ÓÁ«¬ 2 H°¢¤£¬°¨ M ¶£ n = 1 ¸Ï ¥ ¦î©» −7 H Á¥©¯ D¥â¢¤£¬»¨½ M ©F n = 2 Á Dô¬
−14 Hî Hî Á¥©¯ Dô M¢¤£¬°¨ MôÅ Õ ¹ª³£ ¡©¯ J Â¶ n2 − 10n + 2 »àº³£³¶ç£³ 2 ³ï
¥°Á³£ÌÅ
³£³È¹ºÓ±¬©¯°§³Á¥ ¸¹£¬ »Ï©È¢¤£³Á£Ä©¨ º©¯¾ J ³ ³¨½©¯±Ä©¯°EJ ´©£³±¬¬¤¶£ © ¡© F £Ä©¯î±
¢ J ¨ ©¯¥¸¹º ¥°Á È§³£ ¡©¯ JµÅß
ÕÈÁ¥©¯«È F ©·¢ EJ¯ ¨ ©¯¥¸¹º ¹ª³ £ ¡©¯
Jª©£Á«³£º© 40 » £ Fµ³£¿ ³´©¯ n2 + n + 41 Å Ý©¯È»ëº ¥°Á³³ È§³£ ¾©¯» J
º©¯
JÚ ±Ä© F
£ãº¥»ë±¬©¯»ª³Á« Ã»ï±Ä© ³Ó»¶Ú £³Ä© F º ¢©£©ÁÌ£©¬¢¾ ±Ä¤³î©¤ò ¬¢¤£³¬³¤
»º ¬î©»²¿±³î Å
HÄ ¦×©µ±
¸È¥» M
Exercise 78 XBM ª ¶£¬¨ ©¯ ©Ó¶ç£¬¨½©¯¶ç£ ³£¬»Áß¨½ Ì±¬¥£³¨½
»¨ ©Á«¬çÅ Õ ¶ç£¬¨½©¯ Àç¨ ³¸È©¯à F«©¯G F ë ¬ ¨½© J ¬£³¸È³£¦ H Â¤³£¬ ¬ ¦¢¾ª£³³£
ß©ó
î©« ¬§ Ú±¢¾ î MÝ±´© ³»» F ¢¾ª© Jó°¨½©Á«³¿° XBM ¶£¬¨ ©¯îÅ
XBM »¨ ©Á« ª ¶ç£ © »¨½©Á¥ ©¨ G F test ¨§Á« ªÌ Ýº¥ Hî© ô ¶£³¨
¢ I F © XBM ¢©Á« MôÅ
# define test_width 16
# define test_height 7
static char test_bits [] = {
0 x13 , 0 x00 , 0 x15 , 0 x00 , 0 x93 , 0 xcd , 0 x55 ,
0 xa5 , 0 x93 , 0 xc5 , 0 x00 , 0 x80 , 0 x00 , 0 x60 };
J ±´© Á ³³ ¶£³¨ º¥ XBM ª¬ ©£³ Fµ³°Á GF °¤Á£©¯GF »¤ ð©F
£³± ± Fµ¸È¥î±¬ ¨½© ô³ ¿»ãÄ©¯IJ ¢¤£³Ì±Ä¬³¿º³¨ Hº³£³¿à ÄIF £³Ä© Fó»
º F«©¯î© °¨¢¾ JÝ» ±³ Fµ º ªã¶Ä£³¨ º ¢¤£³Á£©¨ º©¯ã ÄIF¯ ¢¤£³±¬³ º
»¨ ©Á« M¥Å Â £ µ©¨¢¾ª test_width ©F test_height
Þ®ÐÙÍFµÊ³Ò º¸¹ F¯º ©Fó³°Á«
Â¶ º »¨ ©Á«° ¢¤ Ú¬çÅ £¬¨½©¯» J ¹º¬¿ ©¨ ¬À©£³
· ¬È°ºÓ¢¤£³Á£Ä©¨ º J
±Ä© G F¼ ô ±Ä¤³î©¤ ¦¢¤£³¬î¤çÅ test_bits © ©£¬£©ÇµJ ÉIPµÂÐç¶ èÚÉ±¬Ù³ÛÜ©Þ®£ÄÐ©µá ±¬áÜ³ÛñË£¦É ÍäÐºá ©¯
³ ±Ä Fµ¬ ºÝ±ª£¦ÝÂ¶ º 16 7 ¢¤ Ú³²° ºó»¨ ©Á«Å D¥K£ J
Â¶
º ¶£¬¨ 0xd1 d2 ³ ±Ä Fµ¬ ³°Á« ¢¤ Ú¬ ¸È³£³ º £ Fµ³£ £³¸ðK J½£³¸ Å Â £ ±Ä©¯
0x13 ©F 0x00 ³ ±Ä Fµ¿º ã¢¤ µ¬²Â¶âº
£¦³Õ£³¸
¸È¥ª 0x15 ©F 0x00 ©£³ ¶ç£ º
Ä±F £³¸ Ï¬±Å
³£³ò¹¸ ©â¸Ó G F°ÁÌ «© Fµ±³°¨½©¯«³£©¯ ³ ±Ä Fµ¬òºã±Ä§£¦ÓÂ¶ ³°Á«Õ±¤ç±³ D¥
¢¤ µ¬² ¸È»º¥» © £³¸ Å
D¥³£KJµ© Fµ±»¨½©¯
F°Á» di â¶Ä£³¨ º ç¬ {0, ..., 9, a, . . . , f}
¸È³£³ a ¢ f ³î©
Fâ¶£ 10, . . . , 15 Å Õ®©µ±¬¡©¯ ¨ ³³£³ ± FµGF KJ © «© Fµ±³°¨½©¯
³£©¯ » 16d1 + d2 {0, . . . , 255} Å
£µ©¨¿¢¾§ 0x13 ©¯ Dô©¯ 1 15 + 3 = 19 Å
¸ © J ¨ ³³£ ° {0, . . . , 255} ©¯ © ¬» ©£KJ £³î¢¤£¬³¤î©¯ ¸¹»º 8 ¬»ºçÅ 19
¶£ µ©¨¢¾ª ©¯ ¬» ©K£ J £³î¢¤£³³³¤î©¯î 00010011 Å â¢¤ Ú¬¹±ª£¦ ©£³ ¥ ¦î©° IF KJ
£³´© F»Á º¥»® Ä©µ± ¸ ©£ F¯ Ï©
Fó»¤£¢¤£³¬»Á 1 ©¯® ¬ª©µ± ©
F 0 ©¯ ¸È¥Åó¥ Èº £¦³
³§Á«Õ¢¤ µ¬² » £³¸ 1 Â¶ º test »¨ ©Á« ©£³® ¦×©µ± Ó ¦×©µ± ¹¸È¥ È¸È¥ ¦×©µ± È¸È¥
¸È¥ ¹¸È¥çÅóÕ ±Ä¨¢¾ª¬ test »¨ ©Á« ªÌ Ì¿ ô º¥
¬£ ë© ¢¤£³ÄÁ£©¨ xbm.C º ©¯ #include Ó© XBM §ëÂ¶JÚ£ë ±¬± HNJ ¨½©J¿´©£³±¬
ÍÂºÊÌËä ÐÑËÂ¸ÓÉ³è _ ì èµF É³åÌÍ É³É³îÒ©« ¦ª XBM ª¬NM ò©Fº©¯Óñ¢¤º © XBM §¹¶ç£ ºÑ©¨ à»¨ ©Á«
Å ¢¤£³ÁÌ£©¨ ¨ ©Jó¸¹£¬º£¬»Á ª ³î©F«©£ F ñ¢¤îÅ
Â ±Ä©¯çßÂ¶ëº test °¨½©Á« ïºë£³¬Ä»Á XBM ª ©F ºà£³³»Á®£³î©¯GF°¨½©Á«©£³
©¯¹¶»ª¸ÈçÅ
# define rotated_width 7
# define rotated_height 16
static char rotated_bits [] = {
0 x3c , 0 x54 , 0 x48 , 0 x00 ,
0 x04 , 0 x1c , 0 x00 , 0 x1c ,
0 x14 , 0 x08 , 0 x00 , 0 x1f ,
0 x00 , 0 x0a , 0 x15 , 0 x1f };
¿º©¯ï¸Ï ¸ ©D¥ ß»¤³´© F Â¶ «© Fµ±³°¨½©¯ £©¯²ÑÅÕ¥ë» F º
¶ç©µ±³ º©¯·Ä©µ±³ Â¶Ïº £¸òÏ GF¯ ë³£Ä©¯¶ç£â»º ¢¤ Ú¬ Ã¸ò£³àºëªÄ© F°Á ¬»º
Â¶¿º ³» ©K£ J £³î¢¤£³³³¤î©¯î¤ ©£³ ³»Á°Á £³GF«Å
¨½© J
F JÚ£ ¢¤£³ÁÌ£©¨ à¢ ³£º¶£¬¨ º³£ »F¯ Â¶ã»¨ ©Á« ¢¤£³Ì±Ä¬³°Áî©¯ Ì
Â¶ JÚ£ ±¬î±ÄÅ
«©¨¢¾ª¬ ° ±¬"
Fµ ±ª£ » Dô³£¦î Hî£³¢¾ª©µ±Ä ¦×©µ± ¸È»º ¸È¥ ©F
DÌ± D¥³£¦ÑK© M ë±¨¿¢¤°Á ©½¨ »£¬£³£ß°¨½©Á« ±Ä©¯»Á ºß°¨½©Á« Hî º©¯ ëÌ±Ä±³ ¢¤¬ß§³³
£ ¨½£³ã¢¤ Ú¬N M ¬±çÅ
K$
07
'*
$&/S? 8*
( A
%.*6
A BI
*
.6#*
% R7
char
R7
L"
/1J
.
α − 256
9%*
,$&
2
6
,)7
-
#
,
{−128,
. .. ", 127}
A ' :
M
.
. *&
A#"
"5
%*(#*@*
/
8
%/ ."K %2
*L R'(9N. %
$ &"5
%"
%*(J
.
α
+(6 %
Chapter 3
Functions
3.1 A first C++ function
©£Ì ©Á«» ïÁ¥©£Ì Ä©Á«îÅ
º£¬ ¬GF
Ä£îÁ«
Èª©¯ ¡±¬«³
¥ ±³î »¤£ F ±Ä¬ ¶ ±¬î¤ ©¯ © ¨½Ä©¤ ³ ±Ä©¬¢¾Ä×©¯ ©F
Õ
£³Ó¶ ±¬î ©¯ J â©F ¡ IF DÌ Fµó© ¢¤£Á£©¨ »¤ ¾ ¦î©¯ ÌÑÅ ¸È»»
ªÄ©£¬ ¸ © F FÓ¶Ä ±¬¤àJ £ ¢¤£³Á£©¨ ©F ¸ ±´©¯»¡º³¨½Å
©¯ ¦¢¾ª©° ¸ ¶Ä ±¬¤ ±Ä© ±³³¤ºEJ ³ ¨½© Fµ ©Dô©×©« ¦ª ¶£½¨½© J
¢¤£³ÄÁ£©¨ó®©¯·º ç©¨ »¨½ Èº¥£³ Á î¢©£©¯ ±¨¿¢¤×©¯î ©Fó ¬£©£¬¬çÅ
ê Ø Þ®ÐØ ì Ø ÖµÞßÉÍÂÛÜÙ¬Ðçá Ù¬ÐáÜÙÖÚáÜÐçËÂÛÜÊØµÒ³í Ù³ÊÞÀÎÚÖ«ËÂÛÜØµåßÎ Ê¦æ¹ÉÍÂÒÏÛÜÒëÐ®Ô×ÖµØµè ÐçÞßÉ³Ø ËäÐçá ÊÌÎ ÉÍäÐçËÛÜÊØ^»ÒÉ³É
" É³ÙËÛÜÊØ \ é `Äí«ÐØµè ËÇÚÉÄÍÂÉâÐÑÍÂÉÓÞ®ÐØ ìßÊÌËÇÚÉÄÍòÊÌÎÉÍäÐçËÂÛñÊØÚÒ ËÇ ÐÑËòÊ Ù³Ù³Ö«ÍÈÔÜÍÂÉXW ÖÚÉØ ËÂáñì ÛÜØ ÐçÎµÎÚáÜÛÜÙ¬ÐÑËÂÛÜÊØµÒ³é
êØ&àí
ÐÑÍÂÉàÖµÒÉ³è ËÂÊßÉ³ØµÙ¬ÐçÎÚÒÖµá×ÐçËÉëÒÖµÙäÇ ÔÜÍÂÉXW ÖµÉ³Ø ËÂá²ì ÖµÒÉ³è ÊÌÎ ÉÍäÐçËÛÜÊØµÒ³í Þ®Ð ]¥ÛÜØµå Û²Ë

³
±

É¬ÐÒì ËÂÊëÛñØ UÊ ]É·ËÂÇµÉ³Þ Þ®ÐØ ì ËÂÛñÞßÉ³Ò³íæ ÛñËÇÀèµÛ ÉÍÉ³Ø Ë¹ÐÑÍÂåÖµÞßÉ³Ø ËÂÒí ÐçØÚèßÔÜÍÊÞ èµÛ ÉÍÉ³Ø ËÕÎÚÍÊåÌÍäÐÞßÒ³í
_ÚÖ«Ë
ÇÚXÐ U¥ÛÜØµå ËÂÊÍÉÎµÍÂÊåÌÍäÐÞ ËÂÇµÉ³Þ É UÉÍì ËÂÛÜÞßÉé
¸¹º
UÉØÞßÊÌÍÂÉòÛÜÞÀÎ ÊÌÍËäÐçØ ËÂáñìíÔ×ÖÚØµÙËÛÜÊØµÒ¹ÐçÍÂÉ ÖµÒÉ³è®ËÂÊàÒîËîÍÂÖµÙËÂÖ«ÍÂÉÓÐãÎµÍÂÊåÌÍäÐçÞ é êØ ÎµÍäÐçÙËÂÛÜÙ³Éíôá×ÐçÍåÉ
ÎµÍÂÊåÌÍäÐÞßÒ ÙÊØÚÒîÛÜÒîËÕÊÔÞ®ÐØ¯ìàÒÞ®ÐáÜáôÔ×ÖÚØµÙËÛÜÊØµÒ³í É³ÐÙäÇÀÊÔ æ·ÇÚÛñÙ´ÇÒÉÍ UÉÒïÐÏÙáÜÉ¬ÐçÍÂá²ìàèµÉ YÚØÚÉèÀÒÖ _µËäÐÒ ]é
Æ¹ÇÚÛñÒòÞ®Ð ]ÉÒÈÛ²Ë·Ð áÜÊÌËÈÉ¬ÐÒîÛÜÉÍÈËÂÊÍÂÉ³Ðè¤íÚÖµØµèµÉÍÂÒîËÂÐØµè¾íµÐçØÚèÞ®ÐÛÜØ ËÂÐÛÜØÀËÂÇµÉÓÎÚÍÊåÌÍäÐÞ é
SÝÉ ÇÚXÐ UÉÐçáñÍÂÉ¬Ðè«ìÒÉ³É³Ø WôÖµÛñËÂÉ ÐÀØ ÖÚ
Þ _ ÉÍ ÊÔÕÔ×ÖµØµÙËÂÛÜÊØµÒ³íÒÛÜØµÙ³ÉëËÂÇµÉ
Ô×ÖµØµÙËÂÛÜÊØ ÊÔÃÉ UÉÍì
main
ÎµÍÂÊåÌÍäÐÞ ÛÜÒòÐ ÒîÎ É³Ù³ÛÜÐá¤Ô×ÖÚØµÙËÛÜÊØ ^ " ÉÙËÂÛÜÊØ \ é é `Äé
ÍÊåÌÍäÐÞ É³ÞÀÎÚÇÚÐÒÛ ³É³ÒëËÂÇµÉ É³ØµÙ¬ÐçÎÚÒÖµá×ÐçËÂÛñÊØÐÒÎ É³ÙË ÐØµèÒÇµÊ¦æ Ò¿ÇµÊÑæ Ô×ÖµØµÙËÂÛÜÊØµÒ Ù¬Ð
Ø _ É
ÖµÒÉ³è¤é ê»Z
Ë YµÍÂÒîËëèµÉ YÚØµÉ³ÒàÐ Ô×ÖÚØµÙËÛÜÊØ Ô×ÊÌÍàÙ³ÊÞÀÎÚÖ«ËÂÛÜØµå®ËÂÇµÉ UÌÐçáÜÖµÉ e Ô×ÊÌÍ Ð åVÛ UÉ³Ø ÍÂÉ¬ÐçáØôÖµ
Þ _ ÉÍ
ÐØµè®åVÛ UÉ³Ø ÛÜØ ËÉ³åÉÍ ^»ÎÊÒÒVÛ _ÚáñìØµÉ³åÐçËÂVÛ UKÉ `Äé ê»ËïËÂÇµÉ³Ø®Ù¬ÐáÜáñÒËÂÇµÛÜÒ Ô×ÖµbØµÙËÂÛÜÊØ®Ô×ÊÌÍ¹ÒIÉ UÉÍäÐá UÐáÜÖµÉ³ÒïÊÔ b
ÐØµè éÃÆ¹ÇµÉâÙ³ÊÞÀÎÚÖ«eËäÐçËÂÛñÊØÚÒ ÐçÍÂÉÓÎ ÉÍÂÔ×ÊÌÍÂÞßÉèÊ UÉÍ·ËÂÇµ
É QÚÊÐçËÂÛÜØµåàÎ ÊÛÜØ¯ËòØ Öµb
Þ _ ÉÄÍÈË ì Î É
é b
e
// Prog : callpow . C
// Define and call a function for computing powers .
# include < iostream >
// PRE : e >= 0 || b != 0.0
// POST : return value is b^e
double pow ( double b , int e)
{
double result = 1.0;
if ( e < 0) {
// b^ e = (1/ b )^( - e)
b = 1.0/ b ;
e = - e;
}
for ( int i = 0; i < e ; ++ i ) result *= b;
double
return result ;
}
int main ()
{
std :: cout
std :: cout
std :: cout
std :: cout
std :: cout
<<
<<
<<
<<
<<
pow ( 2.0 , -2) < < "\n " ;
pow ( 1.5 , 2) < < "\n " ;
pow ( 5.0 , 1) < < "\n " ;
pow ( 3.0 , 4) < < "\n " ;
pow ( -2.0 , 9) < < "\n " ;
//
//
//
//
//
outputs
outputs
outputs
outputs
outputs
0.25
2.25
5
81
-512
return 0;
}
Program 19: ¢¤£³Á
´± ©¯» ¢ ¸ Å
É³Ô×ÊÌÍÂÉ æ¹ÉòÉIP¥ÎÚá×ÐÛñØ ËÂÇµÉ Ù³ÊØµÙ³ÉÎµËÂÒØÚÉÙ³É³ÒÒÂÐÑÍì ËÂÊâÖÚØµèµÉÍÂÒËäÐØµèÀËÂÇµÛÜÒÕÎµÍÂÊåÌÍÂÐÞ ÛñØßèÚÉÄËäÐÛÜáíáÜÉÄË ÖµÒ
é [ ÊÌÍØµÊØµØµÉ³åÐçËÂÛ UÉÈÉGP¥Î ÊØµÉ³Ø¯ËÂÒ í e
åÉËÐØ ÊUÉÍ UôÛÜÉæ¼ÊÔ æ·ÇÚÐçËÃÛÜÒ åÊÛÜØµåãÊØÛÜØËÂÇµÉÈÔ×ÖµØÚÙÄËÂÛÜÊØ
e b
ÛÜÒòÊ _ÚËÂÐÛÜØµÉ³èóÔÜÍÂÊÞ ËÂÇµÉâÛÜØµÛñËÛ×Ðá UÐáÜÖµÉâÊÔ _ôì Ô×ÊáÜè Þ¿ÖÚpow
á²ËÂÛñÎÚáÜÛÜÙ¬ÐÑËÂÛÜÊØ æ·ÛñËÂÇ é¹Æ¹ÇµÛÜÒ·ÛñÒòæ ÇÚÐçËÈ
ËÂÇµÉ
1
e
b
ºáÜÊ ÊÌÎÀèµÊôÉ³ÒéÕÆ¹ÇµÉ·Ù¬ÐçÒÉ·ÊÔ ØµÉ³åÐçËÛVUÉ
Ù¬ÐØ _ ÉòÇÚÐØµèÚáñÉ³è _ôì¿ËÂÇµÉ·Ô×ÊÌÍÞÖµá×Ð e
ÐÔÜËÉÍ
for
e
b = (1/b)−e
ÛÜØ UÉÍËÂÛÜØµå
ÐØµè ØµÉ³åÐçËÛÜØµå ÛñØ½ËÂÇµÉ
ÒîËäÐçËÉ³ÞßÉ³Ø Ë³íæ¹É Ç XÐ UÉÐØ É WôÖµVÛ UÐáÜÉ³Ø¯ËÏÎµÍÂÊ _ÚáÜÉÞ æ ÛñËÂÇ Ð
Î ÊÒÛ²ËÂÛVUÉÏÉIP¥bÎ ÊØµÉ³Ø¯Ë¬é¹Æ¹ÇÚÉãá×ÐçeËËÉÍ·ÊØµáñì®if
æ¹ÊÌÍ ]ôÒ·ÛÜÔ í ÐØµèóÛÜØµèµÉ³É³è¤íÚØÚÉåÐçËÂVÛ UÉÓÎ Ê¦æ¹ÉÍÂÒ ÊÔ ÐçÍÂÉ
b=0
0
Þ®ÐçËÂÇµÉ³Þ®ÐÑËÂÛÜÙ¬Ðáñáñì ÖµØÚèµÉ YÚØµÉ³è¤é
3.1.1 Pre- and postconditions
UÉ³Ø®Ð UÉÍîì ÒÛñÞÀÎ áñÉïÔ×ÖÚØµÙËÛÜÊØÀÒÇµÊÖµáÜèÀèµÊôÙÖÚÞßÉØ ËÃÛñËÂÒ
ÐØµèÀÛñËÂÒ
í ÛÜØ
¤
¢
³
£

Ä
±

ËÂÇµÉ Ô×ÊÌÍÂÞ ÊÔ Ù³ÊÞßÞßÉ³Ø ËÒ³é Æ¹ÇÚÉòÎÚÍÉ³Ù³ÊØµèµÛñËÂÛÜÊØÀÒÎ É³Ù³ÛVYÚÉ³ÒÕæ ÇÚÐçË ÇÚÐÒÕËÂÊàÇµÊáÜèÀæ ÇµÉ³ØÀËÂÇµÉ Ô×ÖµØµÙËÂÛñÊØßÛÜÒ
Ù¬ÐáñáÜÉ³è¤íÐØµè ËÂÇµÉ¹ÎÊÒîËÂÙ³ÊØµèµÛñËÂÛÜÊØàèµÉ³ÒîÙÍÂÛV_ ÉÒUÐáÜÖµÉ¹ÐØµèÉ É³ÙÄËÕÊÔµËÂÇµÉ¹Ô×ÖµØÚÙÄËÂÛÜÊØ¤é Æ¹ÇÚÛñÒ ÛÜØµÔ×ÊÌÍÂÞ®ÐçËÂÛÜÊØ
ÐáÜáñÊÑæ·Ò ÖÚÒÓËÂÊ ÖÚØµèµÉÍÂÒËäÐØµè ËÂÇµÉ¿Ô×ÖÚØµÙËÛÜÊØ½æ·ÛñËÂÇµÊÖ«ËÓáÜÊôÊ ]ôÛÜØµå ÐÑËÏËÇÚÉÐÙËÖ ÐçáÃÒîÊÖµÍÙ³É³Ù³Ê èÚÉ ËÂÇµÛÜÒÓÛÜØ
ËÂÖ«ÍÂØ ÛÜÒ¹ÐëØµÉ³Ù³É³ÒîÒÂÐçÍì®Ô×ÊÌÍ ]É³ÉÎÚÛÜØµåàËÍäÐçÙ ]ÀÊÔ¡á×ÐÑÍÂåÉÍ ÎµÍÂÊåÌÍäÐçÞßÒ³é êØ Ù³ÐÒÉÓÊÔ ËÂÇµÉ Ô×ÖÚØµÙËÛÜÊØ
íôËÂÇµÉ
pow
ÎµÍÂÉ³Ù³ÊØµèµÛñËÂÛñÊØ
// PRE :
e >= 0 || b != 0.0
ËÂÉ³áñáÜÒëÖµÒàËÂÇÚÐçË ÞÖµÒîË _ÉßØÚÊØµØµÉ³åÐçËÛVUÉÌí ÊÌÍ ^ÛÜÔ ÛÜÒàØµÉ³åÐçËÂÛ UÉ`ãËÂÇÚÐçË
Þ¿ÖµÒîËàÇµÊáÜè¤é ÆÈÇµÉ
e
e
b = 0
Î ÊÒîËÙ³ÊØµèÚÛ²ËÂÛÜÊØ
// POST : return value is b^e
ËÂÉ³áñáÜÒ¹ÖÚÒ¹ËÂÇµÉâÔ×ÖµØµÙËÂÛñÊØ UÐáÜÖµÉÌíµèµÉÎ É³ØµèµÛÜØµåÊØ ËÂÇµÉâÐçÍåÖÚÞßÉØ ËÂÒ³é êØ ËÂÇµÛÜÒ¹Ù¬ÐÒÉÌíµËÂÇµÉÍÂÉÏÛÜÒÈØµÊ É É³ÙË¬é
ÆÈÇµÉ¿ÎÚÍÉ ÐçØÚèÝÎÊÒîËÂÙ³ÊØµèµÛñËÂÛÜÊØµÒ ÒîÎÉ³Ù³ÛÜÔÜì ËÂÇµÉ Ô×ÖµØµÙËÂÛÜÊØ ÛÜØ Ð®Þ®ÐçËÂÇµÉ³Þ®ÐçËÂÛñÙ¬Ðá¾ÒÉ³ØµÒÉÌé ïË YÚÍÒîË
ÒÛÜåÇ¯Ë¬í Ô×ÖµØÚÙÄËÂÛÜÊØµÒëæ ÛñËÇ UÐáÜÖµÉ³Ò
É É³ÙËâèµÊ ØµÊÌË YµËÛñØ ËÂÊóËÂÇµÉ®ÔÜÍäÐÞßÉÄæ¹ÊÌÍ ]ÝÊÔÈÞ®ÐçËÂÇµÉ³Þ®ÐçËÛ
©

F
Ù¬ÐáÔ×ÖÚØµÙËÛÜÊØµÒëæ ÇµÛÜÙäÇ ÊØÚá²ì ÇÚKÐ UL
É UÐáÜÖµÉ³Òé Ö«Ë ÖµÒÛÜØµå ËÂÇµÉßÙ³ÊØµÙ³ÉÎµË ÊÔ
^ " ÉÙ ¢¤£³ÁÌ£©¨ ³î©¯¬
"K(7
S7"K..B#,4 %"K%+*
+*
7.#*
$&7
87
+
*
7*
R
4PK(1M+*
#*
72*
(6#"
* J
22
M7
&$&
.
*L7
@*
/
%B5"
(*@
%,0
.
$
&
-
7.*6M#"
A7
*
7 B#J
.'(*L)
*
\
ËÂÛÜÊØ é \ é `Äí Ð Ô×ÖÚØµÙËÛÜÊØ Ù³ÐØ _ÉßÙ³ÊØµÒÛÜèµÉÍÂÉ³èÐÒ ÐóÞ®ÐçËÂÇµÉ³Þ®ÐÑËÂÛÜÙ¬Ðá Ô×ÖµØµÙËÂÛñÊØ ËÂÇÚÐçË Þ®ÐÑÎ Ò
ÎµÍÂÊåÌÍäÐÞ ÒîËäÐÑËÂÉ³Ò ^»ÛÜÞßÞßÉ³èµÛ×ÐçËÂÉáñì
ËÂÇµÉ Ô×ÖµØµÙËÂÛñÊØÝÙ¬ÐáÜá ` ËÂÊ ÎµÍÂÊåÌÍäÐÞ ÒîËäÐçËÉ³Ò ^ÛñÞßÞßÉ³èµÛ×ÐçËÂÉ³á²ì

ËÂÇµÉãÔ×ÖµØµÙËÂÛÜÊØ Ù¬ÐáÜá `Äé
©ä¶³·£ ØµèµÉÍ ËÇÚÛñÒÀÎ ÊÛÜØ Ë ÊÔ
U¥ÛÜÉÄæëíòËÇÚÉ ÎµÍÂÉ³Ù³ÊØµèµÛñËÂÛñÊØ¼ÒÎ É³Ù³ÛVYÚÉ³Ò®ËÂÇµÉ
ÊÔÏËÂÇµÉ Ô×ÖµØµÙËÂÛñÊØ¾í
µ
F

½
¨

ËÂÇµÉ ÒÉËÊÔòÎµÍÂÊåÌÍäÐÞ ÒîËÂÐçËÂÉ³Ò¿ÛÜØ æ ÇµÛÜÙäÇ ËÂÇµÉ Ô×ÖµØÚÙÄËÂÛÜÊØÞ®Ð³ì _ É®Ù¬ÐçáÜáÜÉ³è¤é êØÙ³ÐÒÉÊÔ
íÃËÂÇµÉ³ÒÉ
ÐçÍÂÉ ÐáÜá ÎµÍÂÊåÌÍäÐÞ ÒîËÂÐçËÂÉ³ÒãÛÜØ æ ÇµÛÜÙäÇ ËÇÚÉ ÐçÍÂåÖµÞßÉ³Ø¯ËÂÒ ÐØµè
ÐçÍÂÉÛÜØ Ð®ÒÖµÛñËäÐ _ÚáÜÉàÍÂÉ³áÜÐçpow
ËÂÛÜÊØ¤éÆÈÇµÉ
Î ÊÒîËÙ³ÊØµèÚÛ²ËÂÛÜÊØ èµÉ³ÒÙÍÂÛ _ É³ÒÓËÂÇµÉÔ×ÖµØµÙËÂÛÜÊØ½ÛñËÂÒÉáÜÔ _ ì½ÒÎ bÉ³Ù³ÛÜÔÜìôÛÜØµeå ÇµÊ¦æ ËÂÇµÉ ^°ÍÂÉ³áÜÉ UÐØ¯ËãÎ ÐçÍËÏÊÔÃËÂÇµÉ `
ÎµÍÂÊåÌÍäÐÞ ÒîËÂÐçËÂÉ åÉËÂÒÏËÍäÐØµÒÔ×ÊÌÍÂÞßÉè¾éëêØÝÙ¬ÐÒÉ¿ÊÔ
í¤ËÂÇµÉ¿ÍÂÉÄËÂÖ«ÍÂØ UÐáÜÖµÉ e æ ÛÜáñá ^»ËÂÉ³ÞÀÎÊÌÍäÐçÍÂÛÜá²aì `
pow
b
_ ÉÓÎÚÖµË·ÐçËòÒÊÞßÉÓÞßÉ³ÞßÊÌÍì áÜÊôÙ¬ÐÑËÂÛÜÊØ¤é
Æ Ê ÒÖµÞßÞ®ÐçÍÂÛ ÉÌíËÂÇµÉÎ ÊÒËÂÙ³ÊØµèµÛñËÂÛÜÊØ ËÂÉáÜáÜÒ ÖµÒßæ ÇÚÐçË ÇÚÐçÎµÎ É³ØµÒÀæ ÇµÉ³Ø ËÂÇµÉóÎµÍÂÉ³Ù³ÊØµèµÛñËÂÛÜÊØÛÜÒ
ÒÂÐçËÛÜÒ YÚÉ³è¤é ØóËÂÇµÉãÊÌËÇÚÉÄÍ ÇÚÐØµè¤íµËÇÚÉÏÎ ÊÒîËÂÙÊØÚèµÛñËÛÜÊØ åÛ UÉ³Ò
Ô×ÊÌÍòËÂÇµÉ
©¯ºÊ"
£ í«ËÂÇµÛÜÒ¹ÛÜÒ
Ù¬ÐÒîÉãæ ÇµÉÍÉãËÂÇµÉãÎµÍÂÉÙ³ÊØµèÚÛ²ËÂÛÜÊØ ÛÜÒ·ØµÊÌËòÒÂÐçËÂÛÜNÒ Y Éè¾é [ ÍÂÊÞ Ð Þ® Ðç ËÂÇµÁÉ³¡Þ®©ÐÑ£ËÂ©ÛÜ¤Ù¬ÐÄáÚß
Î Ê¸ÈÛÜØ¯Ëò
Ô ÌU¥X D¥ÛÜÉæâ
YÚØµÉ Ð Ô×ÖµØµÙËÂÛÜÊØ ÛÜÒòÒîÛÜÞÀÎÚáñìÀØµÊÌËòèÚIÉ Y ØµÉ³èÔ×ÊÌÍ ÐçÍÂåÖµÞßÉ³Ø¯ËÂÒïÊÖ«ËÂÒÛÜèµÉÏÛñËÂÒ¹èµÊÞ®ÐÛÜØ¤é
Æ¹ÇµÉàÙ¬ÐçÍÂÉ³Ô×Öµá ÍÉ¬ÐèµÉÍ ÊÔ " É³ÙËÛÜÊØ \ é ÞßÛÜåÇ¯Ë·ÇÚÐXUÉëÍÂÉ¬ÐçáÜÛ³É³è
ÊÔ ËÂÇµÉ Ô×ÖµØÚÙÄËÂÛÜÊØ
Ù¬ÐØµØµÊÌËL_ É Ù³ÊÌÍîÍÂÉ³ÙË¬é êÔ ÛñÒÀËÂÊôÊ
pow
á×ÐçÍåÉÌíïÔ×ÊÌÍ®ÉIPµÐçÞÀÎ áñÉÌíËÂÇµÉóÙ³ÊÞÀÎÚÖ«ËäÐçËÛÜÊØÞßÛÜåÇ¯Ë Ê UÉÍ QÚÊ¦æëíR_ÚÖ«ËßÒÖµÙäÇ
ÐÑÍÂÉ ØµÊÌËÀÉIP¥Ù³áÜÖµeèµÉ³è _ ì
ËÂÇµÉëÎµÍÂÉ³Ù³ÊØµèµÛñËÂÛñÊØ¾é UÉ³Ø ÛÜÔ ËÂÇµÉÍÉàÛÜÒ ØµÊßÊ UÉÄÍ QÚÊÑæâíËÂÇµZ
É UÌÐçáÜÖµÉëÍäÐØµåÉëÊeÔ ËÂÇµÉëË ìôÎ É
Þ®Ð³ì
ÇÚÐXUÉóÐ ÇµÊáÜÉÐÑË e íÞßÉ¬ÐØµÛÜØµåóËÂÇÚÐçË ËÂÇµÛÜ
Ò UÐáÜÖµÉ®Ù¬ÐØµØµÊÌË _ É®ÍÂÉËÖµÍØÚÉè _ôì ËÂÇµÉ Ô×ÖÚØµdouble
ÙËÛÜÊØ¤é Æ¹ÇµÉ
Î ÊÒîËÙ³ÊØµèÚÛ²ËÂÛÜÊØ ÛÜÒïbËÂÇµÉÍÂÉ³Ô×ÊÌÍÂÉãÛÜÞÀÎµÍÂÉ³Ù³ÛñÒÉÏÐÒÈæ¹ÉáÜáé
êØóËÂÇµÉâÙ³ÊØ¯ËÂÉIPôË ÊÔ ÐçÍÛñËÂÇµÞßÉËÂÛÜÙ ÊÌÎÉÍäÐçËÂÛñÊØÚÒ¹Ê UÉÍ ËÂÇµÉãÔ×ÖÚØµèÚÐÞßÉ³Ø¯ËäÐá Ë ìôÎ ÉÒ³í Û²Ë·ÛÜÒòÊÔÜËÂÉ³Ø
ËÂÉ³èµÛÜÊÖµÒ®ÐØµè¼É UÉ³Ø ÖµØµèµÉ³ÒÛñÍÂÐ _ÚáÜÉËÂÊ æ·ÍÂÛ²ËÂÉóèµÊÑæ·Ø¼ÎµÍÂÉÙ³ÛÜÒÉ ÎµÍÂÉ ÐØµè¼Î ÊÒîËÂÙ³ÊØµèµÛñËÂÛñÊØÚÒ ·Î ÐçÍË®ÊÔ
ËÂÇµÉ®ÎÚÍÊ _ áñÉ³Þ ÛÜÒ ËÂÇÚÐçË¿Ô×ÖÚØµèÚÐÞßÉ³Ø¯ËäÐáË ì Î É³Ò¿Þ®Ð¬ì _ É³ÇÚXÐ UÉèµÛ ÉÄÍÂÉ³Ø Ëáñì ÊØèµÛ ÉÍÂÉ³Ø¯ËÎÚá×ÐçËÔ×ÊÌÍÂÞßÒ³é
Æ¹ÇÚÉÄÍÂÉ³Ô×ÊÌÍÂÉÌí æ¹É½ÊÔÜËÂÉ³Ø Ù³ÊØ YÚØµÉ½ÊÖ«ÍÂÒÉVá UÉ³Ò ËÊÎµÍÂÉ ÀÐØµè Î ÊÒîËÂÙÊØÚèµÛñËÛÜÊØµÒÀËÂÇÚÐçËèµÊ Ù³ÖµÞßÉ³Ø Ë ÊÖ«Í
ÛñØ ËÂÉ³Ø¯ËÂÛÜÊØ¤í _ÚÖµËÃæ¹É ÇÚXÐ UÉ·ËÂÊ ]É³ÉÎ ÛÜØÀÞßÛÜØµèËÇ ÐÑËÛÜØÀÍÉ¬ÐáÜÛñË ìíËÂÇµÉ·Ô×ÖµØµÙËÂÛñÊØßÞßÛÜåÇ Ë
_ ¨½É³©¯ÇÚºXÐ UÉâ¨½èµ©¯Û îÉ±´ÍÂ©¯É³ Ø¯ËÂáñìé
Arithmetic pre- and postconditions.
ËÂÇÚÐçË _ ÊÌËÇ ÎµÍÂÉ
ÐØµè¼Î ÊÒËÂÙ³ÊØµèµÛñËÂÛÜÊØ
Assertions.
// PRE :
" Ê ÔªÐçÍ¬íµÊÖ«ÍÈÎÚÍÉ³Ù³ÊØµèµÛñËÂÛÜÊØµÒ¹ÐçÍÂÉ ÖÚÒË·Ù³ÊÞßÞßÉØ ËÂÒïáÜÛ ]ÉÏÛñØ
e >= 0 || b != 0.0
Æ ÇÚÉÄÍÂÉ³Ô×ÊÌÍÂÉÌíÚÛÜÔ ËÇÚÉÏÔ×ÖµØµÙËÂÛÜÊØ
¹
ÛÜÒÈÙ¬ÐáñáÜÉ³è æ·ÛñËÂÇÐçÍÂåÖµÞßÉ³Ø ËÒ ÐçØÚè ËÂÇÚÐçËU¥ÛÜÊá×ÐÑËÂÉ ËÂÇµÉãÎµÍÂÉ³Ù³ÊØ
pow
èµÛñËÂÛÜÊØ¤íËÂÇµÛÜÒÎ ÐÒÒîÉ³Ò¹ÖÚØµØµÊÌËÂÛÜÙÉ³è¤é Ø®ËÂÇµÉ Òîì¥Ø¯ËäÐÙËÂÛñÙ¬ÐááÜÉIUÉ³áí ËbÇÚÉÄÍÂÉÓÛÜÒïeØµÊÌËÂÇµÛÜØµåâæ¹É Ù¬ÐØ èµÊÐ _ ÊÖ«Ë
í¾Ô×ÊÌÍâÉIP«ÐÞÀÎÚáÜÉÌíæ·ÛÜáÜá Ù³ÊÞÀÎÚÛÜáñÉÌé Ö«ËÏæ¹ÉÙ¬ÐçØÝÞ®Ð ]É ÒÖ«ÍÂÉ
ÛñË ËÇÚÉ¿Ô×ÖµØµÙËÂÛÜÊØ½Ù¬ÐáÜá
pow (0.0, -1)
ËÂÇÚÐçË ËÇÚÛñ
Ò _ áñÖÚØµèµÉÍ ÛñÒ èµÉËÂÉ³ÙËÉ³èÝÐçË ÍÖÚØ¯ËÂÛÜÞßÉÌé ÒÛÜÞÀÎÚáÜÉÏæÈÐ¬ì ËÂÊ®èµÊ®ËÇÚÛñÒ ÖµÒÉ³Ò
é Ø
¯
©
³

¦
£

ÐÒÒîÉÍËÂÛÜÊØ ÇÚÐÒòËÂÇµÉÏÔ×ÊÌÍÂÞ
^ `
æ ÇµÉÍÉ ÛÜÒ Ð
ÎµÍÂÉ³èµÛÜÙ¬ÐÑËÂÉÌíÐØÉIP¥ÎµÍÂÉ³ÒîÒÛÜÊØ ÊÔ Ð Ë ìôÎ Éïæ ÇµÊÒÉ UÐáÜÖµÉ³Ò Ù¬ÐçØ _ É¹Ù³ÊØ UÉÍËÂÉ³èËÂÊ
é
í Ð Ù³ÊØµÒÉXW ÖÚÉØÚÙÉÀÊÔ ËÂÇµÉ ÔªÐÙÄËâËÂÇÚÐçË
ÛÜÒÏØµÊÌËàÐ Ô×ÖµØµbool
ÙËÂÛñÊØ
_ÚÖ«ËàÐ
é Þ®ÐçÙÍÂÊÛñÒëÐ ÎÚÛÜÉÙ³ÉÀÊÔ ÞßÉËÂÐ ºÙ³Ê èµÉ¿ËÂÇÚÐçËãËÂÇµÉ Ù³ÊÞÀÎÚassert
ÛÜáñÉÍÓÍÂÉÎÚá×ÐÙ³ÉÒâæ ÛñËÇÝÐÙËÖ Ðçá
¨
µ
©
³
±
³
£

³Ù Ê èÚÉÓÎµÍÂÛÜÊÌÍÈËÂÊ¿Ù³ÊÞÀÎÚÛÜá×ÐçËÂÛñÊØ¾é
S ÛñËÂÇ ÐÒîÒÉÍËÂÛñÊØÚÒí
¼
Ù¬ÐØ _ ÉÓæ·ÍÛñËËÂÉ³ØÐÒÈÔ×ÊáÜáÜÊ¦æ Ò³é
O ÊóÙÊÞßÞ®ÐÀÛÜÒâÐçáÜáÜÊ¦æ¹É³è ÛÜØ
pow
// PRE : e >= 0 || b != 0.0
// POST : return value is b^e
double pow ( double b , int e)
{
assert ( e >= 0 || b != 0.0);
double result = 1.0;
// the remainder is as before
...
}
ÆÈÇµÉ¿Î Ö«ÍÎÊÒÉ ÊÔÐØ ÐÒÒÉÄÍËÂÛÜÊØÝÛñÒÏËÂÊ ÙäÇÚÉÙ ]½æ ÇµÉËÂÇµÉÍâÐ®Ù³ÉÍËäÐçÛÜØ ÎµÍÂÉ³èµÛÜÙ¬ÐÑËÂÉÇµÊáÜèµÒãÐçËâÐ Ù³ÉÄÍ
ËäÐÛñØ Î ÊÛÜØ¯Ë¬éÀÆ¹ÇµÉÎµÍÂÉ³Ù³ÛñÒÉÀÒÉ³Þ®ÐçØ ËÂÛÜÙÒÏÊÔ¹ÐçØ ÐÒÒÉÍîËÂÛÜÊØÝÛÜÒâÐÒâÔ×ÊáñáÜÊÑæ·Ò³é
ÛÜÒâÉ UÐáÜÖÚÐçËÉ³è¤í ÐØµè
ÛÜÔ ÛñËòÍÂÉËÂÖ«ÍÂØµÒ íÉGP«É³Ù³Ö«ËÂÛñÊØ ÊÔ ËÂÇµÉâÎµÍÂÊåÌÍÂÐÞðËÂÉÄÍÂÞßÛÜØÚÐçËÂÉ³ÒÈÛÜÞßÞßÉ³èµÛ×ÐÑËÂÉ³áñìÀæ·ÛñËÂÇ ÐçØ ÉÄÍÍÂÊÌÍÓÞßÉ³Ò ÒÂÐåÉËÂÉ³áÜáÜÛñØÚå ÖµÒâËÂÇÚÐçËâËÇÚÉÍÂÉ³ÒîÎ ÉÙËÂÛVUÉ®ÐÒîÒÉÍËÂÛñÊØÝæÈÐÒ U¥ÛñÊá×ÐçËÂÉ³è¤é êÔ
ÍÂÉËÖµÍØÚÒ í ÉIP¥É³Ù³Ö
ËÂÛÜÊØ Ù³ÊØ ËÂÛñØôÖµÉ³ÒÏØÚÊÌÍÞ®ÐáÜáñìé¿êØ ÊÖ«ÍâÙ³ÐÒÉÌí ËÇÚÛñÒãÞßÉ¬ÐçØÚÒÓËÂÇÚÐçËÏËÂÇµÉ É UÐáÜÖÚÐçËÛÜÊØ ÊÔËÇÚÉÉIP¥ÎµÍÂÉÒÒÛÜÊØ
áñÉ¬ÐèµÒÈËÊßÐ
é¹Æ¹ÇµÛÜÒÈÞßÛÜåÇ Ë¹ØµÊÌ
Ë _ Éë
Ð UÉÍì Î ÊáÜÛñËÂÉ æÈÐ³ì ÊÔ ËÂÉ³áÜáñÛÜØµå
pow (0.0,-1)
ËÂÇµÉÏÖÚÒîÉÍ·ËÂÇÚÐçË¹ËÂÇµÉãÐçÍÂåÖµÞßÉ³Ø¯£¬ËÂ Òï¤æ¹»É¨½
ÍÂÉÏÛÜáÜ³áÜ£¬É³£³åÐç£ á _ÚÖµËÈËÂÇµÉÓÎ ÊÛÜØ Ë¹æ·ÛÜáÜáÒÖ«ÍÂÉáñìÙÊÞßÉãÐçÙÍÂÊÒÒ³é
ÊÖ¿Ù³ÐØ ÐÑÍÂåÖµÉ ËÂÇÚÐçË ÛñË ÛñÒ Ù³ÊÒîËÂáñìâËÂÊ ËÂÉ³ÒîË ËÇÚÉ¹ÐçÒÒÉÍËÛÜÊØÛñØ¿IÉ UÉÍì Ô×ÖµØµÙËÂÛÜÊØ¿Ù¬ÐáÜá»í ÖµÒîË ËÂÊÓÙ¬ÐçËÂÙäÇ
Þ®ÐÙÄÍÂÊ«í
Ð Ô×Éæ N_ Ð
è ÓÙ¬ÐáÜáñÒ³é ·Ê¦æ¹É UÉÍ³í¯Û²Ë ÛÜÒ Î ÊÒÒÛ _ áñÉËÂÊÓËÂÉ³áÜáôËÇÚÉïÙÊÞÀÎ ÛñáÜÉÍ ËÊÏÛñåØÚÊÌÍÉ ËÂÇµÉ
ÞßÉ¬ÐØµÛÜØµåâËÇ ÐÑËÈÐØÉ³ÞÀÎµË ìÀÎÚÛÜÉ³ÙÉ ÊÔ Ù³Ê èÚÉ ÍÂÉÄÎ áÜÐÙ³É³Ò¹ÛñË¬é Æ¹ÇµÉÓÖÚÒîÖ ÐçáæÈÐ¬ìßËÊ¿assert
åÊ¿ÛÜÒ ËÂÇµÉÍÉ³Ô×ÊÌÍÂÉ
ÐÒÃÔ×ÊáÜáÜÊ¦æ Ò ¡èµÖµÍÛÜØµåÏÙÊôèµÉòèµÉ UÉáÜÊÌÎÚÞßÉ³Ø Ë¬íÌÎÚÖ«ËÐÒÒÉÍîËÂÛÜÊØµÒ É UÉÍì æ ÇµÉÍÂÉÈìÊÖ æÈÐØ Ë ËÂÊ _ ÉÈÒÖ«ÍÂÉÈËÇ ÐÑË
ÒÊÞßÉËÇÚÛñØÚå ÍÂÉ¬ÐáñáñìÝÇµÊáÜèµÒ³é S¼ÇµÉ³Ø ËÂÇµÉßÙ³ÊôèµÉÀÛñÒàÒîËäÐ _ÚáÜÉ ^ÐØµè ØµÊ ÐÒîÒÉÍËÂÛñÊØ UôÛÜÊá×ÐçËÛÜÊØµÒãÒÉÉ³Þ ËÂÊ
ÊôÙÙ³Ö«Í ÐØ¯ì¥ÞßÊÌÍÂKÉ `ÄíôËÂÉ³áñáËÂÇµÉãÙ³ÊÞÀÎÚÛÜáñÉÍ ËÂÊ ÍÂÉ³ÞßÊ UÉ ËÂÇµÉâÐÒîÒÉÍËÂÛñÊØÚÒéÕÆ¹ÇµÉãÞ®ÐçÙ´ÇµÛÜØµÉ á×ÐçØÚåÖÚÐåÉ ÙÊôèµÉ
ÛÜÒ¹ËÂÇµÉ³Ø ÐÒÈÉ Ù³ÛÜÉ³Ø¯Ë ÐÒòÛÜÔ¡ìÊÖ æ¹ÊÖÚáñèØµÉ UÉÍ·Ç XÐ UÉÓæ·ÍÂÛ²ËËÂÉ³Ø ËÂÇµÉâÐÒîÒÉÍËÂÛñÊØÚÒÈÛÜØ ËÂÇµÉ YµÍÂÒîËÈÎÚá×ÐÙ³Éé
Æ Ê ÖµÒÉÏËÂÇµÉ
Þ®ÐÙÍÂÊ«í æ¹ÉãÇÚXÐ UÉãËÂÊÛÜØµÙ³áÜÖµèµÉ ËÂÇµÉãÇµÉ¬ÐèµÉÍ
é
assert
cassert
3.1.2 Function definitions
ÛÜØµÉ³Ò ! ÊÔ ÍÂÊåÌÍäÐçÞ èµÉ YÚØµÉ ÐÈÔ×ÖÚØµÙËÛÜÊØàÙ¬ÐáÜáñÉ³è
é ÆÈÇµÉ Òì¥Ø ËÂÐ PàÊÔ«ÐòÔ×ÖÚØµÙËÛÜÊØëèµÉ YÚØµÛñËÂÛÜÊØ
pow
ÛÜÒòÐÒÈÔ×ÊáÜáÜÊ¦æ Ò³é
^ í í¤éñé²éñí `
Æ¹ÇÚÛñÒ èµÉ YÚØÚÉÒÓÐ Ô×ÖµØÚÙÄËÂÛÜÊØ Ù¬ÐáÜáñÉ³è í ·
æ ÛñËÂÇ
íÐØµè æ·ÛñËÂÇ
íñé´é³é¬í ÊÔ¤Ë ìôÎ ÉÒ íñéÄé¬é³í í¥ÍÂÉ³£³Òî³ÎÉ³Ù£¬ËÂ¼
ÛVU ÉJÑáñ¢ìí« ÐØµè®æ ÛñËÂÇ Ð ¶ç£¬¨½©¯ï©£îÁ¨½³¤º é
¶ ±¬î ³ FJ
" ì¥Ø¯ËäÐÙËÂÛñÙ¬ÐáÜáñìí ÐØµè íñéé¬éí ÐçÍÂÉ Ë ìôÎ ÉÃØÚÐÞßÉ³Ò³í ÐÒ¤æ¹É³áÜá¯ÐÒ íñé´é³é¬í ÐçÍÂÉ·ÛÜèµÉ³Ø ËÂÛ Y ÉÄÍÂ
Ò ^ " É³ÙËÂÛÜÊØ \ é é `Äí¯ÐØµè
ÛñÒïÐ _ÚáÜÊ Ù ]í¥ÐÏÒXÉ WôÖµÉ³ØµÙ³É ÊÔ ÒËäÐçËÂÉ³ÞßÉØ ËÂÒÕÉ³ØµÙ³áÜÊÒÉ³è _ ì
Ù³Ö«ÍÂáñ
ì _µÍäÐÙÉ³Ò ^ " É³ÙËÂÛÜÊØ \ é é `´é
SÝÉãÙ¬ÐØËÂÇµÛÜØ ]®ÊÔ¾ËÂÇµÉãÔ×ÊÌÍÂÞ®Ðçá¤ÐçÍÂåÖµÞßÉ³Ø ËÒ¹ÐÒÈÎÚá×ÐçÙ³É³ÇµÊáÜèµÉÍÂÒ¹Ô×ÊÌÍ·ËÇÚÉãÐÙËÖ Ðçá¤ÐçÍÂåÖµÞßÉ³Ø ËÒ¹ËÇ ÐÑË
ÐçÍÂÉÏÒÖ«ÎµÎÚáÜÛÜÉ³è ^ÊÌ
Í Î ÐÒîÒÉ³
è `¹èµÖ«ÍÂÛÜØµåÐ¿Ô×ÖµØµÙËÂÛñÊØÙ¬Ðáñáé
[ ÖÚØµÙËÛÜÊØ èµÉ YÚØµÛñËÂÛñÊØÚÒÕÞÖµÒîË¹ØµÊÌËÈÐçÎµÎ É³ÐçÍ¹ÛÜØµÒÛÜèµ
É _ÚáÜÊ Ù ]ôÒ³í¥ÊÌËÂÇµÉÍÈÔ×ÖµØµÙËÂÛÜÊØµÒ³íôÊÌÍ¹Ù³ÊØ ËÍÊá ÒîËäÐçËÂÉ ÞßÉ³Ø¯ËÂÒ³é¡ÆÈÇµÉìÏÞ®Ð³ì ÐçÎµÎ É³ÐçÍ¾ÛÜØµÒÛÜèµÉ ØÚÐÞßÉ³ÒîÎ ÐÙ³ÉÒ³í¬ËÂÇµÊÖÚåÇ¤í¦ÊÌÍ¡ÐçË¾åáÜÊ _ ÐáÌÒÙÊÌÎ ÉÌíÑáÜVÛ ]É ÛÜØ
é
callpow.C
µÎ ÍÂÊåÌÍÂÐÞ Þ®Ð³ì¿Ù³ÊØ ËÂÐÛÜØ®ÐØ®ÐçÍ _ Û²ËÍäÐçÍìØôÖµÞ_ ÉÍÕÊÔ Ô×ÖµØÚÙÄËÂÛÜÊØßèµÉ YÚØµÛñËÂÛÜÊØµÒ³í¯ÐçÎµÎ É¬ÐÑÍÂÛÜØµåâÊØµÉ ÐÔÜËÉÍ
ÐØµÊÌËÂÇµÉÍÈæ ÛñËÇÚÊÖ«ËòÐØ¯ì èµÉ³áñÛÜÞßÛñËÂÉÍÒ _ ÉË æ¹É³É³ØóËÂÇµÉ³Þ éÕêØóÔªÐÙÄË¬íÚËÇÚÉÓÎµÍÂÊåÌÍäÐÞ
Ù³ÊØµÒÛÜÒËÂÒ
callpow.C
ÊÔ
Ô×ÖµØµÙËÂÛÜÊØèÚÉIY ØµÛñËÛÜÊØµÒ³í¥ÒÛÜØµÙ³ÉÓËÂÇµÉ
Ô×ÖµØµÙËÂÛñÊØóÛÜÒÈÐ¿Ô×ÖµØµÙËÂÛñÊØ ÐÒ¹æ¹É³áÜáé
¸Ó
main
3.1.3 Function calls
êØ ÍÂÊåÌÍäÐÞ í
ÛÜÒÈÊØµÉãÊÔ Y UÉÏÔ×ÖµØµÙËÂÛÜÊØóÙ³ÐáÜáÜÒ³é [ ÊÌÍÂÞ®ÐçáÜáñìí Ð¿Ô×ÖµØµÙËÂÛÜÊØ ¬Ù ÐáÜá¤ÛÜÒÈÐØ
pow(2.0,-2)
IÉ PôÎÚÍÉ³ÒÒÛÜÊØ¤é Æ¹ÇÚÉ
Òîì¥Ø¯ËäÐ PÝÊÔ¹Ð Ô×ÖµØµÙËÂÛÜÊØ Ù¬çÐ áÜáÃËÇ ÐÑËâÞ®ÐçËÙ´ÇµÉ³ÒÏËÂÇµÉ åÉ³ØµÉÍÂÐáÔ×ÖµØµÙËÂÛÜÊØ èÚIÉ Y ØµÛñËÛÜÊØ
ÔÜÍÂÊÞðÐ _ Ê UÉãÛñÒ·ÐÒÈÔ×ÊáÜáÜÊ¦æ Òé
^ í¤éñé²éñí `
·ÉÍÉÌí íñé¬éé³í Þ¿ÖµÒîË
_ É®ÉIPôÎÚÍÉ³ÒÒÛÜÊØµÒ¿ÊÔòË ìôÎÉ³Òàæ ÇµÊÒÉUÌÐáñÖÚÉÒ¿Ù³ÐØ_ É®Ù³ÊØ UÉÍËÂÉèËÂÊ
ËÂÇµÉ Ô×ÊÌÍÞ®Ðá ÐçÍÂåÖµÞßÉ³Ø¯ËÏË ì Î É³Ò íñéé³é³í éÀÆÈÇµÉ³ÒÉÉIP¥ÎµÍÂÉÒÒÛÜÊØµÒëÐÑÍÂÉËÂÇµÉ
é [ ÊÌÍ
Ä
±
¯
©
»

ÐáÜá Ë ì Î É³ÒãËÇ ÐÑËãæ¹É¥] ØµÊ¦æ ÒîÊÔªÐçÍ¬í¾ËÂÇµÉÙ³ÐáÜáÃÐçÍÂåÖµÞßÉ³Ø¯ËÂÒÓÐÒãæ¹ÉáÜáÃÐÒÏËÇÚÉ¿Ô×ÖµØµÙËÂÛÜÊØÝÙ³ÐáÜá ÛñËÂÒîÉ³áÜÔ ÐÑÍÂÉ
Í UÐáÜÖµÉ³Ò³éÕÆ¹ÇÚÉÓË ìôÎ ÉãÊÔ¡ËÂÇµÉÏÔ×ÖµØÚÙÄËÂÛÜÊØóÙ¬ÐçáÜá ÛÜÒ¹ËÂÇµÉãÔ×ÖµØµÙËÂÛÜÊØ ÒÈÍÂÉÄËÂÖ«ÍÂØË ìôÎ É é
S¼ÇÚÉØóÐàÔ×ÖµØÚÙÄËÂÛÜÊØ®Ù¬ÐáÜá ÛñÒïÉ ÌU ÐçáÜÖÚÐçËÂÉ³è¤íôËÂÇµÉ Ù¬ÐçáÜá ÐçÍÂåÖµÞßÉ³Ø¯ËÂÒïÐçÍÉÏÉIÌU ÐáñÖ ÐÑËÂÉ³èÚ
Y ÍÒîË
^ ÛÜØ ÐØ ÊÌÍÂèµÉÍ
ËÂÇÚÐçË ÛñÒ ÖµØÚÒÎ É³Ù³ÛVÚY É³èZ_ ìëËÂÇµÉ ÒîËäÐØµèÚÐçÍÂè Ä` é ÆÈÇµÉ ÍÂÉ³ÒîÖÚá²ËÂÛÜØµåU ÐáÜÖµÉ³Ò ÐçÍÉ ËÂÇµÉ³Ø ÖÚÒîÉ³è¿ËÊ ÛÜØµÛñËÂÛ×ÐáñÛ³É
ËÂÇµÉÔ×ÊÌÍÞ®Ðá·ÐÑÍÂåÖµÞßÉ³Ø ËÂÒé [ ÛÜØÚÐáÜá²ìíÃËÂÇµÉóÔ×ÖµØµÙËÂÛñÊØ _ Êôè«ì ÛÜÒ ÉG«P É³Ù³Ö«ËÂÉè òÛÜØ ËÂÇµÛÜÒÉI«P É³ÙÖµËÛÜÊØ¤íËÂÇµÉ
Ô×ÊÌÍÂÞ®Ðá ÐçÍÂåÖµÞßÉ³Ø ËÒ
_ É³ÇÚÐKU É áÜÛV] ÉóËÇÚÉÄì¼æ¹ÉÄÍÂÉ U ÐçÍÂÛÜÐ Ú
_ áÜÉ³Ò®èµÉ Ú
Y ØÚÉè ÛÜØ¼ËÂÇµÉ _ ÉåÛÜØµØÚÛñØÚå ÊÔ
í
ÛÜØµÛñËÂÛÜÐáÜÛ³ÉèÀæ ÛñËÂÇ ËÂÇµÉ U ÐáÜÖµÉ³ÒÈÊÔ ËÂÇµÉÏÙ¬ÐáÜá¤ÐÑÍÂåÖµÞßÉ³Ø ËÂÒé
ÆÈÇµÉ ÉIÌU ÐáñÖ ÐÑËÂÛÜÊØÀÊÔ ÐÏÔ×ÖµØÚÙÄËÂÛÜÊØßÙ¬ÐáñáÚËÂÉÄÍÂÞßÛÜØÚÐçËÂÉ³ÒÃÐÒÒÊôÊØ®ÐÒÐÏÍÂÉÄËÂÖ«ÍÂØ®ÒîËäÐçËÂÉÞßÉ³Ø ËÕÛÜÒÕÍÉ¬ÐÙäÇµÉ³è¤í
ÒÉ³É " É³ÙËÂÛñÊØ \ é é é ÆÈÇµÛÜÒ¹ÍÂÉËÂÖ«ÍÂØÒîËäÐçËÂÉÞßÉ³Ø ËÈÞ¿ÖÚÒË
_ ÉãÊÔ¾ËÂÇµÉãÔ×ÊÌÍÂÞ
ÛñÒ·ÐØÉIP¥ÎµÍÂÉ³ÒîÒÛÜÊØ
return
æ ÇµÉÍÉ
ÊÔ ÐàË ìôÎ ÉÏæ ÇµÊÒÉ
UÌÐáñÖÚÉÒ·Ù¬ÐØ _ ÉâÙ³ÊØ UÉÍËÂÉèóËÂÊËÂÇµÉÏÍÂÉËÂÖ«ÍÂØË ì Î É
éµÆÈÇµÉÓÍÂÉ³ÒÖµáñËÂÛñØÚå UÐáÜÖµÉÓÛÜÒïËÂÇµÉ UÐáÜÖµÉÓÊÔ¡ËÇÚÉÓÔ×ÖµØµÙËÂÛÜÊØ Ù¬ÐáÜá»é ÆÈÇµÉÏÉ É³ÙÄË·ÊÔ¾ËÂÇµÉãÔ×ÖµØµÙËÂÛÜÊØ Ù¬Ðáñá ÛÜÒ
èµÉËÂÉÍÞßÛÜØµÉ³è _ôìÀËÂÇµÉ ÊÛÜØ Ë É É³ÙÄËÂÒòÊÔ¤ËÂÇµÉ Ù¬Ðáñá ÐçÍÂåÖµÞßÉ³Ø¯ËïÉ UÐáÜÖÚÐçËÂÛÜÊØµÒ³íôÐØµè ÊÔ¾ÉIP¥É³Ù³Ö«ËÂÛÜØµå
é
ÆÈÇµÉòÔ×ÖÚØµÙËÛÜÊØ _ Ê èµì¿Þ®Ð³ì Ù³ÊØ¯ËäÐÛÜØÀÒîÉ UÉÍäÐáÚÍÂÉËÂÖ«ÍÂØ ÒîËäÐçËÂÉÞßÉ³Ø ËÂÒí_ÚÖ«ËÕÛÜÔØµÊãÍÂÉÄËÂÖ«ÍÂØÀÒîËäÐÑËÂÉ³ÞßÉ³Ø¯Ë
ÛÜÒ¤ÍÂÉ¬ÐÙäÇµÉ³èàèµÖ«ÍÂÛÜØµå¹ËÂÇµÉÕIÉ P¥É³Ù³Ö«ËÂÛÜÊØâÊÔ
íUÐáÜÖµÉÐçØÚèâÉ É³ÙÄË ÊÔ ËÇÚÉÕÔ×ÖµØÚÙÄËÂÛÜÊØâÙ³ÐáÜá¯ÐçÍÂÉÃÖµØÚèµÉ YÚØµÉ³è
^ÖµØµáÜÉ³ÒÒ¹ËÂÇµÉÓÍÂÉËÂÖ«ÍÂØË ì Î ÉãÛñÒ
íÚÒîÉ³É " É³ÙÄËÂÛÜÊØ é é _ É³áÜÊ¦
æ `Äé
void
[ ÊÌÍàIÉ PµÐçÞÀÎ áñÉÌí¡èµÖ«ÍÂÛÜØµå ËÂÇµÉÀIÉ P«ÉÙ³Ö«ËÂÛÜÊØ ÊÔ
ÛñØ
í ÐçØÚè
ÛÜØµÛñËÛ×ÐáÜáñìóÇ XÐ UÉ
b
e
UÐáÜÖµÉ³Ò
ÐØµè
é Æ¹ÇÚÉÒÉ UÐáÜÖµÉ³ÒÐçÍÂÉ Ù´ÇÚÐØµåÉ³è ÛñØ ËÂÇµÉpow(2.0,-2)
ÒîËäÐçËÂÉÞßÉ³Ø Ë ËÂÊ
ÐØµè í _ ÉÔ×ÊÌÍÂÉ
0.5
ËÂÇµÉÒîaÖ _Ú2ÒÉ WôÖµÉ³Ø −2
ËÀáÜÊôÊÌÎ ÒîÉËÂÒ
ËÂÊ
ÛÜØ Û²ËÂÒ YµÍÂÒîËÀÐif
ØµèËÂÊ
ÛÜØ ÛñËÂÒ
ÒÉ³Ù³ÊØµè 2ÐØµè á×ÐçÒîË
ÛñËÂÉÄÍäÐçËÂÛÜÊØ¤éÏÆ¹ÇÚÛñ
Ò UÌÐáñÖÚÉ ÛÜÒ ÍÂÉÄËÂresult
Ö«ÍÂØµÉ³èÝÐØµè 0.5
_ É³Ù³ÊÞßÉÒ ËÂÇµ
É UÐáÜÖµÉ ÊÔ ËÂ0.25
ÇµÉ¿Ô×ÖµØµÙËÂÛñÊØ½Ù¬ÐçáÜá IÉ P¥ÎµÍÂÉÒÒÛÜÊØ
é
pow(2.0,-2)
3.1.4 The type void
êØ í«ËÂÇµÉÍÉÏÛñÒ¹ÐëÔ×ÖÚØµèÚÐÞßÉ³Ø¯ËäÐáÚË ì Î ÉÓÙ¬ÐáÜáñÉ³è
í«ÖµÒÉ³èÐÒïÍÂÉËÂÖ«ÍÂØ®Ë ìôÎ Ó
É Ô×ÊÌÍÈÔ×ÖµØÚÙÄËÂÛÜÊØµÒ ËÂÇÚÐçË
ÊØµáñì®ÇÚÐXUÉëÐØóÉ É³ÙË³ía_ÚÖ«ËòØÚÊbUÌÐçáÜÖµÉÌé " µÖ ÙäÇ Ô×ÖµØµÙvoid
ËÂÛÜÊØµÒÈÐçÍÂÉâÐáÜÒÊ¿Ù¬ÐáÜáñÉ³è
é
D¥ FâÄ¶ ±¬¤
Ò ÐØ ÉIPµÐÞÀÎÚáÜÉíÙ³ÊØµÒÛÜèµÉÍÓËÂÇµÉàÔ×ÊáÜáñÊÑæ·ÛÜØµå ÎµÍÂÊåÌÍäÐÞ ^»ØÚÊÌËÉëËÂÇÚÐçË ËÂÇµÉ ×Ô ÖÚØµÙËÛÜÊØ
Ï
ÍÂÉXW ÖµÛñÍÂÉ³ÒÈØµÊÎµÍÂÉ³Ù³ÊØµèµÛñËÂÛñÊØ¾íôÒÛÜØµÙ³ÉÏÛñËòæ¹ÊÌÍ ]¥ÒÈÔ×ÊÌÍ ÐØ¯ì Ù³ÊÞb_ÚÛÜØÚÐçËÛÜÊØ®ÊÔ
U ÐáÜÖµÉ³Ò `Äé print_pair
int
# include < iostream >
// POST : "( i , j )" has been written to standard output
void print_pair ( int i , int j)
{
std :: cout < < "( " < < i < < " , " < < j < < " )\ n";
}
int main ()
{
print_pair (3 ,4); // outputs
}
(3 , 4)
Æ ÇµÉâË ì Î É
È
Ç çÐ Ò·É³ÞÀÎµË ì UÐáÜÖµÉÏÍäÐØµåÉÌí ÐØµèóËÂÇµÉÍÂÉàÐçÍÉâØµÊÀáÜÛñËÉÍäÐáÜÒíaUÐçÍÂÛÜÐ _ÚáÜÉ³Ò³íµÊÌÍ Ô×ÊÌÍÞ®Ðá
void
Ô×ÖµØµÙËÂÛÜÊØâÐçÍÂåÖµÞßÉ³Ø ËÒ ÊÔ Ë ìôÎ É
é ÆÈÇµÉÍÉÐçÍÉÕÉIPôÎµÍÂÉ³ÒÒÛñÊØÚÒ¾ÊÔË ì Î É
íçËÇÚÊÖµåÇ¤í¦Ô×ÊÌÍ ÉGPµÐÞÀÎÚáÜÉ
void
void
é
print_pair(3,4)
UÊÛÜè Ô×ÖµØµÙËÂÛÜÊØ µ
è Ê É³ÒØµÊÌËóÍÂÉ WôÖµÛñÍÂÉ Ð ÍÂÉËÂÖ«ÍÂØ ÒËäÐçËÂÉ³ÞßÉØ Ë¬í_ÚÖ«Ë ÛñË Þ®Ð³ì Ù³ÊØ¯ËäÐÛÜØ ÍÂÉËÂÖ«ÍÂØ
ÒîËäÐÑËÂÉ³ÞßÉ³Ø¯ËÂÒïæ ÛñËÂÇ
Ê Ô¾Ë ì Î É
í«ÊÌÍòÍÉËÂÖ«ÍÂØ ÒîËäÐÑËÂÉ³ÞßÉ³Ø¯ËÂÒ¹ÊÔ¡ËÂÇµÉÏÔ×ÊÌÍÂÞ
return
void
UÌÐáñÖ ÑÐ ËÂÛÜÊØÊÔ·Ð UÊÛÜèÔ×ÖÚØµÙËÛÜÊØÙ¬ÐçáÜáïËÂÉÍÂÞßÛÜØÚÐçËÉ³Òàæ ÇµÉ³Ø Ð ÍÂÉËÂÖ«ÍÂØ ÒËäÐçËÂÉ³ÞßÉØ Ë¿ÛÜÒ ÍÂÉ¬ÐÙäÇµÉ³è¤í
£
æ µÇ É³Ø ÂË ÇµÉãÉIP¥É³Ù³Ö«ËÂÛÜÊØÊÔ
ÛÜÒ Y ØµÛÜÒîÇÚÉè¾é
3.1.5 Functions and scope
Æ¹ÇÚÉ¿Î ÐçÍÂÉ³Ø¯ËÂÇµÉ³ÒÛ³Éè Î ÐçÍËãÊÔ Ð®Ô×ÖÚØµÙËÛÜÊØÝèµÉ YÚØµÛñËÂÛÜÊØ Ù³ÊØ ËÂÐÛÜØµÒÏËÇÚÉ¿èµÉ³Ù³á×ÐÑÍäÐçËÂÛÜÊØµÒ ÊÔËÇÚÉ¿Ô×ÊÌÍÂÞ®Ðá
ÐçÍÂåÖµÞßÉ³Ø¯ËÂÒ³é [ ÊÌÍ ÐáÜáôÊÔ¥ËÇÚÉÞ íçËÂÇµÉ èµÉ³Ù³áÜÐçÍäÐçËÂÛ UÉÕÍÂÉ³åÛñÊØ¿ÛñÒ ËÂÇµÉ Ô×ÖµØµÙËÂÛÜÊØàèµÉ YÚØµÛñËÂÛÜÊØ¤íçÒÊ ËÂÇµÉ Ô×ÊÌÍÂÞ®Ðá
ÐçÍÂåÖµÞßÉ³Ø¯ËÂÒÏÇ XÐ UÉáÜÊôÙ¬ÐçáÃÒîÙ³ÊÌÎ É ^ " ÉÙËÂÛÜÊØ \ é é `Äé ÆÈÇµÉ ÎÊÌËÂÉ³Ø ËÛ×Ðá ÒÙ³ÊÌÎÉÀÊÔïÐ Ô×ÊÌÍÞ®ÐáÃÐçÍÂåÖµÞßÉ³Ø Ë
èµÉ³Ù³á×ÐÑÍäÐçËÂÛÜÊ
Ø _ É³åÛñØÚÒòÐÔÜËÂÉÄÍ·ËÂÇµÉÏèÚÉÙ³á×ÐçÍäÐÑËÂÛÜÊØóÐçØÚèIÉ P¥ËÂÉØÚèµÒòÖÚØ¯ËÂÛÜáËÂÇµÉãÉ³ØµèóÊÔ ËÂÇµÉâÔ×ÖµØµÙËÂÛñÊØ _ Ê èµìé
Æ¹ÇÚÉÄÍÂÉ³Ô×ÊÌÍÂÉÌí ËÂÇµÉâÔ×ÊÌÍÂÞ®Ðá¾ÐçÍåÖÚÞßÉØ ËÂÒòÐçÍÂÉëØµÊÌ
Ë U¥ÛÜÒÛ _ áñÉãÊÖ«ËÂÒÛñèÚÉÏËÂÇµÉâÔ×ÖµØµÙËÂÛÜÊØ èµÉ YÚØµÛñËÂÛñÊØ¾é S Û²ËÂÇµÛÜØ
ËÂÇµ
É _ Êôè«ìí¥ËÂÇµÉÏÔ×ÊÌÍÂÞ®Ðá¤ÐçÍÂåÖµÞßÉ³Ø¯ËÂR
Ò _ É³ÇÚXÐ UÉãáñVÛ ]É UÐçÍÂÛÜÐ _ÚáÜÉ³ÒïËÂÇÚÐçËòÐçÍÂÉÏáÜÊôÙ³Ðá ËÂÊ é
êØ Î ÐçÍËÂÛñÙ³Öµá×ÐçÍ¬íÙäÇÚÐØµåÉ³ÒÏÞ®ÐèµÉëËÂÊ®ËÂÇµ
É UÐáÜÖµÉ³ÒÏÊÔÔ×ÊÌÍÞ®Ðá ÐçÍÂåÖµÞßÉ³Ø¯ËÂÒ ^áÜÛ ]É¿ÛñØ ËÂÇµÉ¿Ô×ÖµØÚÙÄËÂÛÜÊØ
`ëÐçÍÂÉ îáÜÊÒîË ÐçÔÜËÂÉÍàËÂÇµÉ®Ô×ÖµØÚÙÄËÂÛÜÊØ Ù¬ÐáÜáÐØµè ÇÚXÐ UÉ®ØÚÊ É É³ÙË¿ÊØ ËÇÚL
É UÐáÜÖµÉ³Ò ÊÔ¹ËÂÇµÉßÙ¬ÐáÜá ÐçÍ pow
åÖµÞßÉ³Ø ËÒ³éàÆ¹ÇÚÛñÒ ÛÜÒÓØÚÊÌËÏÒÖ«ÍÎµÍÂÛñÒÛÜØµå«í ÒÛÜØµÙ³É ËÂÇµÉ Ù¬ÐáÜá ÐçÍåÖÚÞßÉØ ËÂÒÓÐçÍÉ Í UÌÐáñÖÚÉÒ³í _ÚÖ«ËÓËÂÊ Þ®Ð ]ÉëËÂÇµÉ
Î ÊÛÜØ¯ËÈÙ³áÜÉ¬ÐçÍ³í«áÜÉËòÖÚÒÈÙ³ÊØµÒÛÜèµÉÍÈËÂÇµÉÏÔ×ÊáÜáÜÊ¦æ ÛÜØµå¿ÐáñËÂÉÄÍÂØÚÐçËÂVÛ UÉ
Ô×ÖµØµÙËÂÛÜÊØóÛñØ
é
main
int main () {
double b = 2.0;
int e = -2;
std :: cout < < pow (b ,e ); // outputs 0.25
std :: cout < < b ;
// outputs 2
std :: cout < < e ;
// outputs -2
callpow.C
return 0;
}
Æ¹ÇÚÉ
UÐáÜÖµÉ³ÒÈÊÔ¡ËÂÇµÉUÌÐÑÍÂÛ×Ð _ÚáÜÉ³Ò ÐØµè èµÉ YÚØµÉ³èóÛÜØ áÜÛÜØµÉ³Ò \ ÒîËäÐ³ì®ËÂÇµÉÏÒÐÞßÉ ËÂÇ«ÍÂÊÖµåÇÚÊÖ«Ë¬íôÒÛÜØµÙ³É
ËÂÇµÉ Ô×ÖµØµÙËÂÛÜÊØ _ Ê èµìÀÊÔ
ÛÜÒ bØµÊÌË¹ÛÜØßeËÂÇµÉ ÒÙ³ÊÌÎÉ ÊÔ¾ËÂÇµÉ³Û²ÍïèµÉ³Ù³á×ÐçÍÂÐçËÂÛÜÊØµÒ³í¯Ô×ÊÌÍïË æ¹Ê ÍÉ¬ÐÒÊØµÒ³é [ ÛñÍÂÒË¬í
pow
ËÂÇµÉ èµÉ YÚØµÛñËÂÛÜÊØ ÊÔ
ÐçÎµÎ É¬ÐÑÍÂÒ
ËÂÇµÉ èµÉ³Ùá×ÐçÍäÐçËÛÜÊØµÒÊÔ ÐØµè ÛÜØßáñÛÜØµÉ³Ò \ íôÒîÊëËÂÇµÉ_ Êôè«ì
³
»

³
£

pow
ÊÔ
Ù¬ÐØµØµÊÌËÓÉ UÉ³Ø _ É¿ÛÜØ ËÂÇµÉ
ÒîÙ³ÊÌÎ É¿ÊÔÃËÂÇµÉ³ÒîÉèµbÉ³Ù³áÜÐçÍäÐçeËÂÛñÊØÚÒé " É³ÙÊØÚè¤í¾É UÉ³ØÝÛñÔÃæ¹É
æ¹Êpow
ÖµáÜè ÞßÊ UÉ ËÂÇµÉ èµÉ³Ù³á×ÐÑÍäÐçËÂÛÜÊØµÒ Ê¢ ÔËÇÚ³¤b
É UÌ©¯ÐÑ ÍÂÛ×Ð _ÚáÜÉ³Ò ÐçØÚè
ËÂÊ ËÂÇµÉ_ É³åÛÜØµØµÛÜØµå ÊÔËÂÇµÉ¿ÎµÍÂÊåÌÍÂÐÞ
b
e
^ _ ÉÔ×ÊÌÍÂÉÏËÇÚÉÓèµÉ YÚØµÛñËÂÛÜÊØ®ÊÔ
í«ÒÊ¿ËÂÇÚÐçËïËÂÇµÉì ÇÚXÐ UÉãåáÜÊ _ ÐáÒÙ³ÊÌÎ É `´í¥ËÂÇµÉ³ÛñÍÈÒÙ³ÊÌÎ ÉÓæ¹ÊÖµáÜè IÉ P«Ù³áñÖÚèµÉ
ËÂÇµ
É _Êôè«ì¿ÊÔ
í ÒÛñØÚÙÉòËÂÇÚpow
Ðç
Ë _ Êôè«ì¿ÛÜÒÃÛÜØ ËÂÇµÉ¹Î ÊÌËÂÉ³Ø¯ËÂÛ×ÐáµÒÙÊÌÎ É³ÒÊÔ ÍÂÉ³èµÉ³Ù³á×ÐÑÍäÐçËÂÛÜÊØµÒ ÊÔËÂÇµÉÈØÚÐÞßÉ³Ò
ÐçØÚè
^»ËÂÇµÉãpow
Ô×ÊÌÍÞ®Ðá¤ÐçÍÂåÖµÞßÉ³Ø¯ËÂÒ `Äí¥ÒîÉ³É " É³ÙÄËÂÛÜÊØ \ é é é
b
e
Ö«ËïËÂÇµÉ åÉ³ØµÉÍäÐáÒÙ³ÊÌÎ É ÍÂÖµáÜÉ³ÒÊÔ " ÉÙËÂÛÜÊØ \ é é ÐáÜáÜÊ¦æ Ô×ÖµØµÙËÂÛñÊ
Ø _ÊôèµÛÜÉ³ÒËÊàÖÚÒîÉ Ø ÐçÞßÉ³Ò ÊÔ
åáÜÊ _ Ðá¥ÊÌÍÕØÚÐÞßÉ³ÒîÎ ÐÙÉÈÒîÙ³ÊÌÎ É ËÂÇµÉÈÎµÍÂÊåÌÍÂÐÞ ÊØ Î ÐåÉ Fµ Ô×ÊÌÍÕIÉ P«ÐÞÀÎÚáÜÉïÖµÒÉ³Ò
ÐÒ ÒÖµÙ´Ç
std::cout
ÐßØÚÐÞßÉÌé òÉÍÂÉ¿ÛÜÒÏÐ®ÙÊØ ËÍÂÛ UÉ³è½ÎµÍÂÊåÌÍäÐçÞ ËÇ ÐÑËÓèÚÉÞßÊØÚÒËÍäÐçËÂÉÒ ÇÚÊ¦æ ÐßÎÚÍÊåÌÍäÐÞ Þ®Ð³ì ÞßÊôèµÛÜÔÜìóÐ
^º
Ð UÐçÍÂÛÜÐ _ÚáÜÉÃæ·ÇµÊÒÉèµÉ³Ù³áÜÐçÍäÐçËÂÛñÊØâÇÚÐÒ åáñÊ _Ðçá¯ÒîÙ³ÊÌÎ É `Ä"
é S ÇµÛÜáñÉÕÒÖµÙäÇ Ù³ÊØµÒîËÍÂÖµÙËÛÜÊØµÒ
Áª¥Ð³ì ©¯_ a É¿
Dô©Öµ£¬ÒÉ³©«Ô×Öµ ¬§áÕ ÛñØ Ù³ÉÍîËäÐÛÜØ½Ù¬ÐÒÉ³Òí¾ËÂÇµÉì ÖµÒÖÚÐáÜá²ì Þ®Ð ]ÉàËÂÇµÉàÎÚÍÊåÌÍäÐÞ áÜÉ³ÒÒÓÍÂÉ¬Ðçè Ð _ áñÉÌí ÒÛÜØµÙ³É ËÂÇµÉ
Þ®
É É³ÙË ÊÔ Ð¿Ô×ÖµØµÙËÂÛñÊØóÙ¬ÐáÜá Þ®Ð³ìßËÂÇµÉ³Ø _ É³Ù³ÊÞß
É UÉÍì ØµÊØ ºáÜÊ Ù¬Ðáé
# include < iostream >
int i = 0; // global variable
void f ()
{
++ i ;
}
// in the scope of declaration in line 3
int main ()
{
f ();
std :: cout < < i < < "\n" ; // outputs 1
return 0;
}
" ÛÜØµÙ³ÉãËÇÚÉãÔ×ÊÌÍÂÞ®Ðá¤ÐçÍåÖÚÞßÉØ ËÂÒ¹ÊÔ Ð Ô×ÖµØÚÙÄËÂÛÜÊØóÇÚÐXUÉâáÜÊ Ù¬Ðá¤ÒÙ³ÊÌÎÉÌíÚËÇÚÉÄìÐçáÜÒÊ ÇÚÐXUÉëÐçÖµËÊÞ®ÐçËÂÛÜÙ
ÒîËÂÊÌÍÂÐåÉâèµÖ«ÍäÐçËÂÛñÊØ¾é¹Æ¹ÇµÛÜÒ ÞßÉ¬ÐçØÚÒ¹ËÂÇÚÐçËòæ¹ÉëåÉË Ð î ÔÜÍÂÉ³ÒÇ
ß
ÒÉË ÊÔ Ô×ÊÌÍÂÞ®Ðá¡ÐçÍÂåÖµÞßÉ³Ø ËÒòÉ UÉÄÍìóËÂÛÜÞßÉ
ËÂÇµÉâÔ×ÖµØµÙËÂÛñÊØóÛÜÒÈÙ¬ÐáÜáÜÉè¾í¥æ·ÛñËÂÇóÞßÉÞßÊÌÍì®ÐÒÒîÛÜåØµÉ³èËÂÊ¿ËÂÇµÉ³Þ ÊØµáñì ÖµØ ËÂÛñá¤ËÇÚÉÏÍÂÉ³ÒîÎÉ³ÙËÂÛVUÉãÔ×ÖµØÚÙÄËÂÛÜÊØ
Ù¬Ðáñá¤ËÉÍÂÞßÛÜØÚÐçËÂÉÒ³é
O ÐÞßÉ³Ò¾èµÉ³Ù³á×ÐÑÍÂÉ³èëÛÜØµÒÛñèÚÉ ËÂÇµÉÕÔ×ÖÚØµÙËÛÜÊØ _ Ê èµìÏÞ¿ÖµÒî"
Ë _ÉÕèµÛÜÒîËÛÜØµÙË¾ÔÜÍÂÊÞ ËÂÇµÉÕØ ÐçÞßÉ³Ò¤ÊÔ«ÐáÜáÔ×ÊÌÍÂÞ®Ðá
ÐçÍÂåÖµÞßÉ³Ø¯ËÂÒ³í¤ÖµØµáÜÉ³ÒÒÓËÂÇµÉì½ÐçÎµÎ É¬ÐçÍãÛÜØ Ð ØµÉ³ÒîËÉ³è _ÚáÜÊôÙ ]é¿Æ¹ÇµÛÜÒãÞ®Ð ]É³ÒÓÒÉ³ØµÒÉÀÒÛñØÚÙÉÊÌËÂÇµÉÍîæ ÛÜÒÉí¾Û²Ë
æ¹ÊÖµáÜè _ ÉßÎ ÊÒÒVÛ _ÚáÜÉÀËÊ½ÛñÍîÍÂÉ UÊôÙ³Ð _Úáñì ÇµÛÜèµÉÀËÂÇµÉ®Ø ÐçÞßÉßÊÔ Ð Ô×ÊÌÍÂÞ®ÐáïÐÑÍÂåÖµÞßÉ³Ø Ë¬é½Æ¹ÇÚÉÄÍÂÉ³Ô×ÊÌÍÂÉÌíÃæ¹É
Ù¬ÐØµØµÊÌËÈæ·ÍÂÛñËÉ
int f ( int i)
{
int i = 5; // invalid ; i hides formal argument
return i ;
}
æ ÇµÛÜáñÉ ËÂÇµÉãÔ×ÊáÜáñÊÑæ·ÛÜØµå¿ÛñÒòØµÊÌË¹ÍÂÉ³Ù³ÊÞßÞßÉØÚèµÉ³è_ÚÖ«Ë·áÜÉåÐáé
int f ( int i)
{
{
int i = 5; // ok ; i is local to nested block
}
return i ; // the formal argument
}
Æ ÇÚÉïáÜÐçËËÂÉÍ Ô×ÖÚØµÙËÛÜÊØÛñÒ ËÂÇµÉïÛÜèµÉ³Ø ËÛñË ìíÌÒÛÜØµÙ³É ËÂÇµÉïÒÙÊÌÎ É¹ÊÔµËÇÚÉ¹èµÉ³Ù³áÜÐçÍäÐçËÂÛñÊØ
¹
ÛÜÒ áÜÛÜÞßÛ²ËÂÉ³è
int i = 5
ËÂÊ¿ËÂÇµÉãØµÉ³ÒîËÉ³è _ÚáÜÊôÙ ]é
Ô×ÖµØÚÙÄËÂÛÜÊØ ÛñËÂÒÉáÜÔ¹ÐáÜÒÊ ÇÚÐÒ ÐóÒîÙ³ÊÌÎ ÉÌíÃÐØµè ËÂÇµÉßÔ×ÖµØÚÙÄËÂÛÜÊØ Ù¬ÐØ ÊØµáñì _ É
Ù¬ÐáñáÜÉ³èæ·ÛñËÂÇµÛÜØÛ²ËÂÒ·ÒÙÊÌÎ ÉÌéòÆÈÇµÉëÒÙÊÌÎ ÉëÊÔ ÐÔ×ÖµØµÙËÂÛñÊØ ÛÜÒ·Ê _µËäÐÛñØÚÉè _ôìÙ³ÊÞb_ÚÛÜØµÛÜØµå ËÂÇµÉâÒÙÊÌÎ É³Ò ÊÔ
ÐáÜá«Û²ËÂÒ
éÃÆ¹ÇµÉÈÎ ÐçÍËÕÊÔ ËÇÚÉÈÔ×ÖµØµÙËÂÛÜÊØèÚÉIY ØµÛñËÛÜÊØb_É³Ô×ÊÌÍÂÉ
ÛñÒÐÓèµÉ³Ù³á×ÐçÍÂÐçËÂÛÜÊØ¤í _ÚÖ«Ë
FµÄÐ³±¬ì ×©_ £É¿
©¯Ô×îÖÚØµ¤Ù ËÛÜÊØ èµÉ³Ù³áÜÐçÍäÐçËÂÛñÊØÚÒ ËÇ ÐÑËãÇÚÐXUÉ ØµÊ ÒÖ _ ÒîÉXWôÖµÉ³Ø¯Ë
ËÂÇµÉÍÂÉ¿Þ®
é¿ÆÈÇµÛÜÒÏÛÜÒÓÛÜØ ÙÊØ ËÍäÐçÒîË
ËÂZ
Ê UÐçÍÂÛ×Ð _ áñÉ³Òæ ÇµÉÍÉ É UÉÍì®èµÉ³Ù³áÜÐçÍäÐçËÂÛñÊØßÛÜÒïÐçË ËÇÚÉ ÒÐÞßÉ·ËÂÛÜÞßÉ·ÐâèµÉ YÚØµÛñËÂÛÜÊØ¤é Ô×ÖµØµÙËÂÛñÊØ Þ®Ð³ìb_ É
èµÉ³Ù³á×ÐÑÍÂÉ³èóÒIÉ UÉÍäÐá¤ËÂÛÜÞßÉ³Òí _ÚÖµËòÛñËÈÙ¬ÐØ _ ÉãèµÉ YÚØµÉ³èóÊØµÙ³ÉãÊØµáñìé
ÆÈÇµÉàÔ×ÊáÜáñÊÑæ·ÛÜØµå¿ÎµÍÂÊåÌÍäÐçÞ í Ô×ÊÌÍ GÉ PµÐÞÀÎÚáÜÉÌíÚèµÊôÉ³Ò·ØÚÊÌË Ù³ÊÞÀÎÚÛÜáÜÉÌí«ÒÛñØÚÙÉëËÂÇµÉëÙ¬ÐáÜá¡ÊÔ ÛÜØ
ÛÜÒ
f
main
ØµÊÌË·ÛñØ ËÂÇµÉÏÒÙ³ÊÌÎ ÉãÊÔ é
Function declarations.
f
# include < iostream >
int main ()
{
std :: cout < < f (1); // f undeclared
return 0;
}
int f ( int i ) // scope of f begins here
{
return i ;
}
Ö«Ë
æ ÉàÙ¬ÐØ ÎÚÖ«Ë ÛñØ ËÂÊÀËÂÇµÉàÒÙÊÌÎ ÉàÊÔ
¹
_ôì ÐèµèµÛÜØµåßÐÀèµÉ³Ù³áÜÐçÍäÐçËÂÛñÊØ _ É³Ô×ÊÌÍÂÉ
í ÐØµè ËÇÚÛñÒ
f
main
main
ì¥ÛñÉ³áÜèµÒÈÐUÐáÜÛÜè ÎµÍÂÊåÌÍäÐÞ é
# include < iostream >
int f ( int i ); // scope of f begins here
int main ()
{
std :: cout < < f (1); // ok , call is in scope of f
return 0;
}
int f ( int i)
{
return i ;
}
ê ØßËÂÇµÉ·ÎµÍÂÉ U¥ÛñÊÖÚÒÕÎÚÍÊåÌÍäÐÞ íæ¹É Ù³ÊÖµáÜèßåÉËÍÂÛÜèßÊÔËÂÇµÉ ÉIPôËÍäÐâèµÉ³Ùá×ÐçÍäÐçËÛÜÊØ _ôì ÒÛÜÞÀÎÚáñì èµÉ YÚØµÛÜØµå
_ É³Ô×ÊÌÍÂÉ
í _ÚÖ«ËïÒÊÞßÉËÂÛñÞßÉ³Ò³íÒÉÎ ÐçÍäÐÑËÂÉ Ô×ÖµØÚÙÄËÂÛÜÊØ®èµÉ³Ù³á×ÐçÍÂÐçËÂÛÜÊØµÒÕÐçÍÂÉ ÛÜØµèµÉ³É³è®ØµÉ³Ù³É³ÒîÒÂÐçÍìé*¹ÊØ
f
ÒÛÜèµÉÍÃË æ¹Êâmain
Ô×ÖµØµÙËÂÛÜÊØµÒ ÐØµè ÒîÖÚÙäÇÀËÂÇÚÐçË ÛñÒÕÙ¬ÐçáÜáÜÉ³è¿ÛÜØ ËÂÇµÉÈÔ×ÖµØÚÙÄËÂÛÜÊØ _ Ê è«ì ÊÔ íôÐØµè ÛÜÒ Ù¬ÐáÜáÜÉè
ÛÜØ ËÂÇµÉÏÔ×ÖµØµÙËÂÛñÊØ _ Ê èµfìÀÊÔ é g SÝÉ
ËÂgÊ¿èÚÉIY ØµÉÓÊØµÉãÊÔ¤ËÂÇµÉÓË æ¹Ê¿Ô×ÖµØµÙËÂÛÜÊØµÒRYµfÍÂÒîË ^ í«fÒÂÐ³ìa`Äí_ÚÖ«Ë
ÒÛÜØµÙ³Éãæ¹ÉëÙ¬ÐáÜá æ ÛñËÇÚÛñØËÂÇµgÉ _ Ê èµì © ÊD¥Ô í ÞÖµÒîËòÇÚXÐ UÉ Ð èµÉ³Ù³á×ÐçÍÂÐçËÂÛÜÊØ
ËÂÇµÉëèµfÉ YÚØµÛñËÂÛÜÊØóÊÔ
g
f g
¶£
é
f
3.1.6 Procedural programming
" ÊÔªÐÑÍ¬í æ¹ÉÇÚÐXUÉb_É³É³Ø Ð _ÚáÜÉàËÂÊ îáÜÛVUÉ æ·ÛñËÂÇµÊÖ«Ë Ô×ÖµØÚÙÄËÂÛÜÊØµÒÓÊØÚá²ì ÒÛÜØµÙ³É ËÂÇµÉ ÎµÍÂÊåÌÍÂÐÞßÒ ËÂÇÚÐçËÓæ¹É
ÇÚXÐ UÉâæ·ÍÂÛ²ËËÂÉ³Ø ÐÑÍÂÉãÎµÍÂÉËîË ìÒÛÜÞÀÎÚáÜÉÌé Ö«Ë É UÉ³Ø ÒÊÞßÉÏÊÔ ËÂÇµÉ³ÒÉëÒîÛÜÞÀÎÚáÜÉÓÊØÚÉÒ·æ¹ÊÖµáÜè _ É³ØµÉ YµË·ÔÜÍÂÊÞ
Ô×ÖµØµÙËÂÛÜÊØµÒ³é ¹ÊØµÒÛñèÚÉÄÍ¿ÐÒ ÐØ ÉIPµÐÞÀÎÚáÜÉËÂÇµÉ®ÎµÍÂÊåÌÍÂÐÞ
ÔÜÍÊÞ P¥ÉÍÂÙ³ÛÜÒîÉ
é êØ ËÇÚÛñÒ
perfect.C
IÉ P¥ÉÍÂÙ³ÛÜÒîÉÌí æ¹É ÇÚXÐ UÉßÐÒ ]Éè ìÊÖ ËÂÊ Y ØµèÝËÂÇµÉ¿Î ÉÍÂÔ×É³ÙÄË ØôÖµÞb_ÉÍÂÒ
_ ÉË æ¹É³É³Ø
ÐçØÚè í Ô×ÊÌÍàÐ åÛVUÉ³Ø
1
n
ÛÜØ«ÎÚÖ«ËâØ ÖÚ
Þ _ ÉÍ éßÆ¹ÇÚÉÒÊáÜÖ«ËÂÛÜÊØ½ÒÊ ÔªÐçÍâÖÚÒîÉ³ÒëÊØµÉ N_ÚÛÜå ^áñÊôÊÌÎ æ·
ÛñËÂÇµÛÜØ Ð áÜÊ ÊÌÎ `
µ
F

¦
ª

¢
n
ËÂÇÚÐçËãÛÜØ ËÂÖ«ÍÂØ ÙÊØ ËäÐÛñØÚÒÓË æ¹Ê
ÒîËäÐÑËÂÉ³ÞßÉ³Ø¯ËÂÒ³é ·áñËÂÇµÊÖµåÇ ÛÜØ ËÇÚÛñÒãÙ¬ÐçÒÉÌí ËÂÇµÉ N_ Ûñå áÜÊ ÊÌÎ ÛÜÒÏÒîËÂÛÜáñá
if
ÒÞ®ÐáñáÚÉØÚÊÖµåÇßËÂÊ _É·ÍÂÉ¬Ðèßæ·ÛñËÂÇµÊÖ«ËèµÛ Ù³ÖµáñËÂÛÜÉ³Òí ÛñËèµÊôÉ³Ò
Ø Ë ÍÂÉ¬Ðáñáñì ÍÂÉ QÚÉ³ÙË ËÇÚÉ·áÜÊåÛÜÙ¬ÐáÚÒîËÍÖÚÙÄËÂÖ«ÍÂÉ
ÊÔïËÂÇµÉ®ÒÊáÜÖ«ËÂÛÜÊØ¤é ØµÙ³Éßæ¹É®åÉËàËÂÊóËÍÂÛ²Î áñÉÀÊÌ
Í WôÖÚÐè«ÍÂÖ«ÎÚáÜÉÀáÜÊ ÊÌÎÚÒ³í ËÂÇµÉÀÎµÍÂÊåÌÍäÐÞ _É³Ù³ÊÞßÉ³Ò UÉÄÍì
ÇÚÐçÍÂèËÊ Ô×ÊáÜáÜÊ¦æëé
Ö«Ëïæ ÇÚÐçË ËÂÇµÉ áÜÊåÛÜÙ¬ÐçáÒîËÍÖÚÙÄËÂÖ«ÍÂÉ ÊÔ¾ËÇÚÉ ÒîÊáÜÖ«ËÂÛÜÊØ [ ÊÌÍ¹É UÉÍì Ø ÖÚÞ _ ÉÍ i _ ÉÄË æ¹ÉÉ³Ø 1 ÐçØÚè
íôæ¹ÉÓÇÚKÐ UÉ ËÊ
¥ÐØµè ËÂÊ èµÊàËÂÇµÉ á×ÐçËËÉÍ¬í æ¹É ÇÚXÐ UÉ ËÂÊ
n
test whether i is perfect
ÐçØÚè Ù´ÇµÉ³Ù ]¿æ ÇµÉËÂÇµÉÍÃÛñËÃÛÜÒÃXÉ W Ö ÐçáµËÊ éÃÆÈÇ ÖµÒ³íæ¹ÉòÇÚXÐ UÉòË compute
æ¹ÊÏÙ³áÜÉ¬ÐçÍthe
áñì èµsum
É YÚØÚÉofè
all proper divisors of i
ÒÖ _µËäÐÒ ]ôÒïËÂÇÚÐçËïËÂÇµÉ ÎÚÍÊåÌÍäÐÞ ÇÚÐÒïËÂÊ ÒÊVá UÉÓÔ×ÊÌÍÈÉ UÉÍîì Ø Öµb
Þ i _ ÉÄÍ í¥ÐØµèÛñËïÛñÒ _É³ÒîËïËÂÊ¿É³ØµÙ¬ÐçÎÚÒÖµá×ÐçËÉ
ËÂÇµÉ³ÒÉ ÛÜØ ËÊ Ô×ÖµØµÙËÂÛÜÊØµÒ³é ÍÂÊåÌÍÂÐÞ \ ÒÇµÊÑæ·Ò ÇµÊ¦æ ËÇÚÛñÒÛÜÒ¿èµÊØµi ÉÌé O ÊÌËÉ ËÂÇÚÐçË¿ËÂÇµÉ ÎµÍÂÊåÌÍÂÐÞ ÛÜÒ
ØµÊÑæ ÐáÜÞßÊÒËÕÒîÉ³áÜÔ ºGÉ P¥ÎÚá×ÐØÚÐçËÂÊÌÍîì ËÂÇµÉÈÎ ÊÒîËÂÙ³ÊØµèµÛñËÂÛñÊØÚÒ Ù¬ÐçØ ÞßÊÌÍÉòÊÌÍáÜÉ³ÒÒ èµÛñÍÉ³ÙËÂáñ
ì _ ÉÈÍÂÉ¬Ðè®Ê ËÂÇµÉ
Ô×ÖµØµÙËÂÛÜÊØ Ø ÐçÞßÉ³Ò³é
// Program : perfect2 .C
// Find all perfect numbers up to an input number n
# include < iostream >
// POST : return value is the sum of all divisors of i
//
that are smaller than i
unsigned int sum_of_proper_divisors ( unsigned int i)
{
}
unsigned int sum = 0;
for ( unsigned int d = 1; d < i ; ++ d)
if ( i % d == 0) sum += d ;
return sum ;
// POST : return value is true if and only if i is a
//
perfect number
bool is_perfect ( unsigned int i )
{
return sum_of_proper_divisors ( i ) == i;
}
int main ()
{
// input
std :: cout < < " Find perfect numbers up to n =? ";
unsigned int n;
std :: cin > > n;
// computation and output
std :: cout < < " The following numbers are perfect .\ n" ;
for ( unsigned int i = 1; i <= n ; ++ i )
if ( is_perfect ( i )) std :: cout < < i < < " ";
std :: cout < < "\ n";
return 0;
}
Program 20: ¢¤£³ÄÁ ¢ ³£º¶ç±¬ Å
èµÞßÛñËËÂÉèÚá²ìíËÂÇµÉÎµÍÂÊåÌÍäÐçÞ ÛÜÒÏáÜÊØµåÉÍÏËÂÇÚÐØ
í"_ÚÖ«ËâÛ²ËâÛÜÒÏÞßÊÌÍÂÉ ÍÂÉ¬ÐèÚÐ _ÚáÜÉí ÐØµèÝÛ²Ë
ÇÚÐÒÈÒÛÜÞÀÎÚáÜÉÍ¹Ù³ÊØ¯ËÍÂÊá QÚÊÑæâéÃêØÎ ÐçÍËÂÛÜÙÖÚáÜÐçÍ¬íôËÂÇµÉÏèµperfect.C
ÊÖa_ÚáÜÉÓáÜÊ ÊÌÎóÇÚÐÒÈèµÛÜÒÂÐçÎµÎ É¬ÐÑÍÂÉ³è¤é
ÆÈÇµÉÃá×ÐçÍÂåÉÄÍ Ð ÎµÍÂÊåÌÍäÐçÞ åÉÄËÂÒ³íçËÇÚÉ ÞßÊÌÍÂÉ ÛÜÞÀÎÊÌÍËäÐØ¯ËÛÜÒ¾ÛñË ËÂÊòÒÖ _ èµÛVU¥ÛÜèµÉ ÛñË¾ÛÜØ¯ËÂÊ¹ÒÞ®ÐáÜáÒÖ _ÚËÂÐÒ ]¥Òí
ÛÜØ½ÊÌÍÂèµÉÍÏØÚÊÌËÓËÂÊ áÜÊÒÉàËÍÂÐÙ ]½ÊÔ æ ÇÚÐçËÏÛÜÒÓåÊÛÜØµå®ÊØ ÛñØ ËÂÇµÉ ÎµÍÂÊåÌÍäÐÞ ÊØ ËÂÇµÉ¿æ·ÇÚÊáñÉ ËÂÇµÛÜÒÓÛÜÒ ËÂÇµÉ
Î ÐçÍäÐèµÛÜåÞ íôÐØµèÛÜØ íÚÛñËÈÛÜÒ¹ÍÂÉ¬ÐáÜÛ³É³è æ·ÛñËÂÇ Ô×ÖµØÚÙÄËÂÛÜÊØµÒ³é
¢¤£³Ì±ÄI F £©¯¤¢¤£Á£©¨ ¨»Á
ÆÈÇµÉàÎµÍÂÊôÙÉ³èµÖµÍÂÐá ÎµÍÂÊåÌÍäÐÞßÞßÛÜØµå Î ÐçÍäÐèµÛÜåÞðÛÜÒÓØÚÊÌËÓÒîÊ ÒÉáÜÔ ºÉ UôÛÜèµÉ³Ø ËÏÐçÒÏÛ²ËÓÞ®Ð¬ìÒÉ³É³Þ ËÂÊ èÚÐ¬ìé
Æ¹ÇÚÉ YÚÍÒîË·ÎµÍÂÊåÌÍäÐçÞßÞßÛÜØµåàá×ÐØµåÖÚÐåÉãËÂÇÚÐç
Ë _ É³Ù³ÐÞßÉëÐÙÙ³É³ÒÒÛ _ áñÉãËÂÊ®Ð¿åÉ³ØµÉÍäÐá ÐÖµèÚÛñÉ³ØµÙ³ÉâÒÛñØÚÙÉâËÂÇµÉ
Þ _ ÊáÜÛñÙ ØµÒîËÍÖÚÙÄËÂÛÜÊØ Ê èµÉ `Äé
ÒïæÈÐÒ " ê ^ É³åÛÜØµØµÉ!Í Ò áÜá ÎÚÖ«ÍÎ ÊÒîÉ ì¥
B
A
S
I
C
êØ " ê ÓíÚËÂÇµÉÍÂÉëæ¹ÉÍÉàØÚÊßÔ×ÖµØµÙËÂÛñÊØÚÒ ÛÜØ ÊÌÍÂèµÉÍ ËÊßIÉ P«É³ÙÖµËÉ ÐÀÙ³ÊôèµÉëÔÜÍäÐçåÞßÉ³Ø ËòÍÂÉ³ÒîÎ ÊØµÒÛ _ áñÉ
Ô×ÊÌÍ Ð ÒîaÖ _µËäÐçÒ ]íÃìÊÖÇÚÐèËÂÊ ÖµÒÉ®ËÂÇµÉ
ÒîËÂÐçËÂÉ³ÞßÉ³Ø¯
Ë ^»æ Û²ËÂÇÐ áÜÛñØÚÉßØ Öµb
Þ _ ÉÄGÍ ` # ÊÌÍ
ÛÜØ
GOTO
GOSUB
Þ®ÐØ¯ìßèµÛ×ÐáÜÉ³ÙÄËÂ%Ò #¼ËÂÊ ÖµÞÀÎ ËÂÊ¿ËÂÇÚÐçËÈÙ³ÊôèµÉÌíµÐçØÚè ËÂÇµÉ³
Ø ÖµÞÀÎ _ ÐÙ ] ÖµÒÛÜØµå ÐçØÚÊÌËÇÚÉÄÍ
^
í
GOTO RETURN
ÍÂÉ³ÒÎ É³ÙËÂÛ UÉ³áñaì `´é Æ¹ÇµÉÍÂÉ³ÒîÖÚá²Ë æÈÐÒÊÔÜËÂÉ³Ø ÍÂÉ³Ô×ÉÍîÍÂÉ³è ËÂÊ ÐÒ
í¹èµÖµÉËÂÊ ËÇÚÉÙ³ÊØ ËÍÊá
QÚÊÑæ ÞßÉ¬ÐØµèµÉÍÂÛÜØµåàáÜVÛ ]ÉÓZ
Ð _ ÊÛñáÜÉ³è®ÒîÎ ÐåÇµÉËËÂÛ ÊØ ÐëÎÚá×ÐçËÂÉÌé ) ¢ÊÌ©ÍÉ³ÁÊ U¬ÉÄºÍ¬í« ÎÚ±Í ÊFµåÌ ÍäÐÞßÞßÉÍÂÒÕÊÔÜËÂÉ³ØóèµÛÜèµ
Ø Ë
!
ËÂÇµÛÜØ ]ßÛÜØ ËÂÉÍÂÞßÒïÊÔ¡Ù³áÜÉ¬ÐÑÍÂáñìßèµÉ YÚØµÉ³èóÒÖ _µËäÐÒN]¥Ò³í«ÒÛñÞÀÎ á²ì _ É³Ù³ÐÖµÒÉÏËÇÚÉÓá×ÐçØÚåÖÚÐåÉ èµÛÜè ØµÊÌËòÒîÖµÎµÎ ÊÌÍîË
ÛñË¬éÃÆ¹ÇµÛÜÒòÖµÒÖÚÐáñáñìßáÜÊ¦æ¹ÉÍÂÉ³è ËÂÇµÉãÙ³Ê èµÉ W Ö ÐçáÜÛñË ì®É UÉ³Ø Ô×Ö«ÍËÂÇµÉÍ³é
É³ÒîÎÚÛñËÂÉÓËÂÇµÛÜÒ³í " ê æÈÐÒ·ÐØóÉIPôËÍÂÉ³ÞßÉáñì®ÒÖµÙ³Ù³É³ÒîÒÔ×Öµá ÎµÍÂÊåÌÍäÐÞßÞßÛÜØµåëáÜÐØµåÖ ÐçåÉÌéê»ËÈÍÂÉ¬ÐçÙ´ÇµÉ³è
ËÂÇµÉßÎ É¬Ð ] ÊÔÈÛñËÂÒëÎÊÌÎ Öµá×ÐÑÍÂÛñË ìÝÛÜØ ËÂÇµÉ®á×ÐçËÂÉ Ò ÐØµè É³ÐçÍÂáñì Òëæ ÇµÉ³Ø ËÇÚÉÀÎµÍÂÊÖµè ÊÑæ·ØÚÉÄÍÂÒ
ÊÔ ËÂÇµZ
É YÚÍÒîËÏÇÚÊÞßÉâÙ³ÊÞÀÎÚÖµËÉÍÂZ
Ò ^ºÐçÞßÊØÚåÀËÇÚÉÞ ËÂÇµÉ ÐÖ«ËÂÇµÊÌÍÂÒ `·ÙÍÂÉ³ÐçËÂÉ³è ÎÚÍÊåÌÍäÐÞßÒ ÊÔÃÔªÐÛñÍÂá²ìóÇµÛÜåÇ
Ù³ÊÞÀÎÚáÜIÉ P¥ÛñË ìßÛÜØ " ê Óé
3.1.7 Arrays as function arguments
SÝÉÈÇÚXÐ UÉ·ÒÉ³ÉØßÛÜØ " É³ÙÄËÂÛÜÊØ \ é é \ ËÂÇÚÐçËÕÐØßÐçÍÍÂÐ¬ì¿Ù¬ÐçØÚØµÊÌË_ ÉÈÛÜØµÛñËÂÛ×ÐçáÜÛ³É³è ÔÜÍÂÊÞ ÐØµÊÌËÂÇµÉÍÕÐçÍÍÂÐ¬ìí¯ÐØµè
ËÂÇµÛÜÒÃÛÜÞÀÎÚáÜÛÜÉ³Ò¡ËÂÇÚÐçË ÐÑÍÍäÐ³ì¥ÒÕÇ ÐXUÉÈËÂÊâÍÂÉ³ÙÉ³ÛVUÉ·ÒÎ É³Ù³Û×Ðçá ÐÑËËÂÉ³Ø¯ËÂÛÜÊØÀÛÜØ¿ËÂÇµÉ·Ù³ÊØ¯ËÂÉIPôË ÊÔÔ×ÖµØµÙËÂÛÜÊØµÒ³é Æ¹ÇµÉ
ÖµÒÖÚÐá YµÍÂÒîËÈÒîËÉÎÛñØÐëÔ×ÖµØµÙËÂÛÜÊØ®Ù¬ÐçáÜáÉ UÐáÜÖÚÐçËÂÛñÊØ ^°ËÂÇµÉ Ù¬ÐáÜá ÐÑÍÂåÖµÞßÉ³Ø ËÂÒ ÐçÍÂÉ É UÐáÜÖÚÐçËÂÉè¾í¥ÐØµè ËÂÇµÉ³Û²Í
UÐáÜÖµÉ³ÒòÐçÍÂÉÏÖÚÒîÉ³èóËÂÊ ÛñØÚÛ²ËÂÛ×ÐáÜÛ ³ÉòËÂÇµÉÏÔ×ÊÌÍÂÞ®Ðá¤ÐçÍåÖÚÞßÉØ ËÂÒ ` Ù¬ÐçØ Ë _ ÉãèµÊØµÉÓæ ÛñËÂÇÐçÍÍäÐ³ìôÒ³é
VÛ UÉØ½ËÂÇµÛÜÒ³í Û²ËÓÞßÛÜåÇ Ë _É¿ÒîÖµÍîÎÚÍÛÜÒÛÜØµåßËÂÇÚÐçËÓÔ×ÊÌÍÂÞ®Ðá ÐçÍÂåÖµÞßÉ³Ø ËÒ ÊÔ ÐçÍîÍäÐ³ì Ë ì Î É¿ÐçÍÂÉ¿ÐáÜáñÊÑæ¹É³è¤é
[ ÊÌÍ·GÉ PµÐÞÀÎÚáÜÉÌí æ¹ÉãÙ³ÊÖµáÜèèÚÉÙ³á×ÐçÍÂÉãÐ Ô×ÖÚØµÙËÛÜÊØ
// PRE : a [0] ,... ,a[n -1] are elements of an array
// POST : a [i ] is set to value , for 0 <= i < n
void fill_n ( int a [] , int n , int value );
ËÂÊóÒîÉË¿ÐáÜáÃÉ³áÜÉ³ÞßÉØ ËÂÒÏÊÔÈÐØ ÐçÍÍäÐ³ì½ËÂÊóÒÊÞßÉbYaP¥É³è UÌÐçáÜÖµÉÌéÆ¹ÇµÉÀÙ³ÊÞÀÎÚÛÜáÜÉÄÍ¬í¡ÇµÊ¦æ¹É UÉÍ¬í ÛÜØ ËÂÉÄÍÂØÚÐáÜáñì
ËÂÇµÛÜÒïËÂÊ¿ËÂÇµÉãÙ³ÊÞÀÎÚáÜÉÄËÂÉ³áñì ÉXWôÖµÛVUÐáÜÉ³Ø¯Ë·èµÉ³Ùá×ÐçÍäÐçËÛÜÊØ
© F ³º
// PRE : a [0] ,... ,a[n -1] are elements of an array
// POST : a [i ] is set to value , for 0 <= i < n
void fill_n ( int * a , int n , int value );
Æ¹ÇÚÉßÒÐÞßÉßÐè ÖµÒîËÂÞßÉ³Ø¯Ëâæ¹ÊÖµáÜè ÇÚÐçÎµÎ É³Ø Ô×ÊÌÍàËÂÇµÉßÔ×ÊÌÍÂÞ®Ðá ÐÑÍÂåÖµÞßÉ³Ø Ë
í ÒÂÐ³ìí ÞßÉ¬ÐØµÛÜØµå
ËÂÇÚÐçËòËÂÇµÉëÐçÍÍÂÐ¬ìáÜÉ³ØµåÌËÂÇóÛñÒ ÛÜåØµÊÌÍÂÉ³è¤é ÊÖ Ù³ÊÖµáÜèóÛÜØ ÔªÐçÙË ^áÜÉåÐáÜáñìí_ÚÖ«Ëint
W ÖÚÛ²ËÂa[5]
ÉâÙ³ÊØµÔ×ÖµÒÛÜØµåáñì `ÈÇÚÐXUÉ
Ð Ô×ÊÌÍÂÞ®Ðá¤ÐçÍÂåÖµÞßÉ³Ø¯Ë
í ÐçØÚè ËÂÇµÉ³ØóÖµÒÉëÐçØ ÐçÍÍäÐ³ì ÊÔ áñÉ³ØµåÌËÂÇ
ÐÒÈÙ¬ÐáÜá¾ÐçÍÂåÖµÞßÉ³Ø¯Ë¬é
int a[5]
10
ÆÈÇµÉÓÞßÊÌÍäÐáÚÛÜÒ ËÂÇÚÐçË ÛÜØßÍÂÉ¬ÐáÜÛ²Ë ìí¯ØµÊ Ô×ÊÌÍÂÞ®ÐáÐÑÍÂåÖµÞßÉ³Ø ËÂÒÕÊÔ ÐÑÍÍäÐ³ìÀË ìôÎ É ÉGP«ÛÜÒîË³í«ÐØµè®ÛÜØ®ÊÌÍÂèµÉÍïËÂÊ
XÐ UÊÛÜèÙÊØÚÔ×ÖµÒÛñÊØ¾íµÛñËÈÛÜÒ _ ÉËîËÂÉÍòØÚÊÌË¹ËÂÊ¿ÎµÍÂÉËÂÉ³ØµèóÊÌËÇÚÉÄÍæ ÛÜÒîÉÌé
êÔòæ¹ÉÀæÈÐØ¯Ë ËÂÊ _ ÖµÛÜáñè Ð Ô×ÖµØµÙËÂÛÜÊØ ËÂÇÚÐçËàæ¹ÊÌÍ ]ôÒàæ ÛñËÂÇ ÐçÍîÍäÐ³ì¥Ò³í æ¹ÉßËÂÇµÉÍÂÉ³Ô×ÊÌÍÂÉ®ÇÚÐXUÉ ËÊ ËÇÚÛñØa]
Ð _ ÊÖ«ËÕÐáñËÉÍÂØÚÐçËÂVÛ UÉïæÈÐ³ì¥ÒÃÊÔÎ ÐÒÒîÛÜØµåÏËÂÇµÉòÐçÍÍäÐ³ì¿ËÂÊÓËÂÇµÉ·Ô×ÖµØµÙËÂÛÜÊØ¤é Ø Ê _ U¥ÛñÊÖÚÒ æÈÐ³ì¿ÛÜÒÕÒîÖÚååÉÒîËÂÉ³è
_ôì ËÂÇµÉ¿èµÉ³Ù³á×ÐçÍÂÐçËÂÛÜÊØ ÊÔ
Ð _ Ê UÉ æ¹É¿Î ÐÒîÒâÐÀÎ ÊÛÜØ¯ËÂÉÍÓËÂÊ®ËÂÇµb
É YµÍÂÒËãÉáÜÉ³ÞßÉ³Ø¯Ë¬í¤ÐáÜÊØµå®æ ÛñËÂÇ
fill_n
ËÂÇµÉàØôÖµ
Þ _ ÉÍ·ÊÔÃÉáÜÉ³ÞßÉ³Ø¯ËÂÒ³é èµÛ ÉÍÂÉ³Ø¯Ë Î ÊÒÒÛ _ ÛñáÜÛñË ì ÛÜÒ·ËÂÊßÎ ÐÒÒ
Î ÊÛÜØ ËÉÍÂÒ³í ÊØÚÉëËÊ®ËÇÚÉ YÚÍÒîË
É³áÜÉÞßÉ³Ø Ë¬í ÐçØÚè ÐóÎ ÐÒîË »ËÂÇµÉ ºÉ³Øµè Î ÊÛÜØ ËÉÍ¬é½ÆÈÇµÛÜÒ ÐáñÒÊ ÖÚØµÛ W ÖµÉ³áñìÝèµ¸ÓÉ³Òî ÙÍÂVÛ _ ÉÒ ËÇÚÉ ÐçÍîÍäÐ³ìé êØ _ ÊÌËÂÇ
UÐçÍÂÛ×ÐØ¯ËÂÒ³íµæ¹ÉâÞ®Ð³ì ÐÙËÂÖÚÐáñáñì Ù´ÇµÊ ÊÒÉãËÂÇµÉëÙ³ÐáÜá¾ÐçÍÂåÖµÞßÉ³Ø¯ËÂÒòÛñØ ÒÖµÙ´Ç Ð æÈÐ¬ì ËÂÇÚÐçËòËÂÇµÉìèµÉ³ÒîÙÍÂVÛ _ ÉâÐ
Ù³ÊØ¯ËÂÛÜåÖµÊÖµÒ
ÊÔËÂÇµÉ ÊÌÍÂÛÜåÛÜØÚÐáÚÐçÍÍÂÐ¬ìéÃÆ¹ÇµÛÜÒ åÉ³ØµÉÍäÐçáÜÛ ¬ÐçËÛÜÊØÀÛÜÒÎÊÒÒVÛ _ÚáÜÉòÒÛÜØµÙ³É ËÂÇµÉ ÐçÍîÍäÐ³ì
ÛñËÂÒîÉ³áÜÔ¾ØÚIÉ UÉÍ Ðç¾Îµ Î ©É³£¬Ðç£ÍÂ© ÒòJ ÐÒ·ÐçØ ÐçÍÂåÖµÞßÉ³Ø¯Ë¬é
ïË¡ËÂÇµÛÜÒ¤Î ÊÛÜØ¯Ë¬í¦ÛñË¡ÒÉ³É³ÞßÒ¾áÜVÛ ]ÉÃÐ¹Þ®ÐçËËÂÉÍ¾ÊÔ¯ËäÐÒîËÂÉ æ ÇµÛÜÙäÇëÊÔ¯ËÂÇµÉÕË æ¹Ê UÌÐçÍÛ×ÐØ ËÒ¡ÛÜÒ¤ÎµÍÂÉ³Ô×ÉÍÂÐ _ÚáÜÉ _ÚÖ«Ë
ÛÜÔ ìÊÖ ËÂÇµÛÜØ ]óÐ _ ÊÖ«Ë ÇµÊÑæ ËÂÇµÉëÔ×ÖµØÚÙÄËÂÛÜÊØ
ÛÜÒ ØÚÐçËÂÖ«ÍäÐáñáñì ÛÜÞÀÎÚáÜÉ³ÞßÉ³Ø¯ËÂÉ³è ÛÜØ _ ÊÌËÂÇ UÌÐÑÍÂÛ×ÐØ¯ËÂÒ³í
HB>
%*
6#
$*&L
75 BI
-72 A6#"
*
:
I
/
% -2*
/
$
O2"5
8
%)'26
*
2!
&0
fill_n
$
#6%"
:#*
/.
. I/
/L
M
,BI
. -
J
/L.*
#/
"5
PS*
$
'(9$
%"K26I *
N$
,*
% O /C7
(Q$
)
N
*'(
/
! æ¹É ÒÉÉ ÐãèµÛ ÉÍÂÉ³ØµÙ³ÉÌé ·ÉÍÉ·ÛÜÒÐãÎµÍÂÊåÌÍäÐÞ ÂË ÇÚÐçË èµÉ YÚØµÉ³ÒïÐçØÚèßÖµÒÉÒËÂÇµÉ·Ë æ¹Ê UÐçÍÂÛ×ÐØ¯ËÂÒ^°ËÂÇµÉ ÒÉ³Ù³ÊØµè
ÊØµÉ®ÛÜÒ ÖµÒîËÙ¬ÐáñáÜÉ³è
ÒîÛÜØµÙ³ÉßËÂÇµÉÍÂÉ ÛÜÒ ÚØ Ê `âËÂÇÚÐçË¿ØÚÐçËÂÖ«ÍäÐáÜá²ì ÍÂÉ³ÒîÖÚá²Ë¬íÕåÛ UÉ³Ø ËÂÇµÉ®ÍÂÉ³ÒîÎÉ³ÙËÂÛVUÉ
fill
n
Ô×ÊÌÍÂÞ®Ðá¤ÐÑÍÂåÖµÞßÉ³Ø ËÂÒé
# include < iostream >
// Program : fill . C
// define and use two functions to fill an array
// PRE : a [0] ,... ,a[n -1] are elements of an array
// POST : a[i ] is set to value , for 0 <= i < n
void fill_n ( int * a , int n , int value ) {
// iteration by index
for ( int i = 0; i < n ; ++ i )
a[ i ] = value ;
}
// PRE : [ first , last ) is a valid range
// POST : * p is set to value , for p in [ first , last )
void fill ( int * first , int * last , int value ) {
// iteration by pointer
for ( int * p = first ; p != last ; ++ p)
*p = value ;
}
int main ()
{
int a [5];
fill_n (a , 5 , 0); // a == {0 , 0 , 0 , 0 , 0}
fill ( a , a +5 , 1); // a == {1 , 1 , 1 , 1 , 1}
return 0;
}
Program 21: ¢¤£³ÄÁ
» ×Å
êØ ËÇÚÉYÚÍÒîË UÐçÍÂÛ×ÐØ¯Ë¬í æ¹É·ÛñËÂÉÍäÐÑËÂÉ ÊUÉÍÈÐáÜáÚÛÜØµèµÛÜÙ³É³Ò ÛñØÀËÂÇµÉ ÒÉË
í ÐØµèßæ¹É åÉË ËÂÇµÉ
{0, . . . , n − 1}
ÐçÍÍÂÐ¬ìÝÉ³áÜÉ³ÞßÉ³Ø¯ËÂÒ _ ìÝÍäÐØµèµÊÞ ÐÙÙ³É³ÒÒ³é êØ ËÂÇµÉßÒÉ³ÙÊØÚè UÐçÍÂÛ×ÐØ¯Ë¬í æ¹ÉßÛñËÉÍäÐçËÂÉ Ê UÉÍ¿ÐçáÜá Ðèµè«ÍÂÉ³ÒîÒÉ³Ò
ÛÜØ ËÂÇµÉ
í¡ÐçØÚè æ¹É åÉËÓËÂÇµÉÐÑÍÍäÐ³ì É³áÜÉ³ÞßÉ³Ø¯ËÂÒ_ôì èµÉÍÉ³Ô×ÉÍÂÉ³ØµÙ³ÛñØÚå«é UÐáÜÛÜè
Ä
£

«
Á

[first, last)
ÍäÐØµåÉ Ù³ÊØ ËÂÐÛÜØµÒ¿ËÇÚÉÐèµè«ÍÂÉ³ÒÒÉÒÀÊÔ Ð ^»Î ÊÒÒÛ _ á²ì É³ÞÀÎµË ì ` ÒîÉËÀÊÔ ÙÊØÚÒîÉ³Ù³Ö«ËÂVÛ UÉÐçÍÍäÐ³ì É³áÜÉ³ÞßÉ³Ø¯ËÂÒ³í
æ ÇµÉÍÉËÂÇµÉ¿Ç ÐçáÜÔ×ÊÌÎ É³ØÝÛñØ ËÂÉÍ UÌÐáÃØµÊÌËäÐçËÂÛñÊØ
ÞßÉ³ÐØµÒÓËÇ ÐÑËãËÂÇµÉ¿ÍäÐØµåÉÛñÒãåVÛ UÉØ _ôì
[first, last)
ËÂÇµÉUÐáÜÖµÉ³ÒÊÔ
í
íé¬é³é³í
éÃêØßÊÌËÂÇµÉÍæ¹ÊÌÍèÚÒí
ÛñÒ ÐÏÎ ÐÒË ËÂÇµÉ É³ØµèÀÎ ÊÛÜØ¯ËÂÉÍ
first first+1
last-1
last
Ô×ÊÌÍÈËÂÇµÉâÒÖ _ ÐçÍÍäÐ³ì èµÉ³ÒÙÍÛV_ É³è _ ì®ËÂÇµÉÏÍäÐçØÚåÉé
Òïæ¹ÉÓÇ ÐXUÉÏÐáñÍÉ¬Ðè«ì®ÐçÍÂåÖµÉ³è ÛÜØ " É³ÙËÂÛñÊØ \ é é í ËÂÇµÉÓÒÉ³ÙÊØÚè UÌÐÑÍÂÛ×ÐØ¯Ë¹ÛÜÞÀÎÚáÜÉ³ÞßÉ³Ø¯ËÂÒÃËÇÚÉ îØÚÐçË Ö«ÍäÐá ®ÛñËÉÍäÐçËÂÛñÊØ½Ê UÉÍâÐØ ÐçÍÍÂÐ¬ìí¾ÐØµè ËÂÇµÉÍÂÉ³Ô×ÊÌÍÂÉÒÉ³É³ÞßÒ ÎµÍÂÉ³Ô×ÉÄÍäÐ _ÚáÜÉÌé Ö«ËÓËÂÇµÉ
ÍÂÉ¬ÐÒîÊØ æ Ç¯ì
ÛñËòÛÜÒÈÛÜØµèÚÉÉ³èóÎµÍÂÉ³Ô×ÉÍäÐ _ áñÉâáÜÛÜÉÒòÒÊÞßÉÄæ ÇµÉÍÂÉÓÉ³áÜÒîÉÌé¹êØ àí ËÂÇµÉÍÉëÐçÍÂÉÏËÂÉ³ÙäÇµØµÛ WôÖµÉ³£³ÒÈ´©¯ËÂÊ Þ®Ð ]ÉÏÔ×ÖµØµÙ !\
ÂË ÛÜÊØµÒïáÜÛ ]É
ÊÌÍ
ÐXUÐÛÜá×Ð _ áñÉ ØµÊÌË¹ÊØÚá²ìßÔ×ÊÌÍòÐçÍÍäÐ³ì¥Òí _ÚÖµËÈÔ×ÊÌÍÈÞ®ÐØ¯ìÀÊÌËÂÇµÉÍ¹ÙÊØ ËäÐÛñØÚÉÄÍÂÒÈÐçË
fill
fill_n
ËÂÇµÉëÒÂÐçÞßÉãËÂÛÜÞßÉéòêØóËÇÚÛñÒ åÉ³ØµÉÍäÐá¡ÒÉËËÛÜØµå«íÚËÂÇµÉëÔ×ÖµØµÙËÂÛñÊØÚÒòæ¹ÊÌÍ ]æ·ÛñËÂÇóÛñËÉÍäÐçËÂÊÌÍÒ³é S ÉâÞ®Ð³ì ËÇÚÛñØa]
ÊÔ ËÂÇµÉ³Þ ÐÒëåÉ³ØµÉÍäÐçáÜÛ ³É³èÝÎÊÛÜØ ËÂÉÄÍÂÒãËÂÊÙ³ÊØ ËäÐçÛÜØµÉÍëÉ³áñÉ³ÞßÉ³Ø ËÒ³í"_ÚÖ«ËâËÂÇµÉÀÊÌÎÉÍäÐçËÂÛñÊØÚÒÓËÂÇÚÐçËãæ¹ÉÀÙ¬ÐØ
ÐÙËÖ ÐçáÜáñìÀÎ ÉÄÍÂÔ×ÊÌÍÂÞ ÊØËÂÇµÉ³ÒÉ Î ÊÛÜØ ËÉÍÂÒ ¿èµÉÎ É³ØµèóÊØËÇÚÉÏÙ³ÊØ¯ËäÐÛÜØµÉÍ¬é
ÆÈÇµÉÍÉ ÐçÍÂÉòÞ®ÐØ ì¿ØÚÐçËÂÖ«ÍäÐáÚÙ³ÊØ¯ËäÐÛÜØµÉÍÂÒÃËÂÇÚÐçËÕèÚÊâØÚÊÌËÊ ÉÍÕÍäÐØµèµÊÞ ÐÙÙ³É³ÒÒËÊâËÂÇµÉ³ÛñÍÉ³áÜÉ³ÞßÉ³Ø¯ËÂÒ³é
[ ÊÌÍ ÒÖµÙäÇàÙ³ÊØ¯ËäÐÛÜØµÉÍÂÒí
ÐÒ Ð _Ê UÉ æ¹Ê
Ø Ë¾æ¹ÊÌÍ ]íçÒÛÜØµÙ³É ËÂÇµÉÕÒÖ _ÚÒÙÍÛñÎµË ÊÌÎ ÉÍÂÐçËÂÊÌÍ¾ÛÜÒ¾ØÚÊÌË XÐ UÌÐçÛÜá Ð _ÚáÜÉ¹Ô×ÊÌÍÃËÂÇµÉ³ÛñÍ Î ÊÛÜØ¯ËÂÉfill_n
ÍÂÒ «é ÆÈÇµÉÈÖÚØµèµÉÍÂá²ì¥ÛÜØµåÓÊÌÎ ÉÍäÐÑËÂÛÜÊØ¿ÊÔÐèµèµÛÜØµåãÛñØ ËÂÉ³åÉÄÍÂÒ ËÂÊâÒÖµÙäÇ Î ÊÛñØ ËÂÉÍÒ ÛÜÒ¹ËÂÇµÉ³ØóØµÊÌËÈèµÉ YÚØÚÉè¾íµÉ³Û²ËÂÇµÉÍ¬é
Ø ËÂÇµÉ ÊÌËÂÇµÉÍÀÇÚÐØµè¤íÕËÇÚÉ æÈÐ³ì æ¹É Ç XÐ UÉ èÚIÉ Y ØµÉ³è Ð Ù³ÊØ¯ËäÐÛÜØµÉÍÛÜØ " É³ÙÄËÂÛÜÊØ \ é é íÕæ¹É
åÖÚÐçÍäÐØ¯ËÂÉ³É³èÀËÇ ÐÑËæ¹É Ù¬ÐçØ Û²ËÂÉÍäÐçËÉ·Ê UÉÍ¹ÛñËÂÒÕÉ³áÜÉ³ÞßÉ³Ø¯ËÂÒ³é ÕìÀÙ³ÊØ UÉ³Ø¯ËÂÛÜÊØ¤íËÂÇµÛÜÒ ÛÜÒÃÍÂÉ¬ÐáÜÛ ³É³èÀËÂÇ«ÍÂÊÖµ©å£³Ç
Î ÊÛÜØ¯ËÂÉÍ ÛÜØµÙÍÉ³ÞßÉ³Ø Ë³í¥ÖµÒÛÜØµå¿ËÇÚÉÓÊÌÎ ÉÄÍäÐçËÂÊÌÍ
éêØÔªÐÙË¬í¥ËÂÇµÉÏÊÌÎ ÉÍäÐÑËÂÛÜÊØ
ÛÜÒÈXÐ UÌÐÛñá×Ð _ÚáÜÉÌí É UÉ³Ø
ÛÜÔ
ÛÜÒ¿ØÚÊÌË ËÂÇµÉ á×ÐÑËËÂÉÍ ÛÜÒ ÍäÐØµèµÊÞ ÐÙ³Ù³ÉÒ++
Ò Ô×ÖµØµÙËÂÛÜÊØÚÐáñÛñË ìÝÔ×ÊÌÍ¿ËÂÇµÉ ÒîÎ++p
É³Ù³Û×ÐáïÍÂÛñåÇ Ë Ç ÐçØÚè ÒÛÜèµÉ
ÊÌÎ p+1
ÉÍÂÐØµè é
ÆÈÇµÉÍÉ³1Ô×ÊÌÍÂÉÌí ËÂÇµÉÀÔ×ÖµØµÙËÂÛÜÊØ
ÐÒ Ð _ Ê UÉÀÇÚÐÒëËÂÇµÉ Î ÊÌËÉ³Ø ËÂÛÜÐáÃËÊóæ¹ÊÌÍ ]ÝÔ×ÊÌÍ ÐçáÜáÙ³ÊØ¯ËäÐÛÜØµÉÍÒ³í
ÒÛÜØµÙ³ÉÏÛñËÈÊØµáñì®ÍÂXÉ W ÖÚÛ²ÍÂÉ³
Ò Î ÊÛÜØ Ëfill
ÉÍ ¿Ô×ÖÚØµÙËÛÜÊØÚÐáÜÛñË ì ËÇ ÐÑË·ÛÜÒ¹Ê ÉÍÂÉ³è _ ìóÐáñá Ù³ÊØ ËäÐçÛÜØµÉÍÈÛñËÂÉÍäÐÑËÂÊÌÍÂÒ³é
Æ ÇµÉÍÂÉ¹ÛÜÒÃÐ ÒÖ _ÚÒîËäÐØ¯ËÂÛ×Ðá¥èµÛ ÉÍÂÉ³ØµÙ³ÉR_ ÉÄË æ¹ÉÉ³ØËÇÚÉïÔ×ÖµØµÙËÂÛñÊØ
¹
ÊØ ËÂÇµÉ¹ÊØÚÉ
pow
ÇÚÐØµè¤íµÐØµè ËÂÇµÉÓÔ×ÖµØÚÙÄËÂÛÜÊØµÒ
ÐØµè
ÊØ ËÂÇµÉÓÊÌËÂÇµÉÍòÇ ÐçØÚè¤é Ù¬ÐáÜá ËÂÊ¿ËÇÚÉÏÔ×ÖµØµÙËÂÛÜÊØ
fill
fill_n
ÇÚÐÒ·ØµÊßÉ É³ÙË¬í ÒÛÜØµÙ³ÉâËÇÚÉãÙ³ÊÞÀÎÚÖ«ËäÐçËÂÛÜÊØµÒ¹ÊØµáñì ÞßÊôèµÛÜÔÜì Ô×ÊÌÍÂÞ®Ðá¾ÐçÍåÖÚÞßÉØ ËUÌÐáñÖÚÉÒ ËÂÇµÉ³ÒîÉ UÌÐçáÜpow
ÖµÉ³Ò
ÐçÍÂÉ îáÜÊôÙ¬Ðçá ËÊËÂÇµÉ Ô×ÖµØµÙËÂÛÜÊØ Ù³ÐáÜáÕÐØµè îèÚÛñÒÂÐçÎµÎ É¬ÐçÍ Ö«Î ÊØ ËÂÉÄÍÂÞßÛÜØÚÐçËÂÛÜÊØ¤éS¼ÛñËÂÇ
í ÛñË! ÒÏËÂÇµÉ
pow
ÊÔ ÐàÔ×ÖµØµÙËÂÛÜÊØóÙ³ÐáÜá ËÂÇÚÐçË¹æ¹ÉâÐÑÍÂÉãÛñØ ËÂÉÍÉ³ÒîËÂÉ³èÛÜØ¤é
Dô©¯
ÈÐçáÜáÜÒ ËÂÊëËÂÇµÉ·Ô×ÖµØµÙËÂÛÜÊØµÒ
ÐØµè
íôÊØÀËÇÚÉ·ÊÌËÂÇµÉÍïÇÚÐØµè¤í ÇÚXÐ UÉ É É³ÙËÂÒ ËÇÚÉÄìÀÞßÊôèµÛÜÔÜì
ËÂÇµÉ UÐáÜÖµÉ³Ò ÊÔÃÐçÍÍÂÐ¬ìÉ³áÜÉ³ÞßÉØ fill
ËÂÒ³íÐØµè ËÂÇµfill_n
É³ÒZ
É UÐáÜÖµÉ³ÒÓÐçÍÂÉ
áñÊôÙ¬Ðá¾ËÂÊÀËÂÇµÉëÔ×ÖµØµÙËÂÛÜÊØ Ù¬ÐçáÜá
é S¼ÇµÉ³Ø

æ¹ÉÓæ·ÍÂÛñËÉ
Mutating functions.
int a [5];
fill (a , a +5 , 0);
ËÂÇµÉàÉ É³ÙËÏÊÔÃËÇÚÉàÉGP¥ÎµÍÂÉ³ÒÒîÛÜÊØ
ÛñÒ ËÂÇÚÐçËÏÐáÜá¡É³áÜÉ³ÞßÉ³Ø¯ËÂÒ ÊÔ ÍÂÉ³Ù³É³Û UÉUÐáÜÖµÉ é
fill (a, a+5, 0)
Æ¹ÇÚÛñÒëÛÜÒâÎ ÊÒîÒÛV_ÚáÜÉ ÒîÛÜØµÙ³ÉÀËÂÇµÉÍÂÉßÐÑÍÂÉßÔ×ÊÌÍÂÞ®ÐáÕÐçÍÂåÖµÞßÉ³Ø¯ËÂÒâÊÔïÎ ÊÛÜØ¯ËÂÉÍãË ì Î ÉÌé S¼aÇµÉ³Ø ËÂÇµÉßÔ×ÖµØÚÙÄËÂÛÜÊ0Ø
Ù¬Ðáñá
ÛÜÒãÉ UÌÐçáÜÖÚÐçËÂÉ³è¤í¤ËÂÇµÉ Ô×ÊÌÍÂÞ®Ðá ÐÑÍÂåÖµÞßÉ³Ø Ë
ÛñÒãÛÜØµÛñËÛ×ÐáÜÛÉ³è æ ÛñËÂÇ ËÂÇµÉ
fill (a, a+5, 0)
first
ÊÔ Ò YµÍÂÒîË ÉáÜÉ³ÞßÉ³Ø¯Ë¬éïêØ ËÂÇµÉâÔ×ÖµØµÙËÂÛÜÊØ _Êôè«ìí«ËÂÇµÉ UÌÐçáÜÖµÉëÐçËòËÂÇµÛÜÒ Ðèµè«ÍÂÉ³ÒîÒ ÛÜÒ·ÞßÊ èµÛVYÚÉ³è
ËÂ© Ç«F ÍÂFÊ£³Öµ¬åÇ ËÇÚÉÏaVá UÐáÜÖµÉ
íÚÐØµèËÇÚÉÏÒÂÐÞßÉÓÇÚÐçÎµÎ É³ØµÒÈÔ×ÊÌÍòËÇÚÉÏÊÌËÂÇµÉÍòÔ×ÊÖ«Í ÐçÍÍÂÐ¬ì É³áñÉ³ÞßÉ³Ø ËÒÈÛñØ ËÂÖ«ÍÂØ¤é
*p
[ ÊÌÍÂÞ®Ðá¹ÐçÍÂåÖµÞßÉ³Ø¯ËÂÒ¿ÊÔòÎ ÊÛÜØ¯ËÂÉÍ¿Ë ìôÎ É ÐçÍÂÉ ËÂÇµÉÍÂÉ³Ô×ÊÌÍÂÉÐ½ÞßÉ¬ÐçØÚÒ ÊÔ ÙÊØÚÒËÍÂÖµÙËÂÛÜØµå Ô×ÖµØÚÙÄËÂÛÜÊØµÒ
æ Û²ËÂÇ UÐáÜÖµÉ ÞßÊôèµÛÜÔÜì¥ÛñØÚåàÉ É³ÙËÒ³é " ÖµÙäÇ Ô×ÖµØµÙËÂÛÜÊØµÒòÐçÍÂÉÏÙ¬ÐáÜáÜÉè
é
¨î©¯»Á
3.1.8 Modularization
Æ¹ÇÚÉÄÍÂÉ ÐçÍÂÉëÔ×ÖµØµÙËÂÛñÊØÚÒÈËÂÇÚÐçË ÐÑÍÂÉëËäÐÛÜáñÊÌÍºÞ®ÐèµÉ Ô×ÊÌÍÓÐÒîÎ É³Ù³Û Y ÙãÎÚÍÊåÌÍäÐÞ íÚÐØµè ÛñË·æ¹ÊÖµáÜè ØµÊÌË Þ®Ð ]É
ÒÉ³ØµÒÉòËÂÊëÖµÒÉ·ËÂÇµÉ³Þ ÛÜØ®ÐØµÊÌËÂÇµÉÍÃÎµÍÂÊåÌÍäÐÞ é ÖµËÕËÂÇµÉÍÂÉ·ÐçÍÂÉ ÐçáÜÒÊëåÉ³ØµÉÍÂÐáÚÎÚÖ«ÍÎ ÊÒîÉ·Ô×ÖµØµÙËÂÛÜÊØµÒÕËÇ ÐÑË
ÐçÍÂÉ¹ÖµÒÉ³Ô×ÖµáµÛÜØ¿Þ®ÐØ ìãÎµÍÂÊåÌÍäÐÞßÒ³é ê»Ë ÛÜÒ Ù³áÜÉ¬ÐÑÍÂáñìëÖµØµèµÉ³ÒÛñÍÂÐ _ÚáÜÉ ËÂÊÓÙ³ÊÌÎôìàËÇÚÉ¹Ù³ÊÌÍîÍÂÉ³ÒîÎ ÊØµèµÛÜØµåÓÔ×ÖµØÚÙÄËÂÛÜÊØ
©¯î í Ð
èµÉ YÚØµÛñËÂÛÜÊØóÛñØ ËÂÊßÐçØ ìÎµÍÂÊåÌÍÂÐÞðËÂÇÚÐçË Ù¬ÐáÜáÜÒÈËÂÇµÉëÔ×ÖµØµÙËÂÛÜÊØ æ·ÇÚÐçË·æ¹ÉëØµÉ³Éè½ÛÜÒ
¨

ÒÖ _ èµVÛ UôÛÜÒÛÜÊØ ÊÔ¡ËÂÇµÉÓÎµÍÂÊåÌÍäÐÞ ÛÜØ¯ËÂÊ ÛÜØµèµÉÎ ÉØÚèµÉ³Ø¯Ë¹ÎÐÑÍËÂÒ³é
! Æ ÇµÉ¿Î ÊÑæ¹ÉÄÍâÔ×ÖµØµÙËÂÛÜÊØ
È
ÔÜÍÂÊÞ ÍÂÊåÌÍÂÐÞ ÛÜÒÏÙ³ÉÍËäÐçÛÜØµáñì åÉ³ØµÉÍäÐçáÃÎÚÖ«ÍÎ Ê ÒÉé êØÝÊÌÍèÚÉÄÍãËÂÊ
pow
Þ®Ð ]ÉàÛñË ÐKUÐÛÜáÜÐ _ÚáÜÉâËÂÊ ÐáÜá ÊÖµÍÓµÎ ÍÂÊåÌÍäÐçÞßÒ³í æ¹É Ù¬ÐØ½ÒÛÜÞÀÎÚáñì ÎÚÖµË ËÇÚÉ Ô×ÖµØµÙËÂÛÜÊØ èµÉ YÚØµÛñËÂÛÜÊØ ÛÜØ¯ËÂÊ®Ð
Ò ÉÎ ÐçÍÂÐçËÂÉÏÒÊÖ«ÍÂÙ³É³Ù³Ê èµÉ YÚáÜÉ
µí ÒÂÐ³ìí«ÛÜØóÊÖ«ÍÈæ¹ÊÌÍ ]¥ÛÜØµå¿èµÛñÍÂÉ³ÙËÊÌÍìé
pow.C
# include < cassert >
// PRE : e >= 0 || b != 0.0
// POST : return value is b^e
double pow ( double b , int e)
{
assert ( e >= 0 || b != 0.0);
double result = 1.0;
if ( e < 0) {
// b^ e = (1/ b )^( - e)
b = 1.0/ b ;
e = - e;
}
for ( int i =0; i <e ; ++ i ) result *= b;
return result ;
}
Program 22: ¢¤£³Á ¢ ¸ Å
ÆÈÇµÉ³Øóæ¹ÉÏÙ¬ÐØÛÜØµÙ³áÜÖµèµÉÓËÇÚÛñÒ YÚáÜÉÏÔÜÍÂÊÞ
// Prog : callpow2 .C
// Call a function for computing powers .
# include < iostream >
# include " pow .C"
ÊÖ«Í·Þ®ÐÛñØ®ÎµÍÂÊåÌÍÂÐÞ ÐÒòÔ×ÊáñáÜÊÑæ·Ò³é
int main ()
{
std :: cout
std :: cout
std :: cout
std :: cout
std :: cout
<<
<<
<<
<<
<<
pow ( 2.0 , -2) < < "\n " ;
pow ( 1.5 , 2) < < "\n " ;
pow ( 5.0 , 1) < < "\n " ;
pow ( 3.0 , 4) < < "\n " ;
pow ( -2.0 , 9) < < "\n " ;
//
//
//
//
//
outputs
outputs
outputs
outputs
outputs
return 0;
}
Program 23: ¢¤£³ÄÁ
ØóÛÜØµÙ³áñÖÚèµÉÓèµÛñÍÂÉ³ÙÄËÂÛVUÉÏÊÔ¡ËÂÇµÉÏÔ×ÊÌÍÂÞ
#include
±Ä©¯° ¢¸
Å
0.25
2.25
5
81
-512
!
áÜÊåÛÜÙ³ÐáÜáñì ÂÍ ÉÎÚá×ÐçÙ³É³ÒßËÂÇµÉ ÛÜØµÙ³áÜÖµèµÉóèµÛñÍÂÉ³ÙËÛVUÉ _ôì ËÂÇµÉ Ù³ÊØ¯ËÂÉ³Ø ËÒ ÊÔÓËÂÇµÉ ÒîÎ É³ÙÛVYÚÉ³è YÚáÜÉé ·ÒîÖ ÐçáÜáñìí
ÛÜÒÈÛÜØ¯ËÂÉÍÎµÍÂÉËÉ³èÍÂÉ³áÜÐçËÂÛVUÉ ËÂÊ¿ËÂÇµÉÏæ¹ÊÌÍ ]ôÛÜØµåèµÛñÍÉ³ÙËÂÊÌÍìé
Æ¹ÇµÉëÙ³Ê èÚÉâÒÉÎ ÐçÍäÐçËÛÜÊØóÞßÉ³ÙäÇÚÐØµÛÜÒÞ ÔÜÍÂÊÞ ËÂÇµÉâÎµÍÂÉ Separate compilation and object code files.
U¥ÛñÊÖÚÒ Î ÐçÍÂÐåÌÍäÐçÎÚÇ ÇÚÐÒ ÊØµÉ¿Þ®Ð ÊÌÍ·èµÍÂÐ¬æ_ ÐÙ ] òËÇÚÉàÙÊÞÀÎ ÛñáÜÉÍ èµÊ É³Ò ØµÊÌË îÒÉ³É ÛñË¬é É³Ô×ÊÌÍÂÉ Ù³ÊÞ
ÎÚÛÜá×ÐçËÛÜÊØ¤í
ÛÜÒàáÜÊåÛñÙ¬ÐáÜáñì Ù³ÊÌÎÚÛÜÉè _ ÐÙ ] ÛÜØ¯ËÂÊ ËÂÇµÉßÞ®ÐÛÜØ YÚáÜÉÌí ÒîÊ ËÂÇµÉßÙ³ÊÞÀÎÚÛÜáñÉÍëÒîËÛÜáÜáÇÚÐÒëËÊ
pow.C
ËÍäÐçØÚÒîá×ÐçËÂÉÃËÂÇµÉ Ô×ÖµØÚÙÄËÂÛÜÊØ èµÉ YÚØÚÛ²ËÂÛÜÊØëÛÜØ¯ËÂÊ Þ®ÐÙäÇµÛÜØµÉÃá×ÐØµåÖÚÐåÉ
ÛñË Ù³ÊÞÀÎÚÛÜáÜÉ³Ò¡ÐòÎµÍÂÊåÌÍäÐçÞ
ËÂÇÚÐçËÈÙ¬ÐáñáÜÒ
éÕÆÈÇµÛÜÒÈÛÜÒÈÐàæÈÐÒîËÂÉÏÊÔ ËÂÛÜÞßÉ ËÇ ÐÑËòÙ³ÐØ _ ÉãÐXUÊÛÜXèµDôÉ³³è £KJÀ
_ôì °¨
é
pow

£
¯
©

Ä
±

¨
¤
¢

¯
©

êØÊÖ«Í¿Ù¬ÐÒÉíÃæ¹Éßæ¹ÊÖµáÜè Ù³ÊÞÀÎÚÛÜáÜÉËÇÚÉ YÚáÜÉ
ÒÉÄÎÐÑÍäÐçËÂÉ³á²ìé S É ÊØµáñì ÇÚÐXUÉßËÂÊ ËÂÉ³áÜáËÇÚÉ
Ù³ÊÞÀÎÚÛÜáÜÉÄÍ ËÂÇÚÐçËÓÛñËÓÒÇµÊÖµáÜè½ØÚÊÌËÏåÉ³ØµÉÍÂÐçËÂÉ¿ÐØ GÉ P«É³pow.C
Ù³Ö«ËäÐ _ áñÉ ÎµÍÂÊåÌÍÂÐÞ ^ÛñËÓÙ¬ÐçØ Ë¬í¤ÒÛÜØµÙ³ÉàËÂÇµÉÍÂÉ ÛÜÒÓØµÊ
Ô×ÖµØµÙËÂÛÜÊØ ` _ÚÖµËòÐØ
YÚáÜÉÌí«Ù¬ÐáñáÜÉ³è
í«ÒÂÐ³ìéÕÆÈÇµÛÜR
Ò YÚáÜÉ Ù³ÊØ¯ËäÐÛÜØµÒïËÂÇµÉÓÞ®ÐÙäÇÚÛñØÚÉ
¥

¬
±
ã

Ä
±

µ
F

main
pow.o
á×ÐØµåÖÚÐåÉ ÛÜØµÒîËÍÖÚÙÄËÂÛÜÊØµÒ ËÂÇÚÐçËâÙÊÌÍÍÂÉ³ÒîÎÊØÚèÝËÊËÂÇµÉ ÒîËÂÐçËÂÉ³ÞßÉ³Ø¯ËÂÒÏÛÜØ ËÂÇµÉ Ô×ÖµØµÙËÂÛñÊØ _ Êôè«ì ÊÔ
é
pow
Æ ÇÚÉ¿ÒÉÎ ÐçÍÂÐçËÂÉ¿Ù³ÊÞÀÎÚÛÜá×ÐçËÂÛñÊØóÙ³ÊØµÙ³ÉÎµËãÛÜÒÓÞßÊÌÍÂÉëÎ Ê¦æ¹ÉÍÂÔ×Öµá ËÂÇÚÐØ½æ¹É¿ÇÚÐKUÉ¿ÒÉ³ÉØ ÒÊ
¹
ÔªÐçÍ ÒîÖµÍîÎÚÍÛÜÒÛÜØµåáñìíÉIUÉ³Ø ÎµÍÂÊåÌÍäÐÞßÒòËÂÇÚÐçËÓÙ¬ÐáÜá ËÇÚÉ Ô×ÖµØµÙËÂÛÜÊØ
Ù¬ÐØ _ É Ù³ÊÞÀÎÚÛÜáÜÉ³è ÒîÉÎ ÐçÍäÐçËÂÉáñìí
æ Û²ËÂÇµÊÖµË]¥ØµÊ¦æ ÛÜØµå®Ð _ ÊÖ«Ë ËÇÚÉ ÒÊÖ«ÍÂÙÉ¿ÙÊôèµÉZYÚáÜÉ
ÊÌÍ ËÂÇµpow
É Ê _ É³ÙÄËÓÙ³ÊôèµÉZYÚáÜÉ
éS¼ÇÚÐçË
pow.C
pow.o
ËÂÇµÉÏÙ³ÊÞÀÎÚÛÜáÜÉÍ¹ØµÉ³É³èµÒÈËÂÊÇÚXÐ UÉÌíµËÂÇµÊÖµåÇ¾í«ÛñÒòÐ èµÉ³Ù³á×ÐÑÍäÐçËÂÛÜÊØ ÊÔ ËÂÇµÉÏÔ×ÖµØÚÙÄËÂÛÜÊØ
é
pow
ÆÈÇµÛÜÒ¹Ô×ÖµØµÙËÂÛÜÊØ èµÉ³Ù³á×ÐÑÍäÐçËÂÛÜÊØßÛñÒ _É³ÒîËïÎÚÖ«ËòÛñØ ËÂÊ ÐàÒÉÎ ÐçÍÂÐçËÂ
É YÚáÜÉÓÐÒ æ¹É³áÜá»é êØÊÖ«ÍÈÙ¬ÐÒÉÌí¥ËÇÚÛñÒ Y áñÉ
íµÒÂÐ³ìíµÛÜÒ UÉÍì ÒÇµÊÌÍË µÛñËÈÙ³ÊØ¯ËäÐÛÜØµÒ ÖÚÒËòËÂÇµÉÏáÜÛÜØµÉ³Ò
Header files.
pow.h
// PRE : e >= 0 || b != 0.0
// POST : return value is b^e
double pow ( double b , int e );
" ÛÜØµÙ³É ËÂÇµÛÜÒ ÛñÒ ËÂÇµÉ îÇÚÉ³ÐèµÉÍ ÓÊÔ«ËÂÇµÉÈÔ×ÖµØµÙËÂÛñÊØ
íÌËÂÇµÉRYÚáÜÉ
ÛÜÒ Ù¬ÐçáÜáÜÉ³è¿Ð
é êØ ËÂÇµÉ
¬Ù ÐáñáÜÛÜØµå ÍÂÊåÌÍÂÐÞ \ í«æ¹ÉãÒÛÜÞÀÎÚáñìÀÍÂÉÄÎ áÜÐÙ³ÉãËÇÚpow
ÉãÛÜØµÙ³áÜÖµÒÛÜÊØ Êpow.h
Ô
_ôì ËÂÇµÉãÛÜ
ØµÙ³Ä©áñÖÚFµÒîÛÜÊ£ ØÊª Ô
í
pow.C
pow.h
ÍÂÉ³ÒîÖÚá²ËÂÛÜØµåÛñØ ËÂÇµÉÏÔ×ÊáÜáÜÊ¦æ ÛÜØµåàÎµÍÂÊåÌÍäÐÞ é
// Prog : callpow3 .C
// Call a function for computing powers .
# include < iostream >
# include " pow .h"
int main ()
{
std :: cout
std :: cout
std :: cout
std :: cout
std :: cout
<<
<<
<<
<<
<<
pow ( 2.0 , -2) < < "\n " ;
pow ( 1.5 , 2) < < "\n " ;
pow ( 5.0 , 1) < < "\n " ;
pow ( 3.0 , 4) < < "\n " ;
pow ( -2.0 , 9) < < "\n " ;
//
//
//
//
//
outputs
outputs
outputs
outputs
outputs
0.25
2.25
5
81
-512
!
return 0;
}
Program 24: ¢¤£³ÄÁ
Ä± ©¯° ¢¸ Å
[ ÍÂÊÞ ËÂÇµÛÜÒ·ÎµÍÂÊåÌÍäÐçÞ í ËÂÇµÉàÙ³ÊÞÀÎÚÛÜáÜÉÄÍ·Ù¬ÐçØ ËÂÇµÉ³Ø åÉ³ØµÉÍäÐÑËÂÉ ÐØ Ê _ É³ÙË Ù³ÊôèµÉZYÚáÜÉ
é
callpow3.o
êØµÒîËÂÉ¬Ðçè ÊÔ ËÂÇµÉâÞ®ÐÙäÇµÛÜØµÉãáÜÐØµåÖ ÐçåÉãÛÜØµÒîËîÍÂÖµÙËÂÛÜÊØµÒÈÔ×ÊÌÍ IÉ P¥É³Ù³Ö«ËÂÛÜØµåÀËÂÇµÉ _Êôè«ìÊÔ
íÚËÂÇµÛÜÒòÊ _ É³ÙË
Ù³Ê èÚÉ Ù³ÊØ ËäÐçÛÜØµÒàÐ
Ô×ÊÌÍàËÂÇµÉÀáÜÊôÙ³ÐçËÂÛÜÊØÝÖµØµèµÉÍàæ ÇµÛÜÙäÇÝËÂÇµÉ³ÒîÉßÛÜØµÒîËÍÂÖµÙpow
ËÂÛñÊØÚÒâÐçÍÂÉ ËÊ _ É
Ô×ÊÖµØµèßÛÜØËÂÇµÉòÉIP«É³Ù¢¾Öµ×ËÂ©µÐ ±_Ú¬áÜÉÈÎµ ÍÂFµÊ³åÌ£ ÍÂÐÞ é ê»ËÕÛÜÒÃÛÜÞÀÎ ÊÌÍËäÐçØ Ë ËÂÊãÖÚØµèµÉÍÂÒËäÐØµè ËÂÇÚÐçË
Ù¬ÐØµØµÊÌË
callpow3.o
_ ÉÓÐØ ÉIP¥É³Ù³Ö«ËäÐ _ÚáÜÉ ÎµÍÂÊåÌÍäÐçÞ ìÉË ÛñË
Ù³ÊØ ËäÐçÛÜØ®Þ®ÐÙäÇÚÛñØÚÉòá×ÐØµåÖÚÐåÉ Ù³Ê èµÉ Ô×ÊÌÍ
í _ÚÖµË
main
µ
F
Ì

Ô×ÊÌÍ·ÐØµÊÌËÂÇµÉÍ·Ô×ÖµØµÙËÂÛñÊØ ËÂÇÚÐçËòÛñËÈØµÉ³É³èµÒ³íµØÚÐÞßÉ³áñì
é
pow
Øµáñì æ·ÇµÉ³Ø ÐØ GÉ P«É³Ù³Ö«ËäÐ _ áñÉ ÎµÍÂÊåÌÍäÐçÞ ñÛ Ò _ÚÖµÛÜáñË ÔÜÍÂÊÞ
í ËÇÚÉ Ê _ ÉÙË
The linker.
Ù³Ê èÚÉ YÚáÜÉ
Ù³ÊÞßÉ³ÒßÛÜØ¯ËÂÊ Î áÜÐ¬ìé
VÛ UÉØ ÐáÜá·Ê _ É³Ù ËYÚáÜÉ³ÒßËÂÇÚÐçË çÐ callpow3.o
ÍÉ ÛñØ UÊáVUÉ³è¤íÈËÂÇµÉ
» ³ £
pow.o
_ÚÖµÛÜáÜèµÒãËÂÇµÉßÉIP¥É³Ù³Ö«ËäÐ _ÚáÜÉÀÎµÍÂÊåÌÍÂÐÞ _ ì åáÜÖµÛÜØµåËÂÊåÉËÇÚÉÄÍ Þ®ÐÙäÇÚÛñØÚÉá×ÐØµåÖÚÐåÉ ÙÊôèµÉßÔ×ÊÌÍ Ô×ÖµØÚÙÄËÂÛÜÊØ
Ù¬ÐáñáÜÒ»^ ÛÜØ
` æ ÛñËÇÀÞ®ÐÙäÇµÛÜØµÉòá×ÐçØÚåÖÚÐåÉ¹Ù³Ê èµÉ Ô×ÊÌÍ ËÂÇµÉ·ÙÊÌÍÍÂÉ³ÒÎ ÊØµèÚÛñØÚåãÔ×ÖµØÚÙÄËÂÛÜÊØL_ Ê èÚÛñÉ³Ò
callpow3.o
^ ÛÜØ
Ä` é Æ ÉÙ´ÇµØµÛÜÙ¬ÐçáÜáñìí ËÂÇµÛÜÒ¿ÛÜÒ èÚÊØµÉ_ôì ÎÚÖ«ËËÂÛÜØµå½ÐáÜáïÊ _ ÉÙËY áñÉ³Ò¿ËÊåÉËÂÇµÉÍÛÜØ ËÂÊ½Ð ÒÛÜØµåáÜÉ
ÉI¥P É³Ù³pow.o
Ö«ËäÐ Ú_ áÜÉ ÚY áÜÉí ÐØµè ô_ ì Y áñáÜÛÜØµå ÎÚá×ÐçÙ³É³ÇµÊáÜèµÉÍÂÒãÔ×ÊÌÍàÔ×ÖµØÚÙÄËÂÛÜÊØ _ Ê èµì½áÜÊ Ù¬ÐçËÂÛÜÊØµÒÓæ ÛñËÇÝËÂÇµÉÀÐÙËÖ Ðçá
áÜÊ Ù¬ÐçËÂÛÜÊØµÒïÛÜØ ËÂÇµÉãÉG«P É³Ù³Ö«ËäÐ _ áñÉÌé
" ÉÎ ÐçÍäÐçËÉ¿ÙÊÞÀÎ Ûñá×ÐçËÂÛÜÊØ ÛÜÒ
U ÉÍì ÖµÒÉ³Ô×Öµáéãê»ËÏÐáÜáñÊÑæ·Ò·ËÂÊ®ÙäÇ ÐçØÚåÉàËÂÇµÉàèµÉ Ú
Y ØµÛñËÂÛÜÊØ ÊÔÃÐßÔ×ÖµØÚÙÄËÂÛÜÊØ
æ Û²ËÂÇµÊÖµË ÇÚÐKôU ÛÜØµåÝËÂÊÝÍÂÉ³ÙÊÞÀÎ ÛñáÜÉóÐÝÒÛñØÚåáñÉÎµÍÂÊåÌÍäÐÞ ËÇ ÐÑËßÙ¬ÐáÜáñÒÀÛñË¬é ·ÒßáÜÊØµå ÐÒÀËÂÇµÉ Ô×ÖµØÚÙÄËÂÛÜÊØ
èµÉ³Ù³á×ÐÑÍäÐçËÂÛÜÊØ½ÍÂÉ³Þ®ÐÛÜØµÒãÖÚØµÙäÇÚÐØµåÉ³è¤í ÛñËëÛñÒëÊØµáñì ËÂÇµÉÀáñÛÜØ ] ÉÍâËÂÇÚÐçËëÇ ÐçÒâËÂÊóæ¹ÊÌÍ ] ÛñØÝËÂÇµÉßÉ³Øµè ÃÐØµè
ËÂÇµÉâáñÛÜØ ] ÉÍ ÛÜÒòÖµÒÖÚÐáÜáñìU ÉÍìÔªÐÒîË¬é¹ê»Ë Ô×ÊáÜáÜÊ¦æ ÒïËÂÇÚÐçË ÒîÉÎ ÐçÍäÐçËÂÉãÙ³ÊÞÀÎÚÛÜá×ÐçËÛÜÊØ ÐáÜÒÊÀÞ®Ð ] É³ÒòÒîÉ³ØµÒÉàÔ×ÊÌÍ
Ô×ÖµØµÙËÂÛÜÊØµÒ¹ËÂÇÚÐçËòÐçÍÂÉÏÒîÎ É³Ù³Û Y ÙÓËÂÊÊØÚÉÓÎµÍÂÊåÌÍäÐÞ ÊØµáñìé
" ÉÎ ÐçÍäÐçËÉ Ù³ÊÞÀÎÚÛÜáÜÐçËÂÛÜÊØ ÍÉ Ú
Q É³ÙËÂÒ ËÂÇµÉ îÙ³ÖµÒîËÂÊÞßÉÍ ôU ÛÜÉæ ÊÔ¹ËÂÇµÉ Ù¬ÐçáÜáÜÛÜØµåÎµÍÂÊåÌÍäÐçÞ ßÐÒ áÜÊØµå
ÐÒãÐßÔ×ÖÚØµÙËÛÜÊØ èµÊôÉÒÓæ·ÇÚÐçËÏÛñËÂÒ ÎµÍÂÉ ·ÐØµè½Î ÊÒËÂÙ³ÊØµèµÛñËÂÛÜÊØ ÎµÍÂÊÞßÛÜÒÉâÛÜØ½ËÂÇµÉ¿ÇµÉ¬ÐèµÉÍ
ÚY áÜÉÌí¤ÛñËÓÛÜÒÏØµÊÌË
ÛÜÞÀÎ ÊÌÍîËäÐØ ËÏËÂÊ ¥] ØµÊ¦æ
Û²ËàèµÊôÉ³ÒëËÂÇµÛÜÒé Ø ËÂÇµÉßÊÌËÂÇµÉÍ ÇÚÐØµè¾í ÛñÔ¹ËÇÚÉßÔ×ÖµØµÙËÂÛñÊØ èµÉ ÚY ØµÛñËÂÛÜÊØ ÛÜÒ
ÇµÛÜèµèµÉ³ØëÔÜÍÂÊÞ ËÂÇµÉ ¬Ù ÐáÜáñÛÜØµå¹¸ ÎµÍÂÊåÌÍÂÐÞ íçÙáÜÉ¬ÐØâÎµÍÂÉ ¤ÐØµèëÎ ÊÒîËÙ³ÊØµèÚÛ²ËÂÛÜÊØµÒ¤ÐçÍÂÉ ÊÔôÙÍÂÛñËÂÛñÙ¬ÐáÛÜÞÀÎ ÊÌÍËÂÐØµÙ³ÉÌí
ÒÛÜØµÙ³ÉÓËÂÇµÉì Þ®Ð³ì _ Ó
É ËÂÇµÉâÊØµáñì®ÛÜØµÔ×ÊÌÍÂÞ®ÐçËÛÜÊØ®ÐXUÌÐÛñá×Ð _ÚáÜÉÓÐ _ ÊÖ«ËÈËÂÇµÉãÔ×ÖµØµÙËÂÛñÊØ ÒR_ É³ÇÚÐXU¥ÛÜÊÌÍ¬é
êÔ ìÊÖ ÇÚÐXUÉàÙ¬ÐçÍÂÉÔ×ÖÚáñáñìóåÊØµÉâËÂÇ«ÍÂÊÖµåÇ æ ÇÚÐçËòæ¹ÉàÇÚÐKUÉàèµÊØµÉëÒÊ®ÔªÐçÍ³í
ÐÔÜËÂÉÍ·ÇÚKÐ UôÛÜØµå åÉ³Ø pow.C
ÉÍäÐÑËÂÉ³è
í«ÒÛÜØµÙ³ÉÏá×ÐçËÂÉÄÍ¬íôËÂÇµÉÓÔ×ÖµØÚÙÄËÂÛÜÊØ èÚIÉ Y ØµÛñËÛÜÊØ®ÛÜÒ¹ØÚÊÌË¹ØµÉ³ÉèÚÉè ÐØ ìôÞßÊÌÍÂÉÌ
é S¼ÇµÉ³ØìÊÖ _ Ö«ì
pow.o
Ù³ÊÞßÞßÉÍÙ³Û×Ðá ÒÊÔÜË æÈÐçÍÉÌí ìÊÖ ÐçÍÂÉãÊÔÜËÂÉ³Ø ÔªÐÙ³Éè æ·ÛñËÂÇËÂÇµÉâÐ _ÚÒÉ³ØµÙ³ÉàÊÔ ÒÊÖ«ÍÂÙ³ÉÙ³ÊôèµÉ YÚáÜÉ³Òí ÒÛñØÚÙÉâËÂÇµÉ
UÉ³ØµèµÊÌÍ¹èµÊôÉ³ÒïØµÊÌË æÈÐØ¯ËïÙ³ÖµÒîËÂÊÞßÉÄÍÂÒËÂÊàÞßÊ èÚÛñÔÜìËÂÇµÉ ÒîÊÖµÍÙ³É³Ù³Ê èÚÉÓÛÜØµÒîËÉ¬ÐèßÊÔ _ÚÖµìôÛÜØµåàÖ«Î èÚÐçËÂÉ³Ò³í ÊÌÍ
ËÂÊèÚÛñÒÙ³ÊUÉÍ·ÇµÊ¦æ ÞÖµÙäÇÞßÊØµÉìßËÂÇµÉì ÇÚXÐ UÉÏÎ ÐÛÜè Ô×ÊÌÍ áÜÊÖµÒîì®ÒîÊÔÜË æÈÐÑÍÂÉâèµÉ³ÒÛñåØ¾é
êØ ÐçÙ¬ÐèµÉ³ÞßÛÜÙ ÒÊÔÜË æÈÐçÍÂÉÌíòXÐ UÌÐÛñá×Ð _ÚÛÜáÜÛñË ì ÊÔÓÒÊÖ«ÍÂÙ³ÉÙ³ÊôèµÉ åÊôÉ³ÒÀæ·ÛñËÂÇµÊÖ«Ë ÒÐ¬ìôÛÜØµå«é êØ ÊÌÍÂèµÉÍßËÂÊ
É UÐáÜÖÚÐçËÂÉïÊÌÍ ÍÉÎµÍÂÊôèµÖµÙ³É ËÂÇµÉ¹Ù³ÊØ ËîÍÂVÛ _ÚÖ«ËÂÛÜÊØàÊÔµÒÖµÙäÇ ÒÊÔÜË æÈÐçÍÂÉïËÂÊ ËÂÇµÉ ÍÂÉÒîÎ É³ÙËVÛ UÉÈÐçÍÂÉ¬Ð ÊÔ«ÍÂÉ³ÒîÉ¬ÐçÍÂÙäÇ¤í
ÛñË·ÛÜÒ ØµÉ³Ù³É³ÒîÒÂÐçÍì ËÂÊßÇÚXÐ UÉàÒÊÖ«ÍÂÙ³ÉÙ³ÊôèµÉÌé UÉ³Ø½ÛÜØ Ù³ÊÞßÞßÉÍÂÙÛ×Ðá¤Ù³ÊØ ËIÉ P¥ËÂÒí
ÒîÊÔÜË æÈÐÑÍÂÉ
³

¢
³

³
£

ÛÜÒãÐè UÌÐçØÚÙÛÜØµå«éÆ¹ÇµÉÞßÊÒËÏÎµÍÂÊÞßÛñØÚÉØ ËÓÒÊÔÜË æÈÐçÍÂÉ ËÂÇÚÐçËÏÙ³ÊÞßÉ³ÒÓæ Û²ËÂÇ ÐçáÜá ÒÊÖ«ÍÂÙ³É³Ù³Ê èµÉ YÚáÜÉ³ÒÓÛÜÒÓËÂÇµÉ
ÊÌÎ ÉÍÂÐçËÂÛÜØµåÒì¥ÒîËÂÉÞ
é Î É³Ø ÒÊÖ«ÍÂÙÉÝÒÊÔÜË æÈÐçÍÂÉÝÙ³ÐØ UÉÍì É Ù³ÛñÉ³Ø ËÂá²ì _ ÉÝÐèÚÐçÎµËÂÉ³è ÐØµè
Ã° Availability of sourcecode.
ìÊÖóÍÂÉ¬ÐçáÜÛ³ ÉÓËÂÇÚÐçËÈæ¹ÉâÙ³ÊÖµáÜèóÛñØÎµÍÂÛÜØµÙ³ÛñÎÚáÜÉÓèµÉ³áÜÉËÉãËÂÇµÉâÒîÊÖµÍÙ³É³Ù³Ê èÚÉ YÚáÜÉ
0
& ,#$
Q$&*H7
"K(7P
'(
7
% 9 92 7N"L *
( #*
! R*
R6> .*6
$&B
7 /
J
! ÛÜÞÀÎµÍÂÊUÉ³èÛñÔ Þ®ÐØ ì®Î É ÊÌÎ áñÉãÙ³ÊØ¯ËÍÂÛV_ÚÖ«ËÂÉÌé
ÖµË·ÒÖµÙäÇ½Ð¿Ù³ÊØ ËîÍÂÛV_ÚÖ«ËÂÛÜÊØÛñÒÈÎ ÊÒîÒÛV_ÚáÜÉÏÊØÚá²ì æ·ÇµÉ³ØóËÂÇµÉ
ÒÊÖ«ÍÂÙ³ÉÙ³ÊôèµÉâÛñÒ·ÐXUÐÛÜá×Ð _ÚÜá Éé
Æ¹ÇµÉßÔ×ÖÚØµÙËÛÜÊØ
æ·ÛÜáÜáÃØµÊÌËb_ÉÀËÂÇµÉ ÊÌØÚá²ìÝÞ®ÐçËÂÇµÉ³Þ®ÐÑËÂÛÜÙ¬Ðá Ô×ÖµØµÙËÂÛÜÊØ ËÂÇÚÐçËàæ¹ÉßæÈÐØ¯Ë
pow
ËÂÊ ÖµÒÉßÛÜØ ÊÖ«ÍàÎµÍÂÊåÌÍäÐÞßÒ³éó
Æ Ê Þ®Ð ]É ËÂÇµÉ®ÐèµèµÛñËÂÛñÊØ ÊÔÈØµÉæ Ô×ÖÚØµÙËÛÜÊØµÒàÉ¬ÐÒîìí æ¹ÉßÙ¬ÐçØ ÎÚÖ«ËàËÂÇµÉ
èµÉ YÚØµÛñËÂÛÜÊØ ÊÔ
^ºÐØµèóÒÛñÞßÛÜá×ÐçÍ¹Ô×ÖµØµÙËÂÛÜÊØµÒ¹ËÂÇÚÐçËÈæ¹ÉâÞ®Ð³ì Ðèµè á×ÐçËÂÉÄÍG` ÛÜØ ËÂÊ ÐÒÛñØÚåáñÉÏÒîÊÖµÍÙ³É³Ù³Ê èÚÉ
pow
ÚY áÜÉ
í ÒÐ¬ìí ÐØµè ËÂÇµÉÈÙ³ÊÌÍÍÂÉÒîÎ ÊØµèµÛÜØµåâèµÉ³Ù³á×ÐÑÍäÐçËÂÛÜÊØµÒ ÛÜØ¯ËÂÊëÐÓÒÛÜØµåáÜÉ¹ÇµÉ¬ÐèµÉÍYÚáÜÉ
é ÆÈÇµÉ
Ê _ Émath.C
ÙËëÙ³Ê èµÉ YÚáÜÉ
ËÂÇµÉ³Ø Ù³ÊØ ËÂÐÛÜØµÒãÞ®ÐçÙ´ÇµÛÜØµÉ¿á×ÐØµåÖÚÐåÉ¿Ù³ÊôèµÉ Ô×ÊÌÍàÐáÜá ÊÖ«ÍëÞ®math.h
ÐçËÂÇµÉ³Þ®ÐÑËÂÛÜÙ¬Ðá
math.o
Ô×ÖµØµÙËÂÛÜÊØµÒ³é
áñËÂÇµÊÖµåÇàØµÊÌË ÒîËÍÂÛñÙËÂáñìâØÚÉÙ³É³ÒÒÂÐÑÍìíÛñË ÛÜÒ åÊ ÊôèàÎµÍäÐÙËÂÛñÙ³ÉËÂÊÓÛÜØµÙ³áÜÖµèµÉ
ÛÜØàËÂÇµÉ _ É³åÛñØÚØµÛÜØµå
ÊÔ
éÓÆÈÇµÛÜÒ ÉØÚÒîÖµÍÉ³ÒÓÙ³ÊØµÒÛÜÒîËÂÉØÚÙÄì _ ÉË æ¹É³É³Ø Ô×ÖµØµÙËÂÛÜÊØ èµÉ³Ù³á×ÐÑÍäÐçËÂÛÜÊmath.h
ØµÒ·ÐØµè½Ô×ÖµØµÙËÂÛÜÊØ èµÉ YÚØµÛ ËÂÛÜÊmath.C
ØµÒïÐçØÚè®ÎÚÖ«ËÂÒ ËÂÇµÉ Ù³ÊôèµÉ ÛÜØ
ÛÜØ¯ËÂÊëËÂÇµÉ ÒÙ³ÊÌÎ É ÊÔ¡ÐáÜá Ô×ÖµØÚÙÄËÂÛÜÊØµÒ èµÉ³Ù³á×ÐÑÍÂÉ³è ÛÜØ
í¥ÒÉ³É
math.C
" É³ÙËÛÜÊØ é é é êØ ÐçáÜá Ô×ÖµØµÙËÂÛñÊL
Ø _ ÊôèµÛÜÉ³ÒÃÛÜØ
í¯æ¹É Ù¬ÐØßËÂÇµÉÍÂÉÔ×ÊÌÍÂÉ Ù¬Ðáñá ËÂÇµÉ·ÊÌËÇÚÉÄmath.h
ÍïÔ×ÖµØÚÙÄËÂÛÜÊØµÒ³í
math.C
æ Û²ËÂÇµÊÖµË¹ÇÚXÐ U¥ÛñØÚå ËÂÊ¿ËÂÇµÛÜØ ] Ð _ ÊÖ«Ë¹æ ÇµÉËÂÇµÉÍÈ
ËÂÇµÉ³ÒÉãÔ×ÖÚØµÙËÛÜÊØµÒÈÇÚXÐ UÉëÐçáñÍÂÉ¬Ðè«ì _ ÉÉ³Ø èµÉ³Ù³á×ÐçÍÉ³è¤é
êØ åÉ³ØµÉÍäÐáíÒÉ UÉÍäÐçá¥Ê _ É³ÙÄË Ù³Ê èÚÉ Y áñÉ³Ò Þ®Ð³
ì _ É ØµÉ³É³èµÉ³è ËÂÊ åÉ³ØµÉÍÂÐçËÂÉïÐØ IÉ P¥É³Ù³Ö«ËäÐ _ÚáÜÉÃÎµÍÂÊåÌÍäÐÞ í
ÐØµè Û²Ë æ¹ÊÖÚáñè _ É¿Ù³Öµ
Þ _ ÉÍÂÒÊÞßÉãËÂÊ ËÉ³áÜá ËÂÇµÉ¿áÜÛÜØ ]ÉÍÏÐ _ ÊÖ«ËÏÐáñá ÊÔÕËÂÇµÉ³Þ éâêØÚÒËÂÉ¬Ðè¤í¤Ê _ É³ÙÄËãÙÊôèµÉ
YÚáÜÉ³ÒÃËÂÇÚÐçËáñÊåÛÜÙ¬ÐáÜá²Z
ì _ É³áÜÊØµåÏËÂÊåÉËÂÇµÉÍÙ¬ÐL
Ø _ É
ÛÜØ ËÊëÐ
é ØµáñìËÇÚÉòØÚÐÞßÉòÊÔËÇÚÛñÒ
F ÇÚXÐ UÉòÐáÜá¥áÜ ³Û _Ú£©ÍÂÐçK£ JÍìëÔ×ÖµØµÙËÂÛñÊØÚÒ XÐ UÐÛÜá×Ð _ áñÉ Ô×ÊÌÍ
áÜVÛ _µÍäÐÑÍìëÞ¿ÖµÒîË ËÂÇµÉ³
Ø _ É¹åVÛ UÉ³ØËÊÓËÇÚÉ¹áÜÛñaØ ]ÉÄÍ ÛÜØ©ÊÌ£Í±¬èÚ¥ÉÄ ÍÃD¥ËG ÊÏ
ËÂÇµÉ·GÉ P«É³Ù³Ö«ËäÐ _ áñÉòÎµÍÂÊåÌÍäÐÞ é êØßÊÖµÍÙ¬ÐÒÉíôæ¹É·ÒÊàÔªÐÑÍ ÇÚXÐ UÉ ÊØµáñì ÊØµÉòÊ _ É³ÙË YÚáÜÉ
ÍÉ³ÒÖµáñËÂÛÜØµå
math.o
ÔÜÍÂÊÞ
í _ Ö«Ë¹æ¹ÉâÙ³ÐØóÒîËÂÛñáÜá _ÚÖÚÛñáÜèÐ áÜVÛ _µÍäÐÑÍL
ì YÚáÜÉ
íµÒÂÐ³ìíµÔÜÍÂÊÞ ÛñË¬é
[ ÛÜmath.C
åÖ«ÍÂÉ ÒÙäÇµÉ³Þ®ÐçËÂÛñÙ¬ÐáÜáñìÝÒÇµÊÑæ·Ò ÇµÊÑæ Ê _ É³ÙËÀÙÊôlibmath.a
èµÉ YÚáÜÉÒ³íïÐ½áñVÛ _µÍäÐçÍì ÐØµè YÚØÚÐáÜá²ì ÐØ IÉ P¥É Ù³Ö«ËäÐ _ÚáÜÉ ÎµÍÂÊåÌÍäÐçÞðÐÑÍÂÉâÊ _µËäÐÛñØÚÉè ÔÜÍÂÊÞðÐ ØôÖµb
Þ _ÉÍÈÊÔ ÒÊÖ«ÍÂÙ³É³ÙÊôèµÉ YÚáÜÉÒ³é
Libraries.
ê»Ë ÛñÒàÙ³áÜÉ¬ÐçÍâËÂÇÚÐçËàæ¹ÉßèµÊ ØµÊÌËàæÈÐØ¯ËëËÂÊ ]É³ÉÎÇµÉ¬ÐèµÉÍYÚáÜÉ³ÒàÐØµè
áÜÛV_µÍäÐÑÍÂÛÜÉ³ÒàÊÔòåÉ³ØµÉÍäÐá¹ÛÜØ¯ËÂÉÍÂÉ³ÒË ÛÜØ ÊÖ«Íæ¹ÊÌÍ ]ôÛÜØµå½èµÛñÍÉ³ÙËÂÊÌÍìíÃÒÛÜØµÙ³É®æ¹É ^ºÐØµèÊÌËÂÇµÉÍÒ ` Þ®Ð³ì ÇÚÐXUÉ
Þ®ÐØ¯ìßæ¹ÊÌÍ ]¥ÛÜØµå¿èµÛñÍÂÉ³ÙËÊÌÍÂÛÜÉ³Ò³
é ·É³ÐèµÉÍYÚáÜÉÒòÐçØÚèáÜÛV_µÍäÐçÍÂÛñÉ³ÒïÒÇµÊÖµáÜè _ ÉâÐçËÈÒÊÞßÉÓÙ³É³Ø¯ËÍäÐá¤ÎÚá×ÐÙ³ÉÌé
SÝÉÓÙ¬ÐØ Þ®Ð ]É ÊÖ«ÍïÎµÍÂÊåÌÍäÐçÞßÒ ÛÜØµèµÉÎ É³ØµèµÉ³Ø Ë ÔÜÍÂÊÞ ËÂÇµÉ áÜÊôÙ³ÐçËÂÛÜÊØßÊÔ¾ÇµÉ¬ÐèµÉÍRYÚáÜÉ³Ò _ ìÀæ·ÍÂÛ²ËÂÛÜØµå
Centralization and namespaces
#include
_ÚÖ«Ë ÛÜØàËÂÇµÛÜÒ Ù¬ÐçÒÉÌíæ¹ÉïÇÚÐKUÉ ËÊÓËÉ³áÜá ËÇÚÉïÙÊÞÀÎ ÛñáÜÉÍ ^»æ ÇµÉ³Øàæ¹ÉïÒîËäÐÑÍË ÛñËG` æ ÇµÉÍÂÉ ËÂÊ ÒîÉ¬ÐçÍÂÙäÇ Ô×ÊÌÍ YÚáÜÉÒ ËÂÊ
_ ÉÕÛÜØµÙ³áÜÖµèµÉ³è¤é Æ¹ÇÚÛñÒ ÛÜÒ¾ÉIµ
P ÐçÙËÂáñìÏËÂÇµÉæÈÐ³ìãËÇ ÐÑË ÇµÉ¬ÐèµÉÍÒ áÜÛV]É
ÔÜÍÂÊÞ ËÂÇµÉÕÒîËäÐØµèÚÐçÍÂèàáÜÛ _ÚÍÂÐçÍì
iostream
ÐçÍÂÉ ÛñØÚÙáÜÖµèÚÉè ¯ËÂÇµÉ³ÛñÍïáñÊôÙ¬ÐçËÛÜÊØµÒ ÐçÍÂÉô] ØµÊÑæ·Ø®ËÊ ËÇÚÉ·Ù³ÊÞÀÎÚÛÜáÜÉÍ¬í¯ÒÊàæ¹É èµÊØ ËïÇÚÐXUÉ ËÂÊàÎµÍÂÊ UôÛÜèµÉ ÐØ¯ì
ÛÜØµÔ×ÊÌÍÂÞ®ÐçËÛÜÊØ®ÇÚÉÄÍÂÉÌé " ÛÜÞßÛÜáÜÐçÍÂáñìíæ¹ÉâÙ¬ÐçØóËÂÉ³áÜá¤ËÂÇµÉãáñÛÜØ ] ÉÍòæ·ÇµÉÍÂÉãËÇÚÉãáÜÛV_µÍäÐçÍÛÜÉ³Òïæ¹ÉâØµÉ³É³è ÐçÍÉãËÂÊb_ É
Ô×ÊÖµØµè¾é åÐÛñØ¾íôÔ×ÊÌÍ·ËÂÇµÉ
ÌU ÐçÍÛÜÊÖµÒÈáñÛVµ_ ÍäÐçÍÂÛÜÉÒÈÊÔ¾ËÂÇµÉãÒËäÐØµèÚÐçÍÂèóáÜÛ Ú_ ÍÂÐçÍìí ËÇÚÉÏÙ³ÊÞÀÎÚÛÜáÜÉÄÍR¥] ØµÊÑæ·ÒÈËÇÚÛñÒ³é
S É æÈÐØ ËÓËÂÊ®ÍÂÉ³Þ®ÐÑÍ ó
Ý
] ËÂÇÚÐçË
ÛñÒÏØµÊÌËÏØµÉ³Ù³É³ÒîÒÂÐçÍÂÛÜá²ì ËÂÇµÉØÚÐÞßÉëÊÔÐÀÎ Ç¯ì¥ÒÛñÙ¬Ðá ÚY áÜÉ ËÂÇµÉ
Þ®ÐçÎµÎÚÛÜØµåàÊÔ
Â
Ë
Ê

Ù
Â
Ë
Ú
Ö

Ð
á
Y
Ú
áÜÉ³Ò¹ÛÜÒÈÛÜÞÀÎÚáÜÉ³ÞßÉØ ËäÐçËÛÜÊØ èµÉ ÚY ØµÉ³è¤é
[ ÛÜØÚÐáÜá²ìí¯Û²Ë ÛÜÒÕåÊôÊôèÎÚÍÂÐÙËÂÛÜÙÉòËÂÊëÎÚÖ«ËïÐáñá Ô×ÖµØµÙËÂÛÜÊØµÒÕÊÔ¤ÐãáÜÛVµ
_ ÍäÐÑÍì¿ÛÜØ ËÂÊëÐãØÚÐÞßÉ³ÒîÎ ÐÙÉÌí ÛñØßÊÌÍÂèµÉÍ
ËÂÊ®ÐXU ÊÛÜè Ù³á×ÐçÒÇµÉ³Ò æ ÛñËÇóÖµÒÉÍèÚÉÙ³á×ÐçÍÂÉè½ØÚÐÞßÉ³ÒíÒÉ³É " ÉÙËÂÛÜÊØ \ é é é ÉËÓÖµÒ ÖµÒÉëËÂÇµÉëØÚÐÞßÉ³ÒîÎ ÐÙÉ
ÇµÉÍÂÉÌé
ifm
·ÉÍÂÉ ÐçÍÉ®ËÇÚÉ®ÇµÉ¬ÐèµÉÍÐØµè ÛÜÞÀÎÚáÜÉ³ÞßÉ³Ø¯ËäÐçËÂÛÜÊØ YÚáÜÉ³Ò
Ð Øµè
ËÂÇÚÐçËàÍÉ³ÒÖµáñË ÜÔ ÍÂÊÞ
math.C
ËÂÇµÉ³ÒÉãåÖµÛÜèµÉ³áÜÛÜØµÉ³Ò¹Ô×ÊÌÍ ÊÖ«ÍòÛÜØ ËÉ³ØµèÚÉèóáÜÛV_µÍäÐçÍîì®ÊÔ Þ®ÐçËÂÇµÉ³Þ®ÐçËÛÜÙ¬ÐáÚÔ×ÖµØÚÙÄËÂÛÜÊØµÒ ^°ËÂÇÚÐçËÈÙ³Ö«ÍÍÂÉ³Ø¯ËÂáñì ³Ù ÊØ
ËäÐÛñØÚÒ
ÊØµáñì `Äé
pow
math.h
! ÛñØÚÙáÜÖµÒÛÜÊØ
ÛÜØµÙ³áÜÖµÒÛÜÊØ
math.C
math.h
callpow4.C
Ù³ÊÞÀÎÚÛÜáÜÐçËÂÛÜÊØ
Ù³ÊÞÀÎÚÛÜá×ÐçËÂÛñÊØ
math.o
callpow4.o
ÐçÍÂÙäÇµÛVUôÛÜØµå
áÜÛñØa]ôÛÜØµå
áÜÛÜØ ]¥ÛñØÚå
libmath.a
Figure 17:
ª¬ È ¬£©£³î¬ß©FÝµ±³î©« ¦§ë¢¤£Á£©¨çÅ
namespace ifm {
// PRE : e >= 0 || b != 0.0
// POST : return value is b ^e
double pow ( double b , int e );
}
F»Á½¥ Ä±¬â±Ä Fµ
// math .h
// A small library of mathematical functions.
callpow4
Program 25: ¢¤£Á ¨ ©¯ºÅ×
// math .C
// A small library of mathematical functions.
# include < cassert >
# include < IFM / math .h >
namespace ifm {
double pow ( double b , int e)
{
!
assert ( e >= 0 || b != 0.0);
// PRE : e >= 0 || b != 0.0
// POST : return value is b^ e
double result = 1.0;
if ( e < 0) {
// b^e = (1/ b )^( - e )
b = 1.0/ b;
e = - e;
}
for ( int i =0; i < e ; ++ i ) result *= b ;
return result ;
}
}
Program 26: ¢¤£³Á ¨ ©¯ºÅ
[ ÛÜØÚÐáÜá²ìí¯ËÂÇµÉ·ÎµÍÂÊåÌÍÂÐÞ
Ù¬ÐáÜáÜÒÕÊÖ«ÍïáÜÛV_µÍäÐçÍîì Ô×ÖµØµÙËÂÛÜÊØ
éÃê»ËïÛÜØµÙ³áñÖÚèµÉ³ÒÃËÂÇµÉ
callpow4.C
ifm::pow
ÇµÉ¬ÐèµÉÍÚY áÜÉ
ÔÜÍÂÊÞ Ð ÙÉ³Ø ËÍÂÐá¡èµÛñÍÉ³ÙËÂÊÌÍì
é
math.h
IFM
// Prog : callpow4 .C
// Call library function for computing powers .
# include < iostream >
# include < IFM / math .h >
int main ()
{
std :: cout
std :: cout
std :: cout
std :: cout
std :: cout
<<
<<
<<
<<
<<
ifm :: pow ( 2.0 , -2) < < "\ n" ;
ifm :: pow ( 1.5 , 2) < < "\ n" ;
ifm :: pow ( 5.0 , 1) < < "\ n" ;
ifm :: pow ( 3.0 , 4) < < "\ n" ;
ifm :: pow ( -2.0 , 9) < < "\ n" ;
//
//
//
//
//
outputs
outputs
outputs
outputs
outputs
0.25
2.25
5
81
-512
return 0;
}
Program 27: ¢¤£³ÄÁ
±Ä©¯° ¢¸
Å
3.1.9 Using library functions
ÊÖÙ¬ÐçØ ÛÜÞ®ÐåÛÜØµÉ ËÇ ÐÑËæ¹É®æ¹ÉÍÂÉ ØµÊÌËËÂÇµÉYµÍÂÒîË¿ËÂÊ½ÎÚÖ«ËÀÐ Ô×ÖµØµÙËÂÛÜÊØ áÜÛV]É
ÛÜØ ËÊ Ð áÜÛV_µÍäÐçÍîìé
pow
êØµèµÉ³É³è¤íïËÂÇµÉóÒîËäÐØµèÚÐçÍÂè áñÛV_µÍäÐçÍì Ù³ÊØ ËÂÐÛÜØµÒÀÐ Ô×ÖµØµÙËÂÛñÊØ
ËÂÇÚÐçË ÛÜÒÀÉ UÉØ ÞßÊÌÍÂÉ åÉ³ØµÉÍäÐá
std::pow
ËÂÇÚÐØ ÊÖ«ÍÂÒ ÛñË Ù¬ÐØ ÙÊÞÀÎ Ö«ËÂÉ e Ô×ÊÌÍ
ÉGP¥Î ÊØµÉ³Ø¯ËÂÒ é Ù³Ù³ÊÌÍèÚÛñØÚåá²ìíòËÂÇµÉÝÐÑÍÂåÖµÞßÉ³Ø ËÂÒ®ÊÔ
ÐØµè Û²ËÂÒÍÂÉËÖµÍØ UÐáÜÖµÉ bÐçÍÉ ÊÔ¤£³Ë ´ì ©¯Î É
éÃêØ eÊÌÍÂèµÉÍ ËÂÊ ÖµÒÉ ËÂÇµÛÜÒ Ô×ÖµØµÙËÂÛñÊØ¾í¯æ¹É ÇÚÐKUÉ
std::pow
double
!
ËÂÊ ÛÜØµÙ³áÜÖµèµÉÀËÂÇµÉ®ÇÚÉ³ÐèµÉÍ
é Æ¹ÇµÛÜÒ ÇÚÉ³ÐèµÉÍ¿Ù³ÊØ ËäÐçÛÜØµÒàèÚÉÙ³á×ÐçÍäÐÑËÂÛÜÊØµÒëÔ×ÊÌÍ Ð UÐçÍÂÛñÉË ì ÊÔòÊÌËÂÇµÉÍ
cmath
ØôÖµÞßÉÍÛÜÙ¬Ðá Ô×ÖµØµÙËÂÛÜÊØµÒ³é
·ÒÛÜØµåàÔ×ÖµØµÙËÂÛÜÊØµÒïÔÜÍÂÊÞ ËÇÚÉ ÒËäÐØµèÚÐçÍÂè®áÜÛV_µÍäÐçÍîìÀÙ¬ÐØ ÇµÉ³áñÎ ÖµÒ ËÂÊ åÉË¹ÒÇµÊÌÍËÂÉÄÍ¬í _ ÉËîËÂÉÍ¬í¥ÊÌÍ¹ÞßÊÌÍÉ
É Ù³ÛÜÉ³Ø¯Ë ÙÊôèµÉÌí¹æ ÛñËÂÇµÊÖ«ËßÇÚÐXU¥ÛÜØµåÝËÂÊ æòÍÂÛñËÂÉ ÐÝÒîÛÜØµåáÜÉ ØµÉæ áÜÛÜØµÉ _ ì ÊÖ«ÍÂÒÉ³áVUÉÒ³é [ ÊÌÍ®ÉIP«ÐÞÀÎÚáÜÉÌí
Ù³ÊÞÀÎÚÖ«ËÂÛÜØµå Ù¬ÐçØÒîÎ ÉÉ³èóÖ«ÎóÊÖ«Í¹ÎµÍÂÛÜÞ®ÐáÜÛ²Ë ì ËÂÉÒîËÈÛÜØ ÍÂÊåÌÍäÐÞ é ÊÖ ÞßÛñåÇ ËïÇÚÐXUÉ

³
£
Ì

ÍÂÉ¬ÐçáÜÛ ³É³è ËÂÇµÛÜÒ¹ÞÖµÙäÇÉ³ÐçÍÂáÜÛÜÉÄÍ¬í _ÚÖ«Ëòæ·ÇÚÉØ æ¹ÉãÐçÍÂÉÏáÜÊôÊ ]ôÛÜØµåàÔ×ÊÌÍ·ÒÊÞßÉ ÎµÍÂÊÌÎ ÉÄÍ·èµVÛ UôÛÜÒÊÌÍ¹ÊÔ Ð ØÚÐçËÂÖ«ÍäÐá
ØôÖµ
Þ _ ÉÍ
í Û²ËÓÛÜÒ ÒÖ Ù³ÛñÉ³Ø ËÓËÂÊ®ÒÉ¬ÐÑÍÂÙäÇ ÛÜØ ËÂÇµÉàÍäÐØµåÉ
éâêØÚèµÉ³Éè¾í ÛñÔ Ù¬ÐØ _ É
æ·ÍÛñËËÂÉ³Ø ÐnÒëÐßÎµ2ÍÂÊôèµÖµÙË
í¡ËÂÇµÉ³ØÝËÂÇµÉ¿ÒÞ®ÐáÜáñÉÍÓÊÔ {2,
ÐØµ.è . . , Þ¿nÖµÒî}
Ë _ É _ÊÖÚØµèµÉ³è n _ ì n
ÒÛÜØµÙ³ÉÓËÂÇµÉÏèÚÛ U¥ÛÜÒÊÌÍÒòÐÑÍÂÉân
ÛÜØ¯ËÂ=É³åÌÍädd
Ðçáíôæ¹ÉÏÉ UÉ³Ø åÉË·
Ð _ÊÖÚØµè dÊÔ d í ÍÂÊÖµØÚèµÉ³èèÚÊ¦æ Ø¤é
n
n
ÆÈÇµÉ¹ÎµÍÂÛÜÞ®ÐáÜÛ²Ë ìÏËÂÉ³ÒËÙ³ÊÖµáÜè¿ËÂÇµÉÍÂÉ³Ô×ÊÌÍ
É _ É¹æ·ÍÂÛ²ËËÂÉ³Ø¿ÞßÊÌÍÂÉ¹É ÙÛÜÉ³Ø Ëáñì¿ÐÒÃÛÜØ ÍÂÊåÌÍäÐÞ \ íÖµÒÛÜØµå
ËÂÇµÉ Ô×ÖµØµÙËÂÛñÊØ
ÔÜÍÊÞ ËÂÇµÉÀáÜÛ _ÚÍÂÐçÍì
í æ·ÇµÊÒÉÀÐçÍåÖÚÞßÉØ ËâÐØµèÝÍÂÉËÂÖ«ÍÂØÝË ìôÎÉ³ÒàÐçÍÂÉ
std::sqrt
cmath
é
double
// Program : prime2 .C
// Test if a given natural number is prime .
# include < iostream >
# include < cmath >
int main ()
{
// Input
unsigned int n;
std :: cout < < " Test if n >1 is prime for n =? ";
std :: cin > > n;
// Computation : test possible divisors d up to sqrt (n)
unsigned int bound = ( unsigned int )( std :: sqrt (n ));
unsigned int d;
for ( d = 2; d <= bound && n % d != 0; ++ d );
// Output
if ( d <= bound )
// d is a divisor of n in {2 ,... ,[ sqrt (n )]}
std :: cout < < n < < " = " < < d < < " * " < < n / d < < " .\ n";
else
// no proper divisor found
std :: cout < < n < < " is prime .\ n ";
return 0;
}
Program 28: ¢¤£³Á ¢¤£¬»¨
ÆÈÇµÉ·ÎµÍÂÊåÌÍÂÐÞ ÛñÒÕÙ³ÊÌÍîÍÂÉ³ÙË ÛÜÔ
d <= bound
Å
ÒîËÛÜáÜáµÇµÊáÜèµÒÐÔÜËÉÍ ËÂÇµÉ·áñÊôÊÌÎ¾íæ¹É·ÇÚÐXUÉ áÜÉ³ÔÜËÃËÂÇµÉ·áÜÊ ÊÌÎ
_ ³É Ù³ÐÖµÒÉËÂÇµÉ
Ù³ÊØµèµÛñËÂÛÜÊØ
Ç ÐçÒãÔªÐÛñáÜÉ³è¤é¿ÆÈÇµÛÜÒÓÞßÉ¬ÐØµÒ ËÂÇÚÐçËÓæ¹É ÇÚÐXUÉ¿Ô×ÊÖÚØµè Ð
n % d != 0

£
èµÛVU¥ÛñÒÊÌÍ¬éÝêÔ
ÇµÊáÜèµÒÐçÔÜËÂÉÍ ËÂÇµÉ áñÊôÊÌÎ¾í æ¹É ÇÚÐXUÉ®ËÍÂÛÜÉ³èÐáÜáÎÊÒÒÛV_ÚáÜÉßèµÛVUôÛÜÒÊÌÍÂÒ ÒÞ®ÐáñáÜÉÍ
d > bound
ÊÌÍ ÉXôW ÖÚÐá ËÂÊ
^»æ ÇµÊÒÉ UÐáÜÖµÉ ÛÜÒ í ÒÛÜØµÙ³É ËÂÇµÉ ÉIP¥ÎÚáÜÛÜÙÛñË®Ù³ÊØ UÉÄÍÂÒÛÜÊØ ÍÊÖÚØµèµÒ®èµÊÑæ·Ø¾í
ÒÉ³É " É³ÙËÛÜÊØ \ bound
é é `Äí ÒÊ æ¹É®Ù³ÉÍËÂÐÛÜØµáñìÝÇÚÐXUÉ n ØÚÊÌË ÞßÛñÒÒÉ³è ÐØ ì èµÛVUôÛÜÒÊÌÍ¬é Ö«Ëàæ¹É®Ç ÐXUÉ®ËÂÊ _ É Ð
áÜÛñËîËÂáÜÉ¿Ù¬ÐçÍÂÉ³Ô×ÖµáïÇµÉÍÂÉ ÊÖ«Í¿ÐçÍÂåÖµÞßÉ³Ø¯ËÂÒëÐçÒÒÖµÞßÉ ËÂÇÚÐçË
æ¹ÊÌÍ ô] ÒàÙ³ÊÌÍÍÂÉÙËÂáñì Ô×ÊÌÍ Ò ôW ÖÚÐçÍÂÉ³Òé
std::sqrt
[ ÊÌÍßÉI«
P ÐÞÀÎÚáÜÉÌí
Þ¿ÖµÒîË¿ÍÂÉËÂÖ«ÍÂØ
^ Ð½áÜÛñËîËÂáÜÉ®ÞßÊÌÍÂÉ æ¹ÊÖµáÜèµØ Ë¿Ç ÖµÍîËGÄ` í Ú_ Ö«Ë
11
íµÒÂÐ³ìéòstd::sqrt(121)
êØ ËÇ ÐÑË á×ÐçËîËÂÉÍ Ù³ÐÒÉÌí
æ¹ÊÖµáÜè ÇÚÐXU É U Ð áÜÖµÉ
10.99998
(unsigned int)(std::sqrt(121))
í ÐØµèLô_ ìÞ®Ð ¥] ÛñØÚåÓËÂÇµÛÜÒÃÊÖ«Í _ ÊÖµØµè¤í¯æ¹ÉòÞßÛÜÒÒ ËÂÇµÉ·èµÛV¥U ÛñÒÊÌÍ
ÊÔ
í¯ÉÍÍÊØÚÉÊÖÚÒîáñì ÙÊØÚÙáÜÖµèÚÛñØÚå
10
11 121
ÜÛ Ò¹ÎµÍÂÛÜÞßÉÌé
ê»ËëÛÜÒãåÉ³ØµÉÍäÐáñáñì ØµÊÌËâÒÂÐÔ×É ËÂÊ ÍÂÉ³áñì½ÊØ ÒÊÞßÉÎµÍÂÉÙ³ÛÜÒÉÒÉ³Þ®ÐØ¯ËÂÛÜÙ³ÒÏÊÔïáÜÛV_µÍäÐçÍîì½Ô×ÖµØµÙËÂÛñÊØÚÒí É UÉ³Ø
ÛÜÔìÊÖ«ÍãÎ áÜÐçËÂÔ×ÊÌÍÂÞ ÛÜÞÀÎÚáÜÉ³ÞßÉ³Ø¯ËÂÒQÚÊÐçËÂÛñØÚå®Î ÊÛÜØ¯ËëÐÑÍÂÛñËÂÇµÞßÉËÂÛñÙàÐÙ³Ù³ÊÌÍèÚÛñØÚå ËÂÊËÂÇµÉ ê ÒËäÐØµèÚÐçÍÂè
^ÒÉ³É " É³ÙÄËÂÛÜÊØ \ é é `´éãÆ¹ÇµÉ Ò WôÖÚÐçÍÂÉàÍÂÊ ÊÌË Ô×ÖµØµÙËÂÛÜÊØ ÛÜÒ ÒîÎ ÉÙ³Û×Ðá ÛñØ ËÂÇµÉ ÒÉ³ØµÒÉàËÂÇÚÐçË ËÇÚÉ ê ÒîËäÐçØÚèÚÐçÍèßÒîËÂÛÜáÜáÚåÖÚÐçÍäÐØ¯ËÂÉ³É³ÒÕËÂÇµÉ·ÍÂÉ³ÒÖµáñËÊÔ
ËZ
Ê _ É·ËÂÇµ
É QÚÊÐçËÂÛñØÚåÏÎ ÊÛÜØ¯ËØ ÖÚ
Þ _ ÉÍÙ³áÜÊÒÉÒîË
ËÂÊ ËÇÚÉÏÍÂÉ¬Ðá¾Ò W Ö ÐÑÍÂÉâÍÂÊ ÊÌË ÙÊØÚÒîXÉ WôÖµÉ³Ø¯ËÂáñìíÚstd::sqrt
ÊÖµÍ·Ð _ Ê UÉëÛÜÞÀÎÚáÜÉÞßÉ³Ø ËäÐÑËÂÛÜÊØÀÊÔ ËÂÇµÉÏÎÚÍÛÜÞ®ÐáÜÛñË ìËÂÉ³ÒîË·ÛÜÒ
ÒÂÐÔ×Éé Ö«Ë·ÒîÛÜÞßÛÜá×ÐçÍåÖÚÐçÍäÐØ¯ËÂÉ³É³ÒÈèµÊ
ØµÉ³Ù³ÉÒÒÂÐçÍÂÛñáñì ÇÚÊáñèÔ×ÊÌÍòÊÌËÂÇµÉÍ·áñVÛ _µÍäÐçÍì®Ô×ÖµØµÙËÂÛÜÊØµÒ³é

áÜÒÊãÛÜØßÊÖ«Í ÒÉÙ³ÊØµèÀÎÚÍÛÜÞßÉ¹ØôÖµb
Þ _ÉÍ ÐÑÎÚÎÚáÜÛñÙ¬ÐçËÂÛÜÊØ¤í
í¯æ¹É L
è _ÉËËÂÉÍÙ¬ÐáÜá
ÐëÒîËäÐçØÚèÚÐçÍè áÜÛ _ÚÍÂÐçÍìÀÔ×ÖµØµÙËÂÛñÊØ ÛÜØ ÊÌÍÂèµÉÍïËÂÊ ÛÜØµÛñËÛ×ÐáÜÛ ÉòÊÖ«¹ÍÈ£©¯áÜÛÜÒîË ³ÊºÔ¡³Ù ÍÊ³Ò ÒÉ³ è ÊÖµX D¥Ë¹ Ø Öµb
Þ _ ÉÄÍÂÒ³í¥ÛÜØµÒîËÉ¬Ðè
ÊÔ èµÊÛÜØµå¿ÛñËòÊÖµÍÒÉ³Vá UÉ³Ò¹æ Û²ËÂÇóÐ áÜÊôÊÌÎ¾é [ ÊÌÍÈËÂÇµÛÜÒ³í¥æ¹Éãæ¹ÊÖµáÜè®ÍÂÉÎÚá×ÐçÙ³ÉÏËÇÚÉÓË æ¹Ê áÜÛñØÚÉÒ
ËÂÇÚÐçË
121
ÊÔ
for ( unsigned int i = 0; i < n ; ++ i)
crossed_out[i ] = false ;
ÍÂÊåÌÍäÐçÞ
æ ÛñËÂÇ®ËÂÇµÉÏÒÛÜØµåáÜÉÓáÜÛÜØµÉ
std :: fill ( crossed_out , crossed_out + n , false );
Æ ÇÚÉÓÎµÍÂÉ ÐØµè ÎÊÒîËÂÙ³ÊØµèµÛñËÂÛÜÊØµÒ ÊÔ¾ËÂÇµÛÜÒ¹ÒîËäÐØµèÚÐçÍÂè áÜÛV_µÍäÐçÍîìÀÔ×ÖµØµÙËÂÛÜÊØ ÉIPµÐÙÄËÂáñì®Þ®ÐçËÂÙäÇ®ËÂÇµÉãÊØµÉ³ÒÈÊÔ
¹
ÊÖ«Í Ê¦æ Ø
Ô×ÖµØµÙËÂÛñÊØ ÔÜÍÂÊÞ ÐåÉ ! éÈÆ¹ÇµÉZ_ É³ØµÉ YµË ÇµÉÍÂÉàÛñÒ·ØµÊÌË ËÂÇµÉëÒÂÐXUôÛÜØµåÀÊÔ ÊØµÉëáÜÛÜØµÉâÊÔ
fill
Ù³Ê èÚÉ ËÂÇµÛÜÒ ÒÂÐXU¥ÛñØÚå¹èµÊôÉÒ¡ØµÊÌË¤É UÉ³ØâÉIP«ÛÜÒË¬íçÒîÛÜØµÙ³É æ¹ÉÕÐçèÚèµÛñËÛÜÊØÚÐáÜáñìÈÇÚÐXUÉÃËÂÊ
#include <algorithm>
ÛÜØ ËÂÇµÉ _ ÉåÛÜØµØÚÛñØÚå ÊÔ ËÇÚÉÓÎµÍÂÊåÌÍäÐÞ é
ÆÈÇµ
É _ É³ØµÉ YµË¹ÛÜÒïËÂÇÚÐçËïæ¹ÉÓÉ³áÜÛñÞßÛÜØÚÐçËÂÉòÎÊÒÒÛV_ÚáÜÉ ÒÊÖ«ÍÂÙÉ³ÒòÊÔ¾ÉÍÍÊÌÍ ^»É UÉ³Ø ÐëËÍÂÛ U¥Û×Ðá áÜÊôÊÌÎ ÇÚÐÒïËÂÇµÉ
Î ÊÌËÂÉØ ËÂÛ×ÐçáÊÔ _ É³ÛÜØµåàæ·ÍÂÊØµå `ÄíôÐØµè ËÂÇÚÐçËæ¹ÉÏÒîÛÜÞÀÎÚáÜÛÜÔÜì¿ËÂÇµÉÏÙÊØ ËÍÂÊá Q Ê¦æ ^ÒîÉ³ÉâÐçáÜÒÊ " É³ÙËÛÜÊØ \ é é `Äé
3.1.10 Details
Default arguments.
" ÊÞßÉ·Ô×ÖµØµÙËÂÛÜÊØµÒÕÇ ÐXUÉ·ËÇÚÉÈÎµÍÂÊÌÎ ÉÍË ìËÂÇÚÐçËÕËÇÚÉÄÍÂÉ ÐçÍÂÉîØ ÐÑËÂÖ«ÍäÐá
UÌÐçáÜÖµÉ³Ò Ô×ÊÌÍ
ÊØµÉ ÊÌÍïÞßÊÌÍÂÉòÊÔ¤ËÇÚÉÛñÍïÔ×ÊÌÍÂÞ®Ðá ÐçÍÂåÖµÞßÉ³Ø¯ËÂÒ³é [ ÊÌÍ¹ÉIP«ÐÞÀÎÚáÜÉÌíæ ÇµÉ³Ø YÚáÜáÜÛÜØµåãÐØ ÐçÍÍäÐ³ìÀÊÔ¤ÖµØµèµÉÍÂáñìôÛÜØµå
Ë ôì Î É
í ËÂÇµÉ UÐáÜÖµÉ
ÛÜÒëÒîÖÚÙäÇ ÐØÚÐçËÂÖ«ÍäÐáUÌÐçáÜÖµÉÌé êØ ÒÖµÙäÇ ÐÙ¬ÐçÒÉÌí ÛñËëÛñÒâÎ ÊÒÒÛ _ áñÉ ËÂÊ ÒîÎ ÉÙ³ÛÜÔÜì
int
0
ËÂÇµÛÜÒLUÐáÜÖµÉ ÐÒ Ð
ãËÂÇµÛÜÒ®ÐáñáÜÊÑæ·ÒËÂÇµÉ Ù¬ÐáñáÜÉÍ®ÊÔÓËÂÇµÉ Ô×ÖµØµÙËÂÛñÊØ¼ËÂÊ ÊÞßÛñËËÂÇµÉ
¶çÐÑ©ÍÂå²ÖµóÞß©É³£îØ ÁÌËë
¨ ÐØµè
¤ áÜÉËàËÇÚÉßÙÊÞÀÎ ÛñáÜÉÍëÛñØÚÒîÉÍËàËÂÇµÉ®èµÉ³ÔªÐÖµáñË UÐáÜÖµÉßÛÜØµÒîËÂÉ¬Ðçè¾é êØ
Ù³ÊÌÍÍÉ³ÒîÎ ÊØµèµÛÜØµå Ù¬ÐFµáÜáï
Ù¬ÐÒîÉâÊÔ¡ËÇÚÉÏÔ×ÖµØµÙËÂÛÜÊØ
ÔÜÍÂÊÞ ÍÂÊåÌÍäÐÞ \ í¥ËÂÇµÛÜÒïæ¹ÊÖµáÜèáÜÊ Ê ] ÐÒÈÔ×ÊáÜáÜÊ¦æ Ò³é
fill
// PRE : [ first , last ) is a valid range
// POST : * p is set to value , for p in [ first , last )
void fill ( int * first , int * last , int value = 0) {
// iteration by pointer
for ( int * p = first ; p != last ; ++ p)
*p = value ;
}
ÆÈÇµÛÜÒÈÔ×ÖµØµÙËÂÛÜÊØ Ù¬ÐØóØµÊ¦æ _ÉãÙ¬ÐçáÜáÜÉ³è æ ÛñËÂÇ É³ÛñËÂÇµÉÍ¹Ë æ¹Ê ÊÌÍÈËÂÇ«ÍÂÉ³ÉâÐçÍÂåÖµÞßÉ³Ø¯ËÂÒ³í«ÐÒÈÔ×ÊáÜáÜÊ¦æ Òé
int a [5];
fill (a , a +5);
// means : fill (a , a +5 , 0)
fill (a , a +5 , 1);
ê ØÀåÉ³ØµÉÍäÐçáíËÂÇµÉÍÂÉÈÙ¬ÐØ _ÉòèµÉ³ÔªÐÖµáñËUÐáÜÖµÉ³Ò Ô×ÊÌÍÐØ ì¿ØôÖµÞ_ ÉÍÃÊÔÔ×ÊÌÍÞ®Ðá«ÐçÍÂåÖµÞßÉ³Ø¯ËÂÒ³í _ÚÖ«ËÕËÇÚÉÒÉ
ÐçÍÂåÖµÞßÉ³Ø¯ËÂÒÏÞÖµÒîË _ ÉÀÐÑËâÙ³ÊØµÒÉÙ³Ö«ËÂÛVUÉ Î ÊÒîÛñËÂÛÜÊØµÒ
ÐçÞßÊØÚå®ËÂÇµÉ
ÐÑÍÂåÖµÞßÉ³Ø ËÂÒí¡Ô×ÊÌÍ
ÒÊÞßÉ é Æ¹ÇµÉßÔ×ÖÚØµÙËÛÜÊØ Ù¬ÐØ ËÂÇµÉ³Ø _ÉÀÙ¬ÐáÜáñÉ³è æ ÛñËÂi,Ç i Ðç+Ø ì 1,Ø . ÖÚ. .Þ, _ kÉÍëÊÔÈÙ³ÐáÜá ÐÑÍÂåkÖµÞßÉ³Ø ËÂÒ _ ÉË æ¹É³É³Ø
ÐiØµè í¥ÐçØÚèËÂÇµÉ·Ù³ÊÞÀÎÚÛÜáÜÉÄÍÃÐÖ«ËÂÊÞ®ÐçËÂÛÜÙ³ÐáÜáñìÏÛÜØµÒÉÍîËÂÒÕËÂÇµÉòèÚÉÔªÐÖµáñË UÐáÜÖµÉ³ÒÕÔ×ÊÌÍËÂÇµÉÈÞßÛÜÒÒÛñØÚåÏÙ¬Ðáñá
i−1
ÐçÍÂåÖµÞßÉ³Ø¯ËÂkÒ³é
Ô×ÖµØµÙËÂÛÜÊØ Þ®Ð³ìÇÚXÐ UÉàÐ ÒîÉÎ ÐçÍäÐçËÂÉâèÚÉÙ³á×ÐçÍäÐÑËÂÛÜÊØóËÂÇÚÐçË·ÒîÎ É³Ù³Û Y ÉÒ èµÉ³ÔªÐÖµáñËÓÐçÍåÖÚÞßÉØ ËÂÒ³íÚáÜVÛ ]ÉãÛÜØ
ËÂÇµÉÏÔ×ÊáÜáÜÊ¦æ ÛÜØµå èµÉ³Ù³á×ÐçÍÂÐçËÂÛÜÊØ ÊÔ
é
fill
// PRE : [ first , last ) is a valid range
// POST : * p is set to value , for p in [ first , last )
void fill ( int * first , int * last , int value = 0);
êØËÇÚÛñÒ ¬Ù ÐÒÉÌíËÂÇµÉÈÐÙËÖ çÐ á¥èµÉ YÚØÚÛ²ËÂÛÜÊØàÞ¿ÖÚÒË ØÚÊÌË ÍÉÎ É¬ÐçË ËÂÇµÉïèµÉ³ÔªÐÖµáñËÃÐÑÍÂåÖµÞßÉ³Ø ËÂÒ ^»ËÂÇµÉÈÐÙËÂÖÚÐáôÍÖÚáñÉ³Ò
ÐçÍÂÉãÐ _ Û²ò
Ë ÞßÊÌÍÉÏÙÊÞÀÎ áñÜÛ Ù¬ÐçËÂÉè¾í _ÚÖµËÈËÂÇµÛÜÒ¹ÛÜÒ¹ËÂÇµÉãÖ«ÎÚÒÇµÊÌËG`Äé
Ô×ÖµØµÙËÂÛñÊØ Þ®Ð³ì Ç XÐ UÉÓÒÉ UÉÄÍäÐá èµÉ³Ù³á×ÐÑÍäÐçËÂÛÜÊØµÒ³í É UÉ³Ø æ ÛñËÇ
ËÂÇµÉ¿ÒÂÐÞßÉ¿èÚÉÙ³á×ÐçÍäÐÑËÂÛVUÉàÍÂÉ³åÛÜÊØµÒb^»ËÂÇµÉ¿á×ÐçËËÂÉÄÍãÛñÒãØµÊÌËâÐáÜáÜÊ¦æ¹É³è Ô×ÊÌÍ UÐçÍÂÛ×Ð _ÚáÜÉÒ³í¤ÒÉ³É " É³ÙÄËÂÛÜÊØ \ é é `Äé
Æ¹ÇÚÉ ÎÚÖ«ÍÎ ÊÒîÉ ÊÔÚÐ·Ô×ÖµØÚÙÄËÂÛÜÊØ èµÉ³Ù³á×ÐçÍÂÐçËÂÛÜÊØëÛñÒ ËÂÊ ÎÚÖ«Ë ÒÖ _ÚÒXÉ WôÖµÉ³Ø¯ËÃÙ³ÊôèµÉ ÛÜØ¯ËÂÊ ËÂÇµÉ Ô×ÖµØµÙËÂÛÜÊØ Ò ÒÙ³ÊÌÎ Éí
ÐØµèËÂÇµÉÍÉãÞ®Ð³ì _ ÉÏÒÉ UÉÍÂÐá¡ÎÚá×ÐÙÉ³Òïæ ÇµÉÍÂÉÏËÂÇµÛÜÒ¹ÛÜÒòØµÉ³ÙÉ³ÒÒÂÐçÍîìé
Ø ËÂÇµÉàÊÌËÂÇµÉÍÓÇÚÐØµè¤íÉ UÉÍîì½Ô×ÖµØµÙËÂÛñÊØ Ù³ÐØ½ÇÚXÐ UÉàÊØµáñìóÊØµÉ èµÉ YÚØµÛñËÂÛÜÊØ¤í ÐØµè ËÇÚÛñÒ ÛÜÒ ËÂÇµÉàÊØµÉ
ÐáÜá Û²ËÂÒòèµÉ³Ùá×ÐçÍäÐçËÛÜÊØµÒ ÍÂÉ³Ô×ÉÍòËÂÊ«é
Function declarations and definitions.
êØ Ô×ÖÚØµÙËÛÜÊØ èµÉ³Ùá×ÐçÍäÐçËÛÜÊØµÒ³í ËÂÇµÉßÔ×ÊÌÍÂÞ®ÐáïÐÑÍÂåÖµÞßÉ³Ø ËàØÚÐÞßÉÒ
Ù¬ÐØ _ÉãÊÞßÛñËîËÂÉ³è¤é
Function
signatures.
íñé´é³é³í
ÆÈÇµÛÜÒ·Þ®Ð ]É³Ò·ÒîÉ³ØµÒÉ ÒÛÜØµÙ³ÉâËÂÇµÉ³ÒîÉàØ ÐçÞßÉ³Ò ÐçÍÉëÊØµáñìØµÉ³ÉèÚÉè½ÛÜØ ËÂÇµÉëÔ×ÖµØµÙËÂÛñÊØ èÚÉIY ØµÛñËÛÜÊØ¤éòÆ¹ÇµÉ
ÛÜÞÀÎ ÊÌÍîËäÐØ ËÈÛÜØµÔ×ÊÌÍÂÞ®ÐÑËÂÛÜÊØ¤í«ØÚÐÞßÉ³áñì èµÊÞ®ÐÛÜØÐØµè ÍÂÐØµåÉëÊÔ ËÂÇµÉëÔ×ÖµØÚÙÄËÂÛÜÊØ¤íÐçÍÉàÐáñÍÂÉ³Ðè«ìÒîÎÉ³Ù³ÛVYÚÉ³è
_ôìËÂÇµÉ ÐçÍÂåÖµÞßÉ³Ø¯ËË ìôÎÉ³ÒïÐØµèßËÂÇµÉ·ÍÂÉËÖµÍØ®Ë ì Î ÉÌé áÜáµËÂÇµÉ³ÒÉ·Ë ì Î É³Ò ËÂÊåÉËÇÚÉÄÍ Ô×ÊÌÍÂÞ ËÇÚÉ
°Á ©¯ £³
ÊÔ¡ËÇÚÉãÔ×ÖµØÚÙÄËÂÛÜÊØ¤é
êØ
í«æ¹ÉãÙÊÖÚáñè ËÂÇµÉÍÂÉÔ×ÊÌÍÂÉâXÉ W ÖµVÛ UÌÐçáÜÉ³Ø Ëáñìßæ·ÍÂÛ²ËÂÉ ËÂÇµÉãèµÉ³Ù³áÜÐçÍäÐçËÂÛñÊØ
math.h
double pow ( double , int );
Æ ÇÚÉãÊØµáñìßÎµÍÂÊ _ÚáÜÉ³Þ ÛÜÒ¹ËÂÇÚÐçËòæ¹ÉÏØµÉ³É³èóËÂÇµÉãÔ×ÊÌÍÂÞ®Ðá¤ÐçÍÂåÖµÞßÉ³Ø¯Ë¹ØÚÐÞßÉ³Ò¹ËÂÊ ÒîÎÉ³Ù³ÛÜÔÜì®ÎµÍÂÉ ÐØµèÎ ÊÒîË ¹
Ù³ÊØµèµÛñËÂÛÜÊØµÒ³í æ Û²ËÂÇµÊÖµË¿åÊÛÜØµå ËÂÊ áñÉ³ØµåÌËÂÇ ì Ô×ÊÌÍÂÞ¿Öµá×ÐçËÂÛÜÊØµÒàÛÜØ UÊáVU¥ÛñØÚåËÂÇµÉYµÍÂÒîËßÐÑÍÂåÖµÞßÉ³Ø Ë ÐØµè
ËÂÇµÉ ÒÉÙ³ÊØµèÐçÍåÖÚÞßÉØ Ë «é Æ¹ÇµÉÍÂÉ³Ô×ÊÌÍÂÉíôæ¹É ÖµÒÖÚÐáÜáñì¿æ·ÍÂÛ²ËÂÉòËÂÇµÉ Ô×ÊÌÍÂÞ®ÐçáÐçÍåÖÚÞßÉØ Ë ØÚÐÞßÉÒÉ UÉ³Ø ÛÜØ
Ô×ÖµØµÙËÂÛÜÊØ èÚÉÙ³á×ÐçÍäÐÑËÂÛÜÊØµÒ³é
\
)óÐØ ì ÊÔ¹ËÂÇµÉßÞ®ÐçËÂÇµÉ³Þ®ÐÑËÂÛÜÙ¬Ðá Ô×ÖÚØµÙËÛÜÊØµÒëËÂÇÚÐçË ÐçÍÂÉ®ÐXUÐÛÜá×Ð _ áñÉ ÊØ ÒÙ³Û Mathematical functions.
³É Ø¯ËÂÛVYÚÙãÎ Ê Ù ]ÉË Ù¬ÐáñÙ³Öµá×ÐçËÂÊÌÍÂÒÈÐçÍÉàÐáÜÒÊ ÐXUÌÐÛñá×Ð _ÚáÜÉÏÔÜÍÂÊÞðËÂÇµÉëÞ®ÐÑËÂÇáÜÛ _ÚÍÂÐçÍì
éÈÆ¹ÇÚÉâÔ×ÊáÜáÜÊ¦æ ÛÜØµå
cmath
ËäÐ _ÚáÜÉ áÜÛÜÒËÂÒÒîÊÞßÉ ÊÔ·ËÇÚÉÞ é áÜáÈÐçÍÂÉÐXUÌÐÛñá×Ð _ÚáÜÉ®Ô×ÊÌÍÀËÂÇµÉ ËÇµÍÉ³É QÚÊÐçËÂÛñØÚå Î ÊÛÜØ¯Ë Ø ÖµÞb_ ÉÄÍË ìôÎÉ³Ò
í
ÐØµè
é
float
double
long
double
ØÚÐÞßÉ
Ô×ÖµØµÙËÂÛñÊØ
std::abs
std::sin
std::cos
std::tan
std::asin
std::acos
std::atan
std::exp
std::log
std::log10
std::sqrt
|x|
ÒÛñØ
Ù³ÊÒ (x)
ËäÐçØ (x)
(x)
ÒîÛÜØ −1
ÙÊÒ −1(x)
ËÂÐØ −1(x)
(x)
e
x
áÜØ
áñÊå x
10
x
x
3.1.11 Goals
Ë¹ËÂÇµÛÜÒ¹Î ÊÛÜØ¯Ë¬íôìÊÖóÒîÇÚÊÖµáÜè é³é³é
¹
Dispositional.
`L_ ÉãÐ _ÚáÜÉÏËÊÉIPôÎÚá×ÐÛÜØ ËÂÇµÉÏÎÚÖ«ÍÎ ÊÒÉÏÊÔ
` ÖµØÚèµÉÍÒîËäÐØµè®ËÂÇµÉ Òîì¥Ø¯ËäÐ P ÐØµè®ÒÉ³Þ®ÐçØ
\ ß
`L¥] ØµÊ¦æ æ·ÇÚÐçË¹ËÂÇµÉÏËÂÉÄÍÂÞ ÎµÍÂÊ Ù³É³èµÖ«ÍäÐá
ÔÜÍÂÊÞ
` ÖµØÚèµÉÍÒîËäÐØµè½ËÇÚÉ Ô×ÖµØµÙËÂÛÜÊØ
ß
pow
ÔÜÍÂÊÞ ÍÂÊåÌÍäÐçÞ \ `L]¥ØµÊ¦æ ËÂÇÚÐçËÓÔ×ÊÌÍÂÞ®Ðá ÐÑÍÂåÖµÞßÉ³Ø ËÂÒ ÊÔ Î
×Ô ÖµØµÙËÂÛÜÊØµÒ¹ÛÜØ ËÂÛÜÙÒ ÊÔ Ô×ÖÚØµÙËÛÜÊØ®èÚÉIY ØµÛñËÛÜÊØµÒ ÐØµè®èµÉ³Ù³á×ÐçÍÂÐçËÂÛÜÊØµÒ
ÎµÍÂÊåÌÍäÐÞßÞßÛñØÚå ãÞßÉ¬ÐØµÒ
ÍÂÊåÌÍäÐÞ ÐØµè½ËÂÇµÉ Ô×ÖµØµÙËÂÛÜÊØµÒ
ÐØµè
fill_n
fill
ÊÛÜØ¯ËÂÉÍ Ë ìôÎÉ¿Ù³ÐØ _ É µÖ ÒÉ³è½ËÂÊßæ·ÍÂÛñËÉ ÐçÍÍäÐ³ì » ÎµÍÂÊôÙ³ÉÒÒÛÜØµå
Ô×ÖÚØµÙËÛÜÊØµÒ³íÚÐØµèÞ¿ÖµËÂÐçËÂÛÜØµåàÔ×ÖµØµÙËÂÛÜÊØµÒ
`L]¥ØµÊ¦æ æ·Ç ì¼ÛñË Þ®Ð ]É³Ò®ÒÉØÚÒîÉ ËÂÊ Ù³ÊÞÀÎÚÛÜáÜÉ Ô×Öµµ
Ø ÙËÂÛÜÊØ èÚIÉ Y ØµñÛ ËÛÜÊØµÒßÒÉÎ ÐçÍäÐçËÉ³áñìí ÐµØ è ËÊ ÎÚÖ«Ë
Ô×ÖÚØµÙËÛÜÊØµÒòÛÜØ¯ËÂÊ¿áÜVÛ _µÍäÐçÍÂÛñÉ³Ò³é
Operational.
^
^
`®åÛ UÉÏË æ¹Ê ÍÂÉ¬ÐÒÊØµÒÈæ Ç ì ÛñËÈÛÜÒÈèµÉ³ÒÛñÍÂÐ _ÚáÜÉ ËÂÊ ÒÖ _ èµÛVUôÛÜèµÉ ÎµÍÂÊåÌÍäÐÞßÒ¹ÛÜØ¯ËÂÊÔ×ÖµØµÙËÂÛñÊØÚÒ Y ØµèóÎµÍÂÉ ïÐØµè ÎÊÒîËÂÙ³ÊØµèµÛñËÂÛÜÊØµÒÈÔ×ÊÌÍ åÛVUÉ³Ø Ô×ÖµØµÙËÂÛñÊØÚÒí æ ÇµÉÍÉâËÂÇµÉâÎµÍÂÉ³Ù³ÊØµèµÛñËÂÛñÊØÚÒÈÒÇµÊÖµáÜè
\ ` Ú
_ ÉãÐÒ
ÐÒ¹Î ÊÒÒÛ _ áñÉÌíµÐØµèËÂÇµÉÓÎ ÊÒîËÂÙ³ÊØµèµÛñËÂÛñÊØÚÒïÒîÇÚÊÖµáÜè _ ÉãÐÒ
ÐÒ¹Î ÊÒÒîÛV_ÚáÜÉ ¸ÓÄ©
£³Á
Y Øµè ÒîìôØ ËäÐÙÄËÂÛÜÙ¬ÐáòÐØµè ÒÉ³Þ®ÐçØ ËÂÛÜÙ³Ðá¹ÉÍÍÂÊÌÍÂÒ ÛÜØ Ô×ÖµØµÙËÂÛÜÊØ èµÉ YÚØµÛñËÂÛÜÊØµÒ³íÐØµè ÛÜØ ÎµÍÂÊåÌÍäÐÞßÒ
` Ú
ËÂÇÚÐçËÈÙ³ÊØ¯ËäÐÛÜØ Ô×ÖµØÚÙÄËÂÛÜÊØóèµÉ ÚY ØµÛñËÂÛÜÊØµÒ
` ÉIÌU ÐáñÖ ÐÑËÂÉÓåÛVU É³ØóÔ×ÖµØµÙËÂÛÜÊØÙ¬ÐáÜá ÉIôP ÎµÍÂÉ³ÒÒÛñÊØÚÒ ®
` æòÍÂÛñËÂÉ ^ Þ¿Ö«ËäÐçËÂÛñØÚå Ó
` Ô×ÖµØÚÙÄËÂÛÜÊØµÒÔ×ÊÌÍåÛVU É³Ø ËäÐÒ ô] Ò³í ÐØµèæòÍÂÛñËÂÉßÎµÍÂÊåÌÍäÐçÞßÒàÔ×ÊÌÍ åÛVU ÉØËäÐÒ ô] Ò
ß
^
^
^
êØÎ ÐçÍËÂÛñÙ³Öµá×ÐçÍ¬í¯ìÊÖ ÒÇµÊÖÚáñè _ ÉãÐ _ÚáÜÉ ËÂÊéé¬é
ËÂÇÚÐçËÈÖµÒÉãÔ×ÖÚØµÙËÛÜÊØµÒ
^
^
`®ÒîÖa_èÚÛ U¥ÛÜèµÉ¹Ð åÛVUÉ³Ø¿ËäÐÒ ] ÛÜØ¯ËÂÊÏÒÞ®ÐáÜá ÒÖ Ú
_ ËÂÐÒ ]¥ÒíôÐØµè æ·ÍÂÛ²ËÂÉÈÐ ÎµÍÂÊåÌÍÂÐÞ Ô×ÊÌÍÕËÇÚÉ¹åÛVUÉØËäÐÒN]
ËÂÇÚÐçËÈÖµÒÉ³ÒòÔ×ÖµØÚÙÄËÂÛÜÊØµÒÈËÊÍÂÉ¬ÐçáÜÛ³É ËÂÇµÉÏÒÖ Ú_ ËÂÐÒ ]¥Ò _ ÖµÛÜáÜèÐ áÜÛVµ
_ ÍäÐÑÍì ÊØóìÊÖµÍ·Î áÜÐçËÂÔ×ÊÌÍÂÞ í«åÛVU ÉØóËÂÇÚÐçËòìÊÖ ÐçÍÉâËÂÊáÜèËÇÚÉãØµÉ³Ù³É³ÒÒÐçÍìóËÂÉÙ´ÇµØµÛÜÙ¬Ðçá
` Ú
èµÉËäÐçÛÜáÜÒ³é
3.1.12 Exercises
Exercise 79
Ð `
_ `
\ Ì^
`
int f ( double i , double j , double k)
{
if ( i > j )
if ( i > k)
return i;
else
return k;
else
if ( j > k)
return j;
else
return k;
}
double g ( int i , int j )
{
double r = 0.0;
for ( int k = i ; k <= j ; ++ k )
r += 1.0 / k;
return r;
}
Exercise 80 Ý©¯à©£®ºë¢¤£³¥ ¦ª³¨ó Hî ¶ © JM ¸¹º º ¶ç»ª¸È°Áà¶ ±¬î¤
©F F½©¬¢«¢¤£³³¢¤£¬©¯Ó¢¤£³ à©F ¢³±F»î¤çÅ
Ð `
bool is_even ( int i)
{
if ( i % 2 == 0) return true ;
}
_ `
^
»F¢¤£³ à©F ¢ ±F¤ò¶ç£ÀºÈ¶»ª¸¹»Áã¶ ±¬î¤çÅ
double inverse ( double x)
{
double result ;
if ( x != 0.0)
result = 1.0 / x;
return result ;
}
^ \ Ìº ^ ³¨ `
Exercise 81 Ý©¯®»Ýº ñ¢¤óÂ¶Ýº®¶°§¸¹»Á ¤
¢ £³Á£©¨
µF î¢³F»Á º » ¥¢¤
¨ ³³£ i
³ ±³£¬ ºßñ¢¤» ¨ ©¯º³¨½©¯±Ä©¯ £¬¨ó §Á £¬»Áâ¢ ³ ¦§ ¥D ³£ ã© F
Fµ£ ¸ÈçÅ
^
`
# include < iostream >
int f ( int i)
{
return i * i ;
}
int g ( int i)
{
return i * f (i ) * f( f(i ));
}
void h ( int i)
{
std :: cout < < g(i ) < < "\ n";
}
int main ()
{
int i ;
std :: cin > > i;
h (i );
return 0;
}
Exercise 82
»Fóº¥£³ã¢¤£³¥ ¦ª³¨ó¿» ºò¶°§¸¹»Á¿¢¤£³ÁÌ£©¨½Å
# include < iostream >
double f ( double x )
{
return g (2.0 * x );
}
bool g ( double x)
{
return x % 2.0 == 0;
}
void h ()
{
^
Ì^
`
std :: cout < < result ;
}
int main ()
{
double result = f (3.0);
h ();
return 0;
}
»¨¿¢¾ ¶ J
Exercise 83
std::pow Å
ºÝ¢¤£³Á£©¨
Ú³£³± ¶Ä£³¨
KJ »Á º ³£©£KJ¶Ä ¬± ^ `
³ ¨ ·
º©¯ JÚ£Õ¢¾ª©¯ ¶ç£¬¨ ºÓ ¬£Ä©£KJ·¶ ±¬î std::sqrt »· D¥³£KJ
£³¬©« ¬§Å £ x © Dô¯© ¡óÂ¶ JÑ¢ double H x 0M ò¸ÏÀ§¬ s(x) ³ßºLD ©¯¡À£³¬ £¬ IF KJ
std::sqrt(
) ë ¶
©¯ Dô©¯ x À©F ¸Ó ©¯ ¨½óº©¯ë¸Ó ¤EJ ¸ º©¯ ¶£
¨½Ï¢ Ä Dô Dô©¯ ε 1/2 ¹º £¬ª©¯ D¥£¬£³£ßç©¯» ¬
Exercise 84
|s(x) − x|
x
ε,
x.
³£¬£³¸£ß±Ä © JÚF ±¬©¨½Á« ©J ¹©¯£³³Á£Ä©¨½¨ ëº ©¯¹¡º±¬
º©¯àß±£¬£±¬º J ¸Ï£ óFµ³£ º¥ó£¬ª©¯ D¥
ç©¯»Áà¢ °¤¹ ¨½ ³£¿ Jô³¨ çIF J £
¢¾×©¯ ¶£¬¨ ¬° ©£KJ ©Fàº©¯ ©¯» D ©¯¡¬ Â¶¹ JÑ¢ unsigned int ©£³ã µ©µ±¬º Jà£³î¢¤£³¬ç³¤î^ ©« ¦ ª `
»º¥¿ J ³³¨½Å HÕ¥àß© ºÄ£KJ Ú³£³±Å M
Exercise 85
©KM
£¬ ©ë¶Ä ±¬
// POST : return value is true if and only if n is prime
bool is_prime ( unsigned int n );
Ë æ ÛÜØßÎµÍÂÛÜÞßÉ³Ò
© F ç º¥ ¶ ±¬î ° © ¢¤£³ÄÁ£©¨ ±¤Èº ¨ ³ ³£ Â¶
» º
£©Á¥ {2, . . . , 10000000} H¸Ó¯¢ ³ ¨°¤NM¥Å
¸¹° ¢¤£¬»¨½½» ©½¢©»£Â¶
¨½ ³£¦ (i, i + 2) ºÂ¶à¸È¥î±¬ ©£³ã¢¤£¬°¨ Å
M ä º ©¬¢«¢¤£³©µ±¬ Â¶Ý©KM º ¬³ H¨ ³ ±³³¤VM ½ º¥À¢¤£³¥ ¦ª³¨
×¶ J
±´© º¥» Â¶ ©ó ¬º³£ß©¬¢«¢¤£³©µ±¬ JÚ © £³¶£³ÄàÀ»¨¿¢¾§¨ ³¤ »»¤³´© F Â¶ÏºÀ
º° IFó° ©KM¥Å
^
`
Exercise 86 ï¶Ä ±¬ pow ° ¹£³ÁÌ£©¨
IF¯ |e| ¨²¢¾±Ä©¯î¤â ±Ä¨¢¤ be Å
¹©Á« ºâ¶Ä ±¬ ³ FJ ¡±¬º©¯àª¬³ ¨²¢¾î±´©¯î¤ ©£³ ¢ ³£º¶ç£¬¨ GF«Å ¨½©J
ç ºÈ¶°§¸¹»Áã¶ç©µ±³îÅ ×¶ e 0 ©F e ©¯À ¬» ©K£ J £³¢¤£³¬³¤î©¯
e=
∞
X
bi 2 i ,
i=0
º³
be =
∞
Y
b2
i
bi
.
i=0
^
`
¬£ ©â¢¤£³ÁÌ£©¨ swap.C º©¯Fµ ¬ © F ±Ä©¯°²à© ¶Ä ±¬ ¶ç£ã°¤³£³±¬©Á
»Áßº Dô©¯ ¬¿Â¶â¸Ó int ¥ ±¬ºçÅ Õ·¢¤£³ÁÌ£©¨ ³
FÀ©D¥àº ¶ç»ª¸¹»Á ³£¬ ±¬ £³Å
Exercise 87
# include < iostream >
// your function definition goes here
int main () {
// input
std :: cout < < "i =? " ;
int i ; std :: cin > > i ;
std :: cout < < "j =? " ;
int j ; std :: cin > > j ;
// your function call goes here
// output
std :: cout < < " Values after swapping : i = " < < i
< < " , j = " < < j < < " .\ n ";
return 0;
}
³£ À© µ©¨¢¾ª£¬ Â¶ º±Ä¨¢¾ª¬IFß¢¤£Á£©¨
i =? 5
j =? 8
Values after swapping : i = 8 , j = 5.
F ¶IJ º·¢¤£³Á£Ä©¨ sort_array.C
¶Ä£³¨
Ú£³±³ç
£³¬Ä»Á ¢¤£Á£©¨ sort_array2.C Fµ ³ ©FÝ±´©¯»®©â¶ ±¬î
Exercise 88
^
`
» Ä ±¬¸ ©J º©¯ïº
// PRE : [ first , last ) is a valid range
// POST : the elements *p , p in [ first , last ) are
//
in ascending order
void sort ( int * first , int * last );
À¢ ³£º¶ç£¬¨ º®£¦°Á Â¶ºó©£¬£©J½°¤Ý©¯±F»Á £ Fµ³£ÌÅ ä ¨½©J ³®³¨¿¢¾»Á HÄ ³
·©¯»ª¸ÓIFÓ¶ç£À¥ KDÌÓ£³Ä©¯¤ M®ß std::sort £à» ¨ ×©£ë³î©
ÛÜÒ F«©£IF ¬£©£KJÏ¶ ±
î¤ß° º ³ FJ Â¶®ºÏ¶Ä ±¬ sort º©¯Ïß ³ Fµ IF«Å ä
©¯°§¸ÓIF ÏºÚÁ
±¨¿¢©£³ßº ±³³ X± J Â¶ JÚ£ sort ¶ ±¬î ¸¹»ºÝº©¯ëÂ¶ std::sort Hî¸È¥±¬Ý©¯
ºÀ Ñ©¨ Ó¢¤£³ à©
F ¢ ³±Ä
F¤®©
F ±Ä© ³G F ©ä¶³£ include<algorithm> M¥Å
£àº¥ Ú³£±³ Õ Fµ¬»£Ä©« ¦§ HÄ ¬Õ ï³£¬±¬ºE J® Ä±¬³Ñ©K£ J M®¹¢ °¤³£à» ±³£
¨ ¤ H ++pM ©¯ëºß¤E J ³¢ ³£Ä©¯î ó¢ »¤£¦ HÄ©¬¢©£¦¶£¨ » »©¯ ©¯©
F ©¯³§Á
¨ ¤ ÈÂ¶ã± £¦ M¥Å ×
¶ J Ä ±±ÄG F ° Fµ°Á¿ JÚ£Ó£¦°Á¹¶ ±³î ©¯ º ¢ ³¤ ©¯
^ Â¶ë ¸Ó` £ °Áã¶ç£®±¤î©» £¦¿º©¯
Fµ® Ï ££©
Fµ¨ ©µ±±³³ HîÀ©¯ ±¬ Å ¥Å M¥Å
Î ÉÍîÎ ÉËÂÖÚÐáòÙ¬ÐáÜÉ³ØµèÚÐçÍ
GF Fµ³³£¬¨° º ¸ÓÄ F«©J H F«©J
ÅÌÅÅ
F«©JM Â¶ © JÝÁ D¥F«©¯ Å ¨½©Jß¶£½µ©¨¿¢¾§ ¸ º©¯ëº ë³£¦» ¸ ©¯»
±Ä©¨½ Fµ¸¹
D¥³¨ ³³£
ï ¬Õ¸È©¯¸ ©¯Óºâ¸Ó F«©J H äÕ¸ ©¯à©ÝÕ¥£¦ F«©JµÅ M
£¿¸È©¯ïãºà¸Ó F«©J Â¶âº º © D¥³£¦ç©K£ J Â¶ëº ¸¹ ±¶çIFµ³£©¯ Èó ³
±³§ ³£©¯IF ÚÁ³ H » © Fµ ¡©¯¬ J ò»ò¸¹° ³ © ©¯£ F«©JµÅ M
Exercise 89
±´©
H F«©J ÅÌÅÅ î©¯
Å ±³ ©¬ »
©KM Õ
Àî©¯ ¿ó¸¹£¬© ¢¤£³Á£©¨ º©¯ëñ¢¤ºº¸ÏÄ F«©J
F«©JM Â¶®© Á Dô³ »¥¢¤ F«©¯Å
XFµ³¤ ¶ J ©¬ Â¶½Ä¡ ¦î©¯ ¸È¥î±¬ J ±´© £³GF ±Ýº¥
®Â¶ ±Ä£¦ ß ¬®©¯°·»F DÌNF¾©¯òÄ¡ ¦î©¯ ³ F ³½¨½©¯° H ¨ ©¯»
º©¯ßºX J ± F Ý£³Ä©¯ IF ¸È»ºÝ¶³¸ » ¬ Â¶ ±Ä Fµ M¥Å äÀÂ¶ ± £¦¢ ³Ä ¦ª
¶ç£½© Ä¡ ¦î©¯ » J £ó³âÝ£³GF ±Ä º³£ó¡ ¦î©¯ ÌçÅH ºÏÁ DÌ»Á ©¸Ó©J
© J º¥»Á · à¡ ¬î©¯ º©¯ JÚ ±Ä³£¦î©»¤ J ÄI F ã Fµ¬³£¬¨ » à¸È¬º³£ß©ÀÁ D¥³
JÚÄ©£À»À© ªÄ©¬¢ JÚÄ©£ M¥Å
M
¬£ ©À¢¤£³ÁÌ£©¨ perpetual_calendar.C º©¯·£³Ä© F¯®© F«©¯·¶Ä£³¨ ºÀ»¥¢¤Ï©F
ñ¢¤º¿ºó±£¬£³³¢ F»Á ¸ÓÄ F«©JµÅ½Õ £Ä©Á«óÂ¶F«©¯¬ º©¯ ºã¢¤£³Á£©¨ ±Ä©
¢¤£³±¬³ ³ F½³î©£¦Ó Ý×©¯³£ º© «© ¡©£KJ H F« ©JM¥Å à¢¤£³ÁÌ£©¨
³ F ±¬± ë¸ò³º³£Ïº »¥¢¤ Ó© § Á¥©¯ F«©¯ ©F ¶È £³ ±¬ îÅ µ©¨¢¾ª
£¬ Â¶¿ºÏ¢¤£Á£©¨ ¨°Á«·ªÌ ôº¥ÑÅ
day =? 13
month =? 11
year =? 2007
Tuesday
À³£¬ ±¬ £³ JÚ£·¢¤£³ÄÁ£©¨ ï°¨¢¾ª³¨ ¤ïºà¡ ¬î©¯ Ì¶Ä£³¨
¢¤·ºã¢¤£Á£©¨ Á«¬º£ ¶£³¨ º¬ò¶ ±¬î¤çÅ
©KM ©¯Õ¶ ±¬î¤ ò©F
^
Ì^
`
Exercise 90 » F ¼
© ³£©£KJ
¹£³Á£©¨
©F ¹
£³ÁÌ£©¨
¶£¨ ¹
£³ÁÌ£©¨
Å
ÚJ £ ¢¾ª©¯ ¶ç£¬¨ ¶£¨ º ª¬ math.h ©F math.C »
ôÅ º¥» ³£©£KJ «Á ³ ³£Ä©¯Ý© ÚÄ±³î©« ¦ªÀ^ ¢¤£³ ÌÁÌ^ £©¨ `
Exercise 91
©KM Â¨¿¢¾§³¨½³¤Óº ¶°§¸¹»Áâ¶ ±¬î ©F ¬Ó»îÅ ¨½©J ¯© ³¨½Àº©¯ º® J¦¢
double ±Ä¨¢¾î¬ã¸È»ºóº ³î©F«©£ F
±¬ Å Å ôÅßÕ ¶Ä ±¬
»¤E J½£³ »£I F ¸Ó£ Ý±Ä£¬£³±¬º J Ï ¶Àºß ´©£³¬³Ï»¤Á¥³£ » » º D ©¯¡ß£©^ Á« `
Â¶¿º JÑ¢ int Å
// POST : return value is the integer nearest to x
int round ( double x );
M Õ
Ï¢³±F»î Â¶¿ºÈ¶ ±³î FµÌ³¿ Ñ©J ¸ò©¯ò©¬¢«¢ ³¤ ¶¿º³£³ ©£³ ¸Ó
´©£³¬³·°¤Á«£¦çÅ ¥¢ Ä±³ ¶IJ º ¬©DÌî£ Â¶ JÚ£¿»¨¢¾ª³¨½³¤î©¯î » ºÓ¢ ³^ ±Ä \ `
F»î Â¶ JÚ£ò¶ ±¬î Å
Ù ` èµè ÐÏèµÉ³Ù³áÜÐçÍäÐçËÂÛñÊØÊÔÚìÊÖµÍÕÔ×ÖµØµÙËÂÛÜÊØ¿ËÂÊãËÂÇµÉRYÚáÜÉ
^ ÍÂÊåÌÍäÐçÞ \ ` ÐØµèßÐ èµÉ YÚØµÛñËÂÛÜÊØ
math.h
ËÂÊ
^ ÍÂÊåÌÍäÐÞ \ `Äé ÖµÛÜáÜèßÐÏáÜVÛ _µÍäÐçÍîì ÔÜÍÂÊÞ ËÂÇµÉ³ÒîÉ·Ë æ¹Ê YÚáÜÉ³Ò³í ÐØµèÀÍÂÉæòÍÂÛñËÂÉÈìÊÖµÍÕËÂÉ³ÒîË
Ô×ÖµØµmath.C
ÙËÂÛÜÊØ ÔÜÍÂÊÞðÐ `ïËÊ Ù¬ÐáñáËÂÇµÉâáÜVÛ _µÍäÐÑÍL
ì UÉÍÂÒîÛÜÊØóÊÔ ËÂÇµÉÏÔ×ÖµØÚÙÄËÂÛÜÊØ
é
^ `
round
ÍÊ É³ÙÄË µ
Ö áÜÉÍ
Exercise 92 ¥»À© º³£ Hî
ÏÌ F ±³²VM Ï¶£¨
H ¹£³¥ ¦ª³¨
http://projecteuler.net/M¥Å
» F ©¯ £©¯ ¨½ ³£¦ a, b < 100 ¶£ È
¸ ¥î±¬ ab ©¯ º
×©£îÁ«¬³±³£³³® ¨ H¨ Â¶ Fµ±³°¨½©¯F§ÁºNMôÅ ¬ã Ñ©JÝ ¢¶Ä£³¤ãº©¯ 9999 »® ãº
©¤¸Ó³£ÌÅ
£¬ ©½¢¤£³ÄÁ£©¨ power_cross_sums.C º©¯ ±Ä¨¢¤³ º ¬³ a ©F b Hî¸¹»º¥»
£³´©¯ ©« ¦ªß»¨½ M¥Å © J ©¯ Fóº ¬³ a, b ¢ 1, 000
3.1.13 Challenges
Exercise 93 °¨¢¾ª¬³®±¨¿¢¤³£Ý¨½ Fµ¬ º©¯¿Ý ³»Á
"
FîG F° ºÄ£³¬î±´©¯à±¨
µè ÉËÉÍÂÞßÛÜØµÛÜÒîËÛÜÙ YÚØµÛñËÂÉ ÐÖ«ËÂÊÞ®ÐçËÂÊØ
¢¤ £Ý±³³ ± º ÐçáñÎÚÇ Ð _ ÉË
H
M¥Å ±¬ © © ¨½©¯ »
Σ Hñ¶ç£ µ©¨¢¾ª Σ = {0, 1}M¥Å®Õ³ ¹©¯ © ¿¬ÓÂ¶
FÒîµËä ÐÑËÂÉ³ Ò GF D¥£ ©
ËÍäÐØµÒÛñËÛÜÊØ Ô×ÖµØµÙËÂÛñÊØ
Q Å ¿¨½©°»ÁÌ£³I F³¤ ¿º
δ:Q
Σ → Q.
À±Ä© D¾©¯ âº¥»Õ¶Ä ±¬©¯Õ¶ç»ª¸ò ¸È³ XD¥³£ δ(q, σ) = q Ã¸Ó F£©¸ ©©£¬£³¸
¶£¨ î©¯ q ³î©¯ qÒîËäÈÐÑªÍ©«ËÂ ÛÜØµ¬ªå®
IF ÒîËä¸¹
ÐçËÉ º σ Å
ÐÙÙ³ÉÎµËÂÛÜØµå ÒîËäÐçËÂÉÒ
s Q ©F©Ý¡ ¦ç¬ F Q Â¶
»

¯
©
°

E
J
ã
º

³
£

©
Å
§Á£³ °ÁÌ£³ Fµî¢¤î±³º ©
¸¹º³î©¯ó³ Q = {0, 1, 2} Å¼Õ³î©£¦°Á ³î©¯
»
F±Ä©¯G F K J © ©£¬£¸ ±¨ »Á °¶Ä£³¨ ¸È³£³ ï©Fºà©µ±±Äî¢¾»Á ³î©¯¬ë©£³ ¨ ©£ ôIF
¸¹º Fµ¡ ¦ª ±»£³±¬ª¬ Hî»º¥®±Ä©¯ òº³£³®¤E J N M¥Å
0
1
0
1
1
0
0
2
1
Figure 18:
Fµ¬³£¬¨° ³±
» ©¨½©¯ H
M
ÝJ ±´© ¸ÓÝ±´©¯»È ±³ © ©¨ ©¯ © ±¨¿¢¤³£ ¨æ¹ÊÌ ÍFµè ¬ ë±Ä© ó»·¢ ³£º ¶£¬¨
© Ä± ¨¢¤î©¯ ¹ ©¨ ¬ J ºò¶°§¸¹»Á Á D¥³ © °¥¢¤
ÐÙ³ÙwÉÎµËÂÒ Σ H ÍÂ É ÉÀ
ÙËÂçÒ ¡³ ±
Â¶ J¯¨½ ²à¶Ä£³¨ º ©¯ ¢¾©« ¬ ΣM º ©¨½©¯ ³»º³£
®£
îÅ
Fµ º¥ ¿º ¸Ï£IF w » ¢¤£³Ì±Ä¬³GF J¯¨½ ³Ï KJ J¯¨½ ³ ³î©£¦»Á » s Å Ý³ XD¥£ º
©¨ ©¯ »à» ¨ ³î©¯ q ©
Fº IJ ¨ ³ » σ ¹ºß©¨½©¯ ¸¹±¬¬¿
³î©¯ q = δ(q, σ) Å Ý³ ©¯»Õ J¯¨½ ³®© D¥ó ³Ä³ ¢¤£Ì±¬IF ãº ©¨½©¯ »³º£
» © ©µ±Ä±î¢¾°Á ³î©¯ q F H» ¸È¥î±¬ ±´©¯ w ©µ±±Äî¢¾I FKM ®£ ° © ´©µ±Ä±î¢¾°Á
³î©¯ q Q \ F H»¸È¥±¬ ±´©¯ w £³ Ä±¬I FKMôÅ
£ µ©¨¿ ¢¾§ ¸ò ¸Ó ¶çG F º¼©¨½©¯ðÂ¶ °Á £³ ¸¹º º ¸Ó£ F w =
0101 Èº ¡³ ± Â¶³î©¯¬º©¯â©£³ ³»Á DÌ»I F » 0, 0, 1, 2, 2 Å Ó¤ç ¡³¤ºE J w »
£³ ±¬G F«Å áÜÐØµåÖ ÐçåÉ
L Â¶Àº ©¨½©¯ ßº ³¿Â¶©µ±Ä±î¢¾G FÝ¸Ó£ F¯çÅ Õ¥À
© HÁ¥³
³£©¯ °E J ° Mó¡ ¦ç¬âÂ¶ ΣçÅ ¬ K£ J Fµ¬³£¬¨° ßºÀ×©Á¾©Á«Â¶º ©¨½©¯
» °ÁÌ£³ Å
ä £¬¤®òº©¯·º¥»¿» · ±¬ © £©°Á« ¶ç£¬¸ ©£ F î©¯
©
F J G F º £¬§Á«
N Fµ´©«Å Hç ³ó ¬³ à¸Ó © F º FµÄ© £¦³ ©
F º ±Ä©¨ ¯¢ ¸¹º © ©¨½©¯
º©¯ë£³´©¯ ¬»V M¥Å Ý±¬×©»¨ º©¯àº ©¨ ©¯ ©µ±±¢¾º µ©µ±¬º J ©¯°¹ºó¸Ó£ F¯óº©¯
©£³ F DÌ» ¦ª K J 3 ¶ J °¤ ³£¢¤£³¬àº ¸ÏI£ F ©¯ © ¬» ©K£ J ¨½ ³£ Hî¸È³£³óº½³¨¿¢¾ J
¸Ó£ F » °¤³£¢¤£³¬G F ©¯ 0MôÅ £ µ©¨¢¾ª 0101 º ³» ©K£ J ¨ ³³£
0 23 + 1 22 + 0 21 + 1 20 = 5,
© Fó°FµGF 5 à F DÌ ¦ª® XJ 3 HÄ©F³ ±Ä £³ ±¬GFKM¥Å Â ¶ç©µ±¬ HÄ©Fóº¥»à º XJ
ÞßºÊ ¿
è ¢¤£³ÌÂ¶ Â¶ £ ±¬×©»¨bM ëº³î©¯½©ä¶³£ ¢¤ £³±¬³Ä »Á w » º½ ¨ ³³£IF ¸¹º
w
3Å
±´© º³£³¶ç£³óÑ©JÝº©¯âº
Â¶ §Á£³ ó©±Ä¨¢¤³£ Hî¸¹»º ©
¬ » ¢¤£³ÁÌ£©¨ M º©¯ ±´© D¥ º Fµ±³»î ¢¤£³¥ ¦ª³¨ Â¶ó±¬Ä± »Á½¸È¬º³£ ©½Á D¥³
¨ ³³£ F DÌ» ¦ªß KJ Å
©£³ ³ª¸ãE J ©¬¢«¢¤£©µ±¬¥»Á º ©µ±¬¾©¯ â ±¬©¯»ª³Á«çÅ £ XDô³£KJ ¾ ¦¬ L Â¶ {0, 1}
¶£¨ º ¶ç»ª¸È°Á¼»³ º³£ F ©
º©¯ ©¯ L ©¯Ýº ×©Á¡©Á« £ ¢¤£³ D¥
\ º©¯À¡±¬ ©
±´© ó¯ Hî¸È¥î±¬¼¸Ó F ³¸
±¨¿¢¤î©¯ ©¯¢ ¸Ó³£ M¥Å
XD¥³ ¨½ ³£óÂ¶
º©¯
³£
©£³ »¨ GF » º»£
©KM L = {w
{0, 1} | w ©¯À©
M L = {w
{0, 1} | w »LF DÌ» ¬§® KJ 5 ¸È³»¤³£¢¤£³³IF ©¯À© ¬° ©£KJ ¨½ ³£ }
± M L = {w
{0, 1} | w ©¯à¨½£³
FKM L = {w
{0, 1} | w Fµ¬ â±¤î©°
³£³¿º©
©F © XD¥³ ¨½ ³£ Â¶ß ¬ }
¬ }
º¥£³±¤Ä±³ D¥ó ¬ }
Exercise 94
"Fµ ¢¤ ªÓ¢ GFó ©Á£¬ FÂ¶ 9 9 ±¬ °² Ã¾ F DÌNFµGF °¤ ¡©£
³ µ ¬½Â¶ 3 3 ±³»½Ä©µ ±³Å ¨½ Á£¬ F ±³» ©£³ ©¯£Ä© FJ »ªIF KJ ¨½ ³£¦ ¬¸Ó³ 1
©F 9 º Á«©¯¹ »¹º£³¨½©° »Á ±Ä¬»½ KJ ¨ ³³£¦ ³¬¸Ó 1 ©F 9 » Ä ±¬
© ¸ ©J º©¯ ¸¹»º¥» Ä©µ±³ £¸ ã±¨ â©F ³ Â¶ ºó±¨¿¢¾§³ IF Á£¬NF ãXD¥³£KJ½ ¨ ³³£
Ì±Ä±³£¦ µ ©µ±¬º J ±Å ³£³ß© µ ©¨¿¢¾§ Â¶À© Fµ ¢¤ ª
1
5
7 4
3 2
9
9
6 7
8
4
2
1
9
7 3
4
7 3
6 5
2
5
6
4
ÉIP«ÐÙÂËÂÝáñì Ê ØµDÌÉ°ÁÀº¹¢¤ § ¹ Ï¨½©J®£KJ® FµIF ±Ä¶Ä£³¨ º©¯£³Ä© FJ °§GFÀ ¨½ ³£¦âº©¯
¨½ ³£ó» ©±Ä©F F«©¯â¶£ © Äî©« ¬§ ³¨¢¾ J ±Ä¬»×Å Õ³ º¥® ¨½ ³£ó»
»ªI Fó»¤ º ±¬° Ó©
F º FµGF ±¬ ¢¤£³Ì±Ä¬³ »¿£î¢ Ä©¯GF«Å Õ³£³ ©£³ß¸Ó »¡©¯¤
¸È³£³Ó¡±¬L
© FµG F ±¬¿¶ç£ ºë±Ä¬»Ú» £³¸ r ±Ä ¨ò ÊÑcæ ©ÊÔÜËÂFëÉØÝ
ÇÚ¨½
³£ êònÒÐÛÜÕ
ÐKU É
è½¢Ë©Ê £¦ìÊ±³Ö ×Ë©Ç £¦ÐÑ J Ë
ÊFÀ
¸ÈÉáÜ®
¢«¢¤ÇÚ£ÐçËÂ©µÉ U±¬É Í¹H ÍÂÉÞ®ÐÛÜØµÒ³íôÇµÊ¦æ¹É UÉÍòÛÜÞÀÎµÍÂÊ _ Ð _ÚáÜÉÌíÞ¿ÖÚÒË
æ Ä©¯ÇµI É³J Ø ©ì
Ö¶ÇÚXÐ °U§Éã
ÛÜÞßºÛÜØÚ Ðç¤
ËÂÉè ³£¦ËÂÇµ§ÉÓ± ÛÜÞÀÎ ÊÒ¨ ÒÛ _ ³áñÉÌí¯©¬æ·
_ ÉÓËÂÇµÉÏËîÍÂÖ«ËÂÇ
M¥Å
¥Å
Å
» ¨½ ³ £¦ F³° ±¬ ¶Ä£³¨ n ©¯£³Ä© FJ ©¬¢«¢ Ä©£À¨ ³¸È³£ß» ºßÑ© ¨ £¸ ã±

¨ Ó£ ³ µÅ¥»¿ Ä±¬³Ñ©£¬ J ¨½
ØôÖµÄÞ©¤_
ÉÍÂºÒ ©¯ º±Ä¬» ©¯¿ ³ »ªI Fó¸¹»º n
° ±¸Ï ©D¥ ¬°¨° ©¯IFÝ©¯° º³£
©¯ »¨¢³ ¬§Å
»òº³£ ±Ä¬» ° º ç©¨ ®£³¸ ¿£° º ç©¨ ±¨ £ » º Ñ©¨ ©£³ ©¯£³´© FJ ¸¹¼ Ó ±Ä¤î©» n Å Ù³É³áÜáÜÒ Á¥©° º ±¬°©¯À °§GF KJ n
º³ Õ° ±Ï¸Ïã©D¥ ¬»¨ » ©¯IF©¯» º³£
¶ç£âºÏ ¨ ³³£ n ¸¹º¥°ÝºÏ£³¸
±Ä ¨ Ó£ µÅ
\ £¬ ©Ý¢¤£³ÄÁ£©¨ sudoku.C º©¯ßî© ô¬©¯ »¥¢¤ © "Fµ ¢¤ §½» ¶ç£¬¨ Â¶ ©
³ ±ÄßÂ¶ 81 ¨½ ³£¦ ³¬¸ÓÄ³ 0 ©F 9 HºãÁ£¬ Fß ¨ ³³£¦ÓÁ Dô³ £³¸ KJ®£³¸ ¸È³£³
0 °Fî±´©¯¬¿© ¨¢¾ J ±¬° M¥Å Õã ¨½ ³£¦â¨§Á«ò àî¢©£Ä©¯IF KJ ¸È¥¬¢©µ±¬ ßº©¯
º Fµ ó¢¤ §È¶£¨ ©« D¥±Ä FÝ±Ä D¥³ î¤ºEJ ³ ± FµGF ô º¥»¿° © §
0 0 0
0 5 0
0 0 6
1 0 0
0 9 0
7 0 0
7 4 0
0 3 2
9 0 0
4 0 0
0 2 0
0 0 0
8 0 0
0 0 0
0 0 9
0 0 0
0 1 0
0 0 5
0 0 4
7 3 0
0 6 5
0 0 7
0 2 0
0 0 4
3 0 0
0 6 0
0 0 0
¢¤£³ÄÁ£©¨ ³ F® ¸ £KJ EDôº ¢¤ ªÀ KJ °Á ¤EJóº ¸Ó ¤³£¦§±

¨ ¬ î J¦¢ FµIF ±³î¤ ¶£³¨ ©« ³ D¥Å ñ¢¤Ó³ F ³ © H°¢©£¦ ©¯» J M ±Ä¨¢¾ª¬IF
Á£¬ F º©¯ò» º³£ ºÀºã¢¤ § £Àº¿ Hî¸ÈJ M ¢©£¦©¯
» ¸È¥î±¬ ¤³£¦ªÌ±
¨ ¬ î J¦¢ FµIF¡±¬î¤ ©¬¢«¢¾ J © J ¨½£³ H´©Á¥©» ³¨¢¾ J ±Ä¬»
³ F ³»
Fî±´©¯I FÝ XJ½ºF°ÁÌ 0M¥Å
Â º®©« ³ D¥ µ©¨¢¾ª òº ñ¢¤ÏÂ¶®© ±£¬£³Ä±¬ ¢¤£³Á£Ä©¨ ¸¹° ³ ºÀ
3 9 2
8 5 7
1 4 6
1 8 5
4 9 6
7 3 2
7 4 6
1 3 2
9 5 8
4 7 9
5 2 8
6 1 3
8 5 1
6 7 3
2 4 9
6 2 3
4 1 9
8 7 5
2 8 4
7 3 1
9 6 5
5 6 7
9 2 8
3 1 4
3 9 1
5 6 4
2 8 7
£ ³£ ´© F» Á½º®»¥¢¤Ã¶Ä£³¨ © § »à±Ä© ± D¥³ ³¤ë £³IF°£³±³ëº ³î©F«©£ F
»¥¢¤ º ª ±Ä¤î©» °Á ºß¢¤ § F«©¯î©«Å £ ±¬± ç»Á ¸ò³º³£ © J ¤³£¦§±
¨ ¬ î J¦¢ FµIF¡±¬î¤½©¬¢«¢¾ J âÀ±´© ³ ç¶È ¨ ©»¤î©° HÄ©F ¢ F«©¯NMÀ¶£Ý© J
£¬²¢¾ª (r, c, n) º°¶£¬¨½©¯ ¸È¬º£ n ³»»Õ©¢ ³ ¦ª ±´©FNF«©¯È¶£º ±Ä¬» »
£³¸ £
±¨ ±Å
¸¹°
F»± D¥£º©¯ï¨ © J "
Fµ ç ¢¤ §³âº©¯¹ JÑ¢¤±Ä©¯°EJ ©¬¢«¢ Ä©£ » ³¸È¢©¬¢ ³£¦
±Ä©É³ èµÛÜ Öµ Þ D¥I ÇÚF
F ©£³óº³£³¶ç£³ÝÄ©¯I J À D¥³ ¶ ºX J ©£³óª©« ¬§G F ©¯
Þß
ÐçÍÂK è J J £ ¢¤£³Á£Ä©¨ ©
£
Å
Hint: äòß© FDÌç©« ¬§¿ Ý³¢¾°¨ ò¶ç£ ±î³ X± J ³£ ° ± º¥ ¸È»»ï¤ J½ªÄ© F
¨½£³ ±Ä¨¢¾î±´©¯I F ©
Fó£¬£³£ §¢¤£³ ± FµÅ D¥³Ýº D¥K£ J Ä¨½©¯»ô¢¤£¥ ¦§¨ Ä Õ¡±¬
³¢¾»¨ ©¯¤à¸Ï © D¥®©ó ±´©« ¦§ ±¬â© J¯¸ © JÚÅ
\ \
3.2 Recursion
Ý°£Àç©
¨°£ÌÅ
â©Dô©£¬© Fî±¬ ¨ ï¨½Ä© »Á
Õ¥ÓÄ±¬îÝ°¤£³ F ¡±¬â£±³ £¦ D¥Õ¶ ±¬î¤ Ú¶ ±¬î¤Ïº©¯F°£³±¬º J £â»
F°£³±¬º J ±´©¯»·º¨ó¬ D¥¬ÑÅ
¼¸¹°·Ä º©¯ £±³ £¦ D¥ ¶Ä ±¬¤½©£³ D¥³£KJ
©¯ £©¯Õ° ¨½© Ý
J Ä¡©¯¤ Ó©F º©¯ÓºX J½ªÄ© F½ ±¨¢©µ±¬ ©F £Ä© F«©« ¦ª
±Ä µF ±¬ª ¨ ©¯º³¨½©¯±Ä©¯ ¶ ±³î µF »î¤çÅ ¸¹°à©¯ ¦¢¾ª©°
¸ £³Ä±³£¦Ä ¥D ¶ ±³î ±´©¯» ©£³ ¢¤£³Ì±³³IF ó©F ¸ £³±³ £¦î ±Ä© Hî°
¢¤£¬° ±³²¢¾ª M ³ £³î¢¾×©µ±GF ¸¹º ³£Ä©¯î Å Â º F ÚJ ¸¹° ½¸Ó¼©¬¢
¢¾î±´©¯î¤H £¦°Á ·©F F £©¸¹»ÁÏ¶£©µ±³î©¯²NM º©¯µF ³¨ ¤£©¯ ºÓ¢ ¸Ó³£ £
£³Ä±³£¦Äî Å
¸Ó ©£³¸Ï Å
3.2.1 A warm-up
)óÐØ¯ì Þ®ÐÑËÂÇµÉ³Þ®ÐçËÂÛÜÙ³ÐáµÔ×ÖÚØµÙËÛÜÊØµÒòÐÑÍÂÉÓØ ÐÑËÂÖ«ÍäÐáÜáñì èµÉ YÚØÚÉè
íµÞßÉ¬ÐØµÛÜØµåëËÂÇÚÐçË¹ËÂÇµÉÓÔ×ÖµØµÙ £¦Î áñ ÉÌD¥íÌ¬Ô× ÊÌJ ÍïÐçØ ì
ËÂÛÜÊØ¿ËÂÊ
_ ÉòèµÉ YÚØµÉ³èßÐÑÎÚÎÉ¬ÐçÍÂÒ ÛÜØÛñËÂÒÃÊÑæ·Ø èµÉ YÚØµÛñËÂÛÜÊØ¤é [ ÊÌÍÕÉI£³PµÐç±ÞÀ
íËÂÇµÉòØ ÖµÞb_ ÉÄÍ
n
Ù¬ÐçØÍÂÉ³Ù³Ö«ÍÂÒîÛVUÉ³áñì_ ÉÏèµÉ YÚØÚÉè ÐÒ·Ô×ÊáñáÜÊÑæ·Ò³é
n!
n! :=
ÛñÔ
1,
ÜÛ Ô n 1 é
n (n − 1)!,
n>1
ê
Ø æ¹É Ù¬ÐØ ÐáñÒÊ ÇÚÐXUÉ ÍÂÉÙ³Ö«ÍÂÒÛVUÉóÔ×ÖµØµÙËÂÛñÊØÚÒ óÐ Ô×ÖÚØµÙËÛÜÊØ Þ®Ð³ì Ù¬ÐáÜá¹ÛñËÂÒîÉ³áÜÔºé Æ¹ÇµÛÜÒ¿ÛÜÒ
ØµÊÌËÂÇµÛÜØµå ÉIP¥ÊÌËÂÛÜÙÌíÚÒÛÜØµÙ³ÉëÐÔÜËÂÉÍ ÐáÜáíÚÐ Ô×ÖµØµÙËÂÛÜÊØÙ¬ÐáÜá¾ÛÜÒ ÖµÒîË ÐçØ ÉIP¥ÎµÍÂÉ³ÒîÒÛÜÊØ ËÂÇÚÐçË Ù³ÐØ ÛÜØóÎµÍÂÛÜØµÙ³ÛñÎÚáÜÉ
ÐçÎµÎ É¬ÐÑÍßÐØ¯ìôæ ÇµÉÍÉÛÜØ ËÂÇµÉ Ô×ÖÚØµÙËÛÜÊØ Ò ÒîÙ³ÊÌÎ ÉÌí¹ÐØµè ËÂÇÚÐçË ÒÙ³ÊÌÎ ÉÛÜØµÙ³áÜÖµèµÉ³Ò¿ËÇÚÉÔ×ÖÚØµÙËÛÜÊØ _ Ê èµìé
¥ÛÜØÀÔªÐçÙË¬í ËÇÚÛñÒÕèµÉ YÚØµÛñËÂÛÜÊØ¿ÉIP«ÐÙËÂáñìÞ®ÐçËÂÙäÇµÉ³Ò ËÂÇµÉ
·ÉÍÉ·ÛÜÒÐÏÍÂÉ³ÙÖµÍÒVÛ UÉ·Ô×ÖµØµÙËÂÛñÊØÀÔ×ÊÌÍïÙ³ÊÞÀÎÚÖ«ËÂÛÜØµå
n!
Þ®ÐçËÂÇµÉ³Þ®ÐÑËÂÛÜÙ¬ÐáÚèµÉ YÚØµÛñËÂÛÜÊØ ÔÜÍÂÊÞðÐ _ Ê UÉÌé
// POST : return value is n!
unsigned int fac ( unsigned int n)
{
if ( n <= 1) return 1;
return n * fac (n -1); // n > 1
}
·ÉÍÂÉíµËÇÚÉãÉIP¥ÎµÍÂÉÒÒÛÜÊØ
fac(n-1)
ÛÜÒ·Ð
£³Ä±³£¦ D¥±´©¯»
ÊÔ
fac
é
S¼ÛñËÂÇ ÍÂÉ³Ù³Ö«ÍÂÒÛ UÉÔ×ÖµØµÙËÂÛÜÊØµÒ³í æ¹É¿ÇÚÐXUÉàËÂÇµÉ ÒÂÐÞßÉàÛÜÒÒîÖÚÉ¿ÐÒ æ·ÛñËÂÇ áÜÊôÊÌÎÚÒZ^ " É³Ù ËÂÛÜÊØ \ é é \ ` ÛñËÈÛÜÒòÉ³ÐÒîì®ËÂÊæòÍÂÛñËÂÉÓèµÊ¦æ ØóÔ×ÖµØµÙËÂÛÜÊØ Ù¬ÐáÜáñÒ¹æ·ÇµÊÒÉãÉIÌU ÐáñÖ ÐÑËÂÛÜÊØèµÊ É³ÒòØµÊÌËÈËÂÉÍÂÞßÛñØ ÐÑËÂÉÌé
·ÉÍÉãÛÜÒ¹ËÂÇµÉÏÒÇµÊÌÍËÂÉ³ÒîËòæÈÐ³ì ÊÔ¡ÙÍÂÉ¬ÐÑËÂÛÜØµå ÐçØóÛÜØ Ú
Y ØÚÛ²ËÂÉ ÍÂÉ³Ù³Ö«ÍÂÒîÛÜÊØ ÕèµÉ Ú
Y ØµÉÏËÇÚÉÏÔ×ÖµØµÙËÂÛÜÊØ
Infinite recursion.
T \ void f ()
{
f ();
}
æ ²Û ËÂÇØµÊÝÐÑÍÂåÖµÞßÉ³Ø ËÂÒ¿ÐØµè ÉIUÌÐáñÖ ÐÑËÂÉ ËÇÚÉ ÉIP¥ÎµÍÂÉÒÒÛÜÊØ
é Æ¹ÇµÉ ÍÂÉ³ÐÒÊØ Ô×ÊÌÍ ØÚÊØ»ËÂÉÍÂÞßÛÜØÚÐçËÛÜÊØ
f()
ÛÜÒ®Ù³áÜÉ³ÐçÍ ËÂÇµÉ É UÐáÜÖÚÐçËÂÛñÊØ¼ÊÔ
Ù³ÊØµÒÛÜÒËÂÒ®ÊÔâÐØ É UÐáÜÖÚÐçËÂÛÜÊØ ÊÔ
æ ÇµÛÜÙäÇ ÙÊØÚÒîÛÜÒîËÂÒ®ÊÔãÐØ
f()
É UÐáÜÖÚÐçËÂÛñÊØ ÊÔ
æ·ÇµÛÜÙäÇ¾éÄé³éìÊf()
ÖóåÉËÈËÂÇµÉÓÎ ÛñÙËÂÖ«ÍÂÉÌé
f()
Û ]ÉãÔ×ÊÌÍòáÜÊôÊÌÎÚÒ³í¥ËÇÚÉÏÔ×ÖµØµÙËÂÛÜÊØ èÚIÉ Y ØµÛñËÛÜÊØ®Ç ÐçÒÈËÂÊÞ®Ð ]ÉÓÒÖ«ÍÂÉÓËÂÇÚÐçË¹ÎÚÍÊåÌÍÂÉ³ÒÒÈËÂÊÑæÈÐÑÍÂèµÒ¹ËÉÍÂÞßÛ ØÚÐçËÂÛÜÊØ ÛÜÒÈÞ®ÐèµÉÏÛñØÉ UÉÍîìóÔ×ÖµØµÙËÂÛÜÊØóÙ³ÐáÜáé [ ÊÌÍ·ËÂÇµÉÏÔ×ÖµØÚÙÄËÂÛÜÊØ
Ð _Ê UÉÌí«ËÂÇµÛÜÒÈÛÜÒ¹ËÂÇµÉãÙ³ÐÒÉ É³ÐÙäÇ
fac
ËÂÛÜÞßÉ
ÛÜÒÙ¬ÐáÜáÜÉè®ÍÉ³Ù³Ö«ÍÂÒVÛ UÉáñìíôËÂÇµ
É UÐáÜÖµÉ ÊÔ ËÂÇµÉ Ù¬ÐáÜá ÐÑÍÂåÖµÞßÉ³Ø Ë _ É³Ù³ÊÞßÉ³ÒÒÞ®ÐáÜáÜÉÄÍ¬íôÐçØÚèßæ·ÇÚÉØ
fac
ËÂÇµL
É UÐáÜÖµÉÀÍÂÉ¬ÐÙäÇµÉ³Ò í ØµÊ ÞßÊÌÍÉ ÍÂÉ³Ù³Ö«ÍÂÒîVÛ UÉ®Ù¬ÐáÜáÜÒëÐçÍÂÉßÎ ÉÍÂÔ×ÊÌÍÞßÉ³è ¿æ¹É®ÒÂÐ³ìÝËÂÇÚÐçËàËÇÚÉÀÍÉ³Ù³Ö«ÍÂÒÛÜÊØ
1
N_ ÊÌËËÊÞßÒïÊÖµË «é
3.2.2 The call stack
É!Ë ÒßËÍîì¼ËÊ ÖµØµèµÉÍÂÒîËäÐçØÚè æ ÇÚÐçË®ÉIP«ÐÙËÂáñì ÇÚÐçÎµÎ ÉØÚÒ èµÖ«ÍÂÛÜØµå ËÂÇµÉ½É UÐáÜÖÚÐçËÛÜÊØ ÊÔ
í ÒÂÐ³ìé
Æ¹ÇÚÉ Ô×ÊÌÍÂÞ®Ðá¿ÐçÍÂåÖµÞßÉ³Ø¯Ë
ÛÜÒ ÛÜØµÛñËÂÛ×ÐáñÛ³É³è æ Û²ËÂÇ í¿ÐØµè ÒÛÜØµÙ³É ËÂÇµÛÜÒ ÛÜÒ½åÌÍÉ¬ÐçËÂÉÍ½fac(3)
ËÂÇÚÐØ íàËÂÇµÉ
n
ÒîËäÐÑËÂÉ³ÞßÉ³Ø¯Ë
ÛÜÒ ÉIP¥É³Ù³Ö«ËÂÉ³è 3 ØÚÉGP¥Ë¬éâÆÈÇµÛÜÒYµÍÂÒîËÓÉIUÌÐáñÖ ÐÑËÂÉ³Ò ËÂÇµÉ ÉIP¥ÎµÍÂ1É³ÒîÒÛÜÊØ
return n * fac(n-1);
ÐçØÚè ÛÜØ®ÎÐÑÍËÂÛÜÙ³Öµá×ÐÑÍ ËÂÇµÉÏÍÛÜåÇ Ë¹ÊÌÎ ÉÄÍäÐØµè
é " ÛÜØµÙ³É
ÇÚÐÒUÐáÜÖµÉ í¥ËÂÇµÉ
n * fac(n-1)
fac(n-1)
n-1
2
Ô×ÊÌÍÂÞ®Ðá¤ÐÑÍÂåÖµÞßÉ³Ø Ë ÛñÒÈËÂÇµÉÍÉ³Ô×ÊÌÍÂÉâÛñØÚÛ²ËÂÛ×ÐáÜÛ ³É³èÀæ Û²ËÂÇ é
Ö«Ë·æÈÐÛñË Õæ ÇÚÐçnË·ÛÜ
Ò ËÇÚÉ ÀÔ×ÊÌÍÂÞ®Ðá¾ÐçÍÂåÖµÞßÉ³Ø¯Ë 2 Ö«ËÂÊÞ®ÐÑËÂÛÜÙ ÒîËÂÊÌÍÂÐåÉâèµÖ«ÍäÐçËÂÛñÊØóÛÜÞÀÎÚáÜÛÜÉ³Ò ËÇ ÐÑË
É¬ÐÙäÇ Ô×ÖµØµÙËÂÛÜÊØÙ¬ÐáÜá¤ÇÚÐÒ·Û²ËÂ$
Ò îÊÑæ·Ø ÔÜÍÂÉ³ÒÇ ÛÜØµÒîËÂÐØµÙ³ÉãÊÔ¡ËÂÇµÉãÔ×ÊÌÍÂÞ®Ðçá¡ÐÑÍÂåÖµÞßÉ³Ø Ë¬í¥ÐØµèËÂÇµÉâáÜÛñÔ×ÉËÂÛÜÞßÉ
í æ¹É ËÇÚÉÄÍÂÉ³Ô×ÊÌÍÂÉßåÉË
ÊÔ ËÂÇµÛÜÒëÛñØÚÒËäÐØµÙ³É ÛñÒâËÂÇµÉ ÍÂÉ³ÒÎ É³ÙËÂÛ UÉÀÔ×ÖµØÚÙÄËÂÛÜÊØ Ù¬ÐáÜá»é êØ É UÐáÜÖÚÐçËÛÜØµå
ÐÀØµÉæ ÛñØÚÒËäÐØµÙ³ÉàÊÔ ËÂÇµÉàÔ×ÊÌÍÂÞ®Ðá¡ÐçÍÂåÖµÞßÉ³Ø¯Ë íÊØ ËÂÊÌÎ ÊÔ ËÇÚÉëÎµÍÂIÉ U¥ÛÜÊf(n-1)
ÖµÒ ÛÜØµÒîËäÐçØÚÙÉëÔÜÍÂÊÞ ËÂÇµÉàÙ¬Ðáñá
^°ËÂÇÚÐçËïÇÚÐÒ ØµÊÌË ìÉËïËÉÍÂÞßÛÜØÚÐçËÂÉè `´é Ö«Ë næ ÇµÛÜÙäÇßÛÜØµÒîËÂÐØµÙ³É ÊÔ èÚÊëæ¹É·ÖÚÒîÉ ÛÜØßËÂÇµÉ É UÐáÜÖÚÐçËÂÛÜÊØ
f(3)
ÊÔ
ÖÚÛ²ËÂÉØÚÐçËÖµÍÂÐáÜáñìí ÛñËÏæ ÛñáÜT
á _ É¿ËÂÇµÉØµÉæ ÊØµÉÌí¡ËÇÚÉ¿ÊØµnÉËÂÇÚÐç
Ë N_ ÉáÜÊØµåÒ ËÊ ËÂÇµÉ Ù³ÐáÜá
f(n-1)
éÃÆÈÇµÛÜÒ ÍÖÚáñÉ ÛÜÒïÛÜØ®áÜÛÜØµÉ·æ·ÛñËÂÇßËÂÇµÉÓåÉ³ØµÉÍäÐáÒîÙ³ÊÌÎ É ÍÂÖµáÜÉ³ÒïÔÜÍÂÊÞ " É³ÙÄËÂÛÜÊØ \ é é ËÇÚÉ ÍÉ³áÜÉ UÐØ Ë
f(n-1)
èµÉ³Ù³á×ÐÑÍäÐçËÂÛÜÊØ ÛÜÒòÐáñæÈÐ³ì¥ÒïËÂÇµÉÏÞßÊÒîË¹ÍÂÉ³Ù³ÉØ Ë ÊØµÉÓËÂÇÚÐçËÈÛÜÒÈÒîËÂÛÜáÜá UôÛÜÒVÛ _ÚáÜÉé
ÆÈÇµÉ¿ËÂÉ³ÙäÇµØÚÛñÙ¬Ðá ÍÂÉ¬ÐáñÛ ¬ÐçËÂÛñÊØ½ÊÔÃËÂÇµÛÜÒãÛñÒ UÉÍîì ÒÛñÞÀÎ áñÉÌé UÉÄÍìôËÂÛñÞßÉÐ Ô×ÖµØÚÙÄËÂÛÜÊØ ÛñÒãÙ¬ÐçáÜáÜÉ³è¤í ËÂÇµÉ
Ù¬ÐáñáòÐÑÍÂåÖµÞßÉ³Ø Ë¿ÛÜÒÉ UÌÐçáÜÖÚÐçËÂÉ³è¤í ÐçØÚè ËÂÇµÉ ÍÂÉ³ÒîÖÚá²ËÂÛÜØµå UÐáÜÖµÉ ÛÜÒ¿ÎÚÖ«ËÀÊØ ËÂÇµÉ
æ ÇµÛÜÙäÇ ÛÜÒ
Ä
±
¯
©
°

µ
©
±
ÒÛÜÞÀÎÚáñìßÐàÍÉ³åÛÜÊØóÛÜØ ËÂÇµÉãÙÊÞÀÎ Ö«ËÂÉÄ!Í Ò¹ÞßÉ³ÞßÊÌÍìé
Û ]É ÐÀÒîËäÐÙ ] ÊÔ Î ÐçÎ ÉÍÂÒ·ÊØ ìÊÖ«ÍÓèµÉ³Ò ]íËÂÇµÉàÙ¬ÐáÜá ÒËäÐÙ ] ÇÚÐÒ ËÂÇµÉëÎµÍÂÊÌÎ ÉÍîË ìËÂÇÚÐçË ËÂÇµÉàÊ _ ÉÙË
ËÂÇÚÐçËòÙ¬ÐÞßÉÓá×ÐÒË·ÛÜ
Ò îÊØËÂÊÌÎ «é òÎ ÊØËÉÍÂÞßÛÜØÚÐçËÂÛñÊØ ÊÔ ÐÔ×ÖµØµÙËÂÛÜÊØ Ù¬ÐáÜá»í¥ËÂÇµÉãËÂÊÌÎ Ê _ É³ÙËòÛÜÒ¹ËäÐ ]É³Ø
Ê óËÂÇµÉ ÒËäÐÙ ]®ÐåÐÛñØ¾
é S¼ÇµÉ³ØµÉ UÉÍòÐâÔ×ÖµØµÙËÂÛñÊØ Ù³ÐáÜáÐçÙ³Ù³É³ÒÒîÉ³ÒòÊÌÍ Ù´ÇÚÐØµåÉ³ÒïÛ²ËÂÒ Ô×ÊÌÍÂÞ®ÐáÐçÍÂåÖµÞßÉ³Ø¯Ë¬í Û²Ë
èµÊôÉ³ÒÈÒÊ _ ì ÐÙ³Ù³É³ÒîÒÛÜØµå ÊÌÍòÙäÇ ÐçØÚåÛñØÚå ËÂÇµÉâÙÊÌÍÍÂÉ³ÒîÎÊØÚèµÛÜØµå¿Ê _ É³ÙÄËòÊØ ËÂÊÌÎ ÊÔ¡ËÇÚÉãÒîËäÐÙ ]é
ÆÈÇµÛÜÒÈÇÚÐÒ ÐçáÜá ËÂÇµÉÏÎµÍÂÊÌÎ ÉÄÍËÂÛÜÉ³Ò¹æ¹ÉãæÈÐçØ Ë ÃÉ UÉÍìÔ×ÖµØÚÙÄËÂÛÜÊØóÙ¬ÐçáÜá æ¹ÊÌÍ ]¥ÒÈæ ÛñËÂÇ ÛñËÂÒÈÊÑæ·ØóÛÜØµÒîËäÐçØÚÙÉ
ÊÔ«ËÂÇµÉïÔ×ÊÌÍÂÞ®ÐáôÐçÍÂåÖµÞßÉ³Ø Ë Ìæ ÇµÉ³Ø ÛñË Ù¬ÐçáÜáÜÒ ÐØµÊÌËÂÇµÉÍ Ô×ÖµØµÙËÂÛñÊØ ^»ÊÌÍ ËÂÇµÉïÔ×ÖÚØµÙËÛÜÊØ ÛñËÂÒÉ³áñÔµÍÉ³Ù³Ö«ÍÂÒVÛ UÉáñaì `Äí
ËÂÇµÛÜÒïÛÜØµÒîËÂÐØµÙ³
É _ É³Ù³ÊÞßÉ³Ò ËÉ³ÞÀÎ ÊÌÍäÐçÍÛÜáñì¿ÇµÛÜèµèÚÉØ¾íôÖÚØ¯ËÂÛÜáËÂÇµÉÓØµÉ³ÒîËÂÉèÙ¬ÐáñáÇÚÐÒïËÂÉÍÂÞßÛÜØÚÐçËÉ³è¤é ïËÈËÇ ÐÑË
Î ÊÛÜØ¯Ë¬í ËÇÚÉÓÛÜØµÒîËÂÐØµÙ³É ÍÂÉ¬ÐçÎµÎ É³ÐçÍÂÒ¹ÊØ ËÊÌÎóÊÔ¤ËÂÇµÉ ÒîËäÐÙ ] ÐØµèÐçáÜáÜÊ¦æ ÒËÂÇµÉ ÊÌÍÂÛñåÛÜØÚÐáÔ×ÖµØµÙËÂÛÜÊØ Ù¬ÐáñáËÂÊ
æ¹ÊÌÍ ]®æ Û²ËÂÇÛñËòÐåÐçÛÜØ¤é
4
"5
,$
A+*
8*
%*
%1(76>"
%*( A#*K7
N(
* ,
8!*
%2
N(76>"
.
*J
&( Q
Q*
A(.
27P
7
P
%
M/
%7
*
\
Æ Ð _ÚáÜÉ ÒÇµÊÑæ·ÒãÇµÊ¦æ Ë ÇÚÛñÒÏáñÊôÊ ]¥Ò áñÛV]É Ô×ÊÌÍ
í Ð ÒîÒÖµÞßÛÜØµå®ËÇ ÐÑË ÂË ÇµÉ¿ÍÂÛñåÇ ËÏÊÌÎ ÉÍäÐçØÚè½ÊÔÕËÂÇµÉ
f(3)
Þ¿ÖÚá²ËÂÛñÎÚáÜÛÜÙ¬ÐÑËÂÛÜÊØ ÊÌÎ ÉÍÂÐçËÂÊÌÍ ñÛ Ò ÐáñæÈÐ³ì¥Ò¡É UÐáÜÖÚÐçËÂÉ³èZYµÍÂÒË¬é µÖ ËîËÂÛÜØµå ÐØàÊ _ É³Ù Ë ÊØëËÂÇµÉ ÒËäÐÙ ] ÎÚÖµÒÇµÉ³Ò ÛñË¬íµÐçØÚè ËäÐ ]¥ÛñØÚåàËÂÇµÉÓËÂÊÌÎ Ê _ É³ÙËòÊÔ Î ÊÌÎÚÒ ÛñË¬é
Ù³ÐáÜá ÒîËäÐçÙ ] ^ _ ÊÌËîËÂÊÞ
n: 3
n: 3
n: 3
n: 3
n: 3
n: 3
n: 3
n: 3
n: 3
n: 2
n: 2
n: 2
n: 2
n: 2
←→
n: 1
Table 5:
ËÂÊÌÎ `
É UÐáÜÖÚÐçËÛÜÊØ ÒîÉXWôÖµÉ³ØµÙ³É
fac(3)
n * fac(n-1)
n * fac(2)
n * (n * fac(n-1))
n * (n * fac(1))
n * (n * 1)
n * (2 * 1)
n * 2
3 * 2
6
ÐÙËÂÛñÊØ
ÎÚÖµÒÇ
3
ÎÚÖµÒÇ
2
ÎÚÖµÒÇ
1
ÎÊÌÎ
Î ÊÌÎ
Î ÊÌÎ
±´©¯»ë³î©µ± ½©F
¸ » D¥EDô¬ F £¬»Á © XDô©¯¡©¯
£³¬ ¢ ±¬ D¥LDô©¯ Â¶ n »®©¯¸ ©Jô¿º ³¢
Â¶ fac(3) º
É³Ù¬ÐÖµÒÉ ÊÔëËÇÚÉ½Ù¬ÐáÜáãÒîËäÐÙ ]íãÛÜØ YÚØµÛñËÂÉ ÍÂÉ³ÙÖµÍÒÛÜÊØµÒóèµÊ ØµÊÌË ÊØµáñì¼Ù³ÊØµÒÖµÞßÉ½ËÂÛñÞßÉ Ú_ Ö«Ë ÐáñÒÊ
ÞßÉ³ÞßÊÌÍìé ·ØµáÜÛV]É ÛÜØ YÚØµÛñËÂÉëáñÊôÊÌÎÚÒ³íËÂÇµÉì ÖµÒÖÚÐáÜáñìóáÜÉ¬Ðè ËÂÊÐÀÎµÍÂÊåÌÍäÐçÞ Ð _ ÊÌÍËÂÛÜÊØ½ÐÒÓÒÊ ÊÝ
Ø ÐÒÓËÂÇµÉ
ÞßÉ³ÞßÊÌÍì ÍÂÉ³ÒÉÍ UÉ³è Ô×ÊÌÍ·ËÂÇµÉÏÙ¬ÐáÜá¤ÒîËÂÐÙ ] ÛñÒòÔ×ÖµáÜáé
3.2.3 Basic practice
ÉË¹ÖµÒïÙ³ÊØµÒÛÜèµÉÍ Ë æ¹Ê ÞßÊÌÍÂÉ·ÒîÛÜÞÀÎÚáÜÉÈÍÉ³Ù³Ö«ÍÂÒÛVUÉÓÔ×ÖµØµÙËÂÛÜÊØµÒËÇ ÐÑË¹ÐçÍÂÉ ÒÊÞßÉÄæ ÇÚÐçË ÞßÊÌÍÉ·ÛÜØ¯ËÂÉÍÂÉ³ÒËÂÛÜØµå
ËÂÇÚÐØ
éãÆ¹ÇÚÉÄì ÒÇµÊÑæ ËÂÇÚÐçË ÍÉ³Ù³Ö«ÍÂÒÛVUÉ Ô×ÖµØµÙËÂÛÜÊØµÒÏÐçÍÂÉëÎ ÐçÍîËÂÛÜÙ³Öµá×ÐçÍáñìóÐçÞßÉ³ØÚÐ _ÚáÜÉãËÂÊ Ù³ÊÌÍÍÉ³ÙËÂØµÉ³ÒÒ
fac
ÎµÍÂÊôÊÔ×Ò®
ÊÔÏËÂÇµÉ³ÛñÍßÎ ÊÒîËÙ³ÊØµèÚÛ²ËÂÛÜÊØµÒ³íÈÐØµè ËÂÇµÛÜÒßÞ®Ð ]É³ÒÀËÂÇµÉ³Þ ÐçËËÍÂÐÙËÂVÛ UÉé Ø ËÂÇµÉ ÊÌËÂÇµÉÍ ÇÚÐØµè¾í
æ¹É ÐáÜÒÊ ÒÉ³É ËÂÇÚÐçË ÛñË¿ÛÜÒ É¬ÐÒì ËÊ æòÍÂÛñËÂÉßÛÜØµØµÊôÙÉ³Ø Ë áÜÊôÊ ]ôÛÜØµåóÍÂÉ³Ù³Ö«ÍÂÒîVÛ UÉ Ô×ÖµØµÙËÂÛñÊØÚÒàËÂÇÚÐçËÐçÍÉUÉÄÍì
ÛÜØµÉ Ù³ÛÜÉ³Ø¯ËòËÂÊ¿É UÐáÜÖÚÐçËÂÉé
ÊØÚÒîÛÜèµÉÍãËÇÚÉ ÎµÍÂÊ _ÚáÜÉ³Þ ÊÔ YÚØµèµÛÜØµå ËÂÇµÉåÌÍÂÉ¬ÐçËÂÉÒîËâÙÊÞßÞßÊØ èµÛVUôÛÜÒÊÌÍ
ï
Greatest common divisor.
åÙ³è
ÊÔÚË æ¹ÊãØÚÐçËÂÖ«ÍäÐçá«ØôÖµÞb_ÉÍÂÒ
é Æ¹ÇµÛÜÒ ÛÜÒ èÚÉIY ØµÉ³èÀÐÒ ËÂÇµÉòá×ÐÑÍÂåÉ³ÒîËÃØÚÐçËÂÖ«ÍäÐáôØôÖµÞb_ÉÍ ËÂÇÚÐçË
a, b
èµÛVU¥Ûñ(a,
èÚÉÒ b)_ ÊÌËÂÇ
ÐØµè
æ ÛñËÇÚÊÖ«ËÓÍÂÉ³Þ®
ÐçÛÜØµèÚÉÄÍ¬éßêØÝÎ ÐçÍËÂÛñÙ³Öµá×ÐçÍ¬í¾åÙ³è
åÙè
Ô×ÊÌÍ
a
b
(n, 0) =
(0, n) = n
µ áÜÉËÈÖµÒ·ÐáñÒÊ èµÉ YÚØµÉãåÙè
é
n>0
(0, 0) := 0
ÆÈÇµÉ
Y ØµèµÒÈåÙ³è
í _ ÐÒîÉ³èóÊØËÂÇµÉÏÔ×ÊáÜáÜÊ¦æ ÛÜØµå
È¡±¬ FµÄ©¼©¯Á«£¬»º¥¨
(a, b)
Lemma 1 ×¶ b > 0 ¹º
åÙ³è
(a, b) =
åÙ³è
(b, a
ÞßÊôè
b).
T \ ÉË
Proof.
_ÉâÐ èµÛVU¥ÛÜÒîÊÌÍòÊÔ
k
èµÛVU
a = (a
b)b + a
ÞßÊ è
b
é [ ÍÊÞ
b
ÛñËÈÔ×ÊáÜáÜÊ¦æ ÒïËÂÇÚÐçË
ÞßÊôè
" ÛÜØµÙ³É
b a
a
b
èµÛVU
= (a
.
b) +
k
k
k
a
èµÛVU
a
ÞßÊôè
b
ÐØµè
b/k
b
ÐçÍÂÉÏÛÜØ ËÉ³åÉÍÂÒ³í¥æ¹ÉãåÉË
a
k
⇔
k
.
êØ æ¹ÊÌÍÂèµÒ³í ÛÜÔ ÛÜÒÓÐßèµÛVUôÛÜÒÊÌÍ ÊÔ íËÂÇµÉ³Ø
èµÛVUôÛÜèµÉ³Ò ÛÜÔÐçØÚè Ê Øµáñì ÜÛ Ô èµÛVU¥ÛÜèµÉ³Ò ÞßÊôè éâÆ¹ÇµÛÜÒ
b
k
Þ É¬ÐØµÒ³íÚËÂÇµÉ èÚkÛ ¥U ÛÜÒÊÌÍÒ ÊÔ
ß
ÐçÍÂÉàÉGPµkÐÙËáñìóËÂÇµÉàaèµÛVU¥ÛñÒÊÌÍÂÒ Ê Ô
ÞßÊ è aéÓÈ
Æ ÇµÛÜÒ bÎµÍÂÊUÉ³Ò
a ©F b
b ©F a
b
ËÂÇÚÐçËÈåÙ³è
Ð Øµè åÙ³è
ÞßÊôè
ÐçÍÉâÉXW Ö Ðçáé
(a, b)
(b, a
b)
·ÉÍÂÉÓÛÜÒïËÂÇµÉ Ù³ÊÌÍîÍÂÉ³ÒîÎ ÊØµèµÜÛ Øµå Ô×ÖµØÚÙÄËÂÛÜÊØ Ô×ÊÌÍòÙÊÞÀÎ Ö«ËÂÛñØÚåãËÂÇµÉÏåÌÍÉ¬ÐçËÂÉ³ÒËòÙÊÞßÞßÊØÀèµÛVU¥ÛÜÒîÊÌÍ
Ê Ô¡Ë æ¹Ê
UÐáÜÖµÉ³Ò³íµÐÙÙ³ÊÌÍÂèµÛÜØµåËÂÊ ËÂÇµÉ ÖµÙ³áÜÛÜèµÉ¬ÐçØÐáñåÊÌÍÂÛñËÂÇµÞ é
unsigned int
// POST : return value is the greatest common divisor of a and b
unsigned int gcd ( unsigned int a , unsigned int b)
{
if ( b == 0) return a ;
return gcd (b , a % b ); // b != 0
}
Æ ÇÚÉ ÖÚÙáÜÛÜèµÉ¬ÐØ Ð áÜåÊÌÍÂÛ²ËÂÇµÞ ÛÜÒRUÉÍì ÔªÐÒË¬éS ÉãÙ³ÐØÉ¬ÐçÒÛÜáñìßÙ³ÐáÜá ñÛ Ë¹Ô×ÊÌÍ·ÐØ ì
¹
UÐáÜÖµÉ³Ò
unsigned int
ÊØóÊÖ«ÍÈÎ áÜÐçËÂÔ×ÊÌÍÂÞ ¯í æ ÛñËÂÇµÊÖ«Ë¹ØµÊÌËÂÛÜÙ³ÛñØÚå¿ÐØ¯ìèµÉ³áÜÐ¬ì®ÛñØËÂÇµÉãÉIUÌÐáñÖ ÐÑÂË ÛÜÊØ¤é
[ ÊÌÍãÍÂÉ³Ù³Ö«ÍÂÒîÛVUÉÀÔ×ÖµØµÙËÂÛÜÊØµÒ³í¾ÛñËãÛÜÒãÊÔÜËÉ³Ø UÉÍì É¬ÐÒîì ËÂÊ ÎµÍÂÊUÉ ËÂÇÚÐçË
ËÂÇµÉßÎ ÊÒîËÂÙÊØÚèµÛñËÛÜÊØ ÛÜÒ Ù³ÊÌÍÍÂÉÙË¬í_ôì ÖµÒÛÜØµå ËÂÇµÉ®ÖÚØµèµÉÍÂá²ì¥ÛÜØµå Þ®ÐçËÂÇµÉ³Þ®ÐÑËÂÛÜÙ¬Ðá èµÉ YÚØµÛñËÂÛÜÊØ èµÛñÍÉ³ÙËÂáñì
^áÜÛ ]É
Ô×ÊÌÍ
`Äí ÊÌÍ _ ì ÖµÒÛÜØµå ÒÊÞßÉÔªÐÙËÂÒàËÂÇÚÐçËàÔ×ÊáñáÜÊÑæ ÔÜÍÂÊÞ ËÇÚÉßÞ®ÐÑËÂÇµÉ³Þ®ÐçËÂÛÜÙ³Ðá èµÉ YÚØµÛñËÂÛÜÊØ
n!
fac
^áÜÛ ]É É³ÞßÞ®Ð Ô×ÊÌÍ
`Äé
ÆÈÇµÉãÙ³ÊÌÍÍÂÉ³ÙÄËÂØµÉ³ÒÒ gcd
ÎµÍÂÊôÊÔ¡ÞÖµÒîËòÛÜØ UÊáVUÉëÐàËÂÉÍÂÞßÛñØ ÐÑËÂÛÜÊØßÎµÍÂÊôÊÔºíÚÒÊÀáÜÉÄË! ÒÈÒîËäÐçÍîË·æ Û²ËÂÇ ËÂÇµÛÜÒ ÐØ¯ì
ËÉÍÂÞßÛÜØÚÐçËÂÉÒ³íôÒÛñØÚÙÉÏËÂÇµÉ
UÌÐçáÜÖµÉ ÊÔ¡ËÇÚÉÓÒÉÙ³ÊØµè ÐçÍÂåÖµÞßÉ³Ø¯Ë¹ÛÜÒ_ÊÖÚØµèµÉ³è ÔÜÍÂÊÞ _ É³áÜÊ¦æ
Ù¬ÐáñáËÂÊ
_ôì
ÐØµgcd
èóåÉËÂÒòÒÞ®ÐáÜáñÉÍïÛÜØ É UÉÍì ÍÂÉ³Ù³Ö«ÍÂÒÛ UbÉâÙ¬Ðáñá ^»æ¹ÉãÇÚÐXUÉ ÞßÊôè
`Äé
0
a
b<b
VÛ UÉØóËÂÇµÛÜÒ³íµËÂÇµÉâÙ³ÊÌÍÍÉ³ÙËÂØµÉ³ÒÒ ÊÔ ËÇÚÉÏÎ ÊÒîËÂÙÊØÚèµÛñËÛÜÊØÔ×ÊáÜáñÊÑæ·ÒòÔÜÍÂÊÞ ÉÞßÞ®Ð _ôìÛÜØµèÚÖµÙËÛÜÊØ
ÊØ é [ ÊÌÍ
íµËÂÇµÛÜÒ·ÛñÒ·Ù³áÜÉ³ÐçÍ¬é [ ÊÌÍ
íÚæ¹ÉëÛÜØµèµÖÚÙÄËÂVÛ UÉ³áñì ÐçÒÒÖµÞßÉãËÂÇÚÐçËòËÂÇµÉâÎ ÊÒîËÙ³ÊØµèÚÛ²ËÂÛÜÊØ
=0
b>0
ÛÜÒÓÙ³bÊÌÍÍÉ³ÙËëÔ×bÊÌÍâ
ÐáÜá Ù³ÐáÜáÜÒ ËÂÊ
æ ÇµÉÍÉ¿
ËÂÇµÉ ÒÉ³ÙÊØÚè ÐÑÍÂåÖµÞßÉ³Ø ËÓÇÚÐ
Ò UÐáÜÖµÉ
é " ÛñØÚÙÉ
gcd
b <b
ÞßÊ è ÒÂÐçËÛÜÒ YÚÉ³Ò
íæ¹ÉàÞ®Ð³ìóÐçÒÒÖµÞßÉâËÂÇÚÐçË ËÇÚÉàÙ³ÐáÜá
Ù³ÊÌÍÍÉ³ÙËÂáñìÍÂÉËÂbÖ«ÍÂØµ=Ò
a
b
gcd(b, a % b)
åÙ³è
ÞßÊôè é bÖ«<
Ë _ bì ËÇÚÉÏáÜÉ³ÞßÞ®Ð íôåÙ³è
ÞßÊôè
åÙè
í¥ÒÊ¿ËÂÇµÉâÒîËäÐÑËÂÉ³ÞßÉ³Ø¯Ë
Correctness and termination.
(b, a
(b, a
b)
return gcd (b , a % b );
Ù³ÊÌÍÍÉ³ÙËÂáñì®ÍÂÉÄËÂÖ«ÍÂØµÒòåÙ³è
(a, b)
é
b) =
(a, b)
\ Æ ÇµÉ Ò ÉXW ÖÚÉØÚÙÉ
¹
ÊÔ ÛÜÒÕÊØÚÉ
0, 1, 1, 2, 3, 5, 8, 13, 21, . . .

µ
©

±
³
±
ï

½
¨
³

¦
£

ÊÔÓËÂÇµÉ ÞßÊÒË®ÔªÐÞßÊÖµÒÀÒÉXW ÚÖ ÉØÚÙÉ³ÒÛÜØ¼Þ®ÐçËÂÇµÉ³Þ®ÐÑËÂÛÜÙ³Ò³é [ ÊÌÍÂÞ®ÐçáÜáñìíËÂÇµÉ ÒÉXW ÖµÉ³ØµÙ³É ÜÛ ÒßèµÉ YÚØµÉ³è ÐÒ
Ô×ÊáÜáÜÊ¦æ Òé
Fibonacci numbers.
F0 := 0,
F1 := 1,
Fn := Fn−1 + Fn−2 ,
n > 1.
Æ¹ÇÚÛñÒÏÞßÉ³ÐØµÒ³í¤É UÉÍì½É³áÜÉ³ÞßÉØ ËÓÊÔËÂÇµÉ ÒÉXW ÖµÉ³ØµÙ³ÉÀÛÜÒ ËÂÇµÉ¿ÒÖµÞ ÊÔÃËÂÇµÉ¿Ë æ¹Ê®ÎÚÍÉ U¥ÛÜÊÖµÒÏÊØµÉ³Ò³é [ ÍÂÊÞ
ËÂÇµÛÜÒ èµÉ YÚØµÛñËÂÛñÊØ¾íæ¹É Ù¬ÐçØ ÛñÞßÞßÉ³èµÛ×ÐçËÂÉ³á²ìëæ·ÍÂÛ²ËÂÉ·èµÊ¦æ Ø ÐãÍÂÉ³ÙÖµÍÒÛVUÉ Ô×ÖµØµÙËÂÛñÊØ Ô×ÊÌÍ¹ÙÊÞÀÎ Ö«ËÂÛñØÚå
[ VÛ _ÊØ ÐçÙ³Ù³Û¤ØôÖµÞ_ ÉÍÂÒ³í«åÉÄËËÂÛÜØµå ËÂÉÍÂÞßÛÜØÚÐçËÛÜÊØ®ÐØµèóÙ³ÊÌÍÍÉ³ÙËÂØµÉ³ÒÒòÔ×ÊÌÍ·ÔÜÍÉ³ÉÌé
// POST : return value is the n - th Fibonacci number F_n
unsigned int fib ( unsigned int n)
{
if ( n == 0) return 0;
if ( n == 1) return 1;
return fib (n -1) + fib (n -2); // n > 1
}
ê Ô ìÊÖ æòÍÂÛñËÂÉ·ÐâÎÚÍÊåÌÍäÐÞ ËÊ¿ÙÊÞÀÎ Ö«ËÂÉÈËÂÇµÉ [ ÛV_ÊØ ÐçÙ³Ù³ÛØ ÖÚÞ_ ÉÍ
ÖµÒÛñØÚåëËÇÚÛñÒ Ô×ÖµØÚÙÄËÂÛÜÊØ¤í ìÊÖ
Fn
æ ÛñáÜá«ØµÊÌËÂÛÜÙ³É¹ËÂÇÚÐçËÕÒÊÞßÉæ ÇµÉÍÉ_ ÉÄË æ¹ÉÉ³Ø
ÐØµè
íËÂÇµÉòÎµÍÂÊåÌÍÂÐÞ _É³Ù³ÊÞßÉ³ÒUÉÍìÒáÜÊ¦æëé
= 50
ÊÖÉ UÉØ ØÚÊÌËÛÜÙ³ÉÏÇÚÊ¦æ Þ¿ÖÚÙäÇóÒîáÜÊÑæ¹ÉÄÍ·ÛñË n_ =
ÉÙ³Ê30
ÞßÉ³Òïæ ÇµnÉ³Ø
ìÊÖÛÜØµÙÍÂÉ¬ÐÒîÉ
_ ì ÖµÒîË é
n
1
ÆÈÇµÉ ÍÂÉ¬ÐÒÊØ ÛñÒ¿ËÂÇÚÐçË¿ËÂÇµÉÞ®ÐÑËÂÇµÉ³Þ®ÐçËÂÛÜÙ³ÐáÕèµÉ YÚØµÛñËÂÛñÊØ ÊÔ
èµÊ É³Ò ØµÊÌËáÜÉ¬ÐèËÊÝ
ÐØ É Ù³ÛÜÉ³Ø¯Ë
Fn
ÐáÜåÊÌÍÛñËÂÇµÞ í¬ÒÛÜØµÙ³É¹Ðáñá UÐáÜÖµÉ³Ò
íÐçÍÉÍÂÉÎ É³ÐçËÂÉ³èµáñìãÙ³ÊÞÀÎÚÖ«ËÂÉ³è¤íÒÊÞßÉÕÊÔ«ËÂÇµÉ³Þ ÉIPôËÍÂÉ³ÞßÉáñì
i , i < n−1
ÊÔÜËÂÉ³Ø¤é ÊÖ Ù³ÐØ Ô×ÊÌÍãIÉ PµÐçÞÀÎ FáñÉà
Ù´ÇµÉ³Ù ] ËÂÇÚÐçË ËÂÇµÉ¿Ù¬ÐáÜá ËÂÊ
ÙÊÞÀÎ Ö«ËÂÉÒ
Ðá²ÍÂÉ¬Ðè«ì Ë æ ÛÜÙÉ
fib(50)
F48
^ÊØµÙ³É¿èµÛñÍÂÉ³ÙËáñì ÛÜØ
í ÐçØÚè½ÊØµÙ³É ÛÜØµèµÛñÍÂÉÙËÂáñì ÔÜÍÊÞ
é
ÛÜÒÏÙ³ÊÞÀÎÚÖ«ËÂÉ³è½ËÇµÍÉ³É
ËÂÛÜÞßÉÒ³í
YaUÉÓËÂÛÜÞßfib(n-2)
É³Ò³í¥ÐØµè
É³ÛÜåÇ¯Ë¹ËÂÛÜÞßÉ³
Ò ^èµÊ ìÊÖ ÒÉ³ÉëÐàfib(n-1)
Î ÐçËËÂÉÄÍÂØ `Äé F47
F46
F45
3.2.4 Recursion versus iteration
[ ÍÂÊÞ ÐßÒîËÍÂÛñÙËÂáñìÔ×ÖÚØµÙËÛÜÊØÚÐá Î ÊÛÜØ¯Ë ÊÔU¥ÛÜÉæâíÍÂÉ³ÙÖµÍÒÛÜÊØ ÛÜÒ ÒÖ«Î ÉÍ QÚÖµÊÖµÒ³í ÒÛÜØµÙ³É ÛñË Ù¬ÐØ _É¿ÒîÛÜÞÖ
á×ÐçËÉ³èÀËÂÇ«ÍÂÊÖµåÇ®ÛñËÂÉÍäÐÑËÂÛÜÊØ ^ºÐçØÚè ÐëÙ¬ÐçáÜáÒîËÂÐÙ ]ßÉIP¥ÎÚáÜÛñÙ³ÛñËÂáñì¿Þ®ÐÛñØ ËäÐÛñØÚÉè _ôì ËÂÇµÉ·ÎµÍÂÊåÌÍäÐÞ æ¹É Ù³ÊÖµáÜè
ÒÛÜÞ¿Öµá×ÐçËÂÉòËÂÇµÉãÙ¬ÐçáÜá ÒîËäÐÙ ]®æ Û²ËÂÇóÐØÐçÍÍäÐ³ì `ÄéS ÉãèµÊØ ËÈÇÚÐXUÉÏËÇÚÉÏÞßÉ¬ÐØµÒ ËÂÊ¿ÎµÍÂÊ UÉÓËÂÇµÛÜÒ¹ÇµÉÍÂÉÌí _ÚÖ«Ë
æ¹É æÈÐØ¯Ë ËÂÊ ÒÇµÊÑæ ÛñËïÔ×ÊÌÍïËÂÇµÉ ÍÂÉ³ÙÖµÍÒVÛ UÉ Ô×ÖµØµÙËÂÛÜÊØµÒ ËÂÇÚÐçË æ¹É ÇÚXÐ UÉ ÒÉ³ÉØÛÜØßËÂÇµÉ ÎµÍÂÉ UôÛÜÊÖµÒïÒÉ³ÙËÂÛñÊØ¾é
ÆÈÇµÉ Ô×ÖµØµÙËÂÛÜÊØ
ÛÜÒ UÉÍì®É³ÐÒîìÀËÂÊëæòÍÂÛñËÂÉ·ÛñËÂÉÍäÐÑËÂVÛ UÉ³áñìíÒÛñØÚÙÉ ÛñËïÛÜÒ
é Æ¹ÇµÛÜÒ
gcd
î

´
³

F
³
£

¦
£

¥
D

ÞßÉ¬ÐØµÒ ËÂÇÚÐçË¿ËÂÇµÉÍÂÉÛñÒÊØµáñì ÊØµÉ ÍÂÉÙ³Ö«ÍÂÒVÛ UÉóÙ³ÐáÜáíÐØµèËÂÇÚÐçËÊØÚÉÐçÎµÎ É¬ÐçÍÒÀÐçË¿ËÂÇµÉ UÉÍîì É³Øµè ÊÔ
ËÂÇµÉëÔ×ÖµØµÙËÂÛÜÊØ _ Ê è«ìé Æ ÐÛñá ºÉ³ØµèÍÉ³Ù³Ö«ÍÂÒÛÜÊØ Ù¬ÐçØ _ÉëÍÂÉÎÚá×ÐÙ³Éè _ ì Ð ÒÛñÞÀÎ áñÉãáÜÊ ÊÌÎ ËÂÇÚÐçË Û²ËÂÉÍäÐçËVÛ UÉ³áñì
Ö«Î èÚÐçËÂÉ³Ò ËÇÚÉ Ô×ÊÌÍÞ®ÐáÐçÍåÖÚÞßÉØ ËÂÒ ÖµØ¯ËÂÛÜá ËÇÚÉ ËÂÉÄÍÂÞßÛÜØÚÐçËÂÛÜÊØÙ³ÊØµèµÛñËÂÛÜÊØßÛÜÒïÒÐçËÂÛÜÒ YÚÉ³è¤éÃêØ®ËÂÇµÉ Ù¬ÐÒÉÓÊÔ
íµËÇÚÛñÒÈÖ«Î èÚÐçËÉÏÙÊÌÍÍÂÉ³ÒîÎÊØÚèµÒÈËÂÊËÂÇµÉÓËÍäÐØµÒÔ×ÊÌÍÞ®ÐçËÂÛÜÊØ
ÞßÊôè é
gcd
(a, b) → (b, a
b)
// POST : return value is the greatest common divisor of a and b
unsigned int gcd2 ( unsigned int a , unsigned int b)
{
while ( b != 0) {
unsigned int a_prev = a;
T \ a = b;
b = a_prev % b ;
}
return a ;
}
Ê ÖßÒÉ³É ËÂÇÚÐçËæ¹É åÉËïáÜÊØµåÉÍ ÐØµè®áÜÉ³ÒÒÍÉ¬ÐèÚÐ _ÚáÜÉ·ÙÊôèµÉÌí«ÐØµèßËÂÇÚÐçËÕæ¹É ØµÉ³É³è ÐØßÉIP¥ËîÍäÐ UÌÐÑÍÂÛ×Ð _ÚáÜÉ¹ËÂÊ
ÍÂÉ³ÞßÉÞb_ ÉÄÍ·ËÂÇµÉâÎµÍÂÉ UôÛÜÊÖµÒUÐáÜÖµÉëÊÔ _ É³Ô×ÊÌÍÂÉëËÂÇµÉàÖ«Î èÚÐçËÂÉâÒîËÂÉÎ ÛÜØ ËÂÇµÉàÒîÎÚÛñÍÛñË ÊÔ " É³ÙÄËÂÛÜÊØ \ é é í
æ¹ÉÏÒÇµÊÖÚáñè ËÂÇµÉÍÂÉÔ×ÊÌÍÂÉâÖµÒÉÏËÂÇµÉãÊÌÍÂÛñåÛÜaØÚÐáÍÂÉ³ÙÖµÍÒÛVUÉâÔ×ÊÌÍÞÖµá×ÐçËÛÜÊØ¤é
Ö«ÍãÔ×ÖµØÚÙÄËÂÛÜÊØ
Ô×ÊÌÍÏÙ³ÊÞÀÎÚÖ«ËÂÛÜØµå [ VÛ _ÊØ ÐçÙ³Ù³Û ØôÖµ
Þ _ ÉÍÂÒ ÛñÒ ØÚÊÌË ËÂÐÛÜá É³Øµè ÍÂÉÙ³Ö«ÍÂÒÛVUÉÌí _ÚÖ«ËÓÛñË
fib
ÛÜÒ ÒîËÛÜáÜá É³ÐÒîìóËÂÊßæòÍÂÛñËÂÉëÛ²Ë ÛñËÂÉÍÂÐçËÂVÛ UÉ³á²ìé ÉÞßÉ³b
Þ _ÉÍ·ËÂÇÚÐçË
ÛÜÒ ËÂÇµÉàÒÖµÞ ÊÔ
ÐØµè
é
S É
Ù¬ÐØ ËÂÇµÉÍÉ³Ô×ÊÌÍÂÉëæ·ÍÛñËÂÉãÐ áÜÊôÊÌÎæ ÇµÊÒÉëÛñËÉÍäÐçËÂÛñÊØ Ù³ÊÞÀÎÚÖµËFÉ³nÒ
ÔÜÍÊÞ ËÂÇµÉâÎµÍÂFÉ n−1
UôÛÜÊÖµÒáñì Ù³FÊn−2
ÞÀÎÚÖµËÉ³è
UÐáÜÖµÉ³Ò
ÐçØÚè
ËÇ ÐÑËÈæ¹ÉÏÞ®ÐÛÜØ¯ËäÐÛÜØ®ÛÜØ ËÂÇµiÉ UÐçÍÂÛÜÐ _ÚáÜÉ³Ò FiÐØµè é
Fi−2
Fi−1
a
b
// POST : return value is the n - th Fibonacci number F_n
unsigned int fib2 ( unsigned int n )
{
if ( n == 0) return 0;
if ( n <= 2) return 1;
unsigned int a = 1; // F_1
unsigned int b = 1; // F_2
for ( unsigned int i = 3; i <= n ; ++ i ) {
unsigned int a_prev = a ; // F_ {i -2}
a = b;
// F_ {i -1}
b += a_prev ;
// F_ {i -1} += F_ {i -2} - > F_i
}
return b ;
}
åÐÛÜØ¤í ËÇÚÛñÒëØµÊØ »ÍÂÉ³Ù³Ö«ÍÂÒîÛVUÉLUÉÍÂÒÛñÊØ
ÛÜÒàÒÖ _ÚÒîËäÐØ¯ËÂÛ×ÐáÜá²ì½áñÊØÚåÉÄÍàÐØµè ÞßÊÌÍÂÉ èµÛ Ù³ÖµáñËâËÂÊ
ÖµØµèÚÉÄÍÂÒîËäÐØµè ËÂÇÚÐØ
í _ÚÖµËßËÇÚÛñÒÀËÂÛÜÞßÉfib2
ËÂÇµÉÍÉ ÛÜÒ Ð _ É³ØµÉ YµË
ÛñÒßÞÖµÙäÇ ÔªÐÒîËÂÉÍ³í·ÒÛÜØµÙ³É ÛñË
fib
fib2
Ù³ÊÞÀÎÚÖ«ËÂÉ³ÒëÉIUÉÍì ØôÖµÞ_ ÉÍ
é S ÇµÛÜáñÉ æ¹Éßæ¹ÊÖÚáñè åÌÍÊÑæ Êáñè ÛÜØ æÈÐÛñËÛÜØµå
,i
Ô×ÊÌÍÓËÂÇµÉ Ù¬ÐáÜá
ËÂÊ®ËÂÉFÍÂiÞß
ÛÜØÚÐçËn
ÉÌí µ©µ±¬º J å±ÛVU É³ÒÓÖµÒÏËÂÇµÉ¿ÐØµÒîæ¹ÉÍÏÛÜØ ØµÊ ËÛÜÞßÉÌéb^ ·ØµÔ×ÊÌÍËÂÖ
ØÚÐçËÂÉ³á²ìíµËÇÚÛñÒ·fib(50)
ÐØµÒîæ¹ÉÄÍ Þ®Ð³ì _ ÉâÛñØÚÙÊÌÍÍÂÉ³Ùfib2(50)
Ë³í ÒÛñØÚÙÉ
Ù³ÊÖµáÜèÉIP«Ù³ÉÉ³è ËÂÇµÉ UÐáÜÖµÉãÍäÐçØÚåÉãÊÔ ËÂÇµÉãË ì Î É
F50
é`
unsigned int
êØËÂÇµÛÜÒ¿Ù¬ÐÒÉ æ¹É æ¹ÊÖµáÜè ÎµÍÂÉ³Ô×ÉÍ
Ê UÉÍ
íÒîÛÜÞÀÎÚáñì ÒÛÜØµÙ³É
ÛÜÒàËÂÊôÊ½ÛÜØµÉ ÙÛÜÉ³Ø ËÔ×ÊÌÍ
fib2
fib
fib
ÎµÍäÐÙËÛÜÙ¬Ðá¡ÖÚÒîÉÌéâÆ¹ÇµÉ ÞßÊÌÍÂÉëÙ³ÊÞÀÎÚáÜÛÜÙ³ÐçËÂÉ³èóÔ×ÖµØµÙËÂÛñÊØ½èµÉ YÚØµÛñËÂÛÜÊØ ÊÔ
ÛÜÒÓÐÀÞßÊ èµÉÍäÐçËÂÉÏÎµÍÂÛÜÙ³ÉëËÂÊ
fib2
Î Ð³ì Ô×ÊÌÍòËÂÇµÉãÒÎ É³É³èµÖ«ÎóËÂÇÚÐçËÈæ¹ÉãåÉÄË¬é
3.2.5 Primitive recursion
ÊÖµåÇµáñì ÒÎ É¬Ð ]ôÛÜØµå«í¤Ð®Þ®ÐÑËÂÇµÉ³Þ®ÐçËÂÛÜÙ³Ðá¤Ô×ÖÚØµÙËÛÜÊØ½ÛñÒ ¢¤£¬°¨» D¥®£³Ä±³£¦ D¥ ÛÜÔÃÛñËÓÙ¬ÐØ _ É¿æòÍÂÛñËËÉ³Ø
ÐÒ¹Ð Ô×ÖÚØµÙËÛÜÊØ ÛÜØ ÒÖµÙäÇóÐãæÈÐ³ìßËÂÇÚÐçË ØµÉ³ÛñËÇÚÉÄÍ¹èÚÛ²ÍÂÉ³ÙËÂá²ìÀØµÊÌÍ¹ÛÜØµèµÛñÍÂÉ³ÙËáñì Ù³ÐáÜáÜÒ Û²ËÂÒÉ³áÜÔ æ·ÛñËÂÇ
f
Ù¬Ðáñá¡ÐÑÍÂåÖµÞßÉ³Ø ËÂÒïèµÉÎÉ³fØµèÚÛñØÚå¿ÊØ é [ ÊÌÍòIÉ P«ÐÞÀ
ÎÚáÜÉÌí
f
unsigned int f ( unsigned int n)
\
{
if ( n == 0) return 1;
return f (f(n -1) - 1);
}
ÛÜÒïØµÊÌËÈÐáÜáñÊÑæ¹É³è¤í¯ÒÛÜØµÙ³É ÍÂÉ³Ù³Ö«ÍÂÒîÛVUÉ³áñì®Ù¬ÐçáÜáÜÒ ÛñËÒÉ³áÜÔæ·ÛñËÂÇ ÐàÐçÍÂåÖµÞßÉ³Ø¯ËïèµÉÎ É³ØµèµÛÜØµåàÊÔ éÃÆ¹ÇÚÛñÒïèÚÊ É³Ò
ÞßÉ¬ÐØ ËÂÇÚÐçË ËÂÇµÉ ÖµfØÚèµÉÍáñì¥ÛÜØµåßÞ®ÐçËÇÚÉÞ®ÐçËÂÛÜÙ¬Ðçá Ô×ÖÚØµÙËÛÜÊØ½ÛñÒ ØÚÊÌË ÎµÍÂÛñÞßÛñËÂÛVUÉÏÍÂÉ³Ù³Ö«fÍÂÒÛ UÉÌí ÛñË ÖµÒîË
ÞßÉ¬ ÐØµÒÏËÂÇÚÐçËâæ¹ÉÇ XÐ UÉÀËäÐ ]É³Ø ËÂÇµÉæ·ÍÊØÚå& Ô×ÖµØÚÙÄËÂÛÜÊØ¤é êØÚèµÉ³Éè¾í¡ËÂÇµÉßÐ _ ÊUÉ ÛÜÞÀÎÚáÜÉ³ÞßÉØ ËÂÒ
ËÂÇµÉ¿Þ®ÐçËÂÇµÉ³Þ®ÐçËÂÛñÙ¬Ðá¾Ô×ÖÚØµÙËÛÜÊØÝÒÂÐçËÛÜÒÔÜì¥ÛñØÚå
Ô×ÊÌÍâÐáÜá í ÐçØÚè½ËÂÇµÛÜÒãÔ×ÖµØµÙËÂÛñÊØ f ÛÜÒÏÊ _ UôÛÜÊÖµÒáñì
f(n) = 1
n
ÎµÍÂÛÜÞßÛñËVÛ UÉ·ÍÂÉ³ÙÖµÍÒVÛ UÉÌé
êØËÂÇµÉ É¬ÐçÍÂáñì
ËÂÇ Ù³É³Ø ËÖµÍîìí ÛñËàæÈÐ
Ò _ É³áÜÛñÉ UÉ³è ËÇ ÐÑË¿ËÇÚÉ Ô×ÖµØµÙËÂÛÜÊØµÒàæ ÇµÊÒ
É UÌÐçáÜÖµÉ³ÒÙ³ÐØ ÛÜØ
20
ÎµÍÂÛÜØµÙ³ÛñÎÚáÜ
É _ ÉÀÙÊÞÀÎ Ö«ËÂÉè _ôìÝÐ Þ®ÐÙäÇµÛÜØµÉ ÐÑÍÂÉÀIÉ P«ÐÙËÂáñì ËÂÇµÉ ÎµÍÂÛÜÞßÛ²ËÂVÛ UÉàÍÂÉ³Ù³Ö«ÍÂÒîVÛ UÉÀÊØµÉ³Ò³é êØµèÚÉÉ³è¤í
ËÂÇµÉ Ô×ÖÚØµÙËÛÜÊ
Ø UÐáÜÖµÉ³Ò¿ÊØµÉ Ù³ÊÞÀÎÚÖ«ËÂÉ³Ò ÛÜØ ÎµÍäÐÙËÂÛñÙ³É ^ÛÜØµÙ³áÜÖµèµÛÜØµå åÙ³è
ÐØµè
`ëÙ³ÊÞßÉ®ÔÜÍÂÊÞ
(a, b)
Fn
ÎµÍÂÛÜÞßÛñËVÛ UÉ·ÍÂÉ³ÙÖµÍÒVÛ UÉâÔ×ÖµØµÙËÂÛñÊØÚÒé
ê»ËÈá×ÐçËÂÉÍ¹ËÂÖ«ÍÂØµÉ³è ÊÖ«ËÈËÇ ÐÑËïËÂÇµÉÍÂÉãÐçÍÂÉÓÙ³ÊÞÀÎÚÖ«ËäÐ _ÚáÜÉ Ô×ÖµØµÙËÂÛÜÊØµÒïËÂÇÚÐçËÈÐçÍÉãØµÊÌËïÎµÍÂÛÜÞßÛñËVÛ UÉòÍÂÉÙ³Ö«Í ÒVÛ UÉé ÒÛÜÞÀÎÚáÜÉÀÐçØÚè æ¹É³áñá ]ôØÚÊ¦æ Ø GÉ PµÐÞÀÎÚáÜÉÛÜÒëËÂÇµL
É _ÚÛÜØÚÐçÍì í
± ô³£¬¨ © ¶Ä ±¬ A(m, n)
èµÉ YÚØµÉ³è _ôì

ÜÛ Ô
 n + 1,
ÛñÔ m = 0
A(m, n) =
A(m − 1, 1),
ÛÜÔ m > 0, n = 0 é

A(m − 1, A(m, n − 1)),
m > 0, n > 0
Æ ÇÚÉ ÔªÐÙËËÂÇÚÐçË ËÂÇµÛÜÒÔ×ÖµØµÙËÂÛñÊØ ÛÜÒ¿ØÚÊÌËÎµÍÂÛÜÞßÛñËÂÛ UÉ ÍÂÉ³Ù³Ö«ÍÂÒîÛVUÉ ÍÂÉ WôÖµÛñÍÂÉ³Ò Ð ÎµÍÂÊ ÊÔZ^»ËÂÇÚÐçË æ¹É Úè ÊØ Ë
¹
åÛVUÉÓÇµÉÍÂÉ`´é ÒòÐáñÍÉ¬Ðè«ìßØµÊÌËÂÉ³è Ð _ Ê UÉí«ÛñËÈÛÜÒ¹ØÚÉÙ³É³ÒÒÂÐÑÍì _ÚÖ«ËÈØµÊÌËòÒîÖ Ù³ÛÜÉ³Ø¯ËòËÇ ÐÑËÈËÇÚÛñÒïèÚÉIY ØµñÛ ËÛÜÊØ
ÍÂÉ³ÙÖµÍÒVÛ UÉ³áñì ÖµÒÉ³Ò
æ Û²ËÂÇóÐ¿ÐçÍÂåÖµÞßÉ³Ø¯ËïËÂÇÚÐçËÈèÚÉÄÎ É³ØµèµÒ·ÊØ é
A
A
ê»Ë·Þ®Ð³ì®ØµÊÌ
Ë _ ÉãÛñÞßÞßÉ³èµÛ×ÐçËÂÉ³á²ìÙáÜÉ¬ÐçÍÈËÂÇÚÐçË¹ËÂÇµÉãÙÊÌÍÍÂÉ³ÒîÎÊØÚèµÛÜØµå Ô×ÖµØµÙËÂÛñÊØ
// POST : return value is the Ackermann function value A (m , n)
unsigned int A ( unsigned int m , unsigned int n ) {
if ( m == 0) return n +1;
if ( n == 0) return A (m -1 ,1);
return A (m -1 , A(m , n -1));
}
Ð áñæÈÐ³ìôÒÕËÂÉÍÞßÛÜØÚÐçËÂÉ³Ò³í_ÚÖ«ËP«ÉÍÂÙÛÜÒÉ ÐÒ ]ôÒ ìÊÖ ËÊëÒÇµÊÑæ ËÂÇµÛÜÒ³éÃÆ Ð _ÚáÜÉ áÜÛÜÒîËÒÒÊÞßÉ Ù ]ÉÍÂÞ®ÐØµØ
Ô×ÖµØµÙËÂÛÜÊØ UÐáÜÖµÉ³Òé [ ÊÌÍ
í
áÜÊôÊ ]ôÒ W ÖµÛñËÂÉÈÞßÊôèµÉÍäÐçËÉÌí _ÚÖµËÒîËäÐçÍîËÂÛÜØµåãÔÜÍÊÞ
íËÂÇµÉ
n)
UÐáÜÖµÉ³Ò åÉÄË IÉ PôËÍÂÉ³ÞßÉáñìÓm
á×ÐçÍÂåÉé 3 A(m,
ÊÖëÙ¬ÐØâ
ÒîËÂÛÜáÜáÙÊÞÀÎ Ö«ËÂÉ
íÑÐá²ËÂÇµÊÖÚåÇÓËÂÇµÛÜÒ¤ËäÐ ]É³Ò¡ÒÖ«mÍÎµ=ÍÂÛÜÒ4ÛñØÚåá²ì
A(4, 1)
áÜÊØµå ÐçáñÍÂÉ¬Ðè«ìé ÊÖ
_ É Ð _ áñÉÃËÂÊ·Ù³ÊÞÀÎÚÖµËÉ
ÐÔÜËÉÍ ÐáÜá»í 65536
ÇÚÐÒ îÊØµáñì( ÐçÍÊÖÚØµè

¨
°

Á
«

A(4, 2)
2
−3
èµÉ³Ù³ÛÜÞ®Ðçá¡èµÛÜåÛñËÒ³é Ö«Ë ËÂÇµÉëÙ¬Ðáñá¡ËÂÊ
æ·ÛÜáÜá¤ØµÊÌË ËÂÉÍÞßÛÜØÚÐçËÂÉ æ ÛñËÇÚÛñØ ÐØ ìÊ _ÚÒÉÍ UÐ _ÚáÜÉ
20, 000
A(4,3)
Î ÉÍÛÜÊôè¤é
ê»Ë Ù¬ÐçØàÛÜØàÔªÐÙ"
Ë _ É ÒÇµÊ¦æ ØëËÂÇÚÐçË
åÌÍÂÊ¦æ Ò¡ÔªÐÒîËÂÉÄÍ ËÂÇÚÐØ ÐØ¯ìãÎµÍÂÛÜÞßÛ²ËÂVÛ UÉ ÍÂÉ³ÙÖµÍÒVÛ UÉ Ô×ÖµØµÙËÂÛÜÊØ
A(n, n)
ÛÜØ
^ºÐØµèËÂÇµÛÜÒÈÛÜÒòÐ ÎµÍÂÊ ÊÔ¡ËÂÇÚÐçË
Ù¬ÐØµØµÊÌ
Ë _ ÉÏÎµÍÂÛñÞßÛñËÂVÛ UÉ·ÍÉ³Ù³Ö«ÍÂÒVÛ UKÉ `Äé É³Ù³Ö«ÍÂÒÛÜÊØÛÜÒ·Ð Î Ê¦æ¹ÉÍÂÔ×Öµá
n
A
_ÚÖ«Ë ÐáñÒÊèÚÐØµåÉÍÊÖÚÒ¹ËÂÊ Êáí¥ÒÛÜØµÙ³ÉÏÛñËÈÛÜÒ¹É¬ÐÒì ËÂÊÉ³ØµÙ³Êôèµb
É ^»ËÂÊ Ê `ïÙÊÞÀÎ áñÛÜÙ¬ÐçËÂÉè ÙÊÞÀÎ Ö«ËäÐÑËÂÛÜÊØµÒæ Û²ËÂÇ
UÉÍì Ô×Éæ áÜÛÜØµÉ³ÒÈÊÔ ÙÊôèµÉÌé
T \
n
m
0
1
2
3
0
1
2
3
5
1
2
3
5
13
2
3
4
7
29
4 13 65533 2
Table 6:
65536
¨½
−3 2
3
4
5
9
61
265536
Dô©¯ ³ Â¶
− 3 n
n+1
n+2
2n + 3
2n+3 − 3
2.
2
..
2| {z } −3
n+3
± ô³£¬¨ © È¶ ±³î
3.2.6 Sorting
" ÊÌÍËÂÛñØÚå®ÐÒÉXWôÖµÉ³ØµÙ³É¿ÊÔ UÌÐçáÜÖµÉ³Ò ^ØôÖµÞ_ ÉÍÂÒí ËÂÉIPôËÂÒ³í ÉËÙÌé `óÛÜØ¯ËÂÊ®ÐÒÙ³ÉØÚèµÛÜØµåßÊÌÍÂèµÉÍÓÛñÒ Ð UÉÍì _ ÐÒÛñÙ
ÐØµèÝÛÜÞÀÎÊÌÍËäÐØ¯Ë ÊÌÎ ÉÍäÐÑËÂÛÜÊØ¤é [ ÊÌÍëÉIP«ÐÞÀÎÚáÜÉÌí¤Ð®ÒîÎ É³Ù³Û Y ÙUÐáÜÖµÉÙ³ÐØ _ É¿Ô×ÊÖµØÚè Þ¿ÖµÙäÇ ÔªÐÒîËÉÍëÛÜØ Ð
ÒÊÌÍËÉ³è½ËÂÇÚÐØ ÛÜØ½ÐØ ÖµØÚÒîÊÌÍËÂÉ³è½ÒÉXWôÖµÉ³ØµÙ³É ^ÒÉÉ$P«ÉÄÍÂÙ³ÛÜÒÉ \ `Äé ÊÖ ]¥ØµÊÑæ ËÂÇµÛÜÒ ÔÜÍÂÊÞ èÚÐÛÜáñìáÜÛÜÔ×ÉÌí
ÐØµè ËÂÇÚÐç!Ë Ò¿æ·Ç ì ìÊÖ ÒÊÌÍËìÊÖµ
Í Ò³í ÐØµè æ Ç ì ËÂÇµÉÉØ ËÍÂÛñÉ³Ò ÛÜØ Ð ËÂÉ³áñÉÎÚÇÚÊØµÉ èµÛñÍÂÉ³ÙËÊÌÍì ÐÑÍÂÉ
ÒÊÌÍËÉ³è _ôì ØÚÐÞßÉÌé
SÝÉ ÇÚKÐ UÉ ÐÒ ]Éè ìÊÖ ÛÜØ P«ÉÍÂÙÛÜÒÉ ËÂÊ æ·ÍÛñËÂÉ®ÐóÎµÍÂÊåÌÍäÐÞ ËÂÇÚÐçË ÒîÊÌÍËÂÒÐ åÛ UÉ³ØÒÉ WôÖµÉ³ØµÙ³É
ÊÔÈÛÜØ ËÂÉåÉÍÂÒ * P¥ÉÍÂÙ³ÛÜÒîÉ
æÈÐÒ Ð _ÊÖµË Þ®Ð ]¥ÛÜØµåËÂÇµÛÜÒàÛñØ ËÂÊ Ð Ô×ÖµØµÙËÂÛÜÊØ ËÂÇÚÐçË ÒÊÌÍîËÂÒ¿ÐáÜá Ø ÖÚ
Þ _ ÉÍÒ
èµÉ³ÒÙÍVÛ _ É³è _ ìóÐ åVÛ UÉ³Ø Î ÊÛÜØ ËÉÍ¹ÍäÐØµåÉÌéÕêØ _ ÊÌËÂÇ IÉ P«ÉÍÙ³ÛÜÒÉ³ÒíÚìÊÖóæ¹ÉÍÉãØµÊÌËòÒÖ«ÎÚÎÊÒÉ³èËÂÊèµÊÀÐØ¯ì
É Ù³ÛÜÉ³ØµÙì Ù³ÊØµÒÛÜèµÉÍäÐÑËÂÛÜÊØµÒ³é
·ÉÍÂÉ æ¹É æÈÐçØ Ë ËÂÊÙ¬ÐÑËÂÙäÇ Ö«Î ÊØ ËÂÇµÛÜÒ ÐØµè ÛÜØ UÉ³ÒîËÂÛñåÐçËÂÉ ËÂÇµÉ
ÊÔãËÂÇµÉ½ÒÊÌÍîËÂÛÜØµå
Ä
±

¨
¾
¢
ª

¯ J
ÎµÍÂÊ _ÚáÜÉ³Þ é ÊÖµåÇµáñì ÒîÎÉ¬Ð ]¥ÛñØÚå«í ËÂÇµÉÙ³ÊÞÀÎÚáÜIÉ P«Û²Ë ìóÊÔïÐ®ÎµÍÂÊ _ÚáÜÉÞ ÛñÒãèµÉ YÚØµÉ³è ÐÒÏËÂÇµÉÙÊÞÀÎ áñIÉ P«ÛñË ì
^»ÍÂÖµØ¯ËÂÛÜÞßÉ ` ÊÔÏËÂÇµÉ ÔªÐÒîËÉ³ÒîË ÐáÜåÊÌÍÂÛ²ËÂÇµÞ ËÂÇÚÐçËßÒîÊVá UÉ³ÒßËÂÇµÉ ÎÚÍÊ _ áñÉ³Þ é êØ Ù³ÊÞÀÎÚÖ«ËÂÛÜØµå [ VÛ _ ÊØÚÐÙ³ÙÛ
ØôÖµ
Þ _ ÉÍÂÒÏÛÜØ " ÉÙËÂÛÜÊØ é \ é ÐØµè " É³ÙËÂÛñÊØ é \ é í¤æ¹ÉÇÚKÐ UÉ ÐáñÍÂÉ³Ðè«ì½ÒîÉ³É³ØÝËÂÇÚÐçËÏËÂÇµÉÍÂÖµØ¯ËÂÛÜÞßÉ³ÒÓÊÔ
èµÛ ÉÄÍÂÉ³Ø ËÏÐçáÜåÊÌÍÂÛñËÂÇµÞßÒ¹Ô×ÊÌÍ ËÂÇµÉëÒÂÐçÞßÉâÎµÍÂÊ _ÚáÜÉ³Þ Þ®Ð³ì UÌÐçÍîì ÐÀáÜÊÌË¬é·ÆÈÇµÉàÒÂÐÞßÉâÛÜÒ·ËÍÂÖµÉàÔ×ÊÌÍÓÒÊÌÍîËÂÛÜØµå
ÐáÜåÊÌÍÛñËÂÇµÞßÒ³í ÐÒÈæ¹ÉÏæ ÛÜáñáèµÛÜÒÙ³Ê UÉÍ ÒîÇÚÊÌÍîËÂáñìé
ÉÄË¹ÖÚÒ¹ÒîËÂÐçÍË _ ì ÐØÚÐáñ
ì ÛÜØµåëÊØµÉÏÊÔ ËÂÇµÉ îÊ _ U¥ÛÜÊÖµÒ àÒÊÌÍîËÂÛÜØµå ÐáÜåÊÌÍÂÛ²ËÂÇµÞßÒÃËÂÇÚÐçËìÊÖÞ®Ð³ìÀÇÚXÐ UÉ
Ù³ÊÞßÉ Ö«Îâæ·ÛñËÂÇÏÛÜØ P«ÉÍÙ³ÛÜÒÉ é Æ¹ÇÚÉÃÒÛÜÞÀÎÚáÜÉÒîË ÊØµÉÃËÇ ÐÑË¾ËÇÚÉÕÐÖ«ËÂÇµÊÌÍÂÒ¤Ù¬ÐØãËÂÇµÛÜØ ] ÊÔôÛÜÒ
¨° »¨ YÚ¨ÍÒîË
é VÛ UÉØ ËÂÇµÉÓÒXÉ W ÖµÉ³ØµÙ³ÉãÊÔ UÐáÜÖµÉ³
Ò ^áÜÉË Ò¹ÐÒÒîÖÚÞßÉ ËÇÚÉÄì ÐçÍÂÉÓÛÜØ ËÉ³åÉÍÂÒ `´í
YÚØµ£¦èÚ ÒãËÂÇµÉßÒÞ®ÐáÜáÜÉÒîËâÉáÜÉ³ÞßÉ³Ø¯ËëÊÔ ËÂÇµÉßÒXÉ W ÖµÉ³ØµÙ³É ÕËÇÚÉØ ÛñËâÛÜØ ËÉÍÂÙäÇÚÐØµåÉ³ÒâËÂÇµ¨
ÛÜÒë°ÉáÜ»É³¨
Þß É³¨ Ø¯Ëâæ Û²£¦ËÂ ÇÝËÂÇµÉ
YµÍÂÒîËàÉáÜÉ³ÞßÉ³Ø¯Ë¬éÆ¹ÇµÉßÒXÉ W ÖÚÉØÚÙÉ ØµÊ¦æ ÒËäÐçÍËÂÒëæ·ÛñËÂÇ ËÇÚÉßÒîÞ®ÐáÜáÜÉ³ÒËãÉáÜÉ³ÞßÉ³Ø¯Ë¬í ÐÒëèµÉ³ÒîÛñÍÂÉ³è¤
í _ÚÖ«ËëËÂÇµÉ
ÍÂÉ³Þ®ÐçÛÜØµèÚÉÄÍÕÊÔ ËÂÇµÉ ÒXÉ W ÖÚÉØÚÙÉ ÒîËÂÛÜáñáÚØµÉ³ÉèÚÒÕËÂÊ _ ÉòÒÊÌÍËÂÉè¾é Ö«ËËÇÚÛñÒÕÛÜÒÕèµÊØÚÉòÛÜØ ËÂÇµÉòÒÂÐÞßÉ¹æÈÐ¬ì ËÂÇµÉ
ÒÞ®ÐáñáÜÉ³ÒîËÈÉ³áÜÉÞßÉ³Ø Ë ÐÞßÊØµåËÂÇµÉâÍÂÉÞ®ÐÛÜØµÛÜØµåÊØµÉ³Ò·ÛÜÒ Ô×ÊÖµØµè½ÐçØÚè ÛÜØ¯ËÂÉÍÂÙäÇÚÐØµåÉ³èóæ·ÛñËÂÇóËÂÇµÉ
Ä±
F
É³áÜÉÞßÉ³Ø ËÈÊÔ¡ËÇÚÉÏÒXÉ W ÖµÉ³ØµÙ³ÉÌíÐçØÚè ÒÊÀÊØ¤é
ÒÒÖµÞßÛÜØµå ËÂÇÚÐçË ËÇÚÉ ÒÉ WôÖµÉ³ØµÙ³ÉÛñÒ¾èµÉ³ÒîÙÍÂVÛ _ Éè _ôìÏÐ Î ÊÛñØ ËÂÉÍ ÍÂÐØµåÉ
í
[first, last) ¨ » »¨ ¨
Ù¬ÐØ _ ÉÏÍÂÉ¬ÐáñÛ ³É³èÐÒòÔ×ÊáñáÜÊÑæ·Ò³é
£¦
// PRE : [ first , last ) is a valid range
// POST : the elements *p , p in [ first , last ) are in ascending order
void minimum_sort ( int * first , int * last )
{
\
for ( int * p = first ; p != last ; ++ p ) {
// find minimum in nonempty range described by [p , last )
int * p_min = p ; // pointer to current minimum
int * q = p ;
// pointer to current element
while (++ q != last )
if (* q < * p_min ) p_min = q;
// interchange * p with * p_min
std :: iter_swap ( p , p_min );
}
}
Æ¹ÇÚÉ ÒîËäÐçØÚèÚÐçÍè áÜÛV_µÍäÐçÍîì Ô×ÖµØÚÙÄËÂÛÜÊØ
ÛÜØ ËÉÍÂÙäÇÚÐØµåÉ³Ò¿ËÇÚÉUÐáÜÖµÉ³Ò ÊÔÈËÂÇµÉ Ê _ É³ÙËÂÒ
std::iter_swap
Î ÊÛÜØ¯ËÂÉ³èÝËÂÊ _ ìÝÛñËÒãË æ¹Ê ÐçÍÂåÖµÞßÉ³Ø¯ËÂÒ³é ÆÈÇµÉÍÉßÛÜÒëÐáñÒÊ ÐÔ×ÖµØµÙËÂÛÜÊØ
ËÂÇÚÐçËãæ¹É
std::min_element
Ù³ÊÖµáÜèÝÖµÒÉ¿ËÂÊóåÉËÏÍÂÛÜè ÊÔËÂÇµÉ ÛÜØµØµÉÍãáÜÊôÊÌÎ ¡ÇÚÊ¦æ¹É UÉÍ³í ÒÛÜØµÙ³É æ¹ÉæÈÐØ¯ËãËÊ ÐØÚÐáñì(³É¿ËÇÚÉÔ×ÖµØÚÙÄËÂÛÜÊØ
ÛÜØ èÚÉÄËäÐÛÜáí¯æ¹ÉÏÍÉ³ÔÜÍäÐÛÜØ ÔÜÍÂÊÞ Ù¬ÐáÜáñÛÜØµå ÐØ ì®ØµÊØ ËîÍÂÛVU¥ÛÜÐáÒîËäÐçØÚèÚÐçÍè áÜÛ _ÚÍÂÐçÍìßÔ×ÖµØµÙËÂÛÜÊØ
minimum_sort
ÇµÉÍÂÉÌé
S¼Ç ÐÑË Ù¬ÐçØ æ¹É ÒÂÐ³ìÐ _ ÊÖ«Ë ËÂÇµÉ ÍÂÖµØ¯ËÂÛÜÞßÉßÊÔ
Ô×ÊÌÍßÐ åÛVUÉ³Ø ÍäÐØµåÉ ÆÈÇÚÐçËÛñË
minimum_sort
èµÉÎ É³ØµèµÒ ÊØÀËÂÇµÉÈÎ áÜÐçËÂÔ×ÊÌÍÂÞ íËÂÇµÛÜÒÃÛÜÒÔ×ÊÌÍ ÒÖ«ÍÂÉÌé Ø ÐÏÞßÊôèµÉÍÂØ í¯ËÂÇµÉ·ÐáÜåÊÌÍÂÛñËÇÚÞ æ·ÛÜáÜá«ÍÂÖµØßÞÖµÙäÇ
ÔªÐÒîËÉÍâËÂÇÚÐØ½ÊØ Ð U¥ÛÜØ¯ËäÐåÉ Ù³ÊÞÀÎÚÖ«ËÂÉÍÏÔÜÍÂÊÞ ËÂÇµÉË æ¹É³Ø¯ËÂÛÜÉËÂÇ½Ù³É³Ø¯ËÂÖ«Íìé Æ¹ÇµÉÍÂÉ¿ÛÜÒÏØÚÊ ÒÖµÙäÇ ËÂÇµÛÜØµå
ÐÒ ËÇÚÉ ÍÂÖµØ ËÂÛñÞßÉÌé Ö«Ë ÛÜÔ æ¹ÉëáñÊôÊ ]ÐçË·æ·Ç ÐÑË·ËÂÇµÉàÐáñåÊÌÍÂÛñËÂÇµÞ èµÊôÉ³Ò³íÚæ¹ÉëÙ¬ÐØ YÚØµè Ð ÞßÉ¬ÐÒîÖµÍÉâÊÔ
ÍÂÖµØ ËÛÜÞßÉ ËÂÇÚÐçËÈÛÜÒ¹Î áÜÐçËÂÔ×ÊÌÍÂ
Þ ÛÜØµèÚÉÄÎ É³ØµèµÉ³Ø Ë¬é
èµÊÞßÛÜØÚÐçËÛÜØµå ÊÌÎ ÉÍÂÐçËÂÛÜÊØ ÛÜØ½ËÇÚÉ¿ÒÉ³ØµÒÉ ËÂÇÚÐçËÏÛñËÏÊôÙÙ³Ö«ÍÂÒ UÉÄÍì ÔÜÍÂXÉ WôÖµÉ³Ø¯ËÂáñì èµÖ«ÍÂÛÜØµåÐßÙ¬ÐçáÜá ËÂÊ
ÛÜÒ ËÇÚÉàÙÊÞÀÎÐÑÍÂÛÜÒÊØ
é S ÉàÙ¬ÐØ É UÉ³Ø GÉ PµÐÙËáñìóÙ³ÊÖµØ¯Ë ËÂÇµÉàØôÖµb
Þ _ÉÍ
minimum_sort
*q < *p_min
ÊÔ ÒÖµÙäÇ½Ù³ÊÞÀÎ ÐçÍÛÜÒÊØµÒ³íÚèÚÉÄÎ É³ØµèµÛÜØµåßÊØ ËÇÚÉàØ Öµb
Þ _ ÉÄÍ ÊÔ É³áÜÉ³ÞßÉØ ËÂÒ ËÂÇÚÐçË ÐçÍÂÉëËÂL
Ê _ ÉàÒîÊÌÍËÂÉ³è¤é êØ
ËÂÇµÉ YµÍÂÒîË IÉ P«É³ÙÖµËÛÜÊØ ÊÔ ËÂÇµÉ
ÒîËÂÐçËÂÉ³ÞßÉ³Ø¯Ë¬í«ËÂÇµÉ YµÍÂÒîË ÉáÜÉ³ÞßÉ³Ø¯Ë·nÛÜÒ·Ù³ÊÞÀÎ ÐçÍÂÉ³èæ·ÛñËÂÇ Ðáñá
ÒÖµÙ³Ù³ÉÉ³èµÛÜØµåßÉ³áÜÉ³ÞßÉ³Ø¯ËÂÒ³éïêØóËÇÚwhile
ÉâÒÉ³Ù³ÊØµè IÉ P¥É³Ù³Ö«ËÂÛÜÊØ¤íµËÂÇµÉëÒÉ³Ù³ÊØµè É³áÜÉÞßÉ³Ø ËòÛÜØóÙ³ÊÞÀÎ ÐçÍÉ³èæ Û²n
ËÂÇ −ÐáÜ1á
ËÂÇµÉ
ÒîÖÚÙÙ³É³É³èµÛÜØµå®É³áÜÉ³ÞßÉØ ËÂÒ³í ÐØµè ÒÊßÊØ¤é êØ ËÇÚÉâÒÉ³Ù³ÊØµè ËÂÊ á×ÐÒîË IÉ P«É³ÙÖµËÛÜÊØ ÊÔ ËÇÚÉ
ÒîËäÐÑËÂÉ³nÞß−É³Ø¯2Ë¬í YÚØÚÐáÜáñìí¬æ¹É ÇÚXÐ UÉ ÊØµÉ Ù³ÊÞÀÎ ÐçÍÛÜÒÊØ¤íÑÐØµè ËÇ ÐÑ!Ë Ò¾ÛñË¬T
é S ÉÕËÂÇµÉÍÉ³Ô×ÊÌÍÂÉïÇÚXÐ UÉËÂÇµÉ Ô×ÊáÜáÜwhile
Ê¦æ ÛñØÚå
Observation 1 ÕÈ¶ ±³î minimum_sort £¦ºß© ±óÂ¶ n ¬ª³¨½³¤º¸¹º
1 + 2 + ...n − 1 =
n(n − 1)
2
±¨¿¢©£¬ç¤® ¬¸Ó³¼ ³ ±Äó¬ª³¨ ¤ºçÅ
S¼Ç ìóèµÊ¿æ¹ÉâÒîÎ ÉÙ³VÛ YÚÙ¬ÐáÜá²ì®Ù³ÊÖµØ ËÈËÂÇµÉ³ÒÉâÙÊÞÀÎÐÑÍÂÛÜÒÊØµÒ É³Ù¬ÐçÖÚÒîÉàÐØ ì ÊÌËÂÇµÉÍòÊÌÎ ÉÍäÐÑËÂÛÜÊØÛñÒòÉ³Û ËÂÇµÉÍ Î ÉÍÔ×ÊÌÍÂÞßÉ³è¿ÞÖµÙäÇáñÉ³ÒÒ ÔÜÍÂXÉ W ÖÚÉØ ËÂáñì^»Ô×ÊÌÍÕÉGPµÐÞÀÎÚáÜÉÌíçËÂÇµÉïèÚÉÙ³á×ÐçÍäÐÑËÂÛÜÊØ ÒîËäÐçËÉ³ÞßÉ³Ø Ë
ÛÜÒ GÉ P«É³Ù³Ö«ËÂÉèÊØµáñì
ËÂÛñÞßÉ³Ò `Äí¯ÊÌÍïæ ÛñËÇ®ÐçÎµÎÚÍÊ P¥ÛÜÞ®ÐçËÂÉ³á²ìàËÂÇµÉ ÒÂÐÞßÉ·ÔÜÍXÉ WôÖµÉ³ØµÙìéÆÈÇµÛÜÒ Ùint*
ÊØÚÙÉÍÂØµqÒ =ËÂÇµpÉ
n
ÐÒÒîÛÜåØµÞßÉ³Ø Ë
æ ÇµÛÜÙäÇÝÞ®Ð³ì ÇÚÐçÎµÎ É³ØÝÖ«Î ËÂÊ
ËÂÛÜÞßÉÒ³í¾ÐØµè ËÂÇµÉ ÉIPôÎµÍÂÉ³ÒÒÛñÊØ
p_min = q
n(n − 1)/2
«ËÂÇµÛÜÒ¹ÊØµÉãÛñÒòÉ UÐáÜÖÚÐçËÉ³èÉ UÉØ ÞßÊÌÍÂÉÓÔÜÍÂXÉ WôÖµÉ³Ø¯ËÂáñìí«ØÚÐÞßÉ³á²ì
ËÂÛñÞßÉ³Ò³é
++q != last
n(n − 1)/2 + n
Æ¹ÇÚÉ ËÂÊÌËäÐçáÃØ Öµb
Þ _ ÉÄÍãÊÔÕÊÌÎ ÉÍäÐÑËÂÛÜÊØµÒÓÛÜÒ ËÂÇµÉÍÂÉ³Ô×ÊÌÍÂÉ ÐçËãÞßÊÒîË
Ô×ÊÌÍâÒÊÞßÉ Ù³ÊØ ÒîËäÐçØ ËÂÒ
é [ ÊÌÍ á×ÐçÍÂåÉ íËÂÇµÉÃáÜÛÜØµÉ¬ÐçÍ¾ËÂÉÍÂÞ
ÛÜÒ¾ØµÉ³åáÜÛÜåÛ _ cáñ1É n(n
Ù³ÊÞÀ−Î Ðç1)/2
ÍÂÉ³èâË+
ÊòËÂcÇµ2É nW ÖÚÐè«ÍäÐçËÂÛÜÙ ËÂÉÄÍÂÞ
c1 , c2
n
c n
Úæ¹ÉëÙ¬ÐØ ËÂÇµÉÍÉ³Ô×ÊÌÍÂÉëÙ³ÊØµÙ³áñÖÚèµÉâË2Ç ÐÑË·ËÂÇµÉâËÂÊÌËÂÐá¡Ø Öµb
Þ _ ÉÄÍ ÊÔ ÊÌÎ ÉÍÂÐçËÂÛÜÊØµÒ·ØµÉ³ÉèÚÉè ËÊ
c1 n(n − 1)/2
ÒÊÌÍË Ø Öµb
Þ _ ÉÄÍÂÒ ÛÜÒ¡ÎÚÍÊÌÎ ÊÌÍËÂÛÜÊØÚÐáËÂÊòËÂÇµÉ ØôÖµ
Þ _ ÉÍ ÊÔ«ÙÊÞÀÎÐÑÍÂÛÜÒÊØµÒ _ ÉË æ¹É³É³Ø ÒXÉ W ÖµÉ³ØµÙ³É¹É³áÜÉ³ÞßÉ³Ø¯ËÂÒ³é
n
T \ Æ ÇµÛÜÒ ÛÜÞÀÎÚáÜÛÜÉ³Ò ËÂÇµÉïÔ×ÊáÜáÜÊ¦æ ÛñØÚå ÛñÔ¥ìÊÖÞßÉ¬ÐçÒÖ«ÍÂÉËÂÇµÉÍÂÖµØ¯ËÂÛÜÞßÉÃÊÔµËÇÚÉ æ·ÇµÊáÜÉÒÊÌÍîËÂÛÜØµå ÐáÜåÊÌÍÛñËÂÇµÞ í
È
ËÂÇµÉãÍÉ³ÒÖµáñËÂÛÜØµå¿ËÂÛÜÞßÉ
æ·ÛÜáÜá _ ÉÏÎµÍÂÊÌÎ ÊÌÍËÂÛñÊØ ÐçáËÂÊ ËÂÇµÉãËÛÜÞßÉ
ËÇ ÐÑË·ÛÜÒ _ É³ÛñØÚåÒîÎ É³Ø¯Ë æ Û²ËÂÇ
Ttotal
Tcomp
Ù³ÊÞÀÎ ÐçÍÂÛñÒÊØµÒ _ ÉË æ¹É³ÉØ¼ÒîÉXWôÖµÉ³ØµÙ³É É³áÜÉÞßÉ³Ø ËÂÒé " ÛÜØµÙ³É
ÛÜÒÀÛÜØ ËÂÖ«ÍÂØ ÎµÍÂÊÌÎ ÊÌÍîËÂÛÜÊØÚÐá¹ËÂÊÝËÂÇµÉ
ØôÖµ
Þ _ ÉÍãÊÔïÙ³ÊÞÀÎ ÐçÍÂÛÜÒîÊØÚÒÏÛñËÂÒîÉ³áÜÔºí¡ËÇÚÛñÒâØôÖµÞ_ ÉÍâÛÜÒëÐ åÊôTÊôcomp
è ÛñØÚèµÛÜÙ³ÐçËÂÊÌÍÏÔ×ÊÌÍëËÂÇµÉÀÉ Ù³ÛÜÉ³ØµÙì ÊÔ ËÂÇµÉ
ÐáÜåÊÌÍÛñËÂÇµÞ é
êÔÈìÊÖ ËÂÇµÛÜØ ] Ð _ÊÖµË ÒîÊÌÍËÂÛÜØµå ÞßÊÌÍÂÉßÙ³ÊÞÀÎÚáÜÛñÙ¬ÐçËÂÉ³è UÌÐáñÖÚÉÒ ^»áÜVÛ ]ÉÀØÚÐÞßÉ³ÒàÛñØ ÐóËÂÉáÜÉÎÚÇµÊØÚÉßèµÛ ÍÂÉ³ÙÄËÂÊÌÍaì `´í Ð Ù³ÊÞÀÎ ÐçÍÂÛñÒÊØ _ ÉË æ¹É³ÉØ Ë æ¹ÊßÉ³áñÉ³ÞßÉ³Ø ËÒ·ÞßÛÜåÇ¯Ë·É UÉØ _ ÉÙ³ÊÞßÉâËÂÇµÉëÒÛñØÚåáñÉâÞßÊÒîËòËÂÛÜÞßÉ Ù³ÊØµÒÖµÞßÛÜØµå ÊÌÎÉÍäÐçËÂÛñÊØ¾é¹êØ ÒÖµÙäÇ½ÐÒÙ³ÉØ ÐÑÍÂÛÜÊ«í
Þ®Ð³ì É¬ÐçË Ö«Î½ÐáÜÞßÊÒîËÈÉ UÉÄÍìôËÂÇµÛÜØµåßÊÔ
í
Ttotal
comp
Þ®Ð ]ôÛÜØµå ËÇÚÉÏÙ³ÊÞÀÎ ÐçÍÂÛñÒÊØ®Ù³ÊÖµØ Ë ÐçØÉ UÉ³Ø ÞßÊÌÍTÉã
ÐÑÎÚÎµÍÂÊÌÎµÍÂÛÜÐçËÂÉÓÞßÉ¬ÐÒÖ«ÍÂÉÓÊÔ¡É Ù³ÛñÉ³ØµÙìé
Æ Ê ÙäÇµÉ³Ù ]ËÇ ÐÑËÀÐáñá¹ËÇÚÛñÒ¿ÛñÒ ØµÊÌË ÊØµáñì åÌÍÉì ËÇÚÉÊÌÍìí áÜÉË¿ÖµÒ Þ®Ð ]É ÒÊÞßÉ®IÉ PôÎ ÉÍÂÛñÞßÉ³Ø ËÂÒ¿ÐØµè
ÞßÉ¬ÐÒîÖµÍÉëËÂÇµÉ ËÛÜÞßÉëËÂÇÚÐçËÓÛñË ËäÐ ]É³ÒÏËÊ GÉ P«É³Ù³Ö«ËÂÉ ÐÀÎÚÍÊåÌÍäÐÞ æ Û²ËÂÇ ËÂÇµÉ¿Ô×ÊáñáÜÊÑæ·ÛÜØµå
Ô×ÖµØÚÙÄËÂÛÜÊØ¤í
main
Ô×ÊÌÍ UÌÐçÍÛÜÊÖµÒ UÐáÜÖµÉ³Ò¾ÊÔ é Ò¡ÊÖ«Í ËÂÉ³ÒË Ù¬ÐçÒÉ «í¦æ¹ÉÕÖµÒÉ ËÂÇµÉ Öµ
Þ _ áñÉ³èÓÒXÉ WôÖµÉ³ØµÙ³É
í
n
0, n−1, 1, n−2, ...
ÐØµèÐÔÜËÂÉÍ ÇÚXÐ U¥ÛÜØµå Ù¬ÐçáÜáÜÉ³èàËÂÇµÉïÔ×ÖµØµÙËÂÛÜÊØ
ÔÜÍÂÊÞ Ð _Ê UÉÌíÌæ¹ÉïÙäÇµÉ³Ù ] æ ÇµÉËÂÇµÉÍ æ¹ÉÈØµÊ¦æ
minimum_sort
ÛÜØµèµÉ³É³èóÇÚXÐ UÉãËÇÚÉãÐÒÙÉ³ØµèÚÛñØÚåÒXÉ W ÖÚÉØÚÙÉ
é ÉÒ³í«ËÂÇµÛÜÒÈÎµÍÂÊåÌÍÂÐÞ èÚÊ É³Ò·ÊÌËÇÚÉÄÍòËÂÇµÛÜØµåÒ
ÐçÎ ÐçÍËÕÔÜÍÂÊÞ ËÂÇµÉ ÐÙËÖ ÐçáÚÒîÊÌÍËÂÛÜØµå«í _ Ö«ËÐáÜáÚ0,Ðèµ1,èµ.ÛñËÂ.ÛÜ.Ê,ØÚnÐá«−ÊÌ1ÎÉÍäÐçËÂÛñÊØÚÒÃÐçÍ
É îÙäÇµÉ¬ÐçÎ ëÛÜØ ËÇÚÉòÒÉ³ØµÒÉ·ËÇ ÐÑË
ËÂÇµÉ³ÛñÍëØôÖµb
Þ _ÉÍàÛÜÒëÎµÍÂÊÌÎ ÊÌÍîËÂÛÜÊØÚÐá ËÂÊ
ÐçË ÞßÊÒË ÃÐÙ³Ù³ÊÌÍÂèµÛÜØµå ËÂÊ ÊÖ«Í¿Ð _ Ê UÉ®áÜÛÜØµÉ ÊÔòÐçÍÂåÖµÞßÉ³Ø¯ËÂÒ³í
ËÂÇµÉì ÒÇµÊÖµáÜèËÂÇµÉÍÉ³Ô×ÊÌÍÂÉâØµÊÌËÈÞ®ÐçËËÉÍ¬n
é
int main ()
{
int n = 100000; // number of values to be sorted
int * a = new int [n ];
std :: cout < < " Sorting " < < n < < " integers ...\ n";
// create sequence : 0 , n -1 , 1 , n -2 ,...
for ( int i =0; i < n ; ++ i)
if ( i % 2 == 0) a[ i ] = i /2; else a[i ] = n -1 - i /2;
// sort into ascending order
minimum_sort ( a , a+n );
// is it really sorted ?
for ( int i =0; i <n -1;++ i )
if ( a[ i ] != i ) std :: cout < < " Sorting error !\ n";
delete [] a;
return 0;
}
Æ Ð Ú_ áÜÉ ÒÖµÞßÞ®ÐçÍÂÛÉ³ÒÏËÇÚÉ¿ÍÂÉ³ÒÖµáñËÒ³é [ ÊÌÍâÉ UÉÍì U ÐáÜÖµÉÀÊÔ í
ÛÜÒãËÇÚÉ ØôÖµÞb_ÉÍëÊÔ
åÐÙ³ÊÞÀÎ ÐçÍÛÜÒÊØµÒ
^ 9 Ù³ÊÞÀÎ ÐçÍÛÜÒÊØµÒ `ÄíµÐÙÙ³ÊÌÍÂèµÛÜØµåËÂÊ _ÚÒÉÍ UÌÐçËnÛÜÊØ Gcomp
é¹êØ ÊÌËÂÇµÉÍ·æ¹ÊÌÍÂèµÒ³í
&
-
10
&
!2(2.*
8#*CBI:
+(- 7"
J
6&2*
/
%2/
/ST *
7&7
>*T
%"K2**
( #,%7 1 1*
2!*
Û
Gcomp=
77..B
\\
n
Gcomp
Time (min)
sec/Gcomp
í é
\ í \
é í \
í é
!
\
é
í í \
\
é ÞßÛÜØµÛÜÞ¿ÖµÞºÒÊÌÍË
¨½ ³£Â¶®±Ä¨¢©£¬»¤®©F£¬¤°¨ Â¶
Table 7:
é
ÛñÒÕËÂÇµÉ Ð _ ÒîÊáÜÖ«ËÂÉÈÍÖÚØ¯ËÂÛÜÞßÉ¹ÊÔËÂÇµÉòÎµÍÂÊåÌÍÂÐÞ ÛñØÀÞßÛÜØôÖ«ËÂÉÒÕÐçØÚèßÒîÉ³Ù³ÊØµèµÒ³íôÊØ
Þ®ÐØ¯ì
ÒÉ³ÙÊØÚèµÒÈËÂÇµÉÏÎµÍÂÊåÌÍäÐçÞ ØµÉ³ÉèÚÒÈËÂÊÎÉÍÂÔ×ÊÌÍÂÞ ÊØµÉ ÛÜåÐÙ³ÊÞÀÎ ÐçÍÂÛñÒÊØ¤é
ÆÈÇµÉ·ËÂÐ _ÚáÜÉòÒîÇÚÊ¦æ ÒÃËÂÇÚÐçËÃËÂÇµÉ·ØôÖµ
Þ _ ÉÍÊÔÒÉ³Ù³ÊØµèµÒÎ ÉÄÍ ÛñåÐÙ³ÊÞÀÎ ÐçÍÂÛÜÒîÊØÛÜÒÕÐçÍÂÊÖµØµè é Ô×ÊÌÍ¹ÐáÜá
Ù³ÊØµÒÛÜèµÉÍÉ³è UÐáÜÖµÉ³Ò ÊÔ é Ò ÎµÍÂÉ³èµÛÜÙËÉ³è Ð _ Ê UÉíËÂÇµÉëÍÂÖµØ ËÂÛñÞßÉâÛÜØ ÎµÍäÐÙËÂÛñÙ³ÉëÛÜÒ ËÂÇµÉÍÉ³Ô×ÊÌÍÂÉ ÛÜØµèµÉ³É³è
ÎµÍÂÊÌÎ ÊÌÍËÛÜÊØÚÐáÕËÂÊ ËÂÇµÉßØ nÖÚ
Þ _ ÉÍ ÊÔÈÙ³ÊÞÀÎ ÐçÍÂÛñÒÊØµÒ _ ÉË æ¹É³ÉØÒXÉ W ÖµÉ³ØµÙ³É ÉáÜÉ³ÞßÉ³Ø¯ËÂÒ³é ÆÈÇµÛÜÒàØ Öµb
Þ _ ÉÄÍ
WôÖÚÐè«ÍÂÖ«ÎÚáÜÉ³Ò¹ÔÜÍÂÊÞðÊØµÉãÙ³ÊáñÖÚÞßØ®ËÂÊ ËÂÇµÉâØµIÉ PôË¬íµÐØµèóÒÊÀèµÊ É³ÒÈËÇÚÉÓÍÂÖµØ ËÛÜÞßÉÌé
SÝÉÐáÜÒÊ ÒÉ³É ËÂÇÚÐçËÏÒÊÌÍËÛÜØµå Ø ÖÚ
Þ _ ÉÍÒ ÖÚÒîÛÜØµå
ÐçÎµÎ É¬ÐçÍÒÏËÂ
Ê _ ÉàÎµÍÂÉËË ì ÛÜØµÉ ¨
»

¨
î

¦
£

Ù³ÛÜÉØ Ë¬é í í ÛñÒÈØµÊÌËòá×ÐçÍÂå
É _ ì ËÂÊ èÚÐ¬ì Ò¹ÒîËäÐçØÚèÚÐçÍèÚÒí _ Ö«ËÈËÂÊ ÒîÊÌÍËòËÇ ÐÑËòÞ®ÐçØ ìßØ ÖÚ
Þ _ ÉÍÒÈËäÐ ]É³Ò
ÐçÎµÎ É¬ÐçÍÂÒ ËÂ
Ê _ ÉÈÙ³ÊØµÒîËäÐØ¯Ë¬íæ¹É¹Ù¬ÐØÀÉ UÉØÀÉ³ÒîËÂÛñÞ®ÐçËÂÉ
ÞßÊÌÍÂÉ ËÂÇÚÐØÊØµÉòÇµÊÖ«Í VÛ UÉ³Ø ËÂÇÚÐçË
ËÂÇµÉ ËÂÛñÞßÉ ËÇ ÐÑËëÛñËÏæ¹ÊÖµáÜè ËäÐ ]ÉËÂÊóÒsec/Gcomp
ÊÌÍË í í ØôÖµ
Þ _ ÉÍÂÒí ÒÂÐ³ìé [ ÊÌÍëËÂÇµÛÜÒ³í¾æ¹É èµÉÍÂVÛ UÉ ÔÜÍÂÊÞ
_ ÒîÉÍ UÐçËÂÛÜÊØ ËÂÇµÉ®ÍXÉ WôÖµÛñÍÂÉè Ø ÖÚ
Þ _ ÉÍëÊÔ ÛÜåÐÙ³ÊÞÀÎ ÐçÍÂÛñÒÊØµb
Ò ^ í `âÐØµè ÞÖµáñËÛñÎÚáñì ÛñËàæ Û²ËÂÇ
é éÃÆ¹ÇµÉãÍÂÉ³ÒîÖÚá²ËÂÛÜØµå í ÒÉ³Ù³ÊØµèµÒ ÐçÍÉâÐáñÞßÊÒîËïË æ¹ÊèÚÐ³ì¥Ò³é
ÃÒîÒÉ³Ø ËÛ×ÐáÜáñì ËÇÚÉ½ÒÂÐÞßÉ YÚåÖ«ÍÂÉ³Ò ÍÉ³ÒÖµáñË ÔÜÍÂÊÞ ÍÂÖµØµØµÛÜØµå ÊÌËÇÚÉÄÍ æ¹É³áñá ]ôØÚÊ¦æ Ø ÒÛÜÞÀÎÚáÜÉ ÒÊÌÍîËÂÛÜØµå
ÐáÜåÊÌÍÛñËÂÇµÞßÒ¤áÜVÛ ]É
ÊÌÍ
é ¹ÐØàæ¹Éïèµ
*
Ê _ ÉËËÂÉÄÍ
É³Ò³íÌæ¹ÉïÙ¬ÐçØ¾íÐçØÚèàÍÂÉÙ³Ö«ÍÂÒÛÜÊØ
Ì ¦ª îç£¦
°¤³£¦ îç£¦
ÇµÉ³áñÎÚÒÈÖÚÒ¹ËÂÊèÚÊÛñ ¬Ë ¾
10−9 n(n − 1)/2 Time
ÐÞßÊ èµÉÍÂØ Ó
é
ÛÜÒ
^ÛÜØ ÒîÉ³Ù³ÊØµèµÒ `òèµÛVU¥ÛñèÚÉè _ôì
ÐØµèóËÂÉ³áÜáñÒ·ÖµÒ·ÇµÊ¦æ
sec/Gcomp Time
Gcomp
Æ ÇµÉÎ ÐçÍäÐèµÛÜåÞ _ É³ÇµÛÜØµèëËÂÇµÉ
¹
ÐáÜåÊÌÍÂÛ²ËÂÇµÞ ÛñÒ ËÂÇµÛÜÒ ¾ÛÜÔµÐòÎµÍÂÊ _ÚáÜÉ³Þ ÛÜÒ ^°ËÂÊôÊ `
½
¨
³

î
£
«
Á

¦
£

á×ÐçÍåÉÈËÂÊ _ É·ÒîÊáVUÉ³èÀèµÛñÍÂÉÙËÂáñìíÒÖ _ èµÛVUôÛÜèµÉòÛñËÃÛÜØ¯ËÂÊâÒîÞ®ÐáÜáÜÉÍ ÒîÖa_µÎµÍÂÊ _ÚáÜÉ³ÞßÒ ËÂÇÚÐçËÐçÍÂÉÈÉ¬ÐÒîÛÜÉÍËÊâÒÊáVUÉí
ÐØµè ËÂÇµÉ³Ø ÎÚÖµËòËÂÇµÉëÊ UÉÍäÐáÜá¡ÒÊáÜÖ«ËÂÛÜÊØ ËÂÊåÉËÂÇµÉÍ ÔÜÍÊÞ ËÂÇµÉëÒÊáÜÖ«ËÂÛñÊØÚÒÈÊÔ ËÂÇµÉëÒÖ _µÎµÍÂÊ _ÚáÜÉ³ÞßÒ³éÈÆ¹ÇÚÛñÒ
Î ÐçÍäÐèµÛÜåÞ Ûñ
Ò ]ôØÚÊ¦æ ØÐÒ
é
F DÌN Fµß©

F ± ¡³£
·ÉÍÂÉ®ÛÜÒâÇÚÊ¦æ ËÂÇµÛÜÒãæ¹ÊÌÍ ]¥ÒàÔ×ÊÌÍ¿ÒÊÌÍËÛÜØµå«é ÉÄË ÖÚÒ ÛÜÞ®ÐçåÛÜØµÉ¿ËÂÇÚÐçËâËÂÇµÉßØôÖµb
Þ _ÉÍÂÒâËÂÊ _ ÉßÒîÊÌÍËÂÉ³è
Ù³ÊÞßÉÈÐÒÕÐÏèµÉ³Ù ]ÊÔÙ¬ÐçÍèÚÒí¯æ·ÛñËÂÇËÇÚÉÈØôÖµ
Þ _ ÉÍÂÒ æòÍÂÛñËËÂÉØ ÊØ ËÂÇµÉ³Þ é ÆÈÇµ
É YÚÍÒîËÒËÂÉÎßÛÜÒ ËÂÊÏÎÐÑÍËÂÛñËÂÛñÊØ
ËÂÇµÉ¿èÚÉÙ ] ÛÜØ¯ËÂÊ®Ë æ¹Ê ÒÞ®ÐáñáÜÉÍ èµÉ³Ù ]¥ÒãÊÔÇÚÐáÜÔ ËÂÇµÉ¿ÒÛ ³É¿É¬ÐÙäÇ¤é¿ÆÈÇµÉ³ÒÉ Ë æ¹ÊèµÉ³Ù ]¥ÒâÐçÍÂÉàËÂÇµÉ³ØÝÒîÊÌÍËÂÉ³è
ÛÜØµèµÉÎ É³ØµèµÉ³Ø¯ËÂáñì ÔÜÍÂÊÞ É¬ÐÙäÇ ÊÌËÂÇµÉÍ¬í æ·ÛñËÂÇóËÂÇµÉàÒÂÐçÞßÉâÞßÉËÂÇµÊ è _ÚÖµË·ËÂÇµÉàÞ®ÐÛÜØóÛÜØµåÌÍÂÉèÚÛñÉ³Ø Ë·ÊÔ ËÇÚÛñÒ
ÞßÉËÂÇµÊ è ÙÊÞßÉ³ÒÓÊØÚá²ì ØÚÊ¦æ æ¹É ÇÚXÐ UÉ¿ËÊ ÞßÉÍåÉ ËÇÚÉàË æ¹Ê ÒÊÌÍËÂÉ³è èÚÉÙ ]ôÒãÛñØ ËÂÊ ÊØµÉ¿ÒîÊÌÍËÂÉ³è½èÚÉÙ ]é
Ö«Ë·ËÂÇµÛÜÒÈÛÜÒÈØµÊÌË ÇÚÐçÍÂè Õæ¹ÉÏÎÚÖµËòËÂÇµÉãË æ¹Ê èµÉ³Ù ]ôÒ ÛÜØóÔÜÍÂÊØ¯Ë ÊÔ ÖµÒ ^ _ ÊÌËÂÇóØµÊ¦æ ÇÚXÐ UÉãËÂÇµÉ³Û²Í ÒÞ®ÐáñáÜÉ³ÒîË
Ù¬ÐçÍè½ÊØ ËÂÊÌÎ ` ÐçÒ áÜÊØµå®ÐÒ·ËÂÇµÉÍÉ ÐçÍÂÉàÒîËÂÛñáÜá¡Ù³ÐçÍÂèµÒ ÛÜØ ÊØµÉëÊÌ
Í _ ÊÌËÂÇ ÊÔ ËÂÇµÉëèµÉ³Ù ]¥Ò³íËÇÚÉàÒîÞ®ÐáÜáÜÉÍòÊÔ
ËÂÇµÉòË æ¹ÊâËÊÌÎ®Ù¬ÐçÍÂèµ
Ò ^ÊÌÍÕËÂÇµÉ·ÒÛñØÚåáñÉòÍÂÉ³Þ®ÐçÛÜØµÛÜØµåÓËÊÌÎ®Ù¬ÐçÍÂè `ÃÛñÒÕËäÐ ]ÉØßÊ ÐØµèÀÎÚÖ«Ë Ö«ÎÚÒÛÜèµÉòèµÊ¦æ ØßÊØ
ÐâØµÉæ èÚÉÙ ] ËÂÇÚÐçË ÛÜØÀËÇÚÉ ÉØÚèßÍÂÉÄÎÚÍÉ³ÒÉ³Ø¯ËÂÒ ËÂÇµÉ·ÍÂÉ³ÒîÖÚá²ËïÊÔ ËÂÇµÉ Ê UÉÄÍäÐáÜá ÒÊÌÍËÂÛÜØµåâÎµÍÂÊ Ù³É³ÒÒé [ ÛÜåÖ«ÍÂÉ U¥ÛñÒÖÚÐáÜÛ ³ÉÒ¹ËÇÚÉÏÞßÉÍÂåÉÓÒîËÉÎ¾é
·ÉÍÂÉ ÛÜÒ·ÇÚÊ¦æ
Ù³ÐØ _ÉëÍÂÉ¬ÐáÜÛ ³É³è ÛÜ&
Ø àí¡ÐçÒÒÖµÞßÛÜØµå ËÂÇÚÐçË æ¹ÉàÇÚXÐ UÉ¿ÐÀÔ×ÖµØÚÙÄËÂÛÜÊØ
ËÂÇÚÐçË ÎÉÍÂÔ×ÊÌ¨½
ÍÂÞß³Ò£îÁ«Ë ÇÚÉ¹Ð _£¦ Ê UÉïÊÌÎÉÍäÐçËÂÛñÊØ¿ÊÔµÞßÉÍåÛÜØµå·Ë æ¹ÊÏÒîÊÌÍËÂÉ³è¿ÒXÉ W ÖÚÉØÚÙÉ³ÒÕÛñØ ËÂÊ ÊØµÉïÒÊÌÍËÉ³è
merge
ÒXÉ W ÖµÉ³ØµÙ³ÉÌé
Merge-sort.
// PRE : [ first , last ) is a valid range
T \K
2
1
1
1
4
6
2
2
3
5
Figure 19:
4
6
4
3
2
1
3
2
1
3
5
4
6
3
4
5
6
5
³£îÁ°Á ¸Ó £¦GF Fµ± Â¶ß±Ä©£ F¯°¤Ý ç£¦IF FµÄ±
// POST : the elements *p , p in [ first , last ) are in ascending order
void merge_sort ( int * first , int * last )
{
int n = last - first ;
if ( n <= 1) return ;
// nothing to do
int * middle = first + n /2;
merge_sort ( first , middle ); // sort first half
merge_sort ( middle , last );
// sort second half
merge ( first , middle , last ); // merge both halfs
}
êÔÚËÇÚÉÄÍÂÉïÛÜÒ ÞßÊÌÍÂÉËÇ ÐçØ¿ÊØµÉïÉ³áñÉ³ÞßÉ³Ø Ë ËÂÊ ÒÊÌÍË¬íÌËÇÚÉ¹Ô×ÖµØµÙËÂÛÜÊØ ÒîÎÚáÜÛ²ËÂÒ ËÂÇµÉ ÍäÐØµåÉ
last)
Ð[first,
Øµè
éçÖµÒîË
ÛÜØ¯ËÂÊ Ë æ¹Ê ÍäÐØµåÉ³Ò
ÐØµè
ÊÔµáÜÉ³ØµåÌËÂÇµÒ
last)
ÐÒÐÏÍÂÉ³ÞßÛñØÚèµÉÍ³íÔ×ÊÌ[first,
Í¹ÐØ ì¿ÍÂÉ¬Ðçmiddle)
á Ø ÖÚÞ_ ÉÍ í [middle,
ÛÜÒ ËÂÇµÉ·ÒîÞ®
ÐáÜáÜÉ³ÒËÕÛñØ ËÂÉ³åÉÄÍåÌÍÂÉ³n/2
ÐçËÂÉÍÊÌÍ ÉXW Ö n/2
ÐçáÚËÂÊ ^ ÍÂÊÖµØµèµÉ³è ÖµÎ `ÄíÐØµè
ÛÜÒ·ËÂÇµÉëá×ÐÑÍÂåÉ³ÒîxË·ÛÜØ xËÂ ÉåÉÍ ÒÞ®ÐáñáÜÉÍòÊÌÍ ÉXWôÖÚÐá¡ËÂÊ
^! ÍÊÖÚØµèµÉ³è èµÊÑæ·xØ
`Äxé
êÔ ÛñÒÕÉ UÉØ¾í _ ÊÌËÂÇ UÐáÜÖµx É³Ò
ÐØµ
è ÐçÍÂÉÈÉXWôÖÚÐáµËÂÊ
í ÐØµèÀÊÌËÂxÇµÉÍîæ x ÛÜÒÉíËÇÚÉ YµÍÂÒîËUÌÐçáÜÖµÉ
n
n/2
n/2
n/2
ÛÜÒÈÒÞ®ÐáÜáñÉÍ _ôì ÊØµÉÌé
ÒÛ²ËÂÒØµIÉ PôËóÒîËÂÉÄÎ¡í ËÂÇµÉ ÐáÜåÊÌÍÂÛñËÇÚÞ ÍÂÉÙ³Ö«ÍÂÒVÛ UÉ³á²ì ÒÊÌÍîËÂÒ ËÂÇµÉ½É³áÜÉ³ÞßÉ³Ø¯ËÂÒ èÚÉÒÙÍÂVÛ _É³è _ôì _ ÊÌËÂÇ
ÍäÐØµåÉ³ÒéßêØÝËÂÇµÉ É³Øµè¤í Û²ËâÙ¬ÐçáÜáÜÒÓËÂÇµÉÔ×ÖµØµÙËÂÛÜÊØ
ÊØÝËÂÇµÉ Ë æ¹Ê ÍäÐØµåÉ³Ò³éßêØ ÙÊÞßÞßÉ³Ø ËÂÛñØÚåÀËÇÚÉ
merge
á×ÐçËîËÂÉÍ®Ô×ÖµØµÙËÂÛÜÊØ¤í¹æ¹É ÒËÂÛÜÙ ] ËÂÊ ËÂÇµÉ èÚÉÙ ] ÐØÚÐáñÊåÌìËÂÇÚÐçËßæ¹É ÇÚXÐ UÉ½ÖµÒÉ³è Ð _ Ê UÉÌé êÔÓìÊÖ ÇÚXÐ UÉ
ÖµØµèÚÉÄÍÂÒîËÂÊ Êôè ËÇÚÉÏèµÉ³Ù ]ÞßÉÄÍÂåÛÜØµåëÎµÍÂÊôÙ³ÉÒÒ³íµìÊÖæ ÛÜáñáÎ ÉÍÂÙÉ³VÛ UÉÓËÂÇµÉãèµÉ YÚØµÛñËÂÛÜÊØ ÊÔ
ÐÒ _ É³ÛñØÚå
merge
ÒîËÍÂÐÛÜåÇ ËÔ×ÊÌÍæÈÐçÍÂè¤é
// PRE : [ first , middle ) , [ middle , last ) are valid ranges ; in
//
both of them , the elements are in ascending order
void merge ( int * first , int * middle , int * last )
{
int n = last - first ;
// total number of cards
int * deck = new int [ n ]; // new deck to be built
int * left = first ;
// top card of left deck
int * right = middle ; // top card of right deck
for ( int * d = deck ; d != deck + n ; ++ d )
// put next card onto new deck
\
if
( left == middle )
else if ( right == last )
else if (* left < * right )
else
*d
*d
*d
*d
=
=
=
=
* right ++;
* left ++;
* left ++;
* right ++;
//
//
//
//
left deck is empty
right deck is empty
smaller top card left
smaller top card right
// copy new deck back into [ first , last )
int * d = deck ;
while ( first != middle ) * first ++ = * d ++;
while ( middle != last ) * middle ++ = * d ++;
delete [] deck ;
}
Ò Ô×ÊÌÍ
Ï
í æ¹É æ ÛÜáñá Ù³ÊÖµØ ËÓËÂÇµÉ¿ØôÖµÞb_ÉÍÓÊÔÙ³ÊÞÀÎ ÐçÍÛÜÒÊØµÒ
¨
»

¨
î

¦
£

_ ÉË æ¹É³ÉØ ÒîÉXWôÖµÉ³ØµÙ³É®É³áÜÉÞßÉ³Ø ËÂÒÏËÂÇÚÐçËëÊ Ù³Ù³Ö«Íëæ·ÇÚÉØ Ð ÒÉXW ÖÚÉØÚÙÉ ÊÔ
Ø ÖÚÞ_ ÉÍÒëÛÜÒ _É³ÛÜØµåóÒÊÌÍËÉ³è¤é
n
åÐÛñØ¾í æ¹ÉãÙ¬ÐçØóÐçÍÂåÖµÉÓËÂÇÚÐçË¹ËÂÇµÉ ËÂÊÌËäÐá Ø ÖÚÞ_ ÉÍÈÊÔ¡ÊÌÎ ÉÍäÐçËÛÜÊØµÒïÛÜÒïÎÚÍÊÌÎ ÊÌÍËÂÛÜÊØÚÐá ËÂÊ¿ËÇÚÛñÒÈØ ÖµÞb_ ÉÄÍ
ÊÔÈÙ³ÊÞÀÎ ÐçÍÂÛÜÒÊØµÒ³é [ ÊÌÍ
íÕËÇÚÛñÒëÔªÐÙË¿ÛÜÒàØµÊÌË ÒÊ ÛÜÞßÞßÉ³èµÛ×ÐÑËÂÉÌí¤ËÂÇµÊÖÚåÇ¤í ÐØµè æ¹É®èÚÊØ Ë
½
¨
³

î
£
«
Á

¦
£

ÉIPôÎ É³ÙËãìÊÖ ËÂÊÖµØµèµÉÍÂÒîËÂÐØµèÝÛñËÏØµÊÑæâé Ö«ËãÔ×ÊÌÍâËÂÇµ
É _ É³ØµÉ YµËãÊ
Ô ^ØµÊÌËãÊØÚá²aì `·ËÂÇµÉÒîÙ³ÉÎµËÂÛÜÙ ÍÂÉ¬ÐèµÉÍ³í
æ¹ÉÓæ ÛÜáÜá ÙäÇµÉ³Ù ] ËÂÇµÛÜÒòÔªÐçÙË·GÉ P¥Î ÉÍÛÜÞßÉ³Ø ËÂÐáÜáñì _ É³áÜÊ¦æëí«ÐçÒòæ¹ÉÏèÚÛñèÔ×ÊÌÍ
é
¨
»

¨
î

¦
£

áÜá¤ËÂÇµÉâÙ³ÊÞÀÎ ÐçÍÛÜÒÊØµÒïËäÐ ]ÉâÎÚá×ÐÙÉâèµÖ«ÍÂÛÜØµåËÂÇµÉãÙ¬ÐáÜáÜÒ¹ËÂÊ ËÇÚÉâÔ×ÖÚØµÙËÛÜÊØ
ÐÑË·ËÂÇµÉ UÐçÍÂÛñÊÖÚÒ
merge
áÜÉ UÉáÜÒ ÊÔòÍÂÉ³ÙÖµÍÒÛÜÊØ¤íÒÊ½áÜÉË Öµ
Ò YÚÍÒîË ÙÊÖÚØ¯Ë ËÂÇµÉ Ø ÖÚ
Þ _ ÉÍ ÊÔòÙ³ÊÞÀÎ ÐçÍÂÛÜÒîÊØÚZ
Ò _ÉË æ¹É³É³Ø ÒÉ WôÖµÉ³ØµÙ³É
É³áÜÉÞßÉ³Ø ËÂÒ ËÂÇÚÐçËÕÊØµÉ·Ù¬ÐáñáµËÊ
ÎÉÍÂÔ×ÊÌÍÂÞßÒÃÛÜØÀÊÌÍÂèµÉÍÕËÂÊâÙÄÍÂÉ¬ÐçËÂÉ·ÐãÒÊÌÍîËÂÉ³èßèµÉ³Ù ]ÀÊÔ Ù³ÐçÍÂèµÒÔÜÍÂÊÞ
merge
n
Ë æ¹Ê ÒÊÌÍîËÂÉ³èóèµÉ³Ù ]¥Òé
ê»Ë·ÛñÒòÐÑÎÚÎ ÐçÍÉ³Ø ËÈÔÜÍÂÊÞ ËÂÇµÉÏÔ×ÖÚØµÙËÛÜÊØ _Êôè«ì ^ºÐØµèóÐçáÜÒÊ¿ÔÜÍÂÊÞ ÊÖµÍòÛÜØµÔ×ÊÌÍÂÞ®Ðá èµÉ³ÒÙÍÂÛ²ÎÚËÛÜÊØ ÊÔ¾ËÂÇµÉ
ÞßÉÍÂåÛñØÚåßÎµÍÂÊ Ù³É³ÒÒàÐ _ Ê UKÉ `ÓËÂÇÚÐçË
ÙÊÞÀÎÐÑÍÂÛÜÒÊØ ÛÜÒÏØÚÉÉ³èµÉ³è Ô×ÊÌÍëÉ UÉÍîìÝÙ¬ÐçÍèÝËÂÇÚÐçËÏÛÜÒ
¯
©
â

ÎÚÖ«Ë ÊØ ËÂÇµÉ®ØµÉæ èµÉ³Ù ]é êØµèµÉ³É³è¤í æ¹ÉßÞ®Ð³ì ÇÚKÐ UÉßËÂÊ Ù³ÊÞÀÎ ÐçÍÂÉËÂÇµÉ®Ë æ¹Ê ËÂÊÌÎ Ù¬ÐçÍÂèµÒ ÊÔòËÇÚÉßáñÉ³ÔÜË
ÐØµèÝËÂÇµÉ ÍÂÛñåÇ ËãèÚÉÙ ]ÝÛÜØÝÊÌÍÂèµÉÍëËÊ YÚØÚè ÊÖ«Ëãæ ÇµÛÜÙäÇ Ù¬ÐçÍÂèÝËÊóËäÐ ]É Ê ØµIÉ P¥Ë¬é Ö«ËëÛñÔïÊØÚÉÊÔ ËÂÇµÉ
Ë æ¹Ê èµÉ³Ù ]¥
Ò _ É³Ù³ÊÞßÉÒÈÉÞÀÎÚË ì ^»ËÇÚÛñÒÈÒîÛñËÂÖÚÐçËÂÛÜÊØ®èµÉ YÚØµÛñËÂÉ³á²ì®ÊôÙ³Ù³Ö«ÍÂÒ _ É³Ô×ÊÌÍÉÏËÂÇµÉÏá×ÐÒîËÈÙ¬ÐÑÍÂèóÛÜÒ¹ÎÚÖµËÈÊØ
ËÂÇµÉ ØµÉæ èµÉ³Ù ]a`´í¡æ¹É èµÊ
Ø ËãèµÊóÐØ¯ì Ô×Ö«ÍËÇÚÉÄÍëÙ³ÊÞÀÎ ÐçÍÂÛñÒÊØµÒ³é¿ÆÈÇµÛÜÒÏÞßÉ¬ÐØµÒÓËÂÇÚÐçË
Ù³ÊÞÀÎ ÐçÍÂÛñÒÊØµÒ _ ÉË æ¹É³ÉØ ÒÉ WôÖµÉ³ØµÙ³ÉÏÉ³áÜÉ³ÞßÉ³Ø¯ËÂÒïÐçÍÉ Î ÉÍÂÔ×ÊÌÍÂÞßÉè ÛñØ ÞßÉÄÍÂåÛÜØµåâË æ¹Ê ÒÊÌÍ©¯ËÂÏÉ³è ¨ èµÉ³³Ù ]¥n
Ò¹ÛÜ−
Ø¯ËÂ1Ê
ÊØµÉãÒîÊÌÍËÂÉ³èóèµÉ³Ù ] æ ÛñËÇ
Ù¬ÐÑÍÂèµÒ³é
ØÚÊ¦æ ÛñØÚå ËÂÇµÛÜÒ³í¥æ¹ÉãÙ¬nÐØóØµÊ¦æ ÎµÍÂÊ UÉãÊÖ«Í·Þ®ÐÛñØ ÍÂÉÒÖµáñË¬é
Analyzing merge-sort.
Theorem 2 ÕÃ¶Ä ±¬ merge_sort ç£¦ºâ©À ± Â¶ n
(n − 1)
×áÜÊå
2
1 ¬§¨ ³¤ºÓ¸¹º ©¯Ã¨½³
n
±¨¿¢©£¬ç¤® ¬¸Ó³¼ ³ ±Äó¬ª³¨ ¤ºçÅ
S ÉèµÉ YÚØµÉ
ËÂÊ _ É ËÂÇµÉ
Î ÊÒÒÛ _ áñÉ¿Ø ÖµÞb_ ÉÄÍãÊÔÙ³ÊÞÀÎ ÐçÍÂÛÜÒÊØµÒ_ ÉÄË æ¹ÉÉ³Ø
Proof.
¨Ð ¨ Ù³ÐáÜá ËÂÊ
ÒXÉ W ÖµÉ³ØµÙ³É É³áÜÉ³ÞßÉ³Ø¯ËÂÒ·ËÂTÇÚ(n)
ÐçË Ù¬ÐØ ÊôÙ³ÙÖµÍÓèµ¨½
Ö«ÍÂ©ÛÜØµ»å®
æ ÛñËÇ ÐØ ÐçÍÂåÖµÞßÉ³Ø ËòÍäÐØµåÉ
merge_sort
ÊÔ áÜÉ³ØµåÌËÂÇ é [ ÊÌÍÀGÉ PµÐÞÀÎÚáÜÉÌí
íÒÛÜØµÙ³ÉóÔ×ÊÌÍÀÍÂÐØµåÉ³Ò ÊÔ áÜÉØÚåÌËÇÚÒ
ÐØµè í ØµÊ
n
T (0) = T (1) = 0
0
1
Ù³ÊÞÀÎ ÐçÍÂÛñÒÊØµÒ ÐçÍÂÉâÞ®ÐèµÉÌ
é S É ÐáÜÒÊßåÉË
íÒÛÜØµÙ³ÉàÔ×ÊÌÍÏÐ¿ÍäÐØµåÉàÊÔ áÜÉ³ØµåÌËÂÇ í
T (2) = 1
Î ÉÍÔ×ÊÌÍÂÞßÒòÊØµÉÏÙ³ÊÞÀÎ ÐçÍÂÛÜÒîÊØ ^ÛñØóÞßÉÍÂåÛÜØµåà
Ë æ¹Ê ÒÊÌÍîËÂÉ³è èµÉ³Ù ]¥Ò ÊÔ ÊØÚÉÏÙ¬ÐçÍè É¬ÐÙäÇ ÛÜØ 2ËÊ¨½ÊØµ³£îÉâÁ«Òî ÊÌÍËÂÉ³£¦è
T \
èµÉ³Ù ]ÝÊÔ Ë æ¹ÊÙ³ÐçÍÂèµÒ `Äé êØ Ð ÒîÛÜÞßÛÜá×ÐçÍ·æÈÐ¬ìí¾æ¹ÉÀÙ¬ÐØ ÙÊØ U¥ÛñØÚÙÉ ÊÖ«ÍÂÒÉ³á UÉ³ÒâËÂÇÚÐçË
é®ÆÈÇµÉÍÂÉ
T (3) = 2
ÒîXÉ WôÖµÉ³ØµÙ³É³ÒÈÊÔ¤áÜÉ³ØµåÌËÂÇ Ô×ÊÌÍïæ·ÇÚÛñÙ´Ç®ÊØµÉ Ù³ÊÞÀÎ ÐçÍÂÛñÒÊØßÒÖ Ù³ÉÒ ^°ËÂÇµÉYµÍÂÒîË¹Ù¬ÐçÍè Þ®Ð³ìb_ É ËäÐ ]É³Ø
©Ê
£³ ËÇÚÉ·áÜÉ³ÔÜË èµÉ³Ù ]Àæ·ÇµÛÜÙäÇßÙ³Ê3ØµÒÛÜÒîËÂÒÕÊØµáñì ÊÔ ÊØµÉ Ù³ÐçÍÂè `Äí _ÚÖ«Ë ËÂÇµÉòÞ®Ð P«ÛÜÞ¿ÖµÞ ØôÖµÞ_ ÉÍËÇ ÐÑË èµÉ YÚØµÉ³Ò
ÛÜÒ é
T (3) 2
[ ÊÌÍ åÉ³ØµÉÍäÐçá
í«æ¹ÉãÇÚXÐ UÉãËÂÇµÉÏÔ×ÊáÜáÜÊ¦æ ÛÜØµåëÍÂÉÙ³Ö«ÍÍÂÉ³ØµÙ³ÉâÍÉ³á×ÐçËÂÛñÊØ
n ) + T ( n ) + n − 1.
n
T (n)
T(
2
2
2
^ é `
Æ Ê ÒîÉ³ÉâËÂÇµÛÜÒ³í«áñÉË·ÖµÒòÙ³ÊØµÒÛÜèµÉÍ·Ð¿ÒÉ WôÖµÉ³ØµÙ³ÉëÊÔ ÉáÜÉ³ÞßÉ³Ø¯ËÂÒïËÂÇÚÐçË ÐçÙËÂÖÚÐáÜáñì ÍÂÉXWôÖµÛñÍÉ³ÒòËÂÇµÉÏÞ®Ð P(
ÛÜÞ¿ÖµÞ ØôÖµÞ_ ÉÍ®ÊÔ
Ù³ÊÞÀÎ ÐçÍÂÛÜÒÊØµÒ³é Æ¹ÇÚÛñÒ Ø nÖµÞb_ ÉÄÍ®ÊÔâÙ³ÊÞÀÎ ÐçÍÂÛñÒÊØµÒÀÛÜÒÀËÂÇµÉ ÒÖµÞ ÊÔÓËÂÇµÉ
T (n)
ÍÂÉ³ÒÎ É³ÙËÂÛ UÉòØ ÖÚÞ_ ÉÍÒÕÛÜØÒÊÌÍËÂÛñØÚåÓËÂÇµÉÈáÜÉ³ÔÜËÕÐØµè ËÂÇµÉ¹ÍÂÛÜåÇ ËÃÇÚÐáÜÔºíÎÚáÜÖµÒ ËÂÇµÉòØ ÖµÞb_ ÉÄÍÕÊÔ Ù³ÊÞÀÎ ÐçÍÂÛÜÒîÊØÚÒ
èµÖ«ÍÂÛÜØµåÏËÇÚÉÈÞßÉÍÂåÉ¹ÒîËÉÎ¾é ÆÈÇµÉÈÔ×ÊÌÍÂÞßÉÍÃË æ¹ÊÏØôÖµÞ_ ÉÍÂÒÃÐçÍÉ ^ _ ì ÙÊØÚÒËÍÂÖµÙËÂÛÜÊØ¿ÊÔ
ÐçØÚè
èµÉ YÚØµÛñËÂÛÜÊØÊÔ `ÕÐÑË ÞßÊÒîË
ÐØµè
íæ ÇµÛÜáÜÉ¹ËÂÇµÉ·á×ÐÑËËÂÉÍÕØôÖµÞ_ ÉÍÕmerge_sort
ÛÜÒÐÑË ÞßÊÒîË
n−1
_ôìÀÊÖµÍÎÚÍÉ U¥ÛÜÊTÖµÒ ÙÊØÚÒîÛÜèµÉÍäTÐç(ËÂÛñn/2
ÊØÚÒÃÍÂ)É³åÐçÍèÚÛñTØÚ(å n/2 ) éÃê»ËïÔ×ÊáÜáÜÊ¦æ ÒÃËÂÇÚÐçË
íôËÂÇµÉ ÐÙÄËÂÖÚÐáØ ÖÚÞ
_ ÉÍ
merge
T (n)
ÊÔ ÙÊÞÀÎÐÑÍÂÛÜÒÊØµÒ³í¥Ûñ
Ò _ ÊÖµØµèµÉ³è _ ì ËÇÚÉÏÒÖµÞ ÊÔ ÐáÜáËÂÇ«ÍÂÉÉãØ ÖÚ
Þ _ ÉÍÒ³é
O Ê¦æ æ¹É®Ù¬ÐØ ÎµÍÂÊ UÉßËÂÇµÉ ÐÙËÂÖÚÐá ÒËäÐçËÂÉ³ÞßÉØ ËàÊÔ¹ËÂÇµÉßËÂÇµÉ³ÊÌÍÂÉ³Þ é " ÛÜØµÙ³ÉßËÂÇµÉ
Ðá ¨

î
£
«
Á

¦
£

åÊÌÍÂÛñËÇÚÞ ÛñÒ ÍÂÉÙ³Ö«ÍÂÒVÛ UÉÌíÓÛñË ÛÜÒ ØÚÐçËÂÖ«ÍäÐá ËÇ ÐÑË ËÇÚÉ ÎµÍÂÊôÊÔãÛÜÒ®ÛÜØµèÚÖµÙËVÛ UÉÌé [ ÊÌÍ
í·æ¹É ÇÚXÐ UÉ
n = 1
×áÜÊå
íµÒÊËÂÇµÉãÒîËÂÐçËÂÉ³ÞßÉ³Ø¯Ë¹ÇµÊáÜèµÒòÔ×ÊÌÍ
é
T (1) = 0 = (1 − 1)
[ ÊÌÍ
í¾áÜÉËãÖÚÒãÐ2Ò2ÒîÖÚ ÞßÉ ËÂÇÚÐçËÓËÂÇµÉ¿ÒîËäÐçËÂÉÞßÉ³Ø ËÏÊÔËÂÇµnÉ =
ËÂÇµ1É³ÊÌÍÂÉ³Þ ÇµÊáÜèµÒÓÔ×ÊÌÍ
UÐáÜÖµÉ³ÒÏÛÜØ
n
2
¯
©
»

^»ËÂÇµÛÜÒÛÜÒËÂÇµÉ ÛÜØµèµÖµÙËÂVÛ UÉ Ç¯ìôÎ ÊÌËÇÚÉÒÛÜÒ `Äé [ ÍÂÊÞ ËÇÚÛñÒ Ç ì Î ÊÌËÂÇµÉ³ÒÛÜÒí æ¹É ØµÉ³É³è®ËÂÊ èµÉÍÂVÛ UÉ
{1, . . . , n − 1}
ËÂÇµ
É UÌÐáñÛÜèµÛñË ìßÊÔ¡ËÂÇµÉÏÒîËäÐçËÉ³ÞßÉ³Ø Ë¹Ô×ÊÌÍÈËÂÇµÉÏØôÖµb
Þ _ÉÍ ÛñËÂÒÉáÜÔ ^»ØÚÊÌËÉÓËÇ ÐÑË
`ÄéÕÆÈÇµÛÜÒ
n
n/2 , n/2
1
åÊôÉÒ·ÐÒÈÔ×ÊáÜáÜÊ¦æ Òé
T (n)
n
^
W ÖÚÐçËÂÛÜÊØ é ` `
n
T(
) + T( ) + n − 1
2
2
×áÜÊå
n
×áñÊå n
^ÛÜØµèµÖµÙËÂÛVUÉÓÇ¯ìôÎ ÊÌËÂÇµÉ³ÒîÛÜÒ `
n
n
+(
− 1)
+n−1
(
− 1)
2
2
2
2
2
2
n
×áñÊå
n
×áÜÊå
^ P«ÉÍÂÙÛÜÒÉ `
− 1)(
(
− 1)(
2 n − 1) + (
2 n − 1) + n − 1
2
2 ^
×áñÊå
`
= (n − 2)(
n = n2 + n2 2 n − 1) + n − 1
×áÜÊå
(n − 1)(
2 n − 1) + n − 1
×áÜÊå
= (n − 1)
2 n .
Ò Ô×ÊÌÍ
íáÜÉË ÖµÒ Ù³ÊØµÙ³áÜÖµèµÉïæ ÛñËÂÇ ÒÊÞßÉïÉGP¥Î ÉÍÛÜÞßÉ³Ø ËÒ ËÂÊÏÙ´ÇµÉ³Ù ] æ ÇµÉËÇÚÉÄÍ ËÂÇµÉÈØ ÖµÞb_ ÉÄÍ

¨
»

¦
£

ÊÔ¾Ù³ÊÞÀÎ ÐçÍÂÛÜÒÊØµÒ _ÉË æ¹É³É³ØÒîÉXWôÖµÉ³ØµÙ³ÉÏÉ³áÜÉ³ÞßÉØ ËÂÒïÛÜÒïÛñØÚèµÉ³ÉèóÐëåÊôÊ è ÛñØÚèµÛÜÙ³ÐçËÂÊÌÍïÔ×ÊÌÍ¹ËÂÇµÉ ÍÂÖµØ ËÂÛñÞßÉ ÛÜØ
ÎµÍäÐÙËÛÜÙ³ÉÌéÃÆ¹ÇÚÉÏÍÂÉ³ÒÖµáñËÒòÛÜØÆ Ð _ÚáÜÉ áñÊôÊ ] UÉÍì èµÛ ÉÄÍÂÉ³Ø ËòÔÜÍÂÊÞ ËÂÇµÉãÊØµÉ³ÒòÛÜØóÆ Ð _ÚáÜÉ é
" ÛÜØµÙ³É
ÛÜØµÙ³Ö«ÍÂÒÞÖµÙäÇ®áÜÉ³ÒÒïÙ³ÊÞÀÎ ÐçÍÛÜÒÊØµÒÕËÂÇÚÐØ
í¥ÊÖ«Í¹ÖÚØµÛñËÇµÉÍÂÉ ÛÜÒ
merge_sort
ÖµÒîË
íÌËÂÇµÉØ ÖÚ
Þ _ ÉÍ¡ÊÔ )É³åÐÙÊÞÀÎÐÑÍÂÛÜÒÊØµ
Ò ^ 6 Ù³ÊÞÀÎ ÐçÍÂÛñminimum_sort
ÒÊØµÒ `ÄíçÐÙ³Ù³ÊÌÍÂèµÛÜØµåÈËÂÊ Æ¹ÇµÉ³ÊÌÍÂÉ³Þ \ é
êØâÊÌËMcomp
ÇÚÉÄÍ¡æ¹ÊÌÍÂèµÒ³í
×áñÊå
é 10 ÛÜÒ¤ËÂÇµÉÃÐ _ÚÒÊáÜÖ«ËÂÉ ÍÂÖµØ ËÛÜÞßÉ ÊÔ ËÇÚÉ ÎµÍÂÊåÌÍäÐçÞ í
−6
ËÂÇµÛÜÒ·ËÛÜÞßÉâÛÜØ ÒÉÙ³Mcomp=
ÊØµèÚÒÏÐØµè 10ØµÊÌË (n−1)
ÞßÛÜØ ÖµËÉ³Ò³é 2nÖ« ËÓÐTime
Ò ÛñØ½Æ Ð _ÚáÜÉ í
ËÂÉ³áÜáñÒ ÖµÒ ÇÚÊ¦æ Þ®ÐçØ ì
ÒÉ³ÙÊØÚèµÒÈËÂÇµÉÏÎµÍÂÊåÌÍäÐçÞ ØµÉ³ÉèÚÒÈËÂÊÎÉÍÂÔ×ÊÌÍÂÞ ÊØµÉ ÛÜåÐÙ³ÊÞÀÎ ÐçÍÂÛñÒÊØ¤sec/Gcomp
é
SÝ
É YÚÍÒîË Ê _ÚÒÉÍ UÉ·ËÂÇÚÐçËÃËÂÇµÛÜÒÕáÜÐçËËÂÉÍÕØôÖµ
Þ _ ÉÍ
æ ÛñËÇ í¯æ·ÇµÉÍÂÉòËÇÚÉÈÍäÐçËÂÉòÊÔèµÉ³ÙÍÂÉ³ÐÒÉ
µ
F

±
³
£
Ä

¯
©

_ É³ÙÊÞßÉ³ÒÕÒîÞ®ÐáÜáÜÉÍÃÐçØÚèßÒîÞ®ÐáÜáÜÉÍ³é ØßÊÖµÍÕÎÚá×ÐçËÂÔ×ÊÌÍÂÞ íÌæ¹ÉòÙ¬ÐØßåÊãÖµn
ÎßËÊâÍÂÊÖµåÇµáñì
n = 51, 200, 000
ÐØµè YÚØµè ËÂÇÚÐçË
Ù³ÊØ¯ËÂÛÜØ ÖÚÉÒòáÜÛ ]ÉÏËÂÇµÛÜÒ¹ÛÜØÆ Ð _ÚáÜÉ
í í K í \ í \ é
sec/Gcomp
\
í é
n
Mcomp
Time (sec)
sec/Gcomp
é a
\ í é
é \
í é
í é\
\ é \ ¨½ ³£óÂ¶ß±¨¿¢©£¬¤ß©F £¬ ¤»¨ Â¶
Table 8:
í í é
é ÞßÉÍÂåÉ ÒÊÌÍË
ÆÈÇµÛÜÒ ÒÉÉ³ÞßÒ ËÂÊÓÛÜØµèÚÛñÙ¬ÐçËÂÉ ËÂÇÚÐçË ËÇÚÉïÍÂÖµØ¯ËÂÛÜÞßÉ ÛÜÒ ÎµÍÂÊÌÎ ÊÌÍîËÂÛÜÊØÚÐá ËÊÏËÂÇµÉ¹ØôÖµÞ_ ÉÍ ÊÔÚÙ³ÊÞÀÎ ÐçÍÂÛÜÒîÊØÚÒ
Ê ØµáñìÝÔ×ÊÌÍ UÉÄÍì á×ÐçÍÂåÉ é êÔ¹ìÊÖ ËÂÇµÛÜØ ] Ð _ ÊÖ«ËàÛñË¬í ËÇÚÛñÒëÛÜÒëØµÊÌË ÒÖ«ÍÎµÍÂÛÜÒîÛÜØµå«é ïÇÚÉ³ÐçÎÊÌÎÉÍäÐçËÂÛñÊØÚÒ
ËÂÇÚÐçËÀÐçÍÂÉÎ ÉÄÍÂÔ×ÊÌÍÂÞßÉ³è n ËÂÛÜÞßÉÒ³íÒÂÐ³ìí¹É¬ÐçËÀÖµÎ¼Ð½ÞÖµÙäÇ ÇµÛÜåÇµÉÍÀÔÜÍäÐçÙËÂÛÜÊØ ÊÔ ËÂÇµÉËÊÌËäÐá¹ÍÂÖµØ ËÂÛÜÞßÉ
æ ÇµÉ³Ø
ÛñÒ ÒÞ®ÐáÜáé ÆÈÇµnÛÜÒ ÛÜÒ_É³Ù¬ÐÖµÒÉ
ÛÜÒ ÍÂÉ³á×ÐÑËÂÛVUÉ³áñì á×ÐçÍåÉàÙ³ÊÞÀÎ ÐçÍÂÉ³è ËÂÊßËÂÇµÉ Ö«ÎÚÎÉÍ_ ÊÖµØµè½ÊÔ
n
×áñÊå
ÊØ ËÂÇµÉ Ø ÖµÞb_ ÉÄÍ ÊÔ·nÙ³ÊÞÀÎ ÐçÍÂÛñÒÊØµÒ _ ÉË æ¹É³É³ØÒÉXWôÖµÉ³ØµÙ³É É³áÜÉÞßÉ³Ø ËÂÒé Ö«ËÒîÛÜØµÙ³É
(n − 1)
n
æ¹ÉÓÛÜåØµÊÌÍÂÉ ËÂÇµ2 ÉÓ
ÙäÇÚÉ³ÐçÎóÊÌÎ ÉÄÍäÐçËÂÛÜÊØµÒ ÛñØ ÊÖ«ÍÈÙ³ÊÞÀÎ ÐçÍÂÛÜÒîÊØßÙ³ÊÖµØ Ë¬í¥ËÂÇµÛÜÒïÙÊÖÚØ¯Ë¹ÛÜÒïËÂÊôÊ ÊÌÎµËÂÛÜÞßÛñÒîËÂÛÜÙÈÔ×ÊÌÍ
_ É³Ù³ÊÞßÉÒ
ÒÞ®Ðáñá é ØµáñìÀÐçÒ _ ÉÙ³ÊÞßÉ³
Ò UÉÍìÀáÜÐçÍÂåÉÌíËÂÇµÉòÍÂÐçËÂÛÜÊ _ ÉË æ¹É³É³Ø
ÐØµè
×áñÊå
n
n
n
(n − 1)
n
2
ØµÉ³åáÜÛÜåÛ _ áñÉ ÐØµèæ¹ÉÏÒîËäÐçÍËÈËÂÊÒÉ³ÉãËÇÚÉÓÎµÍÂÉ³èµÛÜÙËÉ³è ÎµÍÂÊÌÎ ÊÌÍîËÂÛÜÊØÚÐáÜÛñË ìé
[ ÊÌÍ
íËÂÇµÛÜÒ·ÎÚÇµÉ³ØµÊÞßÉ³ØµÊØ èÚÊ É³Ò ØµÊÌË ÒÇµÊ¦æ ÒÛÜØµÙ³É
ÛÜÒ ØµÉ³åáÜÛÜåÛ _ áñÉãÙ³ÊÞÀÎ ÐçÍÉ³è
n

¨
»

¨
î
ç

¦
£

ËÂÊ
ÐáñÍÂÉ³Ðè«ì®Ô×ÊÌÍ ÒÞ®Ðáñá é
n(n − 1)/2
n
ÆÈÇµÉãÞßÊÒîËòÎÊÒÛñËÂVÛ UÉÓØµÉæ·Ò·ÊÔ Æ Ð _ áñÉ ÛÜÒÈËÇ ÐÑË
ÛÜÒòÐÙËÂÖÚÐáÜá²ì Ð ÎµÍäÐçÙËÂÛÜÙ¬Ðçá¾ÒîÊÌÍËÂÛÜØµå
merge_sort
ÐáÜåÊÌÍÛñËÂÇµÞ é S ÇµÛÜáñÉ ÛñË®ËäÐ ]É³Ò
ÞßÊÌÍÂÉ ËÂÇÚÐØ Ë æ¹Ê ÇµÊÖ«ÍÂÒ ËÊ ÎµÍÂÊ Ù³É³ÒÒ
minimum_sort
ØôÖµ
Þ _ ÉÍÂÒí
èµÊôÉÒ ËÇÚÉ ÒÂÐÞßÉßÛñØ ÐçÍÂÊÖµØµè ÒÉ³ÙÊØÚèµÒ³é Æ¹ÇµÛÜÒ ÛÜÒ Þ®ÐÛÜØµáñì½èµ1,ÖµÉ 600,
ËÂÊ 000
ËÂÇµÉ
ÔªÐÙË ËÂÇÚÐçË merge_sort
×áÜÊå
ÛÜÒÀÐ ÞÖµÙäÇ ÒÞ®ÐáñáÜÉÍØôÖµb
Þ _ÉÍ ËÂÇÚÐØ
í ËÂÇµÉóØôÖµ
Þ _ ÉÍ ÊÔ
Ù³ÊÞÀÎ ÐçÍÂÛñÒÊ(n
ØµÒÃØµ−É³É1)èÚÉL
è _ô2ì n
^»ËÂÇÚÐçË Ò æ Ç ì æ¹É Òæ ÛñËÂÙäÇµn(n
É³èÀÔÜÍ−
ÊÞ 1)/2ÙÊÞÀÎ ËÂÊ )ÙÊÞÀÎ `´é
minimum_sort
Ø¿ËÂÇµÉïÊÌËÂÇµÉÍ ÇÚÐØµè¤íÌËÂÇµÉ ËÂÛÜÞßÉØµÉ³ÉèÚÉb
è _ ì
Î ÉÍ Ù³ÊÞÀÎ¿ÛÜÒ è«ÍäÐÞ®ÐçËÛÜÙ¬ÐáñáñìÓÇµÛÜåÇµÉÍ
merge_sort
ËÂÇÚÐØ ÛÜØ
Ô×ÊÌÍ
í¥æ¹ÉëÊ _ÚÒÉÍ UÉ Ð¿ÔªÐÙËÂÊÌÍ ÊÔ ÐÑÍÂÊÖµØÚè
éÈê»Ë Þ®Ð³
ì _ É
ÒÖ«ÍÎµÍÂÛÜÒîÛÜØµminimum-sort
å ËÂÇÚÐçËÏËÂÇµÉ ÔªÐÙÄËÂÊÌÍëÛÜn
ÒÓËÂ=ÇµÛÜ1,
Òãá×600,
ÐçÍÂåÉÌ000
í _ÚÖ«ËãËÂÇµÉÔªÐÙËëËÂÇÚÐçËãÛñË
á×ÐçÍåÉÍâÙ³ÐØ 50
_ É GÉ P¥ÎÚá×ÐÛÜØµÉ³è¤é
íÑæ ÛñËÂÇÏÛñËÂÒ¤ÍÂÉ³ÙÖµÍÒVÛ UÉÒËÍÂÖµÙ ÛÜÒ¾ÐÈÞßÊÌÍÉÃÙÊÞÀÎ áñÛÜÙ¬ÐçËÂÉèãÐçáÜåÊÌÍÂÛñËÂÇµÞ ËÇ ÐçØ
merge_sort
ËÂÖ«ÍÂÉÌí¥ËÇÚÉÓIÉ PôËÍäÐëÞßÉ³ÞßÊÌÍîìÀØµÉ³É³èµÉ³è Ô×ÊÌÍÈËÇÚÉÓØµÉæ èµÉ³Ù ]íµminimum-sort
ÉËÙÌéÃÆÈÇµÉÓÎÚÍÛÜÙ³É ËÂÊ Î Ð¬ì®ÛñÒ ËÂÇÚÐçË¹áÜÉ³ÒÒ¹Ù³Ê
Þ Î ÐçÍÂÛÜÒîÊØÚÒ¹Ù¬ÐçØ _ ÉÏèÚÊØµÉÓÎ ÉÍòÒÉ³Ù³ÊØµè¤íµÒÛÜØµÙ³ÉãÐ¿áñÊÌËòÊÔ¾ËÂÛÜÞßÉ ÛÜÒ¹ØµÉ³É³èµÉ³è Ô×ÊÌÍòÊÌËÂÇµÉÍòÊÌÎ ÉÍäÐÑËÂÛÜÊØµÒ³é Ö«Ë
ËÂÇµÛÜÒïÛÜÒÈÐàÞßÊôèµÉÍÂÐçËÂÉ ÎµÍÂÛÜÙ³Éí¥ÒÛÜØµÙ³ÉÓæ¹ÉÓÙ¬ÐØÞßÊÌÍÉ ËÂÇÚÐØ Î Ð³ì®Ô×ÊÌÍ·Û²R
Ë _ôìßËÂÇµÉÏåÐÛÜØ ÛÜØ ËÊÌËäÐáÍÂÖµØ¯ËÂÛÜÞßÉÌé
3.2.7 Lindenmayer systems
êØËÇÚÛñÒY ØÚÐáôÒÉ³ÙËÂÛñÊØæ¹ÉïæÈÐØ Ë ËÊÏÎµÍÂÉ³ÒîÉ³Ø ËÕÐØµÊÌËÂÇµÉÍÃÐçÎµÎÚáÜÛÜÙ³ÐçËÂÛÜÊØàÛÜØ æ ÇµÛÜÙäÇ ÍÂÉ³Ù³Ö«ÍÂÒîÛÜÊØ ÛÜÒ ÎµÍÂÉèÚÊÞ
ÛÜØÚÐØ¯ËÈÐØµè èµÛ Ù³ÖµáñË ËÂÊÐXUÊÛÜè ^ÐØ Û²ËÂÉÍäÐçËÛVUÉUÉÍÂÒÛñÊØ æ¹ÊÖµáÜè ÛÜØµèµÉ³É³è®ÍÂÉXWôÖµÛñÍÉãÐçØ ÉIPôÎÚáÜÛÜÙ³ÛñË ÒîËäÐÙ ]a`´é
Òò
Ð _ÊØôÖµÒ³í«ËÂÇµÛÜÒÈÐçÎµÎÚáÜÛÜÙ¬ÐçËÛÜÊØµÒ áÜÉËÂÒÈÖµÒòè«ÍäÐ³æ _ É¬ÐçÖµËÛÜÔ×ÖµáÎ ÛñÙËÂÖ«ÍÂÉ³Ò³é
ÉÄËÖµZ
Ò YµÍÂÒîË Y P ÐçØ
æ ÇµÛÜÙäÇ ÛÜÒàÒÛÜÞÀÎÚáñì Ð YÚØÚÛ²ËÂÉÀÒÉË¿ÊÔÈÒîì¥Þ_ ÊáÜÒí Ô×ÊÌÍ¿ÉIPµÐÞÀÎÚáÜÉ
¯
©

¾
¢

«
©
³
¬

Σ
é ÉË
èµÉ³ØµÊÌËÂÉàËÂÇµÉ ÒÉËÏÊÔÕÐáÜá
ËÂÇÚÐçËÓæ¹É Ù¬ÐØ½Ô×ÊÌÍÂÞ ÔÜÍÂÊÞ ÒîìôÞb_ ÊáñÒ ÛÜØ é
¸Ó£ F¯
Σ = {F, +, −}
Σ
Σ
[ ÊÌÍ·GÉ PµÐÞÀÎÚáÜÉÌí
é
F + F+ Σ
O IÉ PôË¬í æ¹
É Y P ÐßÔ×ÖµØµÙËÂÛÜÊØ
é
Þ®ÐçÎÚÒ É UÉÍîì½Òì¥
Þ _ Êá ËÂÊ ÐÀæ¹ÊÌÍÂè¤í¤ÐØµè½ËÂÇµÉ³ÒîÉÐÑÍÂÉ
ËÂÇµÉ
é SÝÉãÞßÛÜåÇ ËïPÔ×ÊÌ:Í·ΣIÉ Pµ→
ÐçÞÀΣÎ áñÉ PÇ XÐ UÉÏËÂÇµÉÏÎµÍÂÊ èÚÖµÙËÛÜÊØµÒ
¢¤£³ F ±³î¤
T \
p
p
Figure 20: Õ £¦º§ ³¶ç£³©F½©ä¶³³£Ï¢¤£³Ì±Ä¬³°Á º ±Ä¨¨ ©F ³ ±Ä F + F+
σ
F
+
−
→
P(σ)
→
F + F+
→
+
→
−
[ ÛÜØÚÐáÜá²ìíôæ¹É Y PÐØ
é
íµÔ×ÊÌÍ·ÉIP«ÐÞÀÎÚáÜÉ
° ©¯ ÛÜÒÏ
¸ÏÙ¬Ð£IáñF áÜÉ³sèÝÐ Σ
s=F
Æ ÇµÉ¿ËÍÂÛñÎÚáÜÉ
È
é " ÖÚÙäÇ Ð®Òîì¥ÒîËÉ³Þ åÉ³ØµÉÍÂÐçËÂÉ³Ò
»

µ
F
³

¨
©
J
³

ó
£

ô
J

¨
= (Σ, P, s)
ÐØ½ÛÜØ Y ØµÛñËÉëÒÉXW ÖÚÉØÚÙÉ ÊÔÕæ¹ÊÌÍèÚÒ
ÐÒÓÔ×ÊáÜáñÊÑæ·Ò³éàÆ Ê åÉËÓËÂÇµÉ ØÚÉGP¥ËÏæ¹ÊÌÍÂè
ÔÜÍÂÊÞ
s = w 0 , w1 , . . .
wi
ËÂÇµÉÓÎÚÍÉ U¥ÛÜÊÖµÒ¹æ¹ÊÌÍÂè
íµæ¹ÉãÒÛÜÞÀÎÚáñì ÒÖ _ ÒËÂÛñËÂÖ«ËÂÉÏÐáÜá¤ÒîìôÞb_ÊáÜÒïÛÜØ
_ôì®ËÂÇµÉ³ÛñÍÈÎµÍÂÊôèµÖµÙËÂÛñÊØÚÒé
wi−1
êØóÊÖ«Í·GÉ PµÐÞÀÎÚáÜÉÌíwËÂi−1
ÇµÛÜÒÈìôÛÜÉ³áÜèµÒ
w0
w1
w2
w3
=
=
=
=
..
.
F,
F+F+
F + F + +F + F + +
F + F + +F + F + + + F + F + +F + F + + +
Æ ÇµÉàØµÉIPôË ÒîËÂÉÎ½ÛÜÒÈËÂÊ îè«ÍäÐ³æ ÀËÂÇµÉ³ÒÉãæ¹ÊÌÍÂèµÒ³í ÐØµèóËÂÇµÛÜÒ åÛ UÉ³Ò·ËÂÇµÉâÎÚÛÜÙËÖµÍÉ³Ò·æ¹Éëæ¹ÉÍÂÉãËäÐáV]ôÛÜØµå
È
Ð _ ÊÖ«Ë¬é
êÞ®ÐåÛÜØµÉÓÐàËÂÖ«ÍËÂáÜÉÓÒÛñËîËÂÛÜØµå ÐÑËÈÒÊÞßÉÏÎÊÛÜØ Ë ÊØ Ð¿áÜÐçÍÂåÉÓÎ ÛñÉ³Ù³ÉÏÊÔ Î ÐçÎ ÉÍ¬í«æ·ÛñËÂÇ
ÛñËÂÒÈÇµÉ¬ÐèóÎ ÊÛñØ ËÂÛÜØµå ÛÜØóÒîÊÞßÉãèµÛñÍÂÉÙËÂÛÜÊØ¤íµÒÉ³É [ ÛÜåÖ«ÍÂÉ \ ^áÜÉ³ÔÜËG`´pé Æ¹ÇÚÉÏËÂÖ«ÍËÂáÜÉÏÙ¬ÐØ ÖµØµèµÉÍÂÒîËÂÐØµèËÂÇµÉ
Ù³ÊÞßÞ®ÐØµèµÒ í í ÐØµè
é
ÞßÉ¬ÐØµ&
Ò îÞßÊ UÉ®ÊØµÉ ÒîËÂÉÎ Ô×ÊÌÍæÈÐÑÍÂè
«í
ÞßÉ¬ÐØµÒ&ËÂÖ«ÍÂØ Ù³ÊÖµØ¯ËÂÉÍ
F +
− F
+
Ù³áÜÊ Ù ]ôæ ÛñÒÉ _ ì ÐØ ÐØµåáÜÉÊÔ
èµÉ³åÌÍÂÉ³ÉÒ «í ÐØµè
ÞßÉ¬ÐØµÒËÂÖ«ÍÂØ Ù³áÜÊôÙ ] æ ÛÜÒÉb_ôìÝÐØ ÐØµåáÜÉ ÊÔ
èµÉ³åÌÍÂÉ³ÉÒ «é Æ¹ÇÚÉËÂÖ«ÍËÂáÜÉÙ¬ÐØ 90
ÎµÍÂÊôÙ³ÉÒÒ ÐØ ì ÒXÉ W ÖÚÉ−ØÚÙÉßÊÔÈÒîÖÚÙäÇ Ù³ÊÞßÞ®ÐØµèµÒ³í _ôì ÉGP«É³Ù³Ö«ËÂÛñØÚåóËÇÚÉ90
Þ
ÊØµÉ®ÐÔÜËÂÉÍ¿ÐçØÚÊÌËÇÚÉÄÍ¬é SÝÉ®ÐçÍÂÉßÛñØ ËÂÉÍÉ³ÒîËÂÉ³è ÛÜØ ËÂÇµÉßÍÂÉ³ÒÖµáñËÛÜØµåÎ ÐçËÂÇ ËäÐ ]É³Ø _ôìÝËÇÚÉÀËÖµÍîËÂáÜÉÀÊØ ËÂÇµÉ
ÎÚÛÜÉ³Ù³É¿ÊÔÕÎ ÐçÎÉÍ¬é Æ¹ÇÚÉàÎ ÐçËÂÇ½åÉ³ØµÉÍäÐçËÉ³è _ôì ËÂÇµÉ¿Ù³ÊÞßÞ®ÐØµè ÒXÉ W ÖµÉ³ØµÙ³É
í¡Ô×ÊÌÍãIÉ PµÐçÞÀÎ áñÉÌíÛÜÒ
ÒÇµÊ¦æ Ø ÛÜØ [ ÛñåÖµÍÉ \ ^»ÍÂÛñåÇ GË `Äí ÐáÜÊØµå æ·ÛñËÂÇ ËÇÚÉÎ ÊÒÛñËÂÛñÊØ ÐØµè ÊÌÍÂÛÜÉ³Ø¯ËäÐçFËÂÛñ+
ÊØ F+
ÊÔ ËÂÇµÉ ËÂÖ«ÍËáÜÉÀÐÔÜËÉÍ
ÎµÍÂÊôÙÉ³ÒÒÛÜØµå¿ËÂÇµÉãÙ³ÊÞßÞ®ÐçØÚè®ÒÉ WôÖµÉ³ØµÙ³ÉÌé
ÆÈÇµÉÓËÂÖ«ÍËÂáÜÉ Ù³ÐØ ËÂÇµÉÍÉ³Ô×ÊÌÍÂÉÏåÌÍäÐçÎÚÇµÛÜÙ¬ÐáÜá²ìÀÛÜØ¯ËÂÉÍÎµÍÂÉËÈÐçØ ìßæ¹ÊÌÍÂè åÉ³ØµÉÍäÐçËÂÉè _ôì Ð ÛÜØµèÚÉØÚÞ®Ð³ìÉÍ
ÒîìôÒîËÂÉ³ÞðÊ UÉÍÈËÂÇµÉëÐçáñÎÚÇ Ð _ ÉË
é
Turtle graphics.
{F, +, −}
\!
[ ÊÌÍ
í¹áÜÉË
èµÉ³ØµÊÌËÂÉ ËÇÚÉ æ¹ÊÌÍÂè ÍÂÉ³ÒÖµáñËÛÜØµå
σ
Recursively drawing Lindenmayer systems.
ÔÜÍÂÊÞ
_ôì ËÂÇµÉ ºÔ×ÊÜá è ÒÖ _ÚÒîËÂÛñËÂÖ«ËÂÛñÊØ ÊÔÕÐáÜá îÒ ì¥Þ_ σ
ÊáÜÒ ÐçÙ³ΣÙ³ÊÌÍÂèµÛÜØµw
åßiËÂÊßËÂÇµÉ³ÛñÍ ÎµÍÂÊ èÚÖµÙËÛÜÊØµÒ³éâêØ ÊÖ«Í
ÍÂÖµØµØµÛÜØµσåÀÉIPµÐçÞÀÎ iáñÉÌíÚæ¹ÉàÇÚÐXUÉàÔ×ÊÌÍÓÉIP«ÐÞÀÎÚáÜÉ
ÐØµè +
Ô×ÊÌÍ
w2 = wF2 = F + F + +F + F + +
wi = +
ÐáÜá é
i
ÆÈÇµÉÀÎ ÊÛñØ ËëÛñÒëØµÊÑæ ËÂÇÚÐçËëÙ¬ÐçØ æ¹ÉÀÉGP¥ÎµÍÂÉ³ÒÒ σ ÛñØÝËÂÉÍÂÞßÒëÊÔ ËÇÚÉ
ÒëÊÔ¹ÊÌËÂÇµÉÍàÒì¥Þ_ ÊáÜÒ³í
wi−1
ÐØµèÀËÂÇµÛÜÒÃÛÜÒÕæ·ÇµÉÍÂÉòÍÉ³Ù³Ö«ÍÂÒÛÜÊØßÙÊÞßÉ³ÒÕÛñØ ËÂÊÏÎ áÜÐ¬ìé w" iÖ«ÎµÎ ÊÒÉÈËÂÇÚÐçË
éÃÆÈÇµÉ³ØÀæ¹ÉòÙ¬ÐØ
P(σ) = σ1 σk
Ê _µËäÐÛÜØ σ ÐçÒòÔ×ÊáÜáñÊÑæ·Ò³éS É YµÍÂÒîËòÒÖ _ÚÒîËÂÛñËÖµËÉ _ôì
^ Ô×ÊáÜèóÒÖ _ÚÒîËÂÛñËÖµËÛÜÊØ `Äí¥ÐØµèóÛñØËÂÇµÉ
wi
σ
σ1 σk
ÍÂÉ³ÒîÖÚá²ËÂÛÜØµåãæ¹ÊÌÍÂè
æ¹É ÐçÎµÎÚáñì
º Ô×ÊáñèßÒÖ _ ÒËÂÛñËÂÖ«ËÂÛÜÊØËÂÊàÐáÜáµËÂÇµÉ ÒîìôÞb_ÊáÜÒ³é Æ¹ÇµÛÜÒÒÇµÊ¦æ Ò
σ1 σk
(i − 1)
ËÂÇÚÐçË
σk
1
wσi = wσi−1
.
wi−1
Æ ÇµÛÜÒ Ô×ÊÌÍÂÞ¿Öµá×Ð ÐáñÒÊãÛñÞÀÎ áñÛÜÉ³Ò¡ËÇ ÐÑËÃËÇÚÉ¹è«ÍäÐ³æ ÛñØÚåÓÊÔ σ ÛÜÒ Ê _ÚËÂÐÛÜØµÉ³è_ôì¿ÒÛÜÞÀÎÚáñìâÙ³ÊØµÙ¬ÐçËÂÉ³ØÚÐçËÛÜØµå
È
wi
ËÂÇµÉëè«ÍäÐ³æ ÛñØÚåÒòÔ×ÊÌÍ σ1
íæ¹ÉëÒîÛÜÞÀÎÚáñì Ù³ÊØµÙ¬ÐçËÂÉ³ØÚÐçËÉâËÂÇµÉ
σk é Æ¡ÊßåÉË ËÂÇµÉ ÐÙËÖ Ðçá¡æ¹ÊÌÍÂè
è«ÍäÐ³æ ÛÜØµåÒ¹ÊÔ ÐçáÜá w
íσ µi−1
Ô×ÊÌÍ , . . .ÍÂ,ÖµwØÚi−1
ØµÛÜØµå ËÂÇ«ÍÂÊÖµåÇ ËÂÇµÉÏÒîì¥Þ_ ÊáÜÒ¹ÊÔ¾wËÂÇµi ÉãÛÜØµÛñËÛ×Ðáæ¹ÊÌÍÂè é
wi
ÍÊåÌÍäÐÞ \ ÒÇµ
ÊÑæ·ÒàÇµσÊÑæ ËÇÚÛñÒâæ¹ÊÌÍ ]¥ÒàÔ×ÊÌÍ¿ÊÖ«ÍàÍÂÖµØµØµÛÜØµå ÉIPµÐÞÀÎÚáÜÉ¿æ ÛñËÇ ÎµÍÂÊ èµsÖÚÙÄËÂÛÜÊØµÒ
ÐØµè ÛÜØµÛñËÂÛ×Ðçáæ¹ÊÌÍÂè é " ÛÜØµÙ³É i
Ô×ÊÌÍÏÐáÜá íæ¹FÉâèÚ→
Ê
F + F+, + →
+, − →
−
F
P (+) = +, P i (−) = −
i
ØµÊÌË·ØµÉ³ÉèËÂÊ ÒîaÖ _ÚÒîËÛñËÂÖ«ËÂÉ
ÐØµè
ÐØµèóåÉË
+
−
^ é\ `
wi = wFi = wFi−1 + wFi−1 + .
Æ¹ÇÚÉòÎÚÍÊåÌÍäÐÞ ÐÒîÒÖµÞßÉ³ÒÕËÂÇµÉ ÉGP«ÛÜÒîËÉ³ØµÙ³É ÊÔ¾ÐâáÜÛV_µÍäÐçÍîì
æ·ÛñËÂÇÀÎµÍÂÉ³èµÉ YÚØµÉ³èßËÂÖ«ÍËÂáñÉ·Ù³ÊÞßÞ®ÐØµè
Ô×ÖµØµÙËÂÛÜÊØµÒ
í
^ÙÊÖÚØ¯ËÂÉÍÂÙáÜÊôÙ ]ôæ·ÛÜÒÉ ÍÂÊÌËäÐÑËÂturtle
ÛÜÊØ®æ ÛñËÂÇ ÒÊÞßÉÏÐØµåáÜÉ`ïÐØµè
^ÙáÜÊôÙ ] forward left
right
æ ÛñÒÉÏÍÊÌËäÐçËÂÛÜÊØµÒ æ·ÛñËÂÇÒîÊÞßÉãÐçØÚåáñÉ` ÛÜØØÚÐÞßÉ³ÒÎÐçÙ³É
é
ifm
êØ ËÂÇµÉ èµÊôÙ³ÖµÞßÉ³Ø¯ËäÐçËÂÛñÊØ ÊÔÏËÇÚÉ ÎµÍÂÊåÌÍäÐçÞ í¹æ¹É ÇÚÐXUÉ ÊÞßÛñËËÂÉ³è ËÇÚÉËÍÛVU¥Û×Ðçá½ÎµÍÂÊ èÚÖµÙËÛÜÊØµÒ
í ÐØµè ÛÜØ ÒîÎ ÉÙ³ÛÜÔÜì¥ÛñØÚåÀÐ ÛÜØµèÚÉØÚÞ®Ð³ìÉÍòÒîìôÒîËÂÉ³Þ íµæ¹ÉëÙ¬ÐØ èµÊÀÒÊ®ÐçÒ·æ¹É³áÜá Ãæ¹Éâæ ÛñáÜá
+→
+, − →
−
ÖµÒÖÚÐáÜáñì®ÊØµáñì®áñÛÜÒîË¹ÎµÍÂÊôèµÖµÙËÂÛÜÊØµÒïÔ×ÊÌÍ·Òîì¥
Þ _ ÊáÜÒ ËÂÇÚÐçË·ÐçÍÂÉÏØµÊÌËÈÞ®ÐçÎµÎ É³è ËÊËÂÇµÉ³ÞßÒÉáVUÉ³Ò³é
// Prog : lindenmayer.C
// Draw turtle graphics for the Lindenmayer system with
// production F - > F +F + and initial word F .
# include < iostream >
# include < IFM / turtle >
// POST : the word w_i ^F
void f ( unsigned int i )
if ( i == 0)
ifm :: forward (); //
else {
f(i -1);
//
ifm :: left (90);
//
f(i -1);
//
ifm :: left (90);
//
}
is drawn
{
F
w_ {i -1}^ F
+
w_ {i -1}^ F
+
T \
}
int main () {
std :: cout < < " Number of iterations =? ";
unsigned int n;
std :: cin > > n;
// draw w_n = w_n ( F)
f(n );
return 0;
}
Program 29: ¢¤£³Á »Fµ ¨½©JÚ³£ÌÅ
[ ÊÌÍ ÛÜØ«ÎÚÖ«Ë
n = 14
íµËÇÚÉÓÎµÍÂÊåÌÍäÐÞ æ·ÛÜáÜáÎµÍÂÊ èÚÖµÙ³ÉÓËÂÇµÉâÔ×ÊáñáÜÊÑæ·ÛÜØµåàè«ÍäÐ³æ ÛÜØµå«é
Ò
åÉËÂÒ á×ÐÑÍÂåÉÍ¬í ÂË ÇµÉßÎÚÛÜÙËÂÖ«ÍÂÉßèµÊôÉÒØµÊÌË ÒÉ³ÉÞ ËÂÊ ÙäÇÚÐØµåÉ®ÞÖµÙäÇ ÛñËàÍÂÊÌËäÐÑËÂÉ³Ò³íÃÐØµèÒîÊÞßÉ
ÞßÊÌÍÂÉÈèµn
ÉËäÐÛÜáñÒÃèµÉ UÉáÜÊÌÎ¾í Ú_ Ö«ËïÐçÎ ÐçÍËÕÔÜÍÂÊÞ ËÂÇÚÐçËÕËÇÚÉòÛÜÞÀÎµÍÂÉ³ÒÒîÛÜÊØ ÛÜÒÃËÂÇµÉ·ÒÂÐçÞßÉÌé ÒÒîÖÚÞßÉÈìÊÖßÙ³ÊÖµáÜè
è«ÍäÐ³æ ËÂÇµÉÏÎÚÛÜÙËÖµÍÉÏÔ×ÊÌÍ
éÕÆ¹ÇµÉ³ØóÉXW ÖÚÐçËÂÛÜÊØ ^ é \ `æ¹ÊÖµáÜèåÛ UÉ
n=∞
w∞ = w ∞ + w ∞ + .
Æ¹ÇÚÛñÒ ÛÜÒÓÐ
SÝÉÏÇÚÐXUÉëÐ ¬ ¶ îÄ»¨» ª©£¬» J
òËÂÇµÉ è«ÍäÐ³æ ÛÜØµåßÊÔ
¶£©µ±¬î©¯
w∞
Ù³ÊØµÒÛÜÒËÂÒ ÊÔÃË æ¹Ê®ÍÂÊÌËäÐçËÉ³è½è«ÍäÐ³æ·ÛÜØµåÒ ÊÔÕÛ²ËÂÒÉ³áÜÔºé
S ÉëÙ¬ÐØ ÉIPôËÂÉ³Øµè ËÂÇµÉâèµÉ YÚØµÛñËÂÛÜÊØóÊÔ Ð ÛñØÚèµÉ³ØµÞ®Ð³ìÉÍòÒîì¥ÒËÂÉ³Þ
Additional features.
ÍÂÊÌËäÐÑËÂÛÜÊØÝÐçØÚåáñÉ
ËÂÇÚÐçËâÞ®Ð³ì _É èµÛ ÉÄÍÂÉ³Ø ËâÔÜÍÂÊÞ
ËÂÇÚÐçËÈè«ÍäÐ³æ ÒòÐ¿ÒîØÚαÊ¦æQ Ð ]ÉÏÔ×ÊÌÍ ÛñØµÎÚÖ«Ë
é
n=5
90
ËÂÊ ÜÛ ØµÙ³áÜÖµèµÉâÐ
èµÉ³åÌÍÂÉ³É³Òé Æ¹ÇµÛÜÒâÛñÒâÒÇµÊ¦æ ØÝÛÜØ ÍÂÊåÌÍäÐçÞ \ \
// Prog : snowflake.C
// Draw turtle graphics for the Lindenmayer system with
// production F - > F -F ++F -F , initial word F ++ F ++ F and
// rotation angle 60 degrees .
# include < iostream >
# include < IFM / turtle >
// POST : the word w_i ^F
void f ( unsigned int i )
if ( i == 0)
ifm :: forward (); //
else {
f(i -1);
//
ifm :: right (60); //
f(i -1);
//
ifm :: left (120); //
f(i -1);
//
ifm :: right (60); //
f(i -1);
//
}
}
is drawn
{
F
w_ {i -1}^ F
w_ {i -1}^ F
++
w_ {i -1}^ F
w_ {i -1}^ F
int main () {
std :: cout < < " Number of iterations =? ";
unsigned int n;
std :: cin > > n;
// draw w_n = w_n ^ F ++ w_n ^F ++ w_n ^F
f(n );
// w_n ^ F
T \ \
ifm :: left (120);
f(n );
ifm :: left (120);
f(n );
//
//
//
//
++
w_n ^ F
++
w_n ^ F
return 0;
}
Program 30: ¢¤£³Á ¸
© ôÅ
Æ ÊÀåÉË·ÞßÊÌÍÂÉ
Q ÉGP«ÛV_ÚÛÜáÜÛ²Ë ìí æ¹ÉâÙ¬ÐØ ÐáñÒÊ ÉIPôËÂÉ³Øµè ËÂÇµÉëÐá²Î ÇÚÐ _ÉË ÊÔ ÒîìôÞb_ ÊáñÒ³é [ ÊÌÍ ÉIP«ÐÞÀÎÚáÜÉÌí
æ¹É Þ®Ð³ì ÐèµèßÒîì¥Þ_ ÊáÜÒ æ ÛñËÇÚÊÖ«ËÐØ ì åÌÍäÐçÎÚÇµÛÜÙ¬ÐáÚÛÜØ¯ËÂÉÍÎµÍÂÉËÂÐçËÂÛÜÊØ ËÂÇµΣÉ³ÒÉ ÐçÍÉ ÒîËÂÛÜáñá ÖµÒÉ³Ô×Öµáí¯ËÂÇµÊÖµåÇ¾í
ÒÛÜØµÙ³É ËÂÇµÉì Þ®Ð³ì_ÉóÖµÒÉ³è¼ÛÜØ ÎÚÍÊôèµÖµÙËÂÛÜÊØµÒ³é [ ÊÌÍ®ÉIPµÐÞÀÎÚáÜÉíËÂÇµÉ ÛÜØµèÚÉØÚÞ®Ð³ìÉÍ Òîì¥ÒËÂÉ³Þ æ Û²ËÂÇ
íôÛñØÚÛ²ËÂÛ×Ðáæ¹ÊÌÍè
ÐØµèÎµÍÂÊ èÚÖµÙËÛÜÊØµÒ
Σ = {F, +, −, X, Y}
X →
X + YF +
Y →
−FX − Y
ì¥ÛñÉ³áÜèµÒïËÂÇµÉ
F£©Á¥ ±³£KDô
^
X
w14
íÚÐØµåáÜÉÓÊÔ
Æ¹ÇÚÉÏÙ³ÊÌÍÍÉ³ÒîÎ ÊØµèµÛÜØµåÙÊôèµÉâÛÜÒ¹ÒÇµÊ¦æ ØóÛñØ
90
èµÉ³åÌÍÂÉ³ÉÒ `Äé
ÍÊåÌÍäÐÞ
é
// Prog : dragon .C
// Draw turtle graphics for the Lindenmayer system with
// productions X - > X+ YF + , Y - > - FX -Y , initial word X
// and rotation angle 90 degrees
# include < iostream >
# include < IFM / turtle >
void y ( unsigned int i );
// POST : w_i ^ X is drawn
void x ( unsigned int i ) {
// necessary: x and y call each other
\ \ \
if ( i > 0) {
x(i -1);
ifm :: left (90);
y(i -1);
ifm :: forward ();
ifm :: left (90);
}
//
//
//
//
//
w_ {i -1}^ X
+
w_ {i -1}^ Y
F
+
}
// POST : w_i ^ Y is drawn
void y ( unsigned int i ) {
if ( i > 0) {
ifm :: right (90);
// ifm :: forward ();
// F
x(i -1);
// w_ {i -1}^ X
ifm :: right (90);
// y(i -1);
// w_ {i -1}^ Y
}
}
int main () {
std :: cout < < " Number of iterations =? ";
unsigned int n;
std :: cin > > n;
// draw w_n = w_n ^ X
x(n );
return 0;
}
Program 31: ¢¤£³Á
F£Ä©Á« Å
[ ÛÜØÚÐáÜá²ìíÊØÚÉÈÙ¬ÐØÀÐèµèÀÒîì¥Þ_ ÊáÜÒ æ·ÛñËÂÇ åÌÍÂÐçÎÚÇÚÛñÙ¬Ðá«ÛÜØ¯ËÂÉÍÎµÍÂÉËÂÐçËÂÛÜÊØ¤é ïÊÞßÞßÊØµáñìãÖÚÒîÉ³èßÒîì¥Þ_ ÊáÜÒ
ÐçÍÂÉ ^ ÖÚÞÀÎÊØµÉâÒîËÂÉÄÎ Ô×ÊÌÍæÈÐçÍÂè¤í«ËÂÇµÛÜÒÈèÚÊ É³ÒØ Ë·áÜÉ¬ÐXUÉëÐàËÍäÐçÙ³É`Äí ^°ÍÂÉ³ÞßÉ³Þ_ ÉÍ¹Ù³Ö«ÍÍÉ³Ø Ë ÎÊÒÛñËÂÛÜÊØ `
Ð Øµè f ^ ÖµÞÀÎ _ ÐÙ ½
] ËÊóá×ÐÒîËÏÍÂÉ³ÞßÉÞb_ ÉÄÍÂÉ³è½ÎÊÒÛñËÂÛÜÊØ `Äéê»ËãÛÜÒëÐáñÒÊ[ Ë ìôÎÚÛÜÙ³Ðá ËÂÊ ÐèµèÝØµÉæ Òîì¥Þ_ ÊáÜÒ
æ Û²ËÂÇ ] ËÂÇµÉÏÒÂÐÞßÉÓÛÜØ¯ËÂÉÍÎµÍÂÉËÂÐçËÂÛÜÊØ ÐÒ
F
íµÒÂÐ³ìé
3.2.8 Details
ÛÜØµèÚÉØÚÞ®Ð³ìÉÍ¡ÒîìôÒîËÂÉ³ÞßÒ ÐçÍÂÉØÚÐÞßÉèàÐÔÜËÂÉÍ ËÂÇµÉÓÐØµÛÜÒÇ _ÚÛÜÊáñÊåÛÜÒîË ·ÍÂÛÜÒËÂÛÜèµÉ
ÛÜØµèÚÉØÚÞ®Ð³ìÉÍ ^ \ `Óæ·ÇµÊóÎÚÍÊÌÎ ÊÒÉ³è ËÂÇµÉ³Þ ÛÜØ ËÂÊ ÞßÊ èµÉ³áÕËÂÇµÉÀåÌÍÊÑæòËÂÇ ÊÔ¹ÎÚá×ÐçØ ËÂÒ³é
ÛÜØµèÚÉØÚÞ®Ð³ìÉÍÒîìôÒîËÂÉ³ÞßÒ ^»æ ÛñËÇåÉ³ØµÉÍäÐçáÜÛ ¬ÐçËÛÜÊØµÒ¿ËÂÊ ºèµÛÜÞßÉ³ØµÒÛÜÊØÚÐáÒîÎ ÐÙ³É`ÇÚÐXUÉóÔ×ÊÖµØµè Þ®ÐØ ì
ÐçÎµÎÚáÜÛÜÙ¬ÐÑËÂÛÜÊØµÒ ÛÜØ Ù³ÊÞÀÎÚÖµËÉÍÈåÌÍäÐçÎÚÇµÛÜÙ³Ò³é
Lindenmayer systems.
T \ \ 3.2.9 Goals
Dispositional.
¹Ë¹ËÂÇµÛÜÒ¹Î
ÊÛÜØ¯Ë¬íôìÊÖóÒîÇÚÊÖµáÜè é³é³é
`ßÖµØÚèµÉÍÒîËäÐØµè ËÂÇµÉÀÙ³ÊØµÙ³ÉÎµË ÊÔ ÍÂÉÙ³Ö«ÍÂÒÛÜÊØ¤í ÐØµèÝæ Ç¯ì ÛñËâÞ®Ð ]É³ÒâÒîÉ³ØµÒÉßËÂÊ èµÉ YÚØÚÉÀÐÔ×ÖµØÚÙÄËÂÛÜÊØ
ÂË Ç«ÍÂÊÖµåÇóÛñËÂÒÉáÜÔ É Ð¬æÈÐÑÍÂÉãËÂÇÚÐçËÈËÂÇµÉì èµÊÀØµÊÌË
\ `ßÖµØÚèµÉÍÒîËäÐØµèóËÂÇµÉâÒîÉ³Þ®ÐØ ËÛÜÙ³Ò¹ÊÔ ÍÂÉ³Ù³Ö«ÍÂÒîÛVUÉâÔ×ÖµØµÙËÂÛÜÊØ Ù³ÐáÜáÜÒÈÐØµè _ â
ÐáñæÈÐ³ì¥Ò¹ËÂÉÍÂÞßÛñØ ÐÑËÂÉ `®ÐçÎµÎµÍÂÉ³Ù³Û×ÐÑËÂÉ ËÂÇµÉÏÎ Ê¦æ¹ÉÍòÊÔ ÍÂÉ³ÙÖµÍÒÛÜÊØóÛÜØ ÒÊÌÍËÂÛñØÚåÐØµèóè«ÍäÐ³æ ÛñØÚå ÜÛ ØµèµÉ³ØµÞ®Ð¬ìÉÄÍòÒì¥ÒîËÂÉÞßÒ³é
êØÎ ÐçÍËÂÛñÙ³Öµá×ÐçÍ¬í¯ìÊÖ ÒÇµÊÖÚáñè _ ÉãÐ _ÚáÜÉ ËÂÊéé¬é
Operational.
Y Øµè ÎµÍÂÉ ïÐØµèÎÊÒîËÂÙ³ÊØµèµÛñËÂÛÜÊØµÒïÔ×ÊÌÍòåÛVUÉ³ØÍÉ³Ù³Ö«ÍÂÒÛVUÉãÔ×ÖµØÚÙÄËÂÛÜÊØµÒ
` Ú
` ÎµÍÂÊUÉâÊÌÍòèµÛÜÒîÎµÍÂÊUÉÏËÉÍÂÞßÛÜØÚÐçËÂÛñÊØ®ÐØµèóÙ³ÊÌÍÍÂÉÙËÂØµÉ³ÒÒòÊÔ ÍÂÉ³Ù³Ö«ÍÂÒîÛVUÉâÔ×ÖµØµÙËÂÛÜÊØ Ù¬ÐáÜáñÒ
\ ß
` ËîÍäÐØµÒá×ÐçËÉÍÂÉ³ÙÖµÍÒÛVU ÉßÞ®ÐçËÂÇµÉ³Þ®ÐÑËÂÛÜÙ¬Ðá Ô×ÖÚØµÙËÛÜÊØ èÚÉIY ØµÛñËÛÜÊØµÒãÛÜØ¯ËÂÊ Ô×ÖÚØµÙËÛÜÊØ èÚÉIY ØµÛ ß
^
^
^
ÂË ÛñÊØÚÒ ^
`ßÍÉæ·ÍÂÛ²ËÂÉÏÐ¿åÛVUÉØÍÂÉ³Ù³Ö«ÍÂÒîÛVUÉâÔ×ÖµØµÙËÂÛÜÊØ ÛÜØÛ²ËÂÉÍäÐçËÛVUÉÓÔ×ÊÌÍÂÞ ^ `ßÍÉ³Ù³ÊåØµÛ ³ÉÏÛÜØµÉ ÙÛÜÉ³Ø ËÈÍÂÉ³ÙÖµÍÒVÛ UÉÏÔ×ÖÚØµÙËÛÜÊØµÒ·ÐØµèóÛñÞÀÎÚÍÊ UÉ ËÂÇµÉ³ÛñÍ¹Î ÉÄÍÂÔ×ÊÌÍÂÞ®ÐØµÙ³É ^ `®ÙÊÖÚØ¯ËïËÂÇµÉÓØôÖµb
Þ _ÉÍ¹ÊÔ¤ÊÌÎ ÉÍäÐçËÛÜÊØµÒïÊÔ ÐëåVÛ UÉØ Ë ìôÎÉÏÛñØÐâÍÂÉÙ³Ö«ÍÂÒÛVUÉÏÔ×ÖµØÚÙÄËÂÛÜÊØ Ù¬Ð áÜáí ÖµÒÛÜØµå
ÛÜØµèµÖµÙËÂÛÜÊØ ÐÒ¹ËÂÇµÉãÞ®ÐçÛÜØ ËÊôÊá ^ `ßæòÍÂÛñËÂÉ ÍÂÉÙ³Ö«ÍÂÒVÛ UÉãÔ×ÖÚØµÙËÛÜÊØµÒÈÔ×ÊÌÍ·åVÛ UÉ³ØËÂÐÒ ]¥Òé
3.2.10 Exercises
Exercise 95
Ð `
_ `
»F¿¢¤£³ â©F¿¢ ³±ÄF¤¹¶ç£ º¹¶ç»ª¸È°Á £³±³ £¦ Dôò¶Ä ±¬¤çÅ
bool f ( int n)
{
if ( n == 0) return false ;
return ! f (n -1);
}
void g ( unsigned int n )
{
if ( n == 0) {
std :: cout < < " *";
return ;
}
g(n -1);
g(n -1);
}
^
`
\ \
Ù`
unsigned int h ( unsigned int n , unsigned int b ) {
if ( n == 1) return 0;
return 1 + h ( n / b , b );
}
Exercise 96 È£ D¥
£ F»¢¤£³ D¥ ¶£ © J Â¶ ºß¶»ª¸¹»Á £³Ä±³£¦Ä D¥®¶ ¬± î¤ º©¯ß»
³£¬¨ » ©¯¬·¶ç£©¯»¢ ¦ª ©£Á¨½³¤ºçÅÂ º¥¿º£KJ Ú³£³± ã D¥³£ ¸ ³ Fó
³ ëî© ô³ »¤ ©µ±± ¤ Å»Å
JÚ½ Fâ¢¤£³¬F º©¯ ºZDô©¯ Ï£©Á«ÀÂ¶ unsigned ^ int
\ `
® ¡©¯ Å
Ð `
unsigned int f ( unsigned int n)
{
if ( n == 0) return 1;
return f( f(n -1));
}
_ `
Ù`
// POST : return value is the Ackermann function value A (m ,n )
unsigned int A ( unsigned int m , unsigned int n ) {
if ( m == 0) return n +1;
if ( n == 0) return A (m -1 ,1);
return A(m -1 , A(m , n -1));
}
unsigned int f ( unsigned int n , unsigned int m)
{
if ( n == 0) return 0;
return 1 + f (( n + m ) / 2 , 2 * m );
}
Exercise 97 Ó¤ÄNFµ³£¿ºÈ¶»ª¸¹»Á®£³Ä±³£¦Ä D¥È¶ ±¬î
) Ù ¹

ÐçÍËÂÇ¯ì Ò [ ÖµØµÙËÂÛÜÊØ
Á¥³£¦ Ó©¯² ¸¹ ©¯
M(n) :==
©KM ¹£³ DÌ Fµ ©
Å
Fµ IF½ ©¯» Á¥©¯ D¥»
n − 10, ¶ n > 100
M(M(n + 11)), ¶ n 100.
¶ ±¬î mccarthy º©¯È»¨¢¾ª³¨½³¤º
±
M Ý©¯Ï©£Àº D ©¯¡¬ Â¶ ºò¶°§¸¹»Áã¶ £ò¶Ä ±¬ ±´©¯»
HîEM mccarthy(101)
Hî°»EM mccarthy(100)
Hî°»EM mccarthy(99)
Hî D M mccarthy(91)
©£¦ºJ
±¬î Å
T \ \ ¦¢¾×©»¼¸òJÝºã¶Ä ±¬
±M
»ó±Ä©¯»ªIF
± ©£¦ºJ ±¬
¸È©¯ò»º Dô©¯ Â ¶ M(n) ¶£ © JóÁÌ D¥³ ¨½ ³£ n
£³ ¢¤£³Ä±³³EJ
^
`I^
`
Exercise 98
£¬¿©Fß¬·© ¶ ±¬î º©¯·± ¨¿¢¤¬ ¬» ¨ ©¯ Ä± Ì ±³³¤º
Å ¬ç¨ © JÝ ³ Fµ I F ° Dô©£¬® Dô©¯ª³¤·¸ ©J çÅ £ µ©¨¢¾ª
©KM
n
k
, n, k
n
n!
,
:=
k!(n − k)!
k
£

 0,
n
1,
:=
 n−1
k
£

 0,
n
1,
:=
 n
k
+
k
n−1
k−1
¶ n<k
¶ n = k £ k = 0 ,
, ¶ n > k, k > 0
¶ n<k
¶ n k, k = 0
¶ n k, k > 0
n−1
k k−1
M Ý¥î±³¼Â¶®º º¥£³ Dô©£¬©¤º®ó ¬³âÄGFÀ¶ç£º » ¨¿¢¾§³¨½³¤î©¯î ©F½¸òJ
£îÁ¡óºÄ£³¬î±´©¯» J À ³â Fµ³£¢¤» JÚ£½©£îÁ¨½³¤º X J ±Ä¨¢©£¬°Á ©¯ëªÄ©¯³ë¸Ó
F ³£³³¤ °¨¢¾ª³¨ ³¤î©¯¤ Â¶¿ºò¶ ±³î Å
^
Exercise 99 Â
`G^
`^
`
¸ ¨½© J ¸ ©Jô½±Ä© JÚ ¸¹

³¢¤ ºç¨ ³¸È©¯·¢¾¥ª
³¢¾¥±Ä©¯·©¬¢«¢ ´©£© ± º ³³î © ¨½©¯º³¨ ©¯î±´©¯Ó çÅ
D¥³ ¨½ ©¨½¤ Â¶
¨ J ° ¸ ¨½© J ¸Ó©J ±´© JÚ ¢©£¦¼» ° Á ºó©Dô©»ª©« ¬§ Fµ ¨» ©¯¤
HÄ Ä© ¬ ©
F ±»¤ M
Fµ³ ¨ » ©¯î¤ß°
©£³ HÄ Ä© ¬N M
Å
Å
Å
Å
H±°¤NM¥Å Õ ©¨½¤âÂ¶
Å ¶£ µ©¨¿¢¾§ Ï±Ä© ¸¹ I F °¶ç£ ¸ ©J H Á«¬ °¤Á«³£¦ ª¬ ¸¹±¬½Ý±³¤°¨ ¬ M
(20), (10, 10), (10, 5, 5), (5, 5, 5, 5) Å
E Dô ÀºÓ¢¤£³¥ ¦ª³¨ ¶£®© JóÁ D¥»¥¢¤Ï©¨½¤ K J ¸¹£¬°Á½©¢¤£³Á£Ä©¨ partition
º©¯ Fµ ³ºò¶»ª¸¹»ÁÏ¶ ±¬î HÄ©¯»T
D ©¯¡¬ ³À Fµ³£¦³Ì F ©¯®±³¤°¨ ¬N MôÅ
// PRE : [ first , last ) is a valid nonempty range that describes
//
a sequence of denominations d_1 > d_2 > ... > d_n > 0
// POST : return value is the number of ways to partition amount
//
using denominations from d_1 , ... , d_n
unsigned int partitions ( unsigned int amount ,
unsigned int * first ,
unsigned int * last );
\ \ ÚJ £ ¢¤£³ÄÁ£©¨ µF ¬³£¬¨° ° ¸ ¨ © Jó¸ © JôZJÚ ±Ä©¼ ¸¹
Å
© J £Ï¢¤£³ÄÁ£©¨ Ä± ¨¢¤Àº¿ ¨½ ³£óÂ¶ ¸ © Jô¹¶£
· ©F ^ `
Exercise 100 «
¢ ¢ JÚ¸ ©¤Ã®±³£©µ± à¨½ ³ F J ±³£¬ò±Ä Fµ ¹±¤»³»Á®Â¶ d F°Á»º
³³¸ÏÄ³ 1 ©F 9 Å ¼©D¥ ¨½¬¸ ¶ F
àº©¯µ©µ±¬º J k Â¶ º¬ F°Á»º ©£³
1 çÅ
©KM
M
£¬ã© ¢¤£³Á£©¨ º©¯ Á«³ ³£Ä©¯¬ë©¯»Ì¢ ³ ¦ªâ± µF ¬çÅ®Õ¢¤£³Á£Ä©¨ ³ FÀ±Ä¤î©»
©â¶ ±³î º©¯·EDô¬ºÓ¢¤£³¥ ¦ª³¨ ¶£¿Á D¥ ©î£ Á ¨ ³¤º d ©F k Å
F«©¬¢¾·ºã¢¤£Á£©¨
óº©¯Èã©¯ ñ¢¤ºº¿ ¨ ³³£ Â¶ ¢ ¦ª ± Fµ¬ÑÅ
ô
º¥
£«©¨¢¾ª ò ¶ d = 2 ©F k = 1 òº ñ¢¤ò¨ ©J §Ì
12 13 14 15 16 17 18 19 21 31 41 51 61 71 81 91
There were 16 possible codes .
^
`
³ ¸¹£¬»º¹¶ç»ª¸È°Á®£³±£¦ D¥¹¶ ±³î » £©¯ D¥¹¶ç£¬¨ ©F ¬³¹¸È»º
©¢¤£Á£©¨ ¸È¬º³£ JÚ£ ³£©¯ D¥ Dô³£¦î ±£¬£³Ä±¬îÅ Ý©¯±´© JÚ Ñ ©J©«^ ³`Ië
^ º `
£¬¤°¨ ¬®Â¶ß ³º D ©£¬©¤º¹¶£ D ©¯¡¬ Â¶ n ¯¢ 100 ¹ç©J
Exercise 101
unsigned int f ( unsigned int n)
{
if ( n <= 2) return 1;
return f (n -1) + 2 * f(n -3);
}
Exercise 102 ¶ç»ª¸È°Á¹¶ ±¬î
F¯ã© ¬§¨ ³¤Ã¸È»ºó©¿ÁÌ D¥³ Dô©¯ x ° © ç£¦IF
³ ±Ä Hî ¶ º³£³À»¿Ä ±¬ © ¬ª³¨ ¤VM¥Å
// PRE : [ first , last ) is a valid range , and the elements *p ,
//
p in [ first , last ) are in ascending order
// POST : return value is a pointer p in [ first , last ) such
//
that * p = x , or the pointer last , if no such pointer
//
exists
int * binary_search ( int * first , int * last , int x)
{
int n = last - first ;
if ( n == 0) return last ;
// empty range
if ( n == 1)
if (* first == x)
return first ;
else
return last ;
// n >= 2
int * middle = first + n /2;
T \ \ if (* middle > x ) {
// x can ’t be in [ middle , last )
int * p = binary_search ( first , middle , x );
if ( p == middle )
return last ; // x not found
else
return p ;
} else
// * middle <= x ; we may skip [ first , middle )
return binary_search ( middle , last , x );
}
Ý©¯ º ¨½©¯°¨¨ ¨ ³³£ T (n) Â¶ ±¨¢©£¬¤ ¬¸Ó³ ³ Ä± ¬ §¨ ³¤º
©F x º©¯¿º¥»â¶ ±¬î ¢ £º¶£¬¨óó ¶óº ¨½ ³£ Â¶ ¡³ ± ¬ª³¨ ¤
º » n £KJ
F © ¯¢«¢ ³£ F T (n) º©¯ß ©¯½Á¥Ì F ©¯ ¢ ³ ¦ªÅ H ¨ ©J ^ º `
³î©¯³¨½³¤ëÂ¶ Ú³£³±³» Å M
Exercise 103
^
`
×áñÊå
2
n
2
£ © JÝ ©¯£Ä©¯ ¨½ ³£ n
×áñÊå
2
n
2
=
×áÜÊå
2
2 ï¢¤£³ D¥ ºã¶ç»ª¸È°Á½¸Ï Hî»M¥ ¾©¯»î¬çÅ
n − 1.
Exercise 104 Ý£¬»ã¢¤£³ÄÁ£©¨óº©¯Õ¢¤£³ F¡± £¦ºªÁ£©¬¢¾¥±¬ F£©¸¹»ÁÈ¶£ÀºÈ¶°§¸¹°Á
^ `
»Fµ³ ¨ ©J ³£ J ³³¨ó (Σ, P, s) Å
Ð `
_ `
í
Σ = {F, +, −} s = F + F + F + F
ÐØµè
P
åÛVUÉ³Ø _ ì
F →
FF + F + F + F + F + F − F.
í
Σ = {X, Y, +, −} s = Y
íµÐØµè
P
åÛVUÉ³Ø _ ì
X →
Y+X+Y
Y →
X − Y − X.
[ ÊÌÍ·ËÇÚÉãè«ÍäÐ³æ ÛÜØµå«í«ÖµÒÉãÍÊÌËäÐçËÂÛÜÊØÐØµåáÜÉ
èµÉ³åÌÍÉ³É³Ò ÐçØÚè ÛñØ ËÂÉÍîÎÚÍÉË
ÐØµè
ÐÒ
α = 60
Y
º X
îÞßÊ
U ÉÓÊØÚÉÏÒîËÂÉÄÎ Ô×ÊÌÍæÈÐçÍÂè
«é
Ù` ÛV]É
_ `´í_ Ö«ËÈæ ÛñËÇ ËÂÇµÉÓÎÚÍÊôèµÖµÙËÂÛÜÊØµÒ
X →
X + Y + +Y − X − −XX − Y +
Y →
−X + YY + +Y + X − −X − Y.
\ \
1
2
3
Figure 21: Õ ¸Ï£Â¶ ©
Exercise 105
Æ Ê¦æ¹ÉÍÂÒ·ÊÔ ÐØµÊÛ
¢ ª Hº©¯â±Ä© ©µ±³¡©¯» JÝ ³ ³ Á« ¶ £³¨ ³³¢¾NM
¤
º ¶°§¸¹°Á¥Å Õ³£³ó©£³ º¥£³Ä ¸ÓÌ Fµ³ ¢Áß×©« ³¬ªIF 1, 2, 3 ¹¸È ³£³ º £¦³ ¢ Á F¯
© î©µ± ½Â¶ n F» Ì ¹³î©µ± ôIF ° FµÄ±³£³Ä©¯Ä»Á £ Fµ£óÂ¶ È °Á £³ °Á £³ ¥Å
àÁ«ç©¯ ë £©¤»¶ç³£ºî©µ± Â¶ F ë¿¢Á · KJó¨½ DÌ»Á F ©¯ ©»¨½
¶£¨ â¢Á © º£ÌÅ½Õ £¬§¿º©¯â©¯· »¨½ Ó© ª©£Á«³£ F ¨½©J ó ³¢
Â¶ ©¨½©¯°§£ çÅ £ µ©¨¢¾ª ï¸Ó ± F ³î©£¦ÏX Jó¨ DÌ°Á º³¢¤¨½³F ß ¢ Á 2
Hî¨½ D¥ (1, 2)M º ¨½ D¥¿ºà F ò¶£³¨ ¢Á 1 ¢Á 3 Hî¨½ D¥ (1, 3)M º³ ¨ Dô
ºÀÄ¨½©¯»ª³£ F ¶£¨ ¢Á 2 ¤óº ª©£îÁ¥³£ F ¢ Á 3 Hî¨ Dô (2, 3)M ³±Å
£¬à© ¢¤£³Á£Ä©¨ hanoi.C º©¯ïñ¢¤ºã© ³ ±Ä Â¶·¨ Dô¬ º©¯ FµÌ¬ º £³ °£³I F
£©¤»¶ç³£ ¶£ Á D¥ »¥¢¤ n Å
£ µ©¨¢¾ª ¶ n = 2 º¼©« ³ D¥ » ©¯à ³ ±Ä
(1, 2)(1, 3)(2, 3) » ©¯£Ä© FJ±Ä¨¢¾ª¬ ©
F D¥¬ßºâ¢¤ ªÅ ¹± ºó±Ä£¬£³ ±³ ¬³½Â¶
JÚ£â¢¤£³ÄÁ£©¨ K J½©
FÝ©¯ ªÄ©¯³ ¶ç£ n = 3 ãX J ¨ © ¡©¯°E J £³î¢¤£ F ±³°Á ºß ³ ±Ä^ Â¶`
¨ Dô¬ © ¢¤îÄ± Â¶Ó¢©¬¢ ³£ HÄ£ © ©µ±¬¾©¯ ¸Ó³£Â¶ © ï ¶ JÚ © D¥ N M¥Å
Exercise 106 Ó ºÏÌ±Ä±Ä©¯ÝÂ¶ï¨½© £·¢£¦ºÏXDô³¤º
3.2.11 Challenges
º ä î ©¯©
Ä± ¨¢© J © » ¾¬°²
³± ô³£¦à ³ß±Ä»ª±¬GF » © ©¯ª ³¨½Å £¿º
Ì±Ä±³£ ±¬©¨¢¤ ¤³¥²¢ º
±°§Ä±¬î ±Ä¨¢¤£¬»IF Â ¶
F ³ £³¤ ³î± ³£¦ Ó©Dô©×©« ¦ª° ¢©µ± ©Á¥¬ Â¶ D¥ ³î± ³£¦
Ä©µ±³Å
Ý³Ý ¬ J¯»Á © ¢©µ± ©Á¥ J ±´© Ãç ¸È¥ î±³ß³î± ³£¦ò» ±¤î©»¤ÑÅ®ã±¨¢© J
¤ J ÁÌ¡©£©¤Ä¬ º©¯ Ä©µ±³ ¢©µ± ©Á« ±¤î©»¤ D¥ F ³£³³ ¤ ³± ô³£¦çÅ ¬ë ©¯¨½
º©¯ Ä©µ±¬À¢ ¦ªà³§Ä±¬î¼Â¶ D¥ F ³£³¤ò³± ô³£¦ë» ¡©¯» Jó ¬E J ³ß±Ä¤î©» I F
» © J Á D¥ ¢©µ± ©Á«Å ¸ ¨½© J¢©µ± ©Á«¬L
Fµ JÚÝ G Fó ¬ J © D¥³£Ä©Á« » £ Fµ³£
ó © Dô ©¯» ºÀ³± ô³£¦
£óº ±Ä©¯ Â¶
¸¹º
î± ô£¦ à©Ý ³¸È¢©¬¢³£Ý±¬×©»¨½I F H´ ©¯G F
±¤Ä»Á © ¨½©¯ºÀ¢¤£³Â¶³³£ M®º©¯¹º¥ã ¨½ ³£¿ 763 Å ¸ F F º ¢¤£³Â¶¬£ß©£¬£¬ D¥
©¯·º©¯ò ¨½ ³£ Ó©
Fó Ï±£¬£³Ä±¬
£¬ ©¢¤£³ÁÌ£©¨ º©¯±¨¿¢¤¬ º © D¥³£©Á¥ ¨ ³³£½Â¶ë¢©µ± ©Á¥¬ º©¯ J I F
¬ JÀ¶ç£ © ±Ä»ª±¬ Â¶ß n Å H © »¨¿¢¾§ ±¬Ä± J ³ F Á«¬ à¢©µ± ©Á«
© Dô³£©Á«À ¶ n = 5M¥Å Ý©¯
Fµ JÚÝÁ«¬ ¶ç£ n = 555
T \ \
Note: Â
£ Fµ³£ß D¥Àº¥» ±¬©¯°§Á«° ©ó¨ ©¯º³¨½©¯±Ä©¯°EJ Fó¸ ©J JÚ GF
¨½ Ä©¯î± ¸ãªIFÁ«½Â¶â¢¤£³¥ ©« ³ J º£KJµÅ Ã¶ç£½ £¿¢¤ £¢ ¬ »Ï»©¯²ç
³·©F¸ ©D¥¿¸ÈJ J £Ï¢¤£³ÄÁ£©¨ »ß±£¬£³Ä±¬îÅ
Exercise 107 Ã°Fµ³ ¨½©JÚ³£® J ³³¨ ±Ä©
©¯ ÀçIF F£Ä©¸ H » £³´©¯»³î± M ¢¾ª©¤º
¸¹ºßº Á£¸òºë¢¤£³±¬³ Ä»¨ª©¯GFß KJºDô©£¬îò»³£©¯î¤ÑÅ £Óº¥» ¸ÓXD¥£ º³£³
¨³ ³ß©¢³ ³ J Â¶ß± £³Ä©¯°Á ¬£© ±³¬çÅ ³òàº£³¶£ ³¤© ±Ä £Fµ»¶ç©È¬
{F, +, −} Â¶âIJ ¨ ³ã¸¹º µIF®¨ ´© »Áó©
FÀ ¸ Σ = {F, +, −, [, ], f} Å¿Õë J¯¨½ [
Fµ GF ó©D¥ º·¶ç»ª¸È°ÒîÁ ËäÐç ËÂÉ¹Òî±¬ËÂ ÐÙ ] º ±³£¬£³¤ ³î©¯ Â¶º £¦ºª H°¢ ©F
F»£±¬M »ï¢¤È½³¢ Â¶Óº
¸È¥±¬ » » »©¯» Jó³¨¿¢¾ JÚÅ®ÕÏ J¯¨½ ³ ] ç¬º
º¿³î©¯ Â¶àº¿£¦ºª® Ä©µ± º® ¹¶ F½ ³¢ Â¶àº¿³î©¯¿³î©µ± ©
F £³³¨½ D¥¬
º ³¢ ³î©¯Ó¶Ä£³¨ º î©µ± Å Õ¥ß¨ Ä±¬© Ä¨ ±Ä© I F £³³¨ ¨½ ³£Ý©±Ä³£¦î©»
³î©¯ ©Fó£³¬ £¬ ó»·×©¯³£ÌÅ
£ µ©¨¢¾ª ¶º £³î©¯î ©Áª » 45 Fµ Á£³³ Àº ¸Ó£ F FF[+F][−F] ¢¤£ F ±Ä¬
º F£Ä©¸È°ÁÝÂ¶ §Á£
Å
p
Figure 22: Õ £¦º§ ©ä¶³³£Ï¢¤£³Ì±Ä¬³°Á º±Ä¨¨ ©F ³ ±Ä FF[+F][−F]
¥» FµÌ¬ë ¹ªÌ ® ôÀ©D¥³£KJó³¢¾¥³±Ä©¯GF¿¢¾×©¤RJ ³ · ¬¹ ¶ J ¶£ß«©¨¢¾ª £KJ
ºã¢¤£ F ±¬
F →
FF + [+F − F − F] − [−F + F + F]
¸ º° ©¯¸Ï£IF F Õ£î©¯î©Áª 22 FµÁ£¬ ò©F ¶ £â£©¯î¤ J ¸¹° â¸ò©¯
¹
¸Ó ¨½Ä© Å
äë£³³¨ ©»¤ ¦¢¾ª©° ¸È©¯ëº J¯¨½ f ¨½Ä©¤çÅ äë©¯ ºóç©¨½ Ä±¬ß º
³î©¯ Â¶ º®£¦ºªó©¯ F ëµ
±î¢¾Ïº©¯ ZFµÌ¬ ZF£©¸ © » Å ¼±Ä© »¨ ©Á» ®º©¯
f ¨½© ¬º £¦ºª
¨¢ Å
³£³½©£³óºã¶ ±³î¤ Â¶ º ¬£©£KJ turtle º©¯±£¬£³¬ ¢ F º¥ © F F ©¯
¶ ±¬î ©¯ JµÅ jump £³Ä©¯ ¬ f ·¸ò¥»§ save ©
F restore ©£³Ó¶£ [ ©F ] Å Â £ Fµ³£
F£©¸
§Á£
ï¸Ó±´© º³£³¶ç£³À ºÈ¶°§¸¹°Áó³î©¯¨ ³¤ºçÅ
ifm :: forward (2);
ifm :: save ();
ifm :: left (45);
\ ifm :: forward ();
ifm :: restore ();
ifm :: save ();
ifm :: right (45);
ifm :: forward ();
ifm :: restore ();
³£³
JÚ Äãº©¯ ¸Ó ±Ä©®¢¤£³ DÌ Fµ © °¤Á«³£â forward ³»»Á¿Ã¸ ¨
¸Ó ¸ ©¤ ¨ D¥ò¶£¬¸ ©£ F Hîº Fµ¶Ñ©òº©¯·¸Ó ©¯¸ ©J ¿IF½ ³¶ç£³ » 1MôÅ
© JÀ³¢¾
¸ ³£³ ±¨ ³ßº ±³ ©¯»ª³Á« ¸¹£¬» ©½ £¦º§ßÁ£©¬¢¾¥±¬â¢¤£³ÁÌ£©¨ amazing.C º©¯
¸¹° Ì± £ Ì± ¼
Âðº³£ ¸Ó£ F¯ ó¸Ó ©£³ ©¯ °Á½¶ç£º ¨ ³´© ¶
£³Ä©¯î±
¸È©¯XD¥£½¢¤î±³£³ º©¯J ±´© ¢¤£ F ±Ä »Á¼º £³Ä±³£¦ D¥ F£©¸¹»Á
±¬¨ ³¢ Â¶óº £¦ºª Á£©¬¢¾¥î±³ ±¨ ¨½©F¯ó»¤£ F ±ÄIF ç ¶Ñ©£ Hº³£³ ©£³ »
¨ £ ±Ä¨¨½©
F¯ º©¯½©£³ ¨ £ £ §¬¼±Ä¨¨ Ý ³½ £ £¦º§ ¬£©£KJ ³³¢¾ ©¯
Σ = {F, +, −, [, ], f}M¥Å
×¶ J º¥»
º©¯ JÚ ±Ä© ¡ ¬¨ ß© ±£©¬¢«¢ J ¢¤£³ÁÌ£©¨ ©
F ³»»â´© £¬¶Ä» ¢ »¤º
JÚ ££¬§Á«îÅ ò¸Ó±Ä¤â J £ ¢£¦º¨½©¤³¥¢ Á D¥ J £ ¬³
Chapter 4
Compound Types
\ \ \
4.1 Structs, or Plain Old Data
³£¬
¨ ¨½ ³£¦ Â¶

àç³£ GFµ
Fµ¬³£¬¡±¬£ÌÅ
±¬ Ï©Ý©ÁÌÁ£³ Á¥©¯±³ª©¯³·º©¯Ã©¯ò ¿ ¤î©¯î±ZF«©¯î©
J¦¢ ®¢ »¤³£½ ¨ ³¨ ³³£ £¬ ±¬ ¿
H£ ©£¬£©J Â¶ ±¬ JÑ¢ ³NM £ £³¶£³³ ± ©F ©¯
G F®±³¢ J ´©¯³Ä°Á ¨½³¤ï³¢ ³£©¯£ë©F ë³£ Fµ IF
Ä±¬î ¢©£©ÁÌ£©¬¢¾ ÓÂ¶¿º ¤î©F«©£IF H ¤
î©F«©£IFKM
ÒîËîÍÂÖµÙËÂÒ
Ä±¬î Ã¸Óâ¸ ¸
©£³ÓGFÀ Á£³¯¢F«©¯î© ©F ¥ ¦î©°
ÊÌ Î ³ÉÄ¸ ÍäÐç ËÂJÑÊÌ¢ Í ³Ê U¸¹
ÉÍÂáñºÊ Ðèµ©¬ÛÜ¢«Øµ¢¾å î±Ä©¯ î¢±³ ±ã¶Ä ±¬ ©¯» JÚÅ ¸¹»»ò©¯ ¸
±Ä© ¬ ¢¼° ¨ © »Á ³¸ JÑ¢ ³ ´©¯ J ©F °¤ D¥
Å
Â½º¥
" Ö«ÎÚÎÊÒÉ¿æ¹É æÈÐØ¯ËÏËÊÖµÒÉ
ÛñØÝÐßÎÚÍÊåÌÍäÐÞ íÛé ÉÌéñí¤Ø ÖÚÞ_ ÉÍÒ ÊÔÕËÂÇµÉ¿Ô×ÊÌÍÂÞ
£

¯
©

½
¨
³

¦
£

í æ ÇµÉÍÂÉ _ ÊÌËÇ ËÂÇµÉàØ ÖÚÞßÉÄÍäÐçËÂÊÌÍ
ÐØµè½ËÂÇµÉ èµÉ³ØµÊÞßÛÜØÚÐçËÂÊÌÍ ÐÑÍÂÉ ÛÜØ ËÂÉåÉÍÂÒ³é èµÊôÉ³ÒÓØµÊÌË
n/d
d
ÇÚÐXUÉëÐ Ô×ÖÚØµèÚÐÞßÉ³Ø¯ËäÐáË ì Î ÉâÔ×ÊÌÍÈÍäÐçËÂÛñÊnØ Ðçá ØôÖµÞb_ÉÍÂÒ³í«ÒÊ¿æ¹ÉãÇÚÐXUÉã
ËÊ ÛÜÞÀÎÚáÜÉ³ÞßÉØ Ë Û²ËòÊÖ«ÍÂÒîÉ³áVUÉ³Ò³é
SÝÉßÙ³ÊÖµáÜè ÊÔ¹Ù³ÊÖ«ÍÂÒÉßÍÂÉÄÎÚÍÉ³ÒÉ³Ø¯Ë¿ÐÍäÐçËÂÛÜÊØÚÐáÃØôÖµÞ_ ÉÍâÒÛÜÞÀÎÚáñì _ôìÝË æ¹Ê UÐáÜÖµÉ³ÒàÊÔïË ìôÎ É
í
int
_ÚÖ«Ë ËÂÇµÛÜÒ·æ¹ÊÖµáÜè ØµÊÌË_ ÉëÛñØ áñÛÜØµÉâæ ÛñËÇ ÊÖµÍ·Î ÉÍÂÙ³ÉÄÎÚËÛÜÊØ ÊÔ ËÇÚÉëÍÂÐçËÂÛÜÊØÚÐá¾ØôÖµÞ_ ÉÍÂÒ ÐçÒ ÐÀèµÛÜÒîËÂÛÜØµÙË
Þ®ÐçËÂÇµÉ³Þ®ÐÑËÂÛÜÙ¬Ðá Ù³ÊØµÙ³ÉÎµË¬é Æ¹ÇÚÉË æ¹Ê Ø ÖÚ
Þ _ ÉÍÒ
ÐØµè
N_ ÉáÜÊØµåËÂÊåÉËÂÇµÉÍ «í ÐØµè ËÇÚÛñÒëÛÜÒàÐáñÒÊ
n
d
ÍÂÉ QÚÉ³ÙÄËÂÉ³è ÛñØ½Þ®ÐçËÇÚÉÞ®ÐçËÂÛÜÙ¬Ðçá¾ØµÊÌËäÐÑËÂÛÜÊØ ÈËÇÚÉ Òîìôb
Þ _Ê
á Ô×ÊÌÍÓËÂÇµÉ ÒÉËÏÊÔ ÍäÐçËÂÛñÊØ Ðçá ØôÖµ
Þ _ ÉÍÂÒ ÛÜØµèµÛ Ù¬ÐçËÉ³Ò ËÂÇÚÐçË æ¹É ÐçÍÂÉàèµÉ¬ÐçáÜÛÜØµå æ Û²ËÂÇ Ð Þ®ÐçËÂÇµÉ³Þ®ÐçËÛÜÙ¬ÐáË ìôÎÉÌí èµÉ YÚØÚÉè _ ì ÛñËÂ
Ò UÐáÜÖµÉëÍäÐçØÚåÉ
ÛñËÒ
ÐFØ _ É
Ô×ÖµØµÙËÂÛÜÊØÚÐáÜÛ²Ë ì ^ÒîÉ³É " É³ÙËÂÛÜÊØ \ é é `Äé êèµÉ¬ÐçáÜáñìí æ¹Éëæ¹ÊÖµáÜè áñVÛ ]ÉâËÂÊ®åÉËÏ
Ð Ë ìôÎ ÉëËÂÇÚÐçË ©Ù¬
ÖµÒÉ³è áÜVÛ ]ÉëGÉ P«ÛÜÒîËÛÜØµåßÐçÍÂÛñËÇÚÞßÉÄËÂÛÜÙ Ë ìôÎ É³Ò ËÂÇµÉëÔ×ÊáÜáÜÊ¦æ ÛÜØµå¿Î ÛñÉ³Ù³ÉàÊÔ Ù³Ê èÚÉ ^»Ô×ÊÌÍÏÐèµèÚÛñØÚåË æ¹ÊÀÍÂÐçËÂÛÜÊØÚÐá
ØôÖµ
Þ _ ÉÍÂÒ `ïÒîÇÚÊ¦æ ÒÈÇµÊÑæ ËÂÇµÛÜÒ¹Ù³ÊÖµáÜèáÜÊôÊ ]®áÜÛ ]ÉÌé
// input
std :: cout < < " Rational number r :\ n";
rational r ;
std :: cin > > r;
std :: cout < < " Rational number s :\ n";
rational s ;
std :: cin > > s;
// computation and output
std :: cout < < " Sum is " < < r + s < < " .\ n" ;
Ê ÉÍÂÒßÒÉIUÉÍäÐá Ù³ÊØµÙ³ÉÄÎÚËÒ Ô×ÊÌÍ®èµÉ YÚØµÛÜØµåÝØµÉæ Ë ì Î É³ÒL_ ÐÒÉ³è¼ÊØ ÉIP«ÛñÒîËÂÛÜØµåÝË ìôÎ É Ò³é êØ¼ËÇÚÛñÒ
ÒÉ³ÙÄËÂÛÜÊØ¤í æ¹ÉòÛÜØ ËÍÊôèµÖµÙ³ÉòËÂÇµÉòÙ³ÊØµÙ³ÉÎµËïÊÔ
é îÒ ËîÍÂÖµÙË ÛÜÒÕÖµÒÉ³èßËÂÊ ÐååÌÍÂÉåÐçËÂÉÈÒÉ UÉ ÍäÐçá UÌÐçáÜÖµÉ³Ò
º Ë ì Î ÌÉ é S¼ÛñËÇ ËÂÇµÛÜÒ³í æ¹É Ù¬ÐØ É¬ÐçÒÛÜáñì ß
ÊÔÓèµÛ ÉÍÉ³Ø ËÀË ì Î É³ÒÀÛÜØ¯ËÂÊ ÊØµÉ UÐáÜÖµÉóÊÔÓ³Ð £¬Øµ¡É±¬æð
Þ ÊôèµÉ³á¹ËÂÇµÉ
Þ ÐçËÂÇµÉ³Þ®ÐÑËÂÛÜÙ¬ÐáË ì Î É ÐÒÓÐÀØµÉæ Ë ì Î É ÜÛ Ø&àé ·ÉÄÍÂÉ ÛÜÒÓÐ æ¹ÊÌÍ ]¥ÛñØÚåÀÎµÍÂÊåÌÍÂÐÞ
®
YµÍÂÒîËòÒîËÂÉÎËÂÊÑæÈÐÑÍÂèËÂÇµÉÏèÚÉÒÛñÍÂÉ³è Î ÛñÉ³³Ù ÉÏÊÔ ³Ù ÊôèµÉãÐ _ ÊUÉÌé
// the new type rational
struct rational {
int n;
int d ; // INV : d != 0
};
// POST : return value is the sum of a and b
rational add ( rational a , rational b)
{
rational result ;
result . n = a.n * b .d + a.d * b.n;
result . d = a.d * b .d;
return result ;
}
int main ()
{
// input
std :: cout < < " Rational number r :\ n";
rational r;
std :: cout < < " numerator =?
" ; std :: cin > > r.n ;
std :: cout < < " denominator =? " ; std :: cin > > r.d ;
std :: cout < < " Rational number s :\ n";
rational s;
std :: cout < < " numerator =?
" ; std :: cin > > s.n ;
std :: cout < < " denominator =? " ; std :: cin > > s.d ;
// computation
rational t = add (r , s );
ËÂÇÚÐçËÓÞ®Ð ]ÉÒ Ð
// Program : userational. C
// Add two rational numbers .
# include < iostream >
\ // output
std :: cout < < " Sum is " < < t.n < < "/" < < t.d < < " .\ n ";
return 0;
}
Program 32: ¢¤£³ÄÁ ç³£©¯î ©¯×Å
\ êØ àí Ð ÒîËîÍÂÖµÙË¿èÚÉIY ØµÉ³Ò¿Ð ØµÉæ Ë ìôÎÉ æ·ÇÚÊÒîÉLUÌÐáñÖÚÉßÍäÐçØÚåÉ®ÛÜÒëËÇÚÉ
ÊÔÃÐ Y P«É³è ØôÖµÞ_ ÉÍ ÊÔ Ë ìôÎ ÉÒ³é êØ½ÊÖ«Í Ù³ÐÒÉÌíæ¹ÉàèµÉ YÚØµÉ ÐÀØµÉæ Ë ìôÎ ÉàØÚÐÞßÉè ©£¦¬ ©¼¢¤£³æ· FÇÚ ÊÒî±¬É
UÐáÜÖµÉãÍäÐçØÚåÉâÛÜÒòËÂÇµÉ ¹ÐçÍËÉ³ÒÛ×ÐØ ÎÚÍÊôèµÖµÙË
í æ·ÇÚÉÄÍÂÉâæ¹ÉëÛÜØ¯ËÂÉÍÎµÍÂÉÄË ÐbUÐrational
áÜÖµÉ
ÐÒòËÂÇµÉ
int int
(n, d)
WôÖµÊÌËÂÛñÉ³Ø Ë
é
n/d
" ÛÜØµÙ³É ËÂÇµÉÍÂÉÓÛÜÒïØµÊàË ìôÎ É Ô×ÊÌÍ¹ËÂÇµÉ èµÉ³ØµÊÞßÛÜØÚÐçËÂÊÌÍÕæ ÛñËÂÇßËÂÇµÉÓÐçÎµÎµÍÂÊÌÎµÍÂÛ×ÐÑËÂÉUÐáÜÖµÉ ÍäÐØµåÉ
í
int\{0}
æ¹É ÒîÎÉ³Ù³ÛÜÔÜìÀËÇÚÉòÍÂXÉ WôÖµÛñÍÉ³ÞßÉ³Ø Ë _ôìÀÐØ®ÛÜØµÔ×ÊÌÍÂÞ®Ðá
í«ÐÏÙ³ÊØµèÚÛ²ËÂÛÜÊØ ËÂÇÚÐçË Ç ÐçÒËÂÊëÇµÊáÜè
Ô×ÊÌÍÐáÜá¥áÜÉåÐá¥Ù³Ê
Þ _ÚÛÜØÚÐçËÂÛÜÊØµÒ ÊÔ dUÌÐç=áÜÖµ0É³Ò³é " ÖµÙäÇßÐØ ÛñØ UÐçÍÂ»Û×Ð Ø¯D Ë ©ÛÜ£¬Ò ©ÛÜØµ¤èµ ÛÜÙ¬ÐçËÉ³è _ôìÐ Ù³ÊÞßÞßÉ³Ø Ë ÒîËäÐçÍËÛÜØµå
æ Û²ËÂÇ
// INV:
Û
] É
µÎ ÍÂÉ ÐçØÚèÏÎ ÊÒîËÂÙÊØÚèµÛñËÛÜÊØµÒÊÔ Ô×ÖµØµÙËÂÛñÊØÚÒ^ÒîÉ³É " ÉÙËÂÛÜÊØ é é `´íÑÛñØ UÐçÍÂÛ×ÐØ¯ËÂÒ¤ÐçÍÂÉÃÐØãÛÜØµÔ×ÊÌÍÂÞ®Ðá
æÈÐ³ìâÊÔôèÚÊ Ù³ÖµÞßÉ³Ø ËÛÜØµå¹ËÇÚÉÃÎµÍÂÊåÌÍäÐÞ ÑËÂÇµÉìëÐÑÍÂÉØµÊÌË¡ÒîËäÐØµèÚÐçÍÂèµÛ³Éè¾íÐØµèëÊÖ«Í æÈÐ³ìãÊÔôæ·ÍÂÛ²ËÂÛÜØµåïËÂÇµÉ³Þ
ÛÜÒÈÊØµÉÏÎ ÊÒîÒVÛ _ÚáÜÉÓÙ³ÊØ UÉØ ËÂÛÜÊØ¤é
ÆÈÇµÉÓË ìôÎ É
ÛÜÒïÍÂÉÔ×ÉÍÍÂÉ³èóËÊ ÐÒòÐ
í ÐçØÚè ÛñË¹Ù¬ÐØ _ ÉÓÖÚÒîÉ³èóáÜÛV]ÉÏÐØ¯ì®ÊÌËÂÇµÉÍ¹Ë ìôÎ É rational
³

¬
£

³
±

Ô×ÊÌÍòIÉ PµÐÞÀÎÚáÜÉí¥ÛñËòÞ®Ð¬ì ÐÑÎÚÎÉ¬ÐçÍòÐÒ¹ÎÐÑÍäÐÞßÉËÂÉÄÍÈË ì Î ÉâÐçØÚè ÍÂÉËÂÖ«ÍÂØË ì Î ÉãÛñØÔ×ÖµØµÙËÂÛÜÊØµÒÈáÜVÛ ]É
é
add
A struct defines a type, not variables.
_ É³åÛñØÚØµÛÜØµå«é Æ¹ÇµÉãèµÉ YÚØµÛñËÂÛÜÊØ
ÉË ÒåÉÄË ÍÂÛÜèÀÊÔ ÊØµÉÈÎ ÊÒÒÛV_ÚáÜÉÈÙ³ÊØµÔ×ÖµÒÛÜÊØÀÍÂÛñåÇ ËÕÔÜÍÂÊÞ ËÂÇµÉ
struct rational {
int n ;
int d ; // INV : d != 0
};
èµÊôÉ³Ò
èµÉ ÚY ØµÉ U ÐçÍÂÛ×Ð _ áñÉ³Ò ÐØµè
ÊÔË ì Î É
í¡ÐáñËÂÇµÊÖµåÇ½ËÂÇµÉ Ë æ¹ÊÞßÛÜèµèµáÜÉëáÜÛñØÚÉÒãáÜÊ Ê ] Üá ÛV]É
èµÉ³Ùá×ÐçÍäÐçËÛÜÊØµÒ Ð Ò æ¹É n ]¥ØµÊÑæ dËÇÚÉÞ é Ðçint
UÐçÍÂÛ×Ð _Ú Üá Éà
ËÂÇµÉÍ¬í¤Ðáñá Ô×ÊÖ«ÍãáñÛÜØµÉ³Ò ËÂÊåÉËÂÇµÉÍÏèµÉ YÚØµÉ¿Ð
Ê Ô

¦
J
¢

ËÂÇµÉ ØÚÐÞßÉ
í Ú_ Ö«ËÈÐçË ËÇ ÐÑË Î ÊÛÜØ¯Ë¬í ØµÉ³Û²ËÂÇµÉÍÈÐ UÐçÍÂÛ×Ð _ÚáÜÉ·ÊÔ¤ËÇ ÐÑËïØµÉæ Ë ìôÎÉÌí¥ØµÊÌÍRUÌÐÑÍÂÛ×Ð _ÚáÜÉ³Ò
rational
ÊÔ¡Ë ì Î É
ÇÚÐXU É
_ É³É³Ø èµÉ Ú
Y ØµÉ³è¤éÆ¹ÇµÉÏË æ¹Ê ÞßÛñèÚèµáÜÉ·áÜÛÜØµÉ³Ò
int
int n ;
int d ; // INV : d != 0
ÒîÎ ÉÙ³ÛÜÔÜì ËÇ ÐÑËïÐØ¯ìßÐÙËÖ Ðçá ÊÔËÂÇµÉ ØµÉæ¼Ë ìôÎ ÉZ^»Ûé ÉÌé ÐçØ ìÀÙÊØÚÙÄÍÂÉËÂÉ ÍäÐÑËÂÛÜÊØÚÐáµØôÖµÞ_ ÉÍG`îÇÚÐÒ ¥

¬
±

^ ÛÜÒ ÍÉÎµÍÂÉ³ÒÉ³Ø¯ËÂÉ³èb_ôìa` Ë æ¹ÊÓÊ _ É³ÙÄËÂÒ ÊÔÚË ì Î É
ËÂÇÚÐçË Ù³ÐØb_ÉòÐçÙ³Ù³É³ÒÒîÉ³è ËÂÇ«ÍÂÊÖµåÇàËÂÇµÉ¹Ø ÐçÞßÉ³Ò ÐØµè
int
n
ôÒîÉ³É·ËÂÇµÉ·ÞßÉÞb_ ÉÄÍÐçÙ³Ù³É³ÒÒ _ É³áÜÊ¦æëé Æ¹ÇµÛÜÒÃÒîÎ É³Ù³Û Y Ù³ÐçËÂÛÜÊØÛÜÒÛÜÞÀÎÊÌÍËäÐØ¯Ë ÛÜÔ æ¹ÉÈæÈÐØ ËÕËÂÊâÛÜÞÀÎÚáÜÉÞßÉ³Ø Ë
d
ÊÌÎ ÉÍÂÐçËÂÛÜÊØµÒ¹ÊØóÊÖ«Í·ØµÉæ Ë ì Î ÉâáÜÛ ]ÉÏÛñØ ËÂÇµÉÏÔ×ÖÚØµÙËÛÜÊØ
é
add
·ÉÍÂÉ·ÛÜÒÐçØ®ÐØÚÐáÜÊåÌì Ô×ÊÌÍËÇÚÉòÒÛñËÂÖÚÐçËÛÜÊØ¤é êÔËÂÇµÉ·ÖµØµÛVUÉÍÒÛñË ìÐèµÞßÛÜØµÛÜÒîËÍäÐÑËÂÛÜÊØëæÈÐØ¯ËÂÒÕËÂÊâÒîÎ É³Ù³ÛñÔÜì
ÇµÊÑæ ÐâÒîËÂÖµèµÉ³Ø ËÈÛÜÒÕÍÂÉÎµÍÂÉ³ÒÉØ ËÂÉ³è ÛÜØ®ËÂÇµÉ³ÛñÍ YÚáÜÉ³ÒíôËÂÇµÉìßÞßÛÜåÇ¯ËÙÊÞßÉ Ö«Î æ·ÛñËÂÇßËÂÇ«ÍÂÉ³É ÎÚÛÜÉ³ÙÉ³Ò¹ÊÔ¾èÚÐçËÂÐ
ËÂÇÚÐçËÐçÍÂÉ ØµÉ³Ù³É³ÒÒÐçÍì Ð ØÚÐÞßÉíÐØ ÛÜèµÉ³Ø ËVÛ YÚÙ¬ÐçËÂÛÜÊØ Ø ÖÚ
Þ _ ÉÍ³í ÐØµè Ð ÎµÍÂÊåÌÍäÐÞ ÊÔ ÒËÂÖµèµìé Æ¹ÇÚÛñÒ
èµÉ YÚØµÉ³ÒËÂÇµÉ Ë ìôÎÉ ëÊÔ¤ÐâÒîËÖÚèµÉ³Ø¯ËïÐØµè ÐáÜáñÊÑæ·ÒÃÔ×ÖµØµÙËÂÛñÊØ ÐçáÜÛñË ì ^»ÍÂÉ³åÛÜÒËÍäÐçËÂÛñÊØ¾íÙäÇÚÐØµåÉ ÊÔÎµÍÂÊåÌÍäÐçÞ
ÊÔ ÒËÂÖµèµìíµÉËÙÌé `àËÂÊ _ ÉÓÍÂÉ¬ÐçáÜÛ ³É³è¤í«áÜÊØµ
å _ É³Ô×ÊÌÍÂÉâÐØ ì ÒîËÖÚèµÉ³Ø¯ËÂÒòÐÙËÂÖÚÐáÜá²ìßÒÇµÊÑæ Ö«Î¾é
*
/K.*C
9 #..$&.*
6,I$
8/
*+BK
BI
M$&
22
A*
6 97@ >#./C%.
"
-B5 7"
+(7.#*
J
\ 4.1.1 Struct definitions.
êØóåÉ³ØµÉÍÂÐáíµÐ ÒîËÍÂÖµÙËòèµÉ YÚØÚÛ²ËÂÛÜÊØ áÜÊôÊ ]ôÒòÐçÒòÔ×ÊáÜáñÊÑæ·Ò³é
éñé²é
struct
G ·ÉÍÉÌí ÜÛ ÒâËÂÇµÉÀØÚÐÞßÉ
Ê Ô ËÂÇµÉßØÚÉÄæ áñì½ÛÜØ ËîÍÂÊôèµÖµÙ³É³è ÒîËÍÖÚÙÄË »^ ËÂÇµÛÜÒëØÚÐÞßÉÞÖµÒîË _ É®ÐØ ÜÛ èµÉ³Ø¯ËÂÛVYÚÉÍ¬í
" É³ÙËÛÜÊØ \ é é `ÄíµÐØµè ñí é³éé¬í ÐÑÍÂÉâØÚÐÞßÉ³Ò¹ÊÔ ÉGP«ÛÜÒîËÛÜØµå¿Ë ìô
Î É³Ò³éïÆ¹ÇµÉ³ÒÉëÐÑÍÂÉâÙ¬ÐçáÜáÜÉ³è ËÂÇµÉ

Fµ£
Ê Ô é Æ¹ÇµÉàÛÜèµÉ³Ø ËÂÛ Y ÉÄÍÂÒ í íñé´é³é¬í I ÐçÍÂÉ ÐçÍÉëËÂÇµÉ
F«©¯î©½¨½³¨½ ³£¦
Ê J Ô¡°ËÇÚÁ Éã ØµJÑÉ¢ æ ³ Ë ìôÎ É é
ÆÈÇµÉ UÌÐçáÜÖµÉóÍäÐØµåÉ ÊÔ ÛÜÒ é ÆÈÇµÛÜÒ®ÞßÉ¬ÐØµÒ³íòÐ UÌÐçáÜÖµÉ ÊÔÓË ìôÎ É ÛÜÒ®ÐØ
...
»ËÂÖ«ÎÚáÜÉ
æ ÇµÉÍÂÉ
é
N
(t , t , . . . , t )
P¥ÛÜÒîËÂÛñ1ØÚå 2Ë ìôÎ É³Ò óNÞßÛÜåÇ Ë _É tiÔ×ÖµØµè ÐçÞßÉ³Ø ËäÐçá Ë ì Î É³Ò³í _ Ö«ËëÐáÜÒÊ ÖÚÒîÉÍºèµÉ YÚØµÉ³è Ë ìôÎ ÉÒ³é [ ÊÌÍëÉIP ÐÞÀÎÚáÜÉÌíÙ³ÊØµÒÛñèÚÉÄÍËÂÇµ
É UÉÙËÂÊÌÍïÒîÎ ÐÙÉ 3 Ê UÉÄÍïËÂÇµÉYÚÉ³áÜè ëé Û UÉ³ØÀËÂÇµÉ·Ë ìôÎ É
ÐÒ Ð _Ê UÉÌí
rational
æ¹ÉÏÙ³ÊÖµáÜèóÞßÊôèµÉ³á 3 ÐÒ·Ô×ÊáñáÜÊÑæ·Ò³é
struct rational_vector_3 {
rational x;
rational y;
rational z;
};
ñá ËÂÇµÊÖµåÇÛ²Ë·Ô×ÊáÜáñÊÑæ·ÒÈÔÜÍÊÞðËÂÇµÉâèµÉ YÚØµÛñËÂÛñÊØ¾íôáÜÉËòÖÚÒòÞ®Ð ]ÉÏÛñËÈÉIP¥ÎÚáÜÛÜÙÛñË ËÂÇµÉÏË ìôÎÉ³Ò ñí é³éé¬í ØµÉ³É³è½ØÚÊÌË_ ÉàËÂÇµÉàÒÂÐÞßÉÌé ·ÉÄÍÂÉàÛÜÒÓÐØ ÉIPµÐçÞÀÎ áñÉ ÈêÔ
ÛÜÒ ËÂÇµÉZUÐáÜÖµÉëÍäÐØµåÉàÊÔ ËÂµÇ ÉëË ìôÎ É
í«æ¹ÉÏÙ¬ÐØóåÉË·Ð UÌÐçÍÛ×ÐØ ËòÊÔ¡ËÇÚÉÓË ìôÎ É 0, 1, . .æ . Ûñ,ËU
Ç UÐáÜÖµÉÓÍäÐØµåÉ
unsigned int
int
{−U, −U + 1, . . . , −1, 0, 1, . . . , U − 1, U}
ÐÒÈÔ×ÊáÜáÜÊ¦æ Ò³é
struct extended_int {
// represents u if n == false and -u otherwise
unsigned int u ; // absolute value
bool
n ; // sign bit
};
Æ ÇÚÉZUÐáÜÖµÉâÍÂÐØµåÉàÊÔ ËÂÇµÛÜÒ·Ë ìôÎ ÉàÛñÒ
¹
í } í _ÚÖ«ËÓáÜÛV]ÉãÛÜØ ËÂÇµÉëÍäÐçËÂÛñÊØ Ðçá¡Ù¬ÐçÒÉÌí
{0, 1, . . . , U} { æ¹É·ÛñØ ËÂÉÍîÎÚÍÉË UÐáÜÖµÉ³ÒÕèµÛ ÉÍÉ³Ø ËÂá²ì Ð
UÌÐçáÜÖµÉ
îÞßÉ¬ÐçØÚÒ
ÛÜÔ
ÐØµè −u ÛÜÔ n = é
(u, n)
u n=
UÉØ ÛÜÔ Ë æ¹Ê®ÒîËÍÂÖµÙËÓèµÉ YÚØµÛñËÂÛÜÊØµÒòÇ ÐXUÉàËÂÇµÉàÒÂÐÞßÉ
^»ËÂÇµÉëÎ ÐçÍËÓÊÔ ËÂÇµÉ
¨
³

¨
³
³

¢
Ä

³
±

´
±
¯
©

èµÉ YÚØµÛñËÂÛÜÊØ½É³ØµÙ³áÜÊÒÉ³è ÛÜØ ÙÖµÍáñì _µÍäÐÙÉ³Ò `Äí ËÇÚÉÄìÝèµÉ YÚØµÉ
Ë ì Î É³Ò³í ÐçØÚè Û²ËëÛÜÒãØÚÊÌËãÎ ÊÒÒÛ _ áñÉ

£
³

ËÂÊ ÍÂÉÎÚá×ÐçÙ³ÉóÊØµÉ Ô×ÊÌÍ®ËÂÇµÉóÊÌËÂÇµÉÍ¬é ïÊØÚÒîÛÜèµÉÍÀËÂÇµÛÜÒ ËîÍÂÛVU¥ÛÜÐá_ÚÖ«Ë®ÛÜØµÒîËÍÖÚÙÄËÂÛVUÉóÉIP«ÐÞÀÎÚáÜÉ æ·ÛñËÂÇ Ë æ¹Ê
ÐçÎµÎ ÐçÍÂÉ³Ø¯ËÂáñì®XÉ W ÖÚÐá¤ÒîËÍÂÖµÙËÂÒòèµÉ YÚØÚÉèóÊ UÉÄÍ ÐØóÉÞÀÎÚË ì®ÒÉÄË·ÊÔ GÉ P«ÛÜÒîËÛÜØµå¿Ë ìôÎÉ³Ò³é
\ struct S {
};
struct T {
};
void foo ( S s ) {}
int main () {
S s;
T t;
foo ( s ); // ok
foo ( t ); // error : type mismatch
return 0;
}
»ê Ë¹ÛÜÒ¹ÐáÜÒîÊëÎ ÊÒÒÛV_ÚáÜÉòËÂÊ ÖÚÒîÉÓÐçÍÍäÐ³ìßÞßÉ³Þ_ ÉÍÒÛÜØ®ÒîËÍÖÚÙÄËÂÒ³é [ ÊÌÍ¹ÉGPµÐÞÀÎÚáÜÉÌíËÂÇµÉYÚÉ³áÜè
ÇÚÐXUÉâèµÛÜÒÙ³ÖµÒÒîÉ³è Ð _ ÊUÉâÙÊÖÚáñèóÐáñËÉÍÂØÚÐçËÂÛVUÉáñìL_ ÉÏÞßÊôèµÉ³áÜÉè áÜÛ ]ÉÓËÇÚÛñÒ³é
3
ËÇ ÐÑË æ¹É
struct rational_vector_3 {
rational v [3];
};
4.1.2 Structs and scope
Æ¹ÇÚÉ ÒÙÊÌÎ ÉàÊÔÐÀÒËÍÂÖµÙËÏÛÜÒ ËÂÇµÉëÎ ÐçÍËÓÊÔ ËÂÇµÉëÎµÍÂÊåÌÍäÐÞ ÛÜØ æ·ÇÚÛñÙ´Ç ÛñË Ù³ÐØ _ ÉàÖµÒÉè ^ ÛÜØ Ð UÐçÍÂÛ×Ð _ áñÉ
èµÉ³Ù³á×ÐÑÍäÐçËÂÛÜÊØ¤íÌÊÌÍÃÐçÒÕÐ Ô×ÊÌÍÞ®Ðá¥Ô×ÖµØµÙËÂÛÜÊØ Î ÐçÍäÐÞßÉËÉÍ Ë ìôÎ Éí¯Ô×ÊÌÍÃÉIP«ÐÞÀÎÚáÜÉ`Äé " ËÍÂÖµÙËÒ_ É Ç ÐXUÉ¹ÒÛÜÞßÛñá×ÐçÍ
ËÂÊ Ô×ÖµØµÙËÂÛÜÊØµÒïÇµÉÍÂÉ ËÂÇµÉ ÒÙ³ÊÌÎÉÏÊÔ¡ÐàÒîËÍÂÖµÙË¹ÛñÒ ËÂÇµÉÏÖµØµÛÜÊØ®ÊÔ¤ËÂÇµÉÏÒîÙ³ÊÌÎ É³Ò¹ÊÔ¡ÐáÜáÛ²ËÂÒ
í
Fµ ±¬×©£©¯¤
æ ÇµÉÍÉâÐ ÒîËÍÂÖµÙËòèµÉ³Ù³á×ÐÑÍäÐçËÂÛÜÊØ ÇÚÐÒ¹ËÂÇµÉâÔ×ÊÌÍÂÞ
struct
Æ¹ÇÚÉÒîËÍÂÖµÙË èÚIÉ Y ØµÛñËÛÜÊØâÛÜÒ ÐòèµÉ³Ù³áÜÐçÍäÐçËÂÛñÊØëÐÒ¾æ¹É³áÜáíÐØµèàÖÚÒîÖ ÐçáÜáñìÓÊØµÉïÐÙËÖ ÐçáÜáñìÓØµÉ³É³èµÒ ËÂÇµÉÕèÚÉIY ØµÛñËÛÜÊØ
ÛÜØãÊÌÍÂèµÉÍ¡ËÊòÖµÒÉÕÐïÒîËÍÂÖµÙË¬é ÆÈÇµÛÜÒ¤ÛÜÒ¤É¬ÐÒìÏËÊòÉIPôÎÚá×ÐÛÜØ ÛÜØãÊÌÍÂèµÉÍ¡ËÊÈËÍÂÐØµÒá×ÐçËÂÉ ÐRUÐçÍÂÛ×Ð _ÚáÜÉ èµÉ³Ùá×ÐçÍäÐçËÛÜÊØ
ÊÔ«ÒîËÍÖÚÙÄË Ë ìôÎ ÉÌíÊÌÍ Ð·Ô×ÖµØµÙËÂÛÜÊØëæ·ÛñËÂÇëÔ×ÊÌÍÂÞ®Ðçá¯Î ÐçÍäÐçÞßÉËÂÉÍÂÒ¾ÊÔ«ÒîËÍÖÚÙÄË Ë ìôÎ É ÛñØ ËÂÊ Þ®ÐçÙ´ÇµÛÜØµÉÃá×ÐØµåÖÚÐåÉÌí
ËÂÇµÉâÙÊÞÀÎ ÛñáÜÉÍ¹ØµÉ³É³èµÒ·ËÂÊ ]¥ØµÊÑæ ËÂÇµÉâÐÞßÊÖµØ ËÈÊÔ ÞßÉ³ÞßÊÌÍìßÍXÉ WôÖµÛñÍÂÉè _ôìÐØ Ê _ É³ÙÄË·ÊÔ ËÇÚÉãÒîËÍÂÖµÙË³é
Ö«Ë ËÂÇµÛÜÒòÛÜØµÔ×ÊÌÍÂÞ®ÐçËÂÛÜÊØ ÛÜÒòÊØÚá²ìÊ _µËäÐÛñØ Ð _ áñÉãÔÜÍÂÊÞðËÂÇµÉâèµÉ YÚØµÛñËÂÛÜÊØóÊÔ ËÇÚÉâÒîËÍÂÖµÙË ÐÒ·áÜÊØµåßÐçÒ·ËÂÇµÉ
Ù³ÊÞÀÎÚÛÜáÜÉÄÍÕÇÚÐÒÕÊØµáñì ÒîÉ³É³Ø ÐãèµÉ³Ù³áÜÐçÍäÐçËÂÛñÊØ ÊÔ í¯ËÂÇµÉòÒîËÍÂÖµÙË ÛÜÒÒÐÛÜèÀËÂÊëÇÚXÐ UÉ
é
° ±¨¢¾ª¬à J¦¢
4.1.3 Member access
¼ÒîËÍÂÖµÙËÛÜÒÕÞßÊÌÍÉ¹ËÇ ÐçØ ËÂÇµÉ ¹
ÐçÍËÂÉÒÛ×ÐØ¿ÎÚÍÊôèµÖµÙËÊÔÛñËÂÒÃÖÚØµèµÉÍÂá²ì¥ÛÜØµåÓË ì Î É³Ò%# Û²ËÊ ÉÍÂÒÒîÊÞßÉ_ ÐÒÛñÙ
Ô×ÖµØµÙËÂÛÜÊØÚÐáÜÛ²Ë ìÊØ ÛñËÒ Ê¦æ Ø ÂË ÇÚÐçË®æ¹É ÉIP¥ÎÚá×ÐçÛÜØ ØµÉIPôË¬é Æ¹ÇµÉ½ÞßÊÒîË®ÛñÞÀÎ ÊÌÍËäÐØ¯Ë ^ºÐçÚØ è ÐáÜÒîÊ ÞßÊÒîË
U¥ÛñÒVÛ _ÚáÜÉ `ïÔ×ÖµØÚÙÄËÂÛÜÊØÚÐáÜÛñË ì®ÊÔ Ð Ò ËÍÂÖµÙË ÛÜÒÈËÂÇµÉàÐÙ³ÙÉ³ÒÒ ËÂÊÀËÂÇµÉãè ÐÑËäÐÞßÉ³Þ_ ÉÍÒ ^»ËÇÚÉ UÌÐñá ÖÚÉÒ
ÛÜØóËÂÇµÉ
ti
\ `´í ÐØµè½ÇµÉÍÉàÛÜÒ æ ÇµÉÍÂÉëËÇÚÉàÛñèÚÉØ ËÂÛVYÚÉÍÒ íñé´é¬é³í I Ù³ÊÞßÉëÛñØ¾é
êÔ
ÛÜÒÈÐØ ÉIP¥ÎµÍÂÉ³ÒîÒÛÜÊØóÊÔ¤Ë ìôÎ É æ ÛñËÂÇUÐáÜÖµÉ
í¥ËÇÚÉØ
#¼ËÂÇµÉ ËÂÇ ³Ù ÊÞÀÎ ÊØµÉ³Ø Ë
(t1 , . . . , tN)
tK
K
Ê Ô Û²ËÂÒUÐáÜÖµÉ # Ù³ÐØ _ ÉâÐÙ³Ù³ÉÒÒÉ³è ÐÒ
N
»ËÂÖ«ÎÚáÜÉ
t = (t1 , . . . , tN )
é ·ÉÍÉÌí
. «ÛÜÒïËÂÇµÉ
^ÒîÉ³ÉâÆ Ð _ÚáÜÉ ÛñØ ËÂÇµÉ ·ÎµÎ É³ØµèµÛ PßÔ×ÊÌÍòÛñËÂÒ¹ÒîÎÉ³Ù³ÛVYÚÙ³Ò `´é
¨

½
¨
³

£
µ
©

±
Ä
±
¬

¢
³

£
¯
©

£
Æ¹ÇÚÉòÙ³ÊÞÀÎ ÊÒÛ²ËÂÉ¹ÉIP¥ÎµÍÂÉ³ÒîÒÛÜÊØ
Û²ËÂÒÉ³áÜÔ ÛÜÒ ÐçØßáVUÌÐçáÜÖµÉÌíôÐçØÚèæ¹É ÒÂÐ³ì
ÛñÒÐØßá UÌÐáñÖÚÉòÛÜÔ
ËÂÇÚÐçË¹ËÂÇµÉÏè ÐÑËäÐ¿ÞßÉÞb_ ÉÄÍ .ÛÜÒ
é
©µ±Ä±¬IF ¶ç£
ÛñØÚÉÒ \ ÐØµè \ ÊÔ ÍÂÊåÌÍäÐÞ \ ÐçÒÒÛÜåØ UÐáÜÖµÉ³Ò·ËÂÊßËÂÇµÉâÍÂÐçËÂÛÜÊØÚÐá¾ØôÖµÞ_ ÉÍÂÒ
ËÂÇ«ÍÂÊÖµåÇ ËÂÇµÉ
r
ÞßÉ³Þ_ ÉÍ ÐÙ³Ù³É³ÒîÒÃÊÌÎÉÍäÐçËÂÊÌÍ³íæ ÇµÛÜáÜÉáÜÛÜØµÉ É³ÞÀÎÚáÜÊ¦ì¥Ò ËÇÚÉïÞßÉÞb_ ÉÄÍ ÐÙ³ÙÉ³ÒÒ ÊÌÎ ÉÍäÐÑËÂÊÌÍ ËÂÊÓÊÖ«ËÎÚÖ«Ë ËÂÇµÉ
UÐáÜÖµÉãÊÔ¡ËÂÇµÉÏÍäÐÑËÂÛÜÊØÚÐá ØôÖµ
Þ _ ÉÍ é Æ¹ÇµÉëÐèµèµÛñËÂÛñÊØ ÐçáÊÖ«ËÎÚÖ«Ë·ÊÔ ÚÛñØÚèµÛÜÙ³ÐçËÂÉ³ÒïËÂÇÚÐçË¹æ¹ÉâÛñØ ËÂÉÍîÎÚÍÉË
/
ËÂÇµÉ \ ËÂÖ«ÎÚáÜÉ
ÐÒÈËÇÚÉ WôÖµÊÌËÂtÛñÉ³Ø Ë
é
(n, d)
n/d
4.1.4 Initialization and assignment
SÝÉãÙ¬ÐØ ÛÜØµÛñËÂÛÜÐáÜÛ³É·Ê _ É³ÙËÒ·ÊÔ ÒîËÍÖÚÙÄË·Ë ìôÎ ÉâÐØµè ÐÒÒÛñåØ UÌÐçáÜÖµÉ³ÒòËÂÊËÂÇµÉ³Þ í ÖµÒîË·áÜÛV]ÉÓæ¹ÉëèµÊÀÛñËòÔ×ÊÌÍ
Ô×ÖµØµè ÐçÞßÉ³Ø ËäÐçáË ì Î É³Ò³é
êØëáÜÛÜØµÉ ÊÔ ÍÂÊåÌÍäÐÞ \ Ô×ÊÌÍ ÉGPµÐÞÀÎÚáÜÉÌí¬ËÂÇµÉ UÐçÍÂÛ×Ð _ÚáÜÉ ÊÔ Ë ìôÎÉ
ÛñÒ¾ÛñØÚÛ²ËÂÛ×ÐáÜÛ³É³è æ Û²ËÂÇ
t
rational
é êØ®ÐÓÒîËÍÂÖµÙË¬í¯ÛÜØµÛñËÂÛÜÐáÜÛ¬ÐÑËÂÛÜÊØëÛÜÒ WôÖµÛñËÂÉÈØÚÐçËÂÖ«ÍäÐáñáñìàèµÊØµÉ
ËÂÇµ
É UÐáÜÖµÉÈÊÔËÂÇµÉ·GÉ P¥ÎµÍÂÉ³ÒÒîÛÜÊØ
add (r, s)
ÞßÉ³
Þ _ ÉÍ »æ ÛÜÒîÉÌí Ûé ÉÌé Ô×ÊÌÍ¿É¬ÐÙäÇèÚÐçËäÐóÞßÉb
Þ _ ÉÄÍëÒÉÎ ÐçÍÂÐçËÂÉ³áñìé ·ØµèµÉÍàËÂÇµÉ®ÇµÊôÊ è¾í ËÇÚÉßèµÉ³Ùá×ÐçÍäÐçËÛÜÊØ
ÒîËäÐÑËÂÉ³ÞßÉ³Ø¯Ë
rational t = add (r , s );
ËÂÇµÉÍÂÉÔ×ÊÌÍÂÉ ÇÚÐÒ ËÂÇµÉ É É³ÙËóÊÔâÛÜØµÛñËÂÛ×ÐáñÛ³ÛÜØµå
^»æ·ÛñËÂÇ¼ËÂÇµÉ YµÍÂÒîËÙ³ÊÞÀÎ ÊØµÉ³Ø¯Ë ÊÔãËÂÇµÉ UÐáÜÖµÉ ÊÔ
` ÐçØÚè
·
^»æ ÛñËÇóËÂÇµÉàÒÉ³Ù³ÊØµè t.n
Ù³ÊÞÀÎ ÊØµÉ³Ø Ë `Äé êØ¯ËÂÉÍÂÉ³ÒËÂÛÜØµåáñìíµËÂÇµÛÜÒ ÐáÜÒîÊÀæ¹ÊÌÍ ]¥Ò æ·ÛñËÂÇ
add (r, s)
" ËîÍÂÖµÙËÂÒßËÂÇµÉÍÂÉ³Ô×ÊÌÍÉ½ÎµÍÂÊU¥ÛÜèµÉ Ð æÈÐ¬ì ÊÔîÔªÐ ]ôÛÜØµå ÐçÍîÍäÐ³ì ÛÜØµÛñËÂÛ×ÐáñÛ¬ÐçËÂÛñÊØÐØµè
ÐçÍÍÂÐ¬ì ÞßÉb
Þ _ ÉÄÍÂÒ³é t.d
ÐÒÒîÛÜåØµÞßÉ³Ø Ë _ôì æòÍäÐçÎµÎÚÛÜØµå ËÂÇµÉÐçÍÍäÐ³ì½ÛÜØ ËÊÐ®ÒîËÍÂÖµÙË³é ·ÉÍÉ ÛÜÒãÐØÝÉIP«ÐÞÀÎÚáÜÉàËÂÊÒîÇÚÊ¦æ æ ÇÚÐçËÓæ¹É
ÞßÉ¬ÐØ¤é
# include < iostream >
struct point {
double coord [2];
};
int main ()
{
point p;
p . coord [0] = 1;
p . coord [1] = 2;
point q = p;
std :: cout < < q. coord [0] < < " "
// 1
\ < < q. coord [1] < < "\n" ;
// 2
return 0;
}
Æ ÇÚÛñÒÏæ¹ÊÌÍ ]ôÒãÒîÛÜØµÙ³É¿ËÇÚÉ¿èÚÐçËäÐßÞßÉ³Þ_ ÉÍÂÒ ÊÔ Ð ÒîËîÍÂÖµÙËâÊ _ É³ÙËãÊôÙ³ÙÖµÎ ì½Ð Ù³ÊØ¯ËÂÛÜåÖµÊÖµÒ ÎÐÑÍËâÊÔÃËÂÇµÉ
¹
Þ®ÐÛÜØ®ÞßÉ³ÞßÊÌÍîìí¥ÐØµèóÒÛñØÚÙÉ ^ÛñØÙ³ÊØ¯ËÍäÐÒîË¹ËÂÊ ÐçÍÍäÐ³ì®Ë ìôÎ ÉÒ `¹ÒËÍÂÖµÙËÈË ì Î É³ÒN]¥ØµÊ¦æ ËÂÇµÉ³ÛñÍòÞßÉ³ÞßÊÌÍì
ÍÂXÉ W ÖµÛñÍÂÉ³ÞßÉ³Ø¯ËÂÒ³é [ ÍÂÊÞ ËÂÇµÛÜÒ³í«ËÂÇµÉãÙ³ÊÞÀÎÚÛÜáÜÉÍÈÙ¬ÐØ YÚåÖ«ÍÂÉãÊÖµËòÇµÊÑæ Þ®ÐØ¯ì®ÞßÉ³ÞßÊÌÍì®Ù³É³áñáÜÒ·ÐÑÍÂÉãËÂÊ _ É
Ù³ÊÌÎÚÛÜÉ³èÔ×ÊÌÍòËÂÇµÉÏÛÜØµÛñËÂÛ×ÐçáÜÛ ¬ÐçËÛÜÊØÀÊÔ ÛÜØ
é
q
point q = p
êØ ËÂÇµÉ ÒÂÐÞßÉàæÈÐ³ì ^ÞßÉ³
Þ _ ÉÍæ·ÛÜÒÉâÛñØÚÛ²ËÂÛ×ÐáÜÛ ¬ÐçËÂÛÜÊØ `Äí«ËÂÇµÉ Ô×ÊÌÍÂÞ®Ðá Î ÐçÍÂÐÞßÉËÂÉÍÒ ÐØµè
ÊÔÕËÂÇµÉ
a
b
Ô×ÖµØµÙËÂÛÜÊØ
ÐçÍÂÉÀÛÜØµÛñËÛ×ÐáÜÛ É³è½ÔÜÍÊÞ ËÂÇµÉ UÌÐçáÜÖµÉ³ÒëÊÔ ÐØµè ËÂÇµÉ UÐáÜÖµÉÀÊÔ
ÛñËÒÉ³áÜÔ
add
r
s
add (r, s)
ÐáÜÒîÊëÍÂÉ³ÒÖµáñËÂÒÔÜÍÂÊÞ ÐØ ÛÜØµÛñËÂÛ×ÐáñÛ ¬ÐçËÂÛñÊØ¿ÊÔ¾Ð ^°ËÂÉ³ÞÀÎ ÊÌÍäÐÑÍaì ` Ê _ É³ÙËæ ÇµÉ³ØßËÂÇµÉ
ÒîËäÐçËÉ³ÞßÉ³Ø Ë ÊÔ
return
ËÂÇµÉÏÔ×ÖÚØµÙËÛÜÊØ
ÛÜÒ¹IÉ P¥É³Ù³Ö«ËÂÉ³è¤é
add
êØµÒîËÂÉ¬ÐèàÊÔ ËÂÇµÉÐ _Ê UÉÕèÚÉÙ³á×ÐçÍäÐÑËÂÛÜÊØÏÒîËäÐçËÂÉÞßÉ³Ø Ë¤ËÂÇÚÐçË¡ÛÜØµÛñËÂÛ×ÐçáÜÛ ³É³Ò æ¹ÉÙ³ÊÖµáÜèëÐáÜÒÊÈÇÚKÐ UÉÕæ·ÍÂÛñËîËÂÉ³Ø
t
rational t ;
t = add ( r , s );
·ÉÍÉÌí
ÜÛ Ò
YµÍÂÒîË¬íÚÐØµèËÇÚÛñÒòèµÉ³ÔªÐÖµáñË ºÛÜØµÛñËÂÛÜÐáÜÛ³ÉÒÕËÂÇµÉÏè ÐÑËäÐ¿ÞßÉÞb_ ÉÄÍÂÒ³é êØóÊÖ«Í
µ
F

ç
¶

¯
©

F
Ù¬ÐÒîÉÌíËÂÇµÉì ÐçÍÉ ÊÔÈË ì Î É
¹Ô×ÊÌÍÔ×ÖÚØµèÚÐÞßÉ³Ø¯ËäÐáË ì Î É³Ò³íÕèÚÉÔªÐÖµáñËºÛñØÚÛ²ËÂÛ×ÐáÜÛ¬ÐçËÂÛÜÊØ½èµÊôÉ³Ò¿ØµÊÌËÂÇµÛÜØµå«í
ÒÊ ËÂÇµÉ UÌÐçáÜÖµÉ³ÒßÊÔÏËÇÚÉ èÚÐçint
ËÂÐÝÞßÉ³Þb_ÉÍÂÒÀÐÑÍÂÉ ÖµØÚèµÉ YÚØµÉ³è ÐÔÜËÂÉÍ èµÉ³ÔªÐÖµáñË ºÛÜØµÛñËÂÛÜÐáÜÛ¬ÐÑËÂÛÜÊØ ^ÒÉ³É ÐáñÒÊ
" É³ÙËÛÜÊØ \ é é `Äé êØ ËÂÇµÉ ØµÉIP¥Ë áÜÛÜØµÉÌí¹ËÂÇµÉ UÌÐçáÜÖµÉ ÊÔ
ÛÜÒ ÐÒÒÛÜåØµÉ³è ËÂÊ í ÐØµè ËÇÚÛñÒ
add (r, s)
t
ÐÒÒîÛÜåØµÞßÉ³Ø ËÈÐåÐçÛÜØÇÚÐçÎµÎ É³ØµÒÈÞßÉ³Þ_ ÉÍ»æ ÛÜÒÉé
t
[ ÊÌÍÀÉ UÉÍì Ô×ÖµØµèÚÐÞßÉ³Ø ËÂÐáòË ìôÎÉ íÃË æ¹Ê ÉIPôÎµÍÂÉ³ÒÒÛñÊØÚÒßÊÔ Ë ìôÎÉ
¬Ù ÐØ _ ÉßËÂÉ³ÒËÂÉ³è Ô×ÊÌÍÉ WôÖÚÐáÜÛñË ìí ÖµÒÛñØÚåóËÇÚÉßÊÌÎÉÍäÐçËÂÊÌÍÒ
ÐØµè
é ê»Ëàæ¹ÊÖµáÜè ËÂÇµÉÍÂÉÔ×ÊÌÍÂÉ®ÒÉ³É³Þ
==
!=
ØÚÐçËÂÖ«ÍäÐá ËÊ ÇÚÐXUÉ®ËÂÇµÉ³ÒÉ®Ë æ¹Ê ÊÌÎ ÉÍÂÐçËÂÊÌÍÂÒ ÐáñÒÊ½Ô×ÊÌÍ¿ÒîËîÍÂÖµÙËÂÒ³íÕÛÜÞÀÎÚáÜÉ³ÞßÉ³Ø¯ËÂÉ³èÝÛÜØ ÒÖµÙäÇ ÐóæÈÐ¬ì ËÇ ÐÑË
ËÂÇµÉì Ù³ÊÞÀÎ ÐçÍÂÉÓÞßÉ³Þ_ ÉÍ æ ÛñÒÉÌé
[ ÊÌÍÂÞ®ÐáÜáñìíËÂÇµÛÜÒ æ¹ÊÖµáÜè_ ÉÓÙ³ÊÌÍÍÂÉÙËÃÛÜÔ
ÐçØÚè
í¥ËÂÇµÉ³Ø æ¹ÉãÇÚÐXUÉ
t = (t1 , . . . , tN)
t = (t1 , . . . tN )
ÛñÔ ÐØµèóÊØµáñì®ÛñÔ
Ô×ÊÌÍ
é
t=t
tK = t K
N
Ö«ËÊÖ«Í®Ë ìôÎ É rational
ÐáñÍÉ¬KÐè«=ì 1,ÒÇµ. Ê¦. æ . ,Ò®
ËÂÇÚÐçË®ËÂÇµÛÜÒßæ¹ÊØ Ëßæ¹ÊÌÍ ] ÖµØµèµÉÍÞßÉÞb_ ÉÄÍæ·ÛÜÒÉ
XÉ W Ö ÐçáÜÛñË ìí æ¹É æ¹ÊÖµáÜè ÉÍîÍÂÊØµÉ³ÊÖµÒáñì Ù³ÊØµÙ³áÜÖµèµÉ ËÂÇÚÐçË
é Æ¹ÇÚÉ½ÎÚÍÊ _ áñÉ³Þ ÛñÒ ËÂÇÚÐçË ËÂÇµÉ
2/3 = 4/6
ÊÔëËÂÇµÉ½Ë ìôÎÉ
èµÊôÉ³ÒóØµÊÌËÙ³ÊÛÜØµÙ³ÛÜèµÉ½æ·ÛñËÂÇ ËÇÚÉ
J¯¤î©µ±¬î±´©¯ Dô©¯ £©Á« ÛÜØó
int int
æ·ÇµÛÜÙäÇóæ¹ÉâÛÜèµÉ³Ø¯ËÂÛÜÔÜì ÎÐçÛñrational
ÍÂÒ
ËÂÇÚÐçË·èÚIÉ Y ØµÉÏËÂÇµÉëÒÂÐÞßÉÏÍäÐçËÂÛÜÊØÚÐá
(n, d)
ØôÖµ³¨ Þ ©_ ¤ÉÍ î±´©¯ Dôé ©¯ £©Á«
n/d
ÆÈÇµÉïÒÂÐçÞßÉÇÚÐçÎµÎ É³ØµÒ¡æ·ÛñËÂÇàÊÖ«Í Ë ìôÎÉ
ÔÜÍÂÊÞ Ð _ Ê UÉ ÒÛñØÚÙÉ _ ÊÌËÂÇëÎ ÐÛñÍÒ
ÐØµè
ÐçÍÂÉ¿ÛÜØ ËÉÍÎµÍÂÉËÂÉè ÐÒ í¡Þßextended_int
Éb
Þ _ ÉÄÍ æ·ÛÜÒÉàXÉ W Ö ÐçáÜÛñË ì æ¹ÊÖµáÜè åVÛ UÉ¿ÖµÒ (0,ÛÜØ ËÂÇµÛÜ)Ò
0
0=0
Ù¬ÐÒîÉÌé (0, )
ØµáñìËÂÇµÉóÛÜÞÀÎÚáÜÉ³ÞßÉ³Ø¯ËÂÊÌÍ ÊÔÏÐ ÒîËÍÖÚÙÄË ]¥ØµÊÑæ·Ò ËÂÇµÉ ÒÉ³Þ®ÐçØ ËÂÛÜÙ³Ð
á UÐáÜÖµÉÍÂÐØµåÉÌíÈÐØµè¼Ô×ÊÌÍßËÇÚÛñÒ
ÍÂÉ¬ÐçÒÊØ¤í ØµÉ³ÛñËÇÚÉÄÍëÎµÍÂÊ UôÛÜèµÉ³ÒëXÉ W ÖÚÐáÜÛñË ì½ÊÌÎ ÉÍäÐçËÊÌÍÂÒâÔ×ÊÌÍ ÒîËîÍÂÖµÙËÂÒ³í ØµÊÌÍ ÐØ¯ìÝÊÌËÂÇµÉÍ ÊÌÎÉÍäÐçËÂÛñÊØÚÒ
_ ÉìÊØµè ËÂÇµÉ¿ÞßÉb
Þ _ ÉÄÍÓÐÙ³Ù³ÉÒÒ³í¤ÛÜØµÛñËÂÛ×ÐçáÜÛ ¬ÐçËÛÜÊØ¤íµÐØµèÝÐÒÒîÛÜåØµÞßÉ³Ø Ë èµÛÜÒîÙ³ÖµÒÒÉ³è Ð _ Ê UÉÌé Î ÉÍäÐçËÛÜÊØµÒ
ËÂÇÚÐçËëÍÉ³ÒîÎ É³ÙÄËàËÂÇµÉÀÒÉ³Þ®ÐØ¯ËÂÛÜÙ¬Ðç
á UÐáÜÖµÉÀÍäÐØµåÉÀÙ³ÐØ _ ÉÀÎµÍÂÊ UôÛÜèµÉ³è _ôìÝËÇÚÉ ÛÜÞÀÎÚáÜÉ³ÞßÉ³Ø¯ËÂÊÌÍ¬í ËÂÇµÊÖµåÇ¾í
ÒÉ³ÉãØµIÉ P¥ËòÒÉ³ÙËÛÜÊØ¤é
What about other operations?
\ ÊÖàÞßÛÜåÇ Ë ÐçÍÂåÖµÉËÂÇÚÐçË É UÉ³Ø¿ÞßÉ³Þ_ ÉÍ»æ ÛÜÒÉ ÛÜØµÛñËÛ×ÐáÜÛ³ÐçËÂÛÜÊØÓÐØµè¿ÐÒÒîÛÜåØµÞßÉ³Ø Ë¡Ù³ÊÖµáÜèZ_ É ÛñØÚÙÊØ ÒÛÜÒËÂÉ³Ø Ëïæ·ÛñËÂÇ®ËÂÇµÉ ÒÉ³Þ®ÐØ¯ËÂÛÜÙ³ÒïÊÔ¤ËÂÇµÉ Ë ìôÎ Éé ÐçËÂÉÍ³íôæ¹É æ ÛÜáÜá ÛÜØµèµÉ³É³è É³ØµÙ³ÊÖµØ ËÂÉÍÈÒÖµÙäÇóÐàÒÛ²ËÂÖÚÐçËÂÛÜÊØ¤í
ÐØµèæ¹ÉÓæ ÛÜáñáÒÇµÊÑæ ÇµÊ¦æ Û²ËòÙ³ÐØ _ ÉÏèµÉ¬ÐáñË¹æ Û²ËÂÇÉ³áñÉ³åÐØ Ëáñìé
4.1.5 User-defined operators
O Éæ Ë ì Î É³Ò ÍÂÉ WôÖµÛñÍÂÉ ØµÉæ ÊÌÎÉÍäÐçËÂÛñÊØÚÒí Ú
_ Ö«Ë æ µ
Ç É³Ø ÛñËïÙ³ÊÞßÉÒ ÂË ÊëËÂÇµÉ ØÚÐÞßÛÜØµåëÊÔ¤ÒÖµÙäÇÊÌÎ É ÍÂÐçËÂÛÜÊØµÒ³í
í¾æ¹É
ÊØµÉáñÉ³ÒÒâØµÛÜÙÉ ÐÒÎ É³ÙËãÊÔ ÍÂÊåÌÍäÐÞ \ îÒ ÇÚÊ¦æ ÒÏÜÛ ØÝáñÛÜØµÉ é Õì èµÉ YÚØÚÛñØÚå®ËÂÇµÉ¿Ô×ÖÚØµÙËÛÜÊØ
add
æ¹ÉÍÂÉâÐ _ÚáÜÉ ËÂÊÎÉÍÂÔ×ÊÌÍÂÞ ËÂÇµÉãÊÌÎÉÍäÐçËÂÛñÊØ
ËÂÇ«ÍÂÊÖµåÇ ËÂÇµã
É ÒîËäÐçËÂÉÞßÉ³Ø Ë
t := r + s
rational t = add (r , s );
êèµÉ¬ÐáÜá²ìí ÇµÊ¦æ¹É UÉÍ¬í«æ¹É æ¹ÊÖµáÜè®áÜÛV]É·ËÊÐçèÚè®ÍäÐçËÛÜÊØÚÐáØ ÖÚÞ_ ÉÍÒïáÜÛV]É·æ¹ÉÏÐèµè ÛÜØ ËÂÉåÉÍÂÒïÊÌÍRQÚÊÐçËÂÛÜØµå Î ÊÛÜØ¯ËÈØôÖµÞb_ÉÍÂÒ³í_ ì ÒîÛÜÞÀÎÚáñìÀæ·ÍÛñËÂÛÜØµå ^ÛÜØÊÖµÍòÙ¬ÐÒîÉ`
rational t = r + s ;
Æ ÇÚÉ
_ É³ØµÉ YµË¹ÊÔ¾ËÂÇµÛÜÒ¹ÞßÛÜåÇ ËïØµÊÌË _ ÉÓÛÜÞßÞßÉ³èµÛ×ÐÑËÂÉ³áñì Ê _ U¥ÛÜÊÖµÒ³í«ÛÜØ®Î ÐçÍËÂÛñÙ³Öµá×ÐçÍïÒÛñØÚÙÉÏËÂÇµÉÓØÚÐÞßÛÜØµåàÊÔ
¹
ËÂÇµÉÏÔ×ÖÚØµÙËÛÜÊØ
ÒÉ³ÉÞßÒ¹ËÊ _ É WôµÖ ÛñËÂÉÓÍÂÉ¬ÐçÒÊØÚÐ _ÚáÜÉ Ú_ Ö«Ë·Ù³ÊØµÒÛñèÚÉÄÍòËÂÇµÉÏÉIP¥ÎµÍÂÉÒÒÛÜÊØ
add
rational t = subtract ( multiply (p , q ) , multiply (r , s ));
ÐØµèóÛñËÒîØÚÐçËÖµÍÂÐá Ù³ÊÖµØ ËÂÉÄÍÎ ÐçÍË
rational t = p * q - r * s;
ËÂÊåÉË ÐçØóÛÜèµÉ¬Ðëæ ÇÚÐçË¹æ¹ÉÏÞßÉ¬ÐØ¤é
ÆÈÇµÉëØÚÐçËÖµÍÂÐá¡ØµÊÌËäÐÑËÂÛÜÊØóÙ¬ÐçØ ÛñØÚèµÉ³Éè _ ÉëÐÙäÇµÛÜÉ UÉè ÈÐ]Éì Ô×É¬ÐçËÖµÍÉëÊÔ ËÂÇµÉ á×ÐØµåÖÚÐåÉÏÛÜÒ
ËÂÇÚÐçË ÞßÊÒîË ÊÔ«Û²ËÂÒ ÊÌÎ ÉÍÂÐçËÂÊÌÍÂÒ ^ÒÉ³ÉïÆ Ð _ÚáÜÉ ÛÜØëËÇÚÉ ·ÎµÎ É³ØµèµÛ PâÔ×ÊÌÍ ËÂÇµÉ Ô×ÖµáÜá¯áÜÛÜÒîËG`¤Ù¬ÐØZ_ É
D¥³ËÂ£¦Ê ª]É³© Ø FµËÂIÊF
ËÂÊ æ¹ÊÌÍ ]®Ô×ÊÌÍÈÊÌËÂÇµÉÍ¹Ë ìôÎ É³ÒÈÐÒïæ¹É³áñáéÃÆÈÇµÛÜÒïÞßÉ¬ÐçØÚÒ ËÇ ÐÑËïæ¹ÉÏÙ³ÐØÖµÒÉÓËÂÇµÉÓÒÂÐÞßÉ ÊÌÎ ÉÍÂÐçËÂÊÌ Íï
ÛÜÞÀÎÚáÜÉ³ÞßÉØ Ë UÐçÍÂÛñÊÖÚÒÈÊÌÎ ÉÍÂÐçËÂÊÌÍÂÒ æ¹
É îÊ UÉÄÍÂáÜÊÐ
è ËÇÚÉÓËÂÊ ]É³Ø¤é
êØ¼ÎµÍÂÛñØÚÙÛñÎÚáÜÉÌíËÂÇµÛÜÒ ÛÜÒÀØµÊÌËÂÇµÛÜØµåÝØµÉæ æ¹É ÐáñÍÉ¬Ðè«
ì ]ôØµÊÑæ ËÇ ÐÑËÀËÂÇµÉ _ÚÛÜØÚÐçÍìÊÌÎ ÉÍäÐçËÊÌÍ
ÛÜÒ
+
XÐ UÌÐçÛÜá×Ð _ÚáÜÉÈÔ×ÊÌÍïÒÉ UÉÄÍäÐáË ìôÎÉ³Ò³íôÔ×ÊÌÍ¹GÉ PµÐÞÀÎÚáÜÉ
ÐØµè
é S¼ÇÚÐçËïÛñÒØµÉæ ÛÜÒËÂÇÚÐçËÕæ¹É Ù¬ÐØ®Ðèµè
É UÉ³Ø ÞßÊÌÍÂÉÊ UÉÍáÜÊÐèµÒëÊØ ÊÖ«ÍàÊÑæ·Ø¤í ÐØµè int
ÒÛÜÞÀÎÚáñì áÜÉdouble
ËãËÂÇµÉÀÙ³ÊÞÀÎÚÛÜáÜÉÄÍ YÚåÖ«ÍÂÉÀÊÖ«ËëÔÜÍÊÞ ËÂÇµÉßÙ¬ÐáÜá
Î ÐçÍäÐÞßÉÄËÂÉÍïË ìôÎ ÉÒòæ ÇµÛÜÙäÇ ÊØÚÉÏÛÜÒÈØµÉ³É³èµÉ³è ÛÜØÐ¿Ù³ÉÄÍËäÐÛÜØÙ³ÊØ ËÂGÉ P¥Ë¬é
êØ Ê UÉÍÂáÜÊÐçèÚÛñØÚå ÐØ ÊÌÎ ÉÍäÐÑËÂÊÌÍ¬í¹æ¹É½Ù¬ÐçØÚØµÊÌË®ÙäÇÚÐØµåÉ ËÇÚÉ ÊÌÎ ÉÄÍäÐçËÂÊÌ!Í Ò®ÐçÍÂÛñË ìí¹ÎµÍÂÉ³Ù³É³èµÉ³ØµÙ³É½ÊÌÍ
ÐÒÒîÊôÙ³Û×ÐÑËÂVÛ U¥Û²Ë ìí _ Ö«Ë æ¹É¹Ù¬ÐØÀÙÄÍÂÉ¬ÐçËÂÉ UÉÍÂÒîÛÜÊØµÒÃÊÔ ÛñË æ ÛñËÇ ÐçÍ _ Û²ËÍäÐçÍìëÔ×ÊÌÍÂÞ®ÐáôÎ ÐçÍäÐçÞßÉËÂÉÍÃÐØµè¿ÍÂÉËÂÖ«ÍÂØ
Ë ìôÎ ÉÒ³é
ÎÉÍäÐçËÂÊÌÍÈÊ UÉÍÂáÜÊÐçèÚÛñØÚåàÛñÒïÒÛÜÞÀÎÚáñì ÐëÒîÎ É³Ù³ÛÜÐáÙ¬ÐÒîÉÏÊÔ
é [ ÊÌÍ¹IÉ P«ÐÞÀÎÚáÜÉÌí

¬
±

¥
D
³

¦
£
ª

F
°

Á
ÇÚXÐ U¥ÛÜØµåâËÂÇµÉ ÒËÍÂÖµÙËÂÒ
ÐØµè
XÐ UÐÛÜá×Ð _ áñÉÌíæ¹É Ù³ÊÖµáÜèßèµÉ³Ù³á×ÐÑÍÂÉ·ËÂÇµÉ Ô×ÊáÜáñÊÑæ·ÛÜØµå
rational
extended_int
Ë æ¹Ê Ô×ÖµØµÙËÂÛñÊØÚÒ ÛñØ½ËÂÇµÉ ÒÂÐÞßÉëÎµÍÂÊåÌÍäÐçÞ í æ Û²ËÂÇµÊÖµË ÙÍÂÉ¬ÐçËÛÜØµå ÐßØÚÐÞßÉàÙ³áÜÐÒÇ Ô×ÊÌÍâÐØ ì Ù¬ÐçáÜá ËÂÊ®ËÂÇµÉ
Ô×ÖµØµÙËÂÛÜÊØ
ÛÜØ ËÇÚÉ ÎµÍÂÊåÌÍÂÐÞ í ËÂÇµÉ Ù³ÊÞÀÎÚÛÜáñÉÍ Ù¬ÐçØ YÚØµè ÊÖ«Ë¹ÔÜÍÂÊÞ ËÂÇµÉÓÙ¬ÐáñáÎ ÐçÍäÐÞßÉËÉÍ Ë ìôÎ É
square
æ ÇµÛÜÙäÇ ÊÔ ËÂÇµÉÏË æ¹ÊÔ×ÖÚØµÙËÛÜÊØµÒ¹æ¹ÉãÞßÉ¬ÐØ¤é
// POST : returns a * a
rational square ( rational a );
// POST : returns a * a
extended_int square ( extended_int a );
\
[ ÖÚØµÙËÛÜÊØ½ÊUÉÍÂáñÊÐèµÛÜØµåßÛÜØ åÉ³ØµÉÍäÐá ÜÛ Ò µ
Ö ÒÉÔ×ÖÚá»í _ÚÖ«ËÓØµÊÌËÓØµÉ¬ÐçÍÂáñì ÐÒ ÖµÒÉ³Ô×ÖµáÃÐÒ ÊÌÎ ÉÍäÐÑËÂÊÌÍÓÊ UÉÄÍ
áÜÊÐèµÛÜØµå«é Æ Ê èµÉ ÚY ØµÉ½ÐØ ÊU ÉÍÂáÜÊÐèµÉ³è ÊÌÎ ÉäÍ ÐÑËÂÊÌÍ¬íòæ¹É ÇÚÐXUÉ ËÂÊ ÖµÒÉ½ËÇÚÉ
éÕêØ®ËÂÇµÛÜÒïØµÊÌËäÐçËÂÛÜÊØ¤íËÂÇµÉÓØ ÐçÞßÉ ÊÔ ËÂÇµÉÏÊÌÎÉÍäÐçËÂÊÌÍ ÛÜÒ¹Ê Ú_ ËÂÐÛÜØµÉ³è ô_ ì®¶ÄÐÑÎÚ Î±¬É³ØµèÚ ÛñØÚ©¯À
åàÛ²³ËÂ¢ Ò ³ËÂ£Ê ©¯]É³Ø £
ËÂ ÊßîËÂ©¯ÇµÉëîÎµ ÍÂÉ Y P
éÏêØ½Ù¬ÐçÒÉ ÊÔÃËÂÇµÉZÚ_ ÛÜØÚÐçÍì ÐèµèÚÛ²ËÂÛÜÊØóÊÌÎ ÉÄÍäÐçËÂÊÌÍÓÔ×ÊÌÍ ËÂÇµÉàË ìôÎ É
í
operator
rational
ËÂÇµÛÜÒ¹áÜÊ Ê ¥] ÒòÐÒòÔ×ÊáñáÜÊÑæ·ÒÈÐØµè ÍÂÉÎÚá×ÐçÙ³É³ÒòËÇÚÉÏÔ×ÖµØµÙËÂÛÜÊØ
é
add
// POST : return value is the sum of a and b
rational operator + ( rational a , rational b)
{
rational result ;
result .n = a .n * b.d + a.d * b. n;
result .d = a .d * b.d ;
return result ;
}
êØ ÍÂÊåÌÍäÐÞ \ í«æ¹ÉÏÙ¬ÐØóØµÊ¦æ ÍÉÎÚá×ÐÙ³ÉÏáÜÛÜØµÉ _ ì
rational t = r + s ; // equivalent to rational t = operator + ( r , s );
·ÉÍÉÌí¾ËÇÚÉ¿Ù³ÊÞßÞßÉ³Ø¯Ë ÍÂÉ³Ô×ÉÍÒâËÂÊ®ËÂÇµÉ ÔªÐÙËÏËÂÇÚÐçËãÐØÝÊÌÎÉÍäÐçËÂÊÌÍÏÙ¬ÐçØ ÐáÜÒÊ _ É
ÛÜØ Ô×ÖµØµÙËÂÛÜÊØÚÐá
µØ ÊÌËäÐçËÂÛñÊØ ÛñØ Ù³ÊØ¯ËÍäÐÒîË³í·ÛñË ÐçÎµÎ É¬ÐÑÍÂÒ®ÛÜØ
ÛñØ
é Æ¹ÇµÉ±Ä©¯Ù³»ªÐIáÜF á ÛÜØ Ô×ÖÚØµÙËÛÜÊØÚÐá
ÛñË É³ÞÀÎÚÇ rÐçÒ+Û³É³sÒ¾ËÂÇµÉïÔªÐÙË ËÂÇÚÐçË ÐØ ÊÌÎ ÉÍäÐçËÊÌÍ
ØµÊÌËäÐçËÂÛñÊØàÙ¬ÐØ _ ÉïÖµÒÉ³Ô×Öµá¥Ô×ÊÌÍ èµÛÜèÚÐÙËÛÜÙÕÎÚÖ«ÍÎ° ÊÒÉ Ò³í ÒÛÜØµîÙ³©¯Éï
ÛÜÒÈÒÛÜÞÀÎÚáñì Ð ÒîÎ É³ÙÛ×Ðá¾Ô×ÖÚØµÙËÛÜÊØ ÚÛñØ ÐØ ÐçÎµÎÚáÜÛÜÙ³ÐçËÂÛÜÊØ¤íôÇµÊ¦æ¹É UÉÍ¬íÚËÂÇµÉãÎ ÊÛñØ ËÈÛÜÒ¹ËÂÊßÐXUÊÛÜèóÔ×ÖµØµÙËÂÛÜÊØÚÐá
ØµÊÌËäÐçËÂÛñÊØÐØµèóÖµÒÉÓËÂÇµÉâÛñaØ Y P ØµÊÌËäÐçËÛÜÊØ¤é
ÆÈÇµÉàÊÌËÂÇµÉÍ·ËÂÇ«ÍÂÉ³É _ ÐÒÛñÙàÐçÍÂÛñËÇÚÞßÉÄËÂÛÜÙÓÊÌÎ ÉÍäÐÑËÂÛÜÊØµÒ ÐçÍÉëÒÛÜÞßÛÜáÜÐçÍ¬íµÐØµè ÇµÉÍÂÉëæ¹ÉàÊØµáñì åÛVUÉâËÂÇµÉ³Û²Í
èµÉ³Ù³á×ÐÑÍäÐçËÂÛÜÊØµÒ³é
// POST : return value is the difference of a and b
rational operator - ( rational a , rational b );
// POST : return value is the product of a and b
rational operator * ( rational a , rational b );
// POST : return value is the quotient of a and b
// PRE : b != 0
rational operator / ( rational a , rational b );
SÝÉßÙ¬ÐçØ ÐçáÜÒÊ ÊUÉÍÂáÜÊÐè ÂË ÇµÉÀÖµØÚÐçÍì
ÊÌÎ ÉÍäÐçËÊÌÍ ÛÜØ Ô×ÖÚØµÙËÛÜÊØÚÐáÕÊÌÎÉÍäÐçËÂÊÌÍëØÚÊÌËÂÐçËÂÛÜÊØ¤í Û²ËëÇÚÐÒ
ËÂÇµÉâÒÐÞßÉÓØ ÐçÞßÉãÐçÒòËÂÇµÉ
_ ÛñØ ÑÐ ÍìUÉÍÂÒÛÜÊØ¤í _ÚÖ«Ë·ÛñËòÇÚÐÒÈÊØÚá²ì ÊØµÉãÛÜØµÒîËÂÉ¬Ðè ÊÔ Ë æ¹ÊÎ ÐçÍäÐçÞßÉËÂÉÍÂÒéêØ
ËÂÇµÉàÔ×ÊáÜáÜÊ¦æ ÛñØÚå¿ÛÜÞÀÎÚáÜÉ³ÞßÉØ ËäÐçËÛÜÊØ¤íôæ¹ÉëÖµÒÉëËÂÇµÉ ^ ÞßÊ èÚÛ Y Éè ` îáÜÊ Ù¬Ðá¡Ù³ÊÌÎôì(ßÊÔ ËÇÚÉàÙ³ÐáÜá¡Î ÐçÍÂÐÞßÉËÂÉÍ
ÐçÒòËÂÇµÉÓÍÂÉËÂÖ«ÍÂØ U ÐáÜÖµÉÌé
a
// POST : return value is -a
rational operator - ( rational a)
{
a .n = - a .n;
return a ;
}
\a
ê Ø ÊÌÍèÚÉÄÍ ËÂÊ Ù³ÊÞÀÎ ÐçÍÂÉÈÍäÐçËÂÛÜÊØÚÐáÚØôÖµÞ_ ÉÍÂÒí æ¹É ØµÉ³É³è®ËÂÇµÉ ÍÂÉ³áÜÐçËÂÛÜÊØÚÐáÚÊÌÎ ÉÍäÐÑËÂÊÌÍÂÒïÐÒÕæ¹É³áÜáé ·ÉÄÍÂÉ
ÛÜÒ¹ËÂÇµÉãÉ WôÖÚÐáÜÛñË ì®ÊÌÎ ÉÄÍäÐçËÂÊÌÍòÐÒòÐØóÉIP«ÐÞÀÎÚáÜÉÌé
// POST : return value is true if and only if a == b
bool operator == ( rational a , rational b)
{
return a .n * b.d == a.d * b .n;
}
4.1.6 Details
ê Ô ËÇÚÉÄÍÂÉëÐçÍÉâÒÉ UÉÄÍäÐá Ô×ÖµØµÙËÂÛÜÊØµÒÈÊÌÍ ÊÌÎ ÉÄÍäÐçËÂÊÌÍÂÒòÊÔ ËÂÇµÉâÒÐÞßÉãØÚÐÞßÉÓÛÜØóÐ
ÎµÍÂÊåÌÍäÐÞ í ËÂÇµÉÙ³ÊÞÀÎÚÛÜáÜÉÍÏÇÚÐÒãËÊ YÚåÖµÍÉ ÊÖ«Ëãæ·ÇµÛÜÙäÇÝÊØµÉÀÛÜÒÏÞßÉ¬ÐØ ËãÛÜØ Ð Ù³ÉÍËÂÐÛÜØÝÔ×ÖµØµÙËÂÛÜÊØ Ù³ÐáÜáé
Æ¹ÇÚÛñÒ¹ÎµÍÂÊ Ù³É³ÒÒÈÛÜÒ¹Ù¬ÐáñáÜÉ³è
ÐØµè ÊØÚá²ìßèµÉÎ É³ØµèµÒòÊØ ËÂÇµÉÏË ì Î É³Ò¹ÊÔ¡ËÂÇµÉÏÙ¬ÐáÜá

¥
D
³

¦
£
ª

F
»

Á
³
£
¬

Î ÐçÍäÐÞßÉÄËÂÉÍÂÒ³é UÉÍÂáñÊÐèµÛÜØµå ÍÉ³ÒÊáÜÖ«ËÂÛÜÊØ ÛÜÒëËÇÚÉÄÍÂÉ³Ô×ÊÌÍÂÉ®èµÊØµÉ®ÐçË Ù³ÊÞÀÎÚÛÜáÜÉ¿ËÂÛÜÞßÉÌé Æ¹ÇµÉÍÂÉ ÐçÍÂÉÀË æ¹Ê
Ù¬ÐÒîÉ³ÒËÂÇÚÐçË æ¹É®ØµÉ³É³èËÊ Ù³ÊØµÒÛÜèµÉÍ ®æ¹É®Ù¬ÐçØ É³ÛñËÂÇµÉÍ¿ÇÚXÐ UÉóÐØ
Ô×ÖµØÚÙÄËÂÛÜÊØÙ³ÐáÜá^áÜÛ ]É
ÛÜØ ÍÂÊåÌÍÂÐÞ \ `ÄíòÊÌÍ Ð Ô×ÖÚØµÙËÛÜÊØ Ù³ÐáÜá ^áÜ¾Û ]©¯É I F
`Äé Æ Ê
¾

¯
©

I
F
add (r, s)
std::sqrt(2.0)
ÎµÍÂÊôÙÉ³ÒÒ ÐçØóÖµØ W ÖÚÐáÜVÛ YÚÉ³è Ô×ÖµØÚÙÄËÂÛÜÊØÙ³ÐáÜá¤ÊÔ¡ËÂÇµÉÏÔ×ÊÌÍÂÞ
Overloading resolution.
^ í¾é²éñé²í
`
ËÂÇµÉ Ù³ÊÞÀÎÚÛÜáÜÉÍ Ç ÐçÒÏËÂÊYÚØµèÝÐßÞ®ÐçËÂÙäÇµÛÜØµåßÔ×ÖµØÚÙÄËÂÛÜÊØ èµÉ³Ùá×ÐçÍäÐçËÛÜÊØ¤é ¹ÐØµèµÛÜèÚÐçËÂÉ³ÒÓÐçÍÉ Ðáñá Ô×ÖµØÚÙÄËÂÛÜÊØµÒ
ÊÔòØ ÐçÞßÉ
ÒîÖÚÙäÇ ËÂÇÚÐçËàËÇÚÉ Ô×ÖµØµÙËÂÛÜÊØ Ù¬ÐáÜáïÛÜÒ ÛÜØ ËÂÇµÉ ÒÙ³ÊÌÎ É ÊÔ·ÒÊÞßÉßèµÉ³Ùá×ÐçÍäÐçËÛÜÊØ ÊÔ f é
f
êØ ÐèµèµÛñËÂÛñÊØ¾í ËÂÇµÉØ ÖÚÞ_ ÉÍÏÊÔÕÔ×ÊÌÍÂÞ®Ðá Î ÐçÍäÐÞßÉÄËÂÉÍÂÒÓÞ¿ÖÚÒËãÞ®ÐÑËÂÙäÇ ËÂÇµÉ¿ØôÖµÞ_ ÉÍÏÊÔÙ¬ÐçáÜá ÎÐÑÍäÐÞßÉ ËÂÉÍÒ³í ÐØµè É¬ÐÙäÇ Ù¬ÐçáÜáÃÎ ÐçÍÂÐÞßÉËÂÉÍÏÞ¿ÖµÒîË _ÉÀÊÔïÐ Ë ì Î Éæ·ÇÚÊÒîÉ UÌÐçáÜÖµÉ³ÒâÙ¬ÐçØ _ É Ù³ÊØ UÉÍËÂÉè ËÊËÂÇµÉ
Ù³ÊÌÍÍÉ³ÒîÎ ÊØµèµÛÜØµåÔ×ÊÌÍÞ®ÐáÎ ÐçÍäÐÞßÉËÉÍ¹Ë ì Î É³Ò³é
êØ Ð WôÖÚÐáÜVÛ YÚÉ³è Ô×ÖµØµÙËÂÛÜÊØóÙ³ÐáÜá ÊÔ ËÂÇµÉÏÔ×ÊÌÍÂÞ
^ æ ÇµÉÍÉ
í¤éñéñé²í
`
ÛÜÒòÐ¿ØÚÐÞßÉ³ÒÎÐçÙ³ÉÌí¥ÊØµáñì®ËÂÇµÛÜÒ¹ØÚÐÞßÉ³ÒîÎ ÐÙÉÏÛñÒòÒÉ³ÐçÍÂÙäÇµÉ³è Ô×ÊÌÍ Ù¬ÐçØÚèµÛÜèÚÐçËÉ³Ò³é
ÆÈÇµÉÍÂÉ ÛÜÒ ÊØµÉ ÒÎ É³Ù³Û×Ðçá ÍÂÖµáÜÉßËÂÇÚÐçË¿ÒÊÞßÉ ËÂÛÜÞßÉÒòÞ®Ð ]ÉÒòËÂÇµÉëáÜÛñÒîË·ÊÔ Ù³ÐØµèÚÛñè ÐÑËÂÉ³Ò·áÜÐçÍÂåÉÍ¬éÈêÔ ÒÊÞßÉãÙ¬ÐáÜá¤Î ÐçÍäÐÞßÉÄËÂÉÍÈË ì Î ÉëÊÔ ÐØ ÖµØaWôÖÚÐáñÛVYÚÉ³è
Ô×ÖµØµÙËÂÛÜÊØ Ù¬ÐáÜáÕÛÜÒàèµÉ YÚØµÉ³èÛÜØ ÐóØÚÐÞßÉ³ÒÎÐçÙ³É
^Ô×ÊÌÍ ÉIP«ÐÞÀÎÚáÜÉ ËÂÇµÉßØÚÐÞßÉ³ÒîÎ ÐÙÉ
`´í ËÇÚÉØ ËÂÇµÉ
std
Ù³ÊÞÀÎÚÛÜáÜÉÄÍÃÐáÜÒÊãÒÉ¬ÐçÍÂÙäÇµÉ³Ò Ô×ÊÌÍÙ¬ÐØµèµÛÜèÚÐçËÂÉ³ÒÃÛÜØ é Æ¹ÇµÛÜÒÃÛÜÒÕÖµÒÉÔ×ÖÚá Þ®ÐÛÜØµáñìàÔ×ÊÌÍÊÌÎ ÉÍäÐÑËÂÊÌÍÂÒÕÐØµèßÐáÜáÜÊ¦æ Ò
ËÂÇµÉ³Þ ËÂÊ _ É®Ù¬ÐáñáÜÉ³è ÖµØ W ÖÚÐáÜVÛ YÚÉ³è ÛÜØ ÛÜØ Y P ØµÊÌËäÐçËÂÛÜÊØ¤é Æ¹ÇµÉ®ÎÊÛÜØ Ë ÊÔÈÖµÒÛÜØµå ÊÌÎ ÉÍäÐçËÊÌÍÂÒàÛÜØ ÛÜØ YaP
ØµÊÌËäÐçËÂÛñÊØÀæ¹ÊÖÚáñ
è _É ÒîÎ ÊÛÜáÜÉè ÛñÔ¤æ¹ÉÓÇÚÐè®ËÂÊàÞßÉ³Ø¯ËÂÛÜÊØ®ÐàØÚÐÞßÉ³ÒîÎ ÐÙ³É·ÒÊÞßÉæ ÇµÉÍÉ ÛÜØ®ËÂÇµÉ ÊÌÎ ÉÍäÐçËÊÌÍ
Ù¬Ðáñáé
Argument-dependent name lookup (Koenig lookup).
Resolution: Finding the best match.
[ ÊÌÍ¾É¬ÐÙäÇëÙ³ÐØµèÚÛñè ÐÑËÂÉ Ô×ÖµØÚÙÄËÂÛÜÊØëÐçØÚèãÉ¬ÐÙäÇëÙ¬ÐçáÜáÌÎ ÐçÍäÐÞßÉËÉÍ¬í¦ÛñË
ÛÜÒ äÙ ÇµÉ³Ù ]É³è®ÇÚÊ¦æ æ¹É³áÜá¥ËÇÚÉÈÙ¬Ðáñá¥ÎÐÑÍäÐÞßÉËÂÉÄÍ Ë ì Î É¹Þ®ÐçËÂÙäÇµÉ³Ò ËÂÇµÉòÙÊÌÍÍÂÉ³ÒîÎÊØÚèµÛÜØµåÓÔ×ÊÌÍÂÞ®Ðá¥Î çÐ ÍÂÐÞßÉËÂÉÍ
Ë ìôÎ Éé®ÆÈÇµÉÍÂÉÀÐçÍÂÉ Ô×ÊÖ«Í W ÖÚÐáÜÛñË ì áÜÉ UÉáÜÒ³í åÊÛÜØµå ÔÜÍÂÊÞ _ ÉÄËËÂÉÍãËÂÊæ¹ÊÌÍÂÒÉí åÛVUÉ³Ø ÛñØÝËÂÇµÉ Ô×Ê áÜáÜÊ¦æ ÛñØÚå
áÜÛÜÒË¬é
\ \
^ `
ÒÂÐçÞßÉÌé
éÕÆ¹ÇÚÉ Ë ìôÎÉ³ÒòÊÔ¤ËÂÇµÉÓÙ¬ÐáÜáÎ ÐçÍÂÐÞßÉËÂÉÍïÐØµè ËÂÇµÉÓÔ×ÊÌÍÂÞ®ÐáÎ ÐçÍÂÐÞßÉËÂÉÍïÐçÍÂÉ ËÂÇµÉ
^ \ `
éÝÆ¹ÇµÉÍÂÉ®ÛÜÒ ÐÎµÍÂÊÞßÊÌËÂÛÜÊØ ÜÔ ÍÂÊÞ ËÂÇµÉßÙ¬ÐáÜáÎ ÐçÍÂÐÞßÉËÂÉÍâË ì Î ÉßËÂÊ ËÂÇµÉ
Ô×ÊÌÍÂÞ®Ðçá ÎÐÑÍäÐÞßÉËÂÉÄÍ¿Ë ì Î ÌÉ é S ÉÇÚÐXUÉóÒÉ³É³Ø ÒîÊÞßÉ IÉ P«ÐÞÀÎÚáÜÉ³Ò Ô×ÊÌÍ ÎµÍÂÊÞßÊÌËÛÜÊØµÒ³í áÜÛV]É ÔÜÍÂÊÞ
ËÂÊ
ÐØµèóÔÜÍÂÊÞ
ËÊ
é
bool
int
^ `
float
double
! "# Æ¹ÇµÉÍÂÉ ÛÜÒ®Ð ÒîËäÐçØÚèÚÐçÍè Ù³ÊØ UÉÍÂÒÛñÊØ ÔÜÍÂÊÞ ËÂÇµÉ Ù¬ÐáÜá
Î çÐ ÍäÐçÞßÉËÂÉÍÈË ìôÎ Ï
É ËÂÊÀËÂÇµÉâÔ×ÊÌÍÂÞ®Ðçá¾Î ÐçÍäÐçÞßÉËÂÉÍ¹Ë ìôÎ ÉéS ÉëÇÚÐKUÉâÒÉ³É³Ø ËÂÇÚÐçË ÐáÜá¤Ô×ÖµØÚèÚÐÞßÉØ ËäÐá
ÐçÍÛñËÂÇµÞßÉËÂÛÜÙòË ì Î ³É ÒÈÙ¬ÐØ _ÉâÙ³ÊØ UÉÍËÂÉ³è ÛñØ ËÂÊ¿É¬ÐÙäÇóÊÌËÂÇµÉÍ_ôì ÒîËäÐçØÚèÚÐçÍèÙ³ÊØ UÉÍÂÒÛÜÊØµÒ³é
^ `$%!&
'() * ! # Æ¹ÇÚÉÄÍÂÉàÛÜÒ Ð ÖµÒÉÄÍºèµÉ YÚØµÉ³è ÙÊØ UÉÍÂÒîÛÜÊØ½ÔÜÍÊÞ ÂË ÇµÉ
Ù¬ÐçáÜáòÎ ÐçÍäÐçÞßÉËÂÉÍÀË ì Î ÉóËÂÊÝËÂÇµÉ Ô×ÊÌÍÂÞ®ÐáÈÎ ÐçÍäÐÞßÉËÉÍ Ë ôì ÎÉÌé SÝÉóæ ÛñáÜáòåÉÄËßËÂÊ ÖµÒÉÍèÚÉIY Øµ³É è
Ù³ÊØ UÉÍÂÒÛñÊØÚÒòÊØµáñì®á×ÐçËÂÉÄÍòÛñØ ËÂÇµÛÜÒ _ ÊôÊ ]é
×Ô ÖµØµÙËÂÛñÊØ ÛÜÒ¹Ù¬ÐáñáÜÉ³è
ËÂÇÚÐØ
æ ÛñËÂÇ ÍÂÉ³ÒîÎÉ³ÙËÈËÂÊÀÐëÎ ÐçÍäÐÞßÉÄËÂÉÍ¬í¥ÛñÔ¡ËÂÇµÉÏÞ®ÐçËÂÙäÇ ËÇ ÐÑË
f
g

£
ÛÜØµèµÖµÙ³É³Ò ÊØ ËÂÇÚÐçË Î ÐçÍÂÐÞßÉËÂÉÍ·ÛÜÒ ÐçË áÜÉ¬ÐÒîË ÐÒ åÊôÊ è ÐÒ ËÂÇµÉëÞ®ÐçËÙ´Ç ÛñØÚèµÖµÙ³É³è _ ì é·êÔ ËÂÇµÉàÞ®ÐçËÂÙäÇ f
g
ÛÜÒ¹ÍÂÉ¬Ðáñáñì _ ÉËËÉÍ¬í ÛÜÒÈÙ¬ÐáÜáñÉ³è
Ô×ÊÌÍòËÇÚÉÓÎ ÐçÍäÐÞßÉÄËÂÉÍ¬é
f
Ô×ÖÚØµÙËÛÜÊØ
ÛÜÒ Ù¬ÐáÜáñÉ³è Ð ³£¬±¬º J ¬º³£ ÛÜÔ ÛñËÓÛÜÒ_ ÉËîËÂÉÍ ËÂÇÚÐØ½ÐØ ì ÊÌËÇÚÉÄÍãÙ³ÐØµèÚÛñè ÐÑËÂÉ ÛÜØ½ÐáÜá
f
³ÏËÂÇÚ¨½Ð©¯Ø ±¬ ÛñØ ÐçËÈáÜÉ¬ÐÒË·ÊØµÉÓÎÐÑÍäÐÞßÉËÂÉÄÍ¬é
g
Î ÐçÍäÐÞßÉÄËÂÉÍÂÒ³í¥ÐØµèóÒîËÍÂÛñÙËÂáñì _ ÉÄË ËÂÉ¬ÍÈ
g
·ØµèµÉÍïËÂÇµÛÜÒ èµÉ YÚØµÛñËÂÛÜÊØ¤íËÂÇµÉÍÂÉÓÛÜÒïÐÑË¹ÞßÊÒîËïÊØµÉ_ É³ÒîË¹Þ®ÐÑËÂÙäÇ¾í_ÚÖ«Ë¹ÛñËïÞ®Ð³ìÀÇÚÐçÎµÎ É³Ø®ËÂÇÚÐçË ËÇÚÉÄÍÂÉ
ÛÜÒòØÚb
Ê _ É³ÒîË·Þ®ÐçËÂÙäÇ¤íµÛÜØæ ÇµÛÜÙäÇóÙ³ÐÒÉãËÇÚÉãÔ×ÖµØÚÙÄËÂÛÜÊØóÙ¬ÐçáÜá¡ÛñÒ
íÚÐØµèËÇÚÉãÙ³ÊÞÀÎÚÛÜáÜÉÍÈÛÜÒîÒÖµÉ³Ò

Á
¡

ÐØóÉÄÍÍÂÊÌÍòÞßÉ³ÒÒÂÐåÉé
·ÉÍÂÉãÛÜÒÈÐØóIÉ P«ÐÞÀÎÚáÜÉÌé ïÊØµÒÛÜèµÉÍÈËÂÇµÉÏË æ¹Ê Ê UÉÍÂáñÊÐèµÉ³èóÔ×ÖµØµÙËÂÛÜÊØ èÚÉÙ³á×ÐçÍäÐÑËÂÛÜÊØµÒ
void foo ( double d );
void foo ( unsigned int u );
êØËÂÇµÉÏÙ³ÊôèµÉâÔÜÍÂÐåÞßÉ³Ø Ë
float f = 1.0 f ;
foo (f );
ËÂÇµÉYµÍÂÒîËÈÊ UÉÄÍÂáÜÊÐè ÛÜÒ¹Ù´ÇµÊÒÉØ¾íµÒÛñØÚÙÉ
Ù¬ÐçØ _ É ÎµÍÂÊÞßÊÌËÂÉ³èßËÂÊ
í_ÚÖµËÈÊØµáñì®ÒîËÂÐØµè ÐÑÍÂè
double
Ù³ÊØ UÉÍËÂÉ³èóËÊ
éÕêØ float
unsigned int
int i = 1;
foo (i );
ËÂÇµÉ Ù¬ÐáÜáÛñÒ¤ÐÞb_ÚÛÜåÖµÊÖµÒ³í¬ÒÛñØÚÙÉ
4.1.7 Goals
Ù¬ÐØ _ÉÃÒËäÐØµèÚÐçÍÂèºÙ³ÊØ UÉÍËÂÉèãËÂÊ _ ÊÌËÂÇ
double
ÐçØÚè
unsigned int
Ê ÛÜØ¯Ë¬íôìÊÖóÒîÇÚÊÖµáÜè é³é³é
`L]¥ØµÊ¦æ ÇµÊ¦æ ÒîËÍÂÖµÙËÒÕÙ¬ÐçØ _ ÉÈÖÚÒîÉ³è ËÂÊâÐååÌÍÂÉ³åÐçËÉÈÒîÉ UÉÍäÐáÚèµÛ ÉÄÍÂÉ³Ø ËÃË ìôÎ ÉÒÕÛÜØ¯ËÂÊÏÊØÚÉÈØµÉæ Ë ìôÎ É
É _ ÉË æ¹É³É³ØÝËÇÚÉÒîì¥Ø¯ËäÐÙËÛÜÙ¬ÐáÕÐØµèÝÒÉ³Þ®ÐçØ ËÂÛÜÙ³ÐáUÌÐçáÜÖµÉÍäÐØµåÉÊÔïÐ
\ `ßÖµØÚèµÉÍÒîËäÐØµèÝËÂÇµÉ èµÛ ÉÍÂÉ³ØµÙ³b
ÒîËÍÂÖµÙË `L]¥ØµÊ¦æ ËÇ ÐÑË Ô×ÖµØµÙËÂÛñÊØÚÒòÐØµèóÊÌÎ ÉÄÍäÐçËÂÊÌÍÂÒ¹Ù¬ÐçØ _ ÉâÊ UÉÍÂáñÊÐèµÉ³è¤é
Dispositional.
¹Ë¹ËÂÇµÛÜÒ¹Î
int
é
Operational.
\
êØÎ ÐçÍËÂÛñÙ³Öµá×ÐçÍ¬í¯ìÊÖ ÒÇµÊÖÚáñè _ ÉãÐ _ÚáÜÉ ËÂÊéé¬é
`®èµÉ YÚØµÉëÒîËîÍÂÖµÙËÂÒ æ ÇµÊÒÉâÒÉ³Þ®ÐØ¯ËÂÛÜÙ¬Ðá UÐáÜÖµÉâÍäÐØµåÉ³Ò ÙÊÌÍÍÂÉ³ÒîÎÊØÚè ËÂÊÀËÇ ÐÑË ÊÔ åÛVUÉ³Ø Þ®ÐçËÂÇµÉ ^
Þ®ÐçËÛÜÙ¬ÐáÒîÉËÂÒ
\ `ßÎµÍÂÊU¥ÛñèÚÉ·èÚÉIY ØµÛñËÛÜÊØµÒÕÊÔ Ô×ÖµØµÙËÂÛÜÊØµÒïÐçØÚè®ÊUÉÍÂáÜÊÐçèÚÉè ÊÌÎÉÍäÐçËÂÊÌÍÒÊØ®ÒîËÍÂÖµÙËÒ³í«ÐÙ³ÙÊÌÍÂèµÛÜØµåâËÂÊ
åÛVU ÉØóÔ×ÖµØÚÙÄËÂÛÜÊØÚÐáÜÛñË ì ^
^
`ßæòÍÂÛñËÂÉ ÎµÍÂÊåÌÍäÐçÞßÒ ËÂÇÚÐçËÈèÚÉIY ØµÉâÐØµèÖµÒÉâÒËÍÂÖµÙËÂÒòÐÙ³Ù³ÊÌÍèÚÛñØÚå¿ËÂÊ¿åÛVUÉ³ØóÔ×ÖµØµÙËÂÛÜÊØÚÐáñÛñË ìé
4.1.8 Exercises
Ý© JÑ¢ Tribool ¶ç£½º¥£ Dô©¯¡IF §ÁÌî± » º¥£³Ä NDô©¯ GF ªÁ± à¸Ó
©D¥ºÀ£¬º Dô©¯ ¬¿£¬ ¡¶Ñ©¯² ©Fó ç ¸È Å
£ßº J¦¢ Tribool È°¨¢¾ª³¨ ¤ ºÀªÁ±Ä©¯ï³¢ ³£©¯£¦
Exercise 108
// POST : returns x AND y
Tribool operator && ( Tribool x , Tribool y );
// POST : returns x OR y
Tribool operator || ( Tribool x , Tribool y );
¸È³£³
H ∧M ©F H ∨M ©£³ Fµ IFÝ©µ±Ä±£ F°Á óºò¶»ª¸¹»Á ¸Ó î ©« ¦ª¬çÅ
^ G` ^ \ `
¶ç©¯
¶Ñ©¯²ç
¶Ñ©¯²ç
ç ¸È ¶Ñ©¯²ç
£¬¡
¶Ñ©¯²ç
Æ É³ÒîËÈìÊÖ«ÍòË ì Î É _
ÊÔ¡ìÊÖ«Í·Ù´ÇµÊÛÜÙÉÌé
∧
¸¹
£¬
¶Ñ©¯²
∨
¶Ñ©¯²ç
¶ç©¯
¶ç©¯
¶ç©¯
¸¹ ¸¹
¸¹ ¸¹
¸¹
£¬
£¬
£¬
ì®æ·ÍÂÛñËÛÜØµå ÐàÎµÍÂÊåÌÍÂÐÞ ËÂÇÚÐçËÈÊÖµËîÎ Ö«ËÂÒïËÂÇµÉ³ÒîÉÏËÍÖµËÇ ËäÐ
¸¹ £¬
ç ¸È £¬¡
ç ¸È £¬¡
£¬¡
£¬¡
_ÚáÜÉÒÈÛñØÒÊÞßÉÓÔ×ÊÌÍÂÞ®ÐçË
© JÑ¢ Z_7 ¶£ ±¨¢¤»Á ¸¹º °¤Á«£¦ ¨½ Fª 7 Å ©¯º³¨ ©¯
±Ä©¯°EJ Èº¥®±£¬£³³¢ F¯ óº » £¬»Á 7 = /7 Â¶à£³¬ F ±¬ª©¯¬ ¨ F § 7 Å
£ßº J¦¢ Z_7 ¹»¨¿¢¾§¨ ³¤ã© F F»î¼©F ¾ ¦£©µ±¬ ³¢ ³£©¯£¦
Exercise 109
// POST : return value is the sum of a and b
Z_7 operator + ( Z_7 a , Z_7 b );
// POST : return value is the difference of a and b
Z_7 operator - ( Z_7 a , Z_7 b );
©µ ±±Ä£ F »Á º·¶ç»ª¸È°Áóî©« ¬§ Hº¥ î©« ¦ª ©¯ Fµ ¬ Ä¡ ¦£©µ±³î
¿ ¨ ³³£ z {0, . . . , 6} ±¬ º©¯ x = y + zM¥Å
x − y ¿º
^
Ì^
\ `
\
+
0
1
2
3
4
5
6
0
0
1
2
3
4
5
6
1
1
2
3
4
5
6
0
2
2
3
4
5
6
0
1
3
3
4
5
6
0
1
2
4
4
5
6
0
1
2
3
Exercise 110 È£³ DNFµ Fµ
J¦¢ rational Å
5
5
6
0
1
2
3
4
6
6
0
1
2
3
4
5
»î¤Ã¶£ëº¶»ª¸¹»Á ¬» ©£KJ ©£¬º¥¨½¬î±³¢ ³£©¯£¦^ \ `G ^ º `
// POST : return value is the difference of a and b
rational operator - ( rational a , rational b );
// POST : return value is the product of a and b
rational operator * ( rational a , rational b );
// POST : return value is the quotient of a and b
// PRE : b != 0
rational operator / ( rational a , rational b );
F î¤È¶£ º·¶°§¸¹°Á ¬ ° ©£KJ £¬ª©¯ ©¯ï³¢ ³£©¯£¦ º
Exercise 111 È£³ D NFµ µ
J¦¢ rational Å Â µF °Á º¥» £KJ £³ ³¢ ³ £©¯£¦¼º©¯ ©£³ ©¯£Ä© FJ Fµ I F«Å
^
\ Ì^
`
// POST : return value is true if and only if a != b
bool operator != ( rational a , rational b );
// POST : return value is true if and only if a b
bool operator > ( rational a , rational b );
// POST : return value is true if and only if a >= b
bool operator >= ( rational a , rational b );
Exercise 112 È£³ DNFµ µ
F »î ¤Ã¶£ëº¶»ª¸¹»Á ¬ » ©£KJ ©£¬º¥¨½¬î±³¢ ³£©¯£¦ º
J¦¢ extended_int H ©Á«
M ©F ¬³Àº¨ ° ©½¢¤£³Á£©¨ Hñ¶ç£Ýº©¯» ±Ä F ³
¬ ¢¶ ¢¤£³ DÌNFµÀ©¼ñ¢¤ ¶ç©µ±³»» Jâ¶£¿º¿ J¦¢ extended_int
·©F ©ã¶Ä ±¬º©¯
©¯³§Á¤ë © extended_int Dô©¯ À©D ©¯¡ÀÂ¶ë J¦¢ intM¥Å â» º ¢¤£³XDîÚ ³£³±
£KJ £³ ± FµÅ
^
\ Ì^
`
\ // POST : return value is the sum of a and b
extended_int operator + ( extended_int a , extended_int b );
// POST : return value is the difference of a and b
extended_int operator - ( extended_int a , extended_int b );
// POST : return value is the product of a and b
extended_int operator * ( extended_int a , extended_int b );
// POST : return value is -a
extended_int operator - ( extended_int a );
Exercise 113 Ó¤ Fµ³£Àºò¶°§¸¹»Á ³âÂ¶¿º¥£³ ¶Ä ±¬¤çÅ
void foo ( double , double )
{ ... }
void foo ( unsigned int , int )
{ ... }
void foo ( float , unsigned int ) { ... }
// function A
// function B
// function C
£ Ä©µ±¬ Â¶º ¶°§¸¹»Áâ¶ ±¬î ±´©¯» FµÄ±³NFµ® ¸È¥î±³ Â¶Àº ¶Ä ±¬¤ H A, B, CM
Ï£³¬ D¥¬ £ Fµ±³ Fµº©¯ãº ±´©¯» ©¨½ ³°Á¡çÅ ¦¢¾×©» JÚ£ Fµ±¤ ¥»
Ú£³±³çà£³ »£³³ZJ ½ß£³Ä© F ºò¢©£©Á£©¬¢¾ D¥£¦§© F°Áß£³¬çî »Ýº
¬î©
Ä±¬î Å
©KM foo(1, 1)
M foo(1u, 1.0f)
±NM foo(1.0, 1)
FKM foo(1, 1u)
NM foo(1, 1.0f)
¶ M foo(1.0f, 1.0)
4.1.9 Challenges
DÌÄ¡©¯
Exercise 114 ¥» ±¬©¯°§Á«ó©¯ó©
Ä± ¨¢¤£ Á£©¬¢¾¥±¬ ©D¥£ÌÅ Ý£¬»½ © ¢¤£³Á£©¨ º©¯
© É¹¢¤ÞßÊô×èµ©¯É³ á ©
F»¨½³¤î ©¯ë¥ ±³îÅ
£ º ç© ô Â¶
æ ©ÛñÍF É³ÔÜÍä¨½ÐÞß
Â ¶ ©
Â¶à©±³¡ âÁ Dô³ KJ GFÁ«¬â° º¥£³ÄIF»¨½³
©¯»ª¸È JÚ
± ±£³¬³ ¬ ¹º¥»
©¯ ¢©µ±çÅ
à¢¤£³ÄÁ£©¨ ³ F ³ ©« ¦ª F£©¸ º ¥ ±³ã» ¢ £¦¢ ±³ D¥ DÌî³¸ð©F ©¯ÏªÄ©¯³
¢¤£³ DÌ Fµàºàç³£ë¸¹ºÝ©ë¢ ¬»» J½Â¶Ï£³î©¯°Á º ß ¥ Ä±¬ © £³F ºàº¥£ß©Ú¬çÅ Õ
F£©¸¹»Á¸¹»
Fµ¸ ¨§Á«¡¶ç£«©¨¢¾ª ªÌ º¥ »
\ Â¤³Ä© FÂ¶©
±³¡ JÚ ¨ ©JÝ
¸ ©¤â½î© ô © º£¿¢¾×©¯ î± NF JÚ ¨½©J £³Ä© F
º ¸¹»£³¶Ä£©¨ ¨ Fµ¬¶£¨ © § J ¨½©J © F F º ¢ ³Ä ¬»» J Â¶ ±´©¯°Á º¼¥ Ä±¬ ¿£©¤³×©¯»Á ¬±Å
º ¬£©£KJ libwindow º©¯ë»½©Dô©×©« ¦ªÝ©¯àº ± £¦
¨ ¢©Á«À½±³£³´©¯ßºÁ£Ä©¬¢¾¥î±Ä©¯ïñ¢¤îÅ
×¶JÚ Fµ
¸ H£Ï© D¥ ¶£îÁ¥º³ M¸ ¿£³î©¯ ©FÏ¢¤£ ±³Õº¥£³ÄIF°¨ ³¤ ©¯
¢ °¤º È³£³ »®© ±³£©¯³ ± £¦ Å
2
î©¯°Á ©à¢ »¤ (x, y)
©£³ F ºÀ£¬°Á° K J © ©Á§ßÂ¶ α Hî£© F ©¤N MÀ£³³º
»ºã¢ °¤ (x , y ) ¸È»º±£ F» ©¯¬
Ù³ÊÒ
ÒÛÜØ
x
α −
x

Ò
Ü
Û
Ø
³
Ù

Ê
Ò α
=
.
y
α
α
y
Â
3
£ Fµ£ £³î©¯ © ¢ °¤ (x, y, z)
©£³ F º z ä©¯ JÚ »¨¿¢¾EJ î¢ z
±³©Á«IF ©F £î©¯ (x, y) ©¯ ©« ³ DôÅ J IJ ¨ ¨ ¬£KJ ZJÚ ±´© Á£ â¸ º¥
¸Ó£ Ì¹¶£Àº º³£ ©µ¬çÅ ³ £©¯¢ ³£¦¢ Ä±¬ Dôâ¢¤£³ Ä±¬îà · Ä©¯IJ ¬ò ¶ JÚ ¸ ©¤·¿¢¤£³ Ä±¬ã© ¢»¤
¤ º z = 0 ª¢¾ª© Hî»¨ ©Á» º©¯º¥» ¢¾ª© º ±Ä¨¢¤£ ±³£³Ä³ º©¯ JÚ
¸ ©¤ F£Ä©¸ M º¥» ¿©I£ F«Å Â¨½©Á° º©¯ v = (vx , vy, vz ) »óº DÌ³¸ ¢ »¤
\ H°¢ Äî Â¶
J
£ J MôÅ vz > 0 ¶ç£ µ©¨¢¾ªâ¨½Ä©¤Ïº©¯ J Ý©£ë»º»Á» ¶Ä£³¤·Â¶Ïº
±³£³ Å Ý³ ÚJ ¤
¢ £³ Ä¬± àº¢ °¤ p = (x, y, z) ¤ º ±³£³Ä³ º°¨½©Á«¿¢ »¤
©¯®±Ì£ F ° ©¯¬
(x − t(vx − x), y − t(vy − y)),
¸È³£³
t=
z
.
vz − z
ÕÏ¢¤£³ ±¬ º¥®¤ J ¸Ó£ Ì ¶ vz = z Å
\
4.2 Type Variants
Ï ¬îXD¥Àº J¦¢ Å
J¯¨
Õ¥Ï±³î ¦¢¾×©»¤â¸Ïß¸ ©J Â¶ ¥ ¦î©° »Á Dô©£¬ ©¤º Â¶à©ÀÁ D¥³ J¦¢ º©¯
©É³D¥Ô×É ÍÂÉ³ºØµÙ³ É Ë çìô©Î¨ É³Ò Dô©¯ £©Á¥¼ ¬ F ³£ °ð±Ä³£¦î©» ¶ ±¬î ©¯ J ©¯¢±¬ºçÅ
³ ©« ¦ªï¶ ±¬î¤âó©µ±±¢¾Ó©F®£³³£¬ EDôï©¯Ê Øµ³Òî
ËË©ì Î F®É³Ò »ó¢©£
±³×©£ ±¬ ©Á« º Dô©¯ ¬óÂ¶ º³»£Ï¶£¬¨ ©¯¤¢©£Ä©¨ ¬³£¦ÑÅ
©¯»ª¸
ë Fµ Dô©¯¡¬ ©¯ ³»Á ¨½ F ©« ¦ª ï° Ä ±¬ © ¸ ©Jóº©¯Èºß±¨
¢¤»§³£ ±´© Fµ³±¬®°§ Á¥©¯ã¨½ F ±´©¯î¤çÅ
¶ç³£³³ ±Ä J¦¢ ¬ ©
F ±¤ î J¦¢ ¬
±´© ±Ä¨½ ¬° I F ©
F ©¯£Ä©¯» J ±¨½ ¯¢ ° »¨¢¾ª³¨½³¤»Á ¶ ±¬î ©¯ J
¶ç£À³£¬¡±¬ºçÅ
4.2.1 Reference types
ÉË ÖµÒ ØµÊÑæ ËÍìãËÂÊ ÛÜÞÀÚ
Î áÜÉ³ÞßÉ³Ø¯Ë ÂË ÇµÉ¹ÐèµèµÛñËÂÛÜÊØâÐÒÒÛÜåØµÞßÉ³Ø¯Ë ÊÌÎ ÉÄÍäÐçËÂÊÌÍ
Ô×ÊÌÍ ËÇÚÉïÒËÍÂÖµÙË
+=
rational
ÔÜÍÂÊÞ ÍÊåÌÍäÐÞ é òÉ ÍÂÉÏÛÜÒ·ÐçØ çÐ ËËÂÉÞÀÎÚË
rational operator += ( rational a , rational b ) {
a .n = a. n * b.d + a. d * b.n ;
a .d *= b .d;
return a ;
}
S¼ÛñËÂÇ ËÂÇµÛÜÒ³í¥æ¹ÉâÙ³ÐØæ·ÍÛñËÂÉ
rational r ;
r.n = 1; r .d = 2;
// 1/2
rational s ;
s.n = 1; s .d = 3;
// 1/3
r += s;
std :: cout < < r .n < < "/ " < < r. d < < "\ n";
ÊÖ¼Þ®Ð³ìÐáñÍÂÉ³Ðè«ì ÒÉ³É ËÂÇÚÐçË ËÂÇµÉóÊÖ«ËÎÚÖµËÀÊÔÓËÇÚÛñÒ æ ÛÜáñáòØµÊÌËL_ ÉóËÇÚÉèÚÉÒÛñÍÂÉ³è
é É³Ù³ÐáÜá
5/6
ÔÜÍÂÊÞ " É³ÙËÛÜÊØ é é æ ÇÚÐçË®ÇÚÐçÎµÎ É³ØµÒßæ ÇµÉ³Ø
^ÉXW ÖµÛVUÌÐçáÜÉ³Ø Ëáñìí
`ÀÛÜÒ
r += s
operator+= (r, s)
É UÐáÜÖÚÐçËÂÉè
ÐçØÚè
ÐÑÍÂÉ É UÐáÜÖÚÐçËÂÉ³è¤íòÐØµè¼ËÇÚÉóÍÉ³ÒÖµáñËÂÛÜØµå UÐáÜÖµÉ³Ò ÐçÍÂÉ ÖµÒÉè ËÂÊ ÛÜØµÛñËÂÛÜÐáÜÛ³É®ËÂÇµÉ
r
s
Ô×ÊÌÍÂÞ®Ðá Î ÐçÍÂÐÞßÉËÂÉÍÒ ÐØµè
ÊÔÃËÂÇµÉÔ×ÖµØµÙËÂÛñÊØ
é¿ÆÈÇµÉUÌÐáñÖÚÉÒãÊÔ ÐØµè
ÐÑÍÂÉØµÊÌË
operator+=
r
s
ÙäÇÚÐØµåÉ³è _ôì ËÂÇµÉÏÔ×ÖÚØµaÙËÛÜÊØÙ¬bÐçáÜáé
·É³ØµÙ³ÉÌí æ·ÛñËÂÇÝËÂÇµÉßÐ _Ê UÉßÛÜÞÀÎÚáÜÉÞßÉ³Ø ËäÐÑËÂÛÜÊØ ÊÔ
í ËÂÇµÉ
ÊÔ ËÂÇµÉßÉIPôÎÚÍÉ³ÒÒÛÜÊØ
operator+=
D
¯
©

ÛñÒ ÛÜØµèÚÉÉ³è
í _ÚÖ«Ë ËÂÇµÉ¹èµÉ³ÒÛñÍÂÉè
í ËÂÇµÉ¹ÛÜØµÙÍÂÉÞßÉ³Ø Ë ÊÔ íèµÊôÉÒÃØµÊÌË ÇÚÐçÎµÎ É³Ø¤é Æ¹ÇÚÐçË! Ò
r += s
r
æ Ç¯ì®æ¹ÉãåÉË
Ð5/6
ÒÈÊÖ«ËÎÚÖµËòÛÜØ ËÂÇµÉâÐ _Ê U ÉÓÎÚÄÛÜ±¬É³ Ù³ÉãÊÔ Ù³Ê èÚÉé
1/2
\
ê ØÊÌÍÂèµÉÍ¹ËÂÊ ÛÜÞÀÎÚáÜÉÞßÉ³Ø Ë
ÎµÍÂÊÌÎ ÉÍÂá²ìí æ¹ÉÓÞÖµÒîË¹ÉØ Ð _ áñÉ Ô×ÖÚØµÙËÛÜÊØµÒ¹ËÊ Ù´ÇÚÐØµåÉ ÂË ÇµÉ
operator+=
UÐáÜÖµÉ³Ò ÊÔ ËÂÇµÉ³ÛñÍ·Ù¬ÐáÜá¤Î ÐçÍäÐÞßÉÄËÂÉÍÂÒ³é " Ö«ÍÎµÍÂÛÜÒÛñØÚåá²ìíµæ¹ÉâèÚÊßØµÊÌË·ØÚÉÉ³è
Ð ØÚÉÄæ Ù³ÊØµÙ³ÉÄÎÚËÓÔ×ÊÌÍ·ËÂÇÚÐçË Ê Ø
ËÂÇµÉÏÔ×ÖÚØµÙËÛÜÊØóÒÛÜèµÉ ¥æ¹ÉÏÒÛÜÞÀÎÚáñìÀØÚÉÉ³è ÐàØÚÉÄæ Ù³ÐçËÂÉ³åÊÌÍì ÊÔ Ë ì Î É³Ò³é
Definition.
êÔ ÛÜÒ·ÐçØ ì®Ë ìôÎ ÉÌí«ËÂÇµÉ³Ø
ÛÜÒ¹ËÂÇµÉãÙÊÌÍÍÂÉ³ÒîÎÊØÚèµÛÜØµå
»^ ÍÉ¬Ðè ÐÒ ÍÂÉ³Ô×ÉÄÍÂÉ³ØµÙ³É ßÊÌÍ ÍÂÉ³Ô×ÉÍÉ³ØµÙ³ÉãËÂÊ `ÄéµêØ
UÐáÜÖµÉ·ÍäÐØµåÉ ÐçØÚèßÔ×ÖµØµÙËÂ£³ÛñÊ¶çØ ³Ðç£³áÜ³ÛñË ìÄ± í ß ÑJ ¢ÜÛ ÒÛÜèµÉ³Ø¯ËÂÛÜÙ¬ÐçáÚËÂÊ é ÆÈÇµÉ èµÛ ÉÄÍÂÉ³ØµÙ³É ÛÜÒÜÛ ØÀËÂÇµÉ ÛÜØµÛñËÂÛÜÐáÜÛ¬ÐÑËÂÛÜÊØ
ÐØµè ÐÒîÒÛÜåØµÞßÉ³Ø Ë¹ÒîÉ³Þ®ÐØ ËÛÜÙ³Ò³é
UÐçÍÂÛ×Ð _ÚáÜÉÈÊÔÍÂÉ³Ô×ÉÄÍÂÉ³ØµÙ³É Ë ìôÎ É ^ºÐáÜÒÊâÙ¬ÐáÜáÜÉèßÐ
` Ù¬ÐØ _ ÉòÛÜØµÛñËÂÛ×ÐçáÜÛ³É³è¿ÊØµáñìÔÜÍÂÊÞ
Ð Ø
ÊÔ¡Ë ì Î É í«ÊÌÍ ÐçØ ì®Ë ìôÎ ÉÏæ ÇµÊÒÉ
UÌÐáñÖÚÉÒòÙ¬ÐçØ £³_¶ç³ÉÏ£³Ù³³Ê Ø ±ÄU ÉÍîËÂÉ³èóËÂÊ éµÆ¹ÇµÉâÛÜØµÛñËÂÛÜÐáÜÛ¬ÐÑËÂÛÜÊØ
Þ®Ð ] Dô³É ©¯Ò¹Ûñ Ëò ÐØ
ÊÔ¡ËÂÇµÉÏáVUÌÐçáÜÖµÉ ÃÐØµÊÌËÂÇµÉÍÈØ ÐçÞßÉÏÔ×ÊÌÍÈËÂÇµÉÏÊ _ É³ÙË_ É³ÇµÛÜØµè ËÂÇµÉãáVUÐáÜÖµÉÌé S ÉâÐáñÒÊ
ÒÂÐ³ì ËÂÚÇ ÐçË¹ËÂÇµÉÏ©¯ÍÂÉ Ô×©¯ÉÍÂÉ³ØµÙ³É
ËÂÊ¿ËÂÇÚÐçËòÊ _ É³ÙË³éÕÆ¹ÇµÉãÔ×ÊáñáÜÊÑæ·ÛÜØµå ÉIPµÐÞÀÎÚáÜÉ ÒîÇÚÊ¦æ Ò¹ËÂÇµÛÜÒ³é
£¶³£¦
int i = 5;
int & j = i ;
// j becomes an alias of i
j = 6;
std :: cout < < i < < "\n" ;
// changes the value of
// outputs 6
i
ÂÍ ÉÔ×ÉÍÂÉ³ØµÙ³É®Ù¬ÐØµØµÊÌË _ ÉÀÙäÇÚÐØµåÉ³èÝËÂÊÍÂÉÔ×ÉÍàËÂÊ ÐØµÊÌËÂÇµÉÍâÊ _ É³ÙÄËàÐÔÜËÂÉÄÍëÛÜØµÛñËÂÛÜÐáÜÛ¬ÐÑËÂÛÜÊØ¤é êÔ æ¹É
á×ÐçËÉÍòÐÒÒÛÜåØ ÒÊÞßÉËÂÇµÛÜØµåàËÂÊ ËÂÇµÉÏÍÉ³Ô×ÉÍÂÉ³ØµÙ³Éí æ¹ÉÏÛÜØÔªÐÙÄË ÐÒîÒÛÜåØËÂÊ ËÂÇµÉ ÍÂÉ³Ô×ÉÄÍÍÂÉ³èóËÂÊb_ôì ÛñË¬é
êØ æòÍÂÛñËÂÛÜØµå
ÛÜØËÇÚÉòÐ _ ÊUÉïÎÚÛÜÉ³Ù³ÉÈÊÔÙ³Ê èµÉÌí¯æ¹É¹ËÂÇµÉÍÂÉÔ×ÊÌÍÂÉòÙäÇÚÐØµåÉ¹ËÂ¥Çµ ÉÄU±¬Ð áÜÖµÉ¹ÊÔ ËÊ íÒÛÜØµÙ³É
i 6
ÛñÒ·ÐØóÐáÜÛ×ÐjÒïÊ=Ô 6 é
j
êØ ËÂÉÄÍÂØÚÐáÜáñìí Ð i UÌÐçáÜÖµÉßÊÔ Ë ì Î É ÛÜÒâÍÉÎµÍÂÉ³ÒÉ³Ø¯ËÂÉ³è _ôìÝËÂÇµÉßÐçèÚè«ÍÂÉÒÒàÊÔïËÂÇµÉÀÊ _ É³ÙÄËàÛñËëÍÂÉÔ×ÉÍÂÒ
ËÂÊ«éÕÆÈÇµÛÜÒÈIÉ PôÎ áÜÐÛÜØµÒïæ Ç ì®æ¹ÉãØÚÉÉ³è ÐØóVá UÐáÜÖµÉ ËÂÊÛÜØµÛñËÂÛ×ÐáñÛ ³É ÐàÍÂÉ³Ô×ÉÍÉ³ØµÙ³ÉâË ìôÎ É UÐçÍÂÛ×Ð _ÚáÜÉíµÐØµèæ Ç¯ì
ËÂÇµÛÜØµåÒ¹áÜVÛ ]É
int & j ;
int & k = 5;
// error : j must be an alias of something
// error : the literal 5 has no address
èµÊ
Ø Ë·æ¹ÊÌÍ ]é Ø ì ÉIPôÎÚÍÉ³ÒÒÛÜÊØ ÊÔ ÍÂÉÔ×ÉÍÂÉ³ØµÙ³ÉàË ìôÎ ÉãÛÜÒ ÐØ á UÌÐáñÖÚÉãÛñËÂÒÉáÜÔºé SÝÉëÙ³ÐØ ËÂÇµÉÍÂÉ³Ô×ÊÌÍÂÉâÖÚÒîÉàÐ
ÍÂÉ³Ô×ÉÄÍÂÉ³ØµÙ³É ËÂÊ ÛÜØµÛñËÛ×ÐáÜÛÉ·ÐØµÊÌËÂÇµÉÍïÍÉ³Ô×ÉÍÂÉ³ØµÙ³É
í _ÚÖµËïËÇÚÉØ æ¹É èµÊØ Ë¹åÉËÈÐâÍÂÉÔ×ÉÍÂÉ³ØµÙ³ÉÏËÂÊ í _ Ö«Ë
r
rr
r
ÐØµÊÌËÂÇµÉÍÈÍÂÉ³Ô×ÉÍÉ³ØµÙ³ÉâËÂÊ ËÂÇµÉãÊ _ É³ÙËÈÍÂÉ³Ô×ÉÍîÍÂÉ³è ËÂÊ_ôì
r
int i = 5;
int & j = i ;
int & k = j ;
// j becomes an alias of i
// k becomes another alias of i
4.2.2 Call by value and call by reference
S¼ÇµÉ³Ø ÐÔ×ÖµØÚÙÄËÂÛÜÊØ ÇÚÐÒàÐ Ô×ÊÌÍÞ®ÐáÃÎ ÐçÍÂÐÞßÉËÂÉÍâÊÔ ÍÂÉ³Ô×ÉÍÉ³ØµÙ³ÉßË ì Î ÉÌí ËÂÇµÉÀÙÊÌÍÍÂÉ³ÒîÎÊØÚèµÛÜØµåóÙ³ÐáÜá Î Ð
ÍäÐÞßÉÄËÂÉÍ Þ¿ÖµÒîË_ÉàÐØ áVUÐáÜÖµÉ Úæ ÇµÉ³ØóËÂÇµÉëÔ×ÖµØµÙËÂÛÜÊØ Ù¬ÐçáÜá¡ÛñÒ É UÐáÜÖÚÐçËÂÉ³è¤íµËÂÇµÉëÛÜØµÛñËÂÛÜÐáÜÛ¬ÐÑËÂÛÜÊØ®ÊÔ ËÂµÇ É
\ Ô×ÊÌÍÂÞ®ÐáµÎ ÐçÍäÐÞßÉÄËÂÉÍ Þ®Ð ]É³ÒÛñËïÐØ ÐáÜÛ×ÐÒÃÊÔ ËÂÇµÉ Ù¬Ðáñá Î çÐ ÍäÐÞßÉÄËÂÉÍ¬é êØ®ËÂÇµÛÜÒÃæÈÐ¬ìí¯æ¹É Ù¬ÐØ®ÛÜÞÀÎÚáÜÉ³ÞßÉØ Ë
Ô×ÖµØµÙËÂÛÜÊØµÒ¹ËÂÇÚÐçËÈÙ´ÇÚÐØµåÉÓËÂÇµÉ UÐáÜÖµÉ³ÒòÊÔ¡ËÂÇµÉ³Û²Í·Ù¬ÐçáÜáÎ ÐçäÍ ÐÞßÉËÉÍÂÒ³é·ÉÍÉãÛÜÒòÐØóÉGPµÐÞÀÎÚáÜÉÌé
void increment ( int & i )
{
++ i;
}
int main ()
{
int j = 5;
increment ( j );
std :: cout < < j < < "\ n" ; // outputs 6
return 0;
}
ê Ô ÐàÔ×ÊÌÍÂÞ®Ðá ÎÐÑÍäÐÞßÉËÂÉÄÍ¹ÊÔ ÐàÔ×ÖµØµÙËÂÛÜÊØ Ç ÐçÒ¹ÍÉ³Ô×ÉÍÂÉ³ØµÙ³ÉÏË ìôÎ ÉÌí¥æ¹ÉÓÇÚÐXUÉ
ÒîÉ ´
±
¯
©
»

´
K
J

¶
³

£
³

Þ®ÐØ¯ËÂÛÜÙ³Ò¤æ ÛñËÇëÍÂÉ³ÒîÎ ÉÙË ËÂÊ ËÂÇÚÐçË Î ÐçÍÂÐÞßÉËÂÉÍ³é WôÖµÛVUÐáÜÉ³Ø¯ËÂáñìíÑæ¹ÉïÒÂÐ³ìëËÂÇÚÐçË æ¹É
ËÇÚÉ Î ÐçÍÂÐÞßÉËÂÉÍ
¢
¯
©
³

_ôì®ÍÂÉÔ×ÉÍÂÉ³ØµÙ³ÉÌé
êÔòËÂÇµÉ Ô×ÊÌÍÂÞ®ÐáÎ ÐçÍÂÐÞßÉËÂÉÍ ÛñÒ¿ØµÊÌË¿ÊÔòÍÉ³Ô×ÉÍÂÉ³ØµÙ³É Ë ì Î ÉÌíÃæ¹É ÇÚÐXUÉ
ÒîÉ³Þ®ÐØ ËÛÜÙ³Ò
Ä
±
¯
©
»

´
X
J
ô
D
¯
©

æ¹ÉâÎ ÐÒÒ ËÇÚÉãÎÐÑÍäÐÞßÉËÂÉÄ
Í _ôì UÌÐçáÜÖµÉÌé ·ØµèµÉÍ Ù¬ÐçáÜá"_ ìÍÂÉ³Ô×ÉÄÍÂÉ³ØµÙ³ÉÌí ËÂÇµÉàÐèµè«ÍÂÉ³ÒîÒ ÊÔR^ÊÌÍÓÐÍÂÉÔ×ÉÍÂÉ³ØµÙ³É
ËÂÊ `¿ËÇÚÉ Ù³ÐáÜáòÎ ÐçÍÂÐÞßÉËÂÉÍ ÛÜÒÀÖµÒÉ³è¼ËÂÊ ÛÜØµÛñËÂÛÜÐáÜÛ ³ÉßËÂÇµÉ Ô×ÊÌÍÂÞ®ÐçáÈÎ ÐçÍäÐçÞßÉËÂÉÍ òÖµØÚèµÉÍ®Ù³ÐáÜá _ôì UÌÐçáÜÖµÉ
ÒÉ³Þ®ÐçØ ËÂÛÜÙÒ³í¥ÛñËòÛÜÒïËÂÇµÉ UÐáÜÖµÉÏÊÔ ËÂÇµÉÏÙ¬ÐáÜá Î ÐçÍäÐÞßÉÄËÂÉÍïËÂÇÚÐçËÈÛÜÒÈÖµÒÉ³è Ô×ÊÌÍòÛÜØµÛñËÂÛ×ÐáñÛ ¬ÐçËÂÛñÊØ¾é
ÆÈÇµL
É _ ÐÒÛñÙ ÍÂÖµáÜÉÀÛÜÒãËÂÊóÎ ÐÒÒàÐÎ ÐçÍäÐçÞßÉËÂÉÍ _ôìÝÍÉ³Ô×ÉÍÂÉ³ØµÙ³É®ÊØµáñì ÛÜÔ ËÂÇµÉÀÔ×ÖµØµÙËÂÛÜÊØ ÛÜØ WôÖµÉ³ÒËÂÛÜÊØ
ÐÙËÖ ÐçáÜáñì¿ØµÉ³É³èµÒËÂÊàÙäÇÚÐØµåÉ·ËÂÇµÉ·Ù³ÐáÜá Î ÐçÍÂÐÞßÉËÂÉ
Í UÐáÜÖµÉÌé êÔËÂÇÚÐçËÛÜÒ ØµÊÌËËÂÇµÉ Ù³ÐÒÉÌí¥Ù³ÐáÜaá _ôì UÐáÜÖµÉòÛÜÒ
ÞßÊÌÍÂR
É QÚIÉ P«Û _ áñÉÌíÒÛÜØµÙ³ÉÈÛñËÕÐáÜáÜÊ¦æ ÒÃÐ á×ÐçÍåÉÍÙá×ÐÒÒÃÊÔÙ¬ÐçáÜá¥Î ÐçÍäÐÞßÉËÉÍÂÒ ^Vá UÐáÜÖµÉ³Ò
Í UÌÐáñÖÚÉÒÕÛÜØµÒîËÉ¬Ðè

F
ÊÔ á UÌÐáñÖÚÉÒòÊØµáñì `Äé
4.2.3 Return by value and return by reference
Æ¹ÇÚÉßÍÂÉÄËÂÖ«ÍÂØ Ë ì Î É®ÊÔ·ÐóÔ×ÖµØµÙËÂÛÜÊØ Ù¬ÐØ _ É®ÐóÍÂÉ³Ô×ÉÍÉ³ØµÙ³É Ë ì Î É ÐÒëæ¹É³áÜáí ÛñØ æ·ÇÚÛñÙ´Ç Ù¬ÐÒîÉ®æ¹É®ÇÚÐXUÉ
ÒÉÞ®ÐØ ËÂÛñÙ³Ò
^ÊÌËÂÇµÉÍæ·ÛÜÒÉÌíôæ¹É
`ÄéÃêÔ¡ËÂÇµÉÏÔ×ÖµØÚÙÄËÂÛÜÊØ ÍÉËÂÖ«ÍÂØµÒ
£³¬ÍÂÉ³ Ô×£¬É ÍÂÉ´ØÚK J ÙÉÌ£í ËÂ¶Çµ³É£³Ô× Öµ±Øµ ÙËÂÛÜÊØ Ù¬ÐáÜá IÉ PôÎÚÍÉ³ÒÒÛÜÊØ ÛñÒâÐØ £³á UÌ¬Ð áñÖÚ£¬É ÛñË KÒJ É³áÜDôÔºí©¯Ð Øµ è æ¹É Ù¬ÐçØÝÖµÒÉÀÛñËÓæ ÇµÉÍÉ UÉÍ
Ðß
Vá UÐáÜÖµÉ³ÒÈÐçÍÂÉãIÉ P¥ÎÉ³ÙËÂÉ³è¤é
ÆÈÇµÛÜÒÕÞßÉ¬ÐØµÒÕËÇ ÐÑËËÂÇµÉ·Ô×ÖµØµÙËÂÛñÊØßÛñËÂÒÉ³áñÔÙäÇÚÊ ÊÒÉ³
Ò ^ _ôìÀÖµÒÛñØÚåãÍÂÉ³Ô×ÉÍÂÉØÚÙÉ Ë ì Î É³Ò ÊÌÍ ØÚÊÌË ` æ·ÇÚÉÄËÂÇµÉÍ
ÛñËÂÒ¤Ù¬ÐçáÜáÌÎ ÐçÍäÐÞßÉËÉÍÂÒ¤ÐØµèãÍÉËÂÖ«ÍÂØ UÐáÜÖµÉÃÐçÍÂÉÃVá UÐáÜÖµÉ³Ò ÊÌÍ¾Í UÐáÜÖµÉ³Ò³é " É³ÙËÂÛÜÊØ \ é é K ÐçØÚè " É³ÙËÂÛñÊØ \ é \ é
èµÊôÙ³ÖµÞßÉ³Ø¯Ë ËÂÇµÉ³ÒÉ¿ÙäÇÚÊÛñÙ³É³ÒÓÔ×ÊÌÍâÒîÊÞßÉàÊÔÕËÂÇµÉ ÊÌÎ ÉÍäÐÑËÂÊÌÍÂÒÓÊØ Ô×ÖµØµèÚÐÞßÉ³Ø ËÂÐá Ë ìôÎ ÉÒ³í _ Ö«ËÏÊØµáñì ØµÊÑæ
æ¹ÉÏÖÚØµèµÉÍÂÒËäÐØµèËÂÇµÉÏÞßÉ³ÙäÇ ÐçØÚÛñÒÞ ËÂÇÚÐçËòÞ®Ð ]É³Ò¹ÒÖµÙäÇ Ù´ÇµÊÛÜÙÉ³ÒÈÎÊÒÒVÛ _ÚáÜÉÌé
Ò Ð·Ù³ÊØµÙÍÉËÂÉïIÉ P«ÐÞÀÎÚáÜÉÌíÑáÜÉË ÖµÒ Ù³ÊØµÒÛÜèµÉÍ ËÂÇµÉ Ô×ÊáÜáÜÊ¦æ ÛÜØµ
å UÉÍÂÒÛÜÊØ ÊÔ«ËÂÇµÉ Ô×ÖµØµÙËÂÛÜÊØ
ËÂÇÚÐçËIÉ PµÐçÙËÂáñì ÞßÊôèµÉ³áÜÒëËÇÚÉ _ É³ÇÚXÐ U¥ÛÜÊÌÍÊÔ·ËÂÇµÉ ÎÚÍÉ ºÛÜØµÙÍÉ³ÞßÉ³Ø ËÊÌÎ ÉÍÂÐçËÂÊÌÍ
ÛñË ÛñØÚÙÄÍÂincrement
É³ÞßÉ³Ø¯ËÂÒÛ²ËÂÒ
++
Vá UÐáÜÖµÉ Î ÐçÍäÐÞßÉËÉÍòÐØµè ÍÂÉËÖµÍØÚÒòÛñËòÐÒòÐØóVá UÐáÜÖµÉÌé
int & increment ( int & i )
{
\ return ++ i;
}
êØ åÉ³ØµÉÍäÐá»í¡æ¹ÉÀÞ¿ÖµÒîËâÞ®Ð ]ÉÒÖ«ÍÂÉËÂÇÚÐçËàÐØ ÉIP¥ÎµÍÂÉ³ÒîÒÛÜÊØ ÊÔ ÍÂÉ³Ô×ÉÍÉ³ØµÙ³ÉÀË ìôÎ É¿ËÂÇÚÐçËâæ¹É¿ÍÂÉËÂÖ«ÍÂØ
ÍÂÉ³Ô×ÉÄÍÂÒ¿ËÊ½Ð
Ê _ É³ÙÄË¬é Æ Ê ÖÚØµèµÉÍÂÒËäÐØµè æ·ÇÚÐçË¿ÐËÂÉ³ÞÀÎ ÊÌÍÂÐçÍì½Ê _ É³ÙË ÛñÒ³í áÜÉË ÖµÒ

©
K
£
J
Ù³ÊØµÒÛÜèµÉÍÈËÂÇµÉÏÔ×ÊáÜáÜÊ¦æ ÛÜØµåàÔ×ÖµØµÙËÂÛÜÊØ¤é
int & foo ( int i)
{
return i ;
}
Æ ÇÚÛñÒÀÛÜÒßÐÒN]¥ÛÜØµåÝÔ×ÊÌÍßËÍÊÖa_ÚáÜÉíïÒÛÜØµÙ³ÉóËÂÇµÉóÔ×ÊÌÍÂÞ®ÐçáÈÎ ÐçÍäÐçÞßÉËÂÉÍ
¹
ÍÖÚØµÒßÊÖ«ËÀÊÔÓÒÙÊÌÎ Éóæ ÇµÉ³Ø ËÂÇµÉ
i
Ô×ÖµØµÙËÂÛÜÊØàÙ¬ÐçáÜá ËÂÉÄÍÂÞßÛÜØÚÐçËÂÉ³Òé Æ¹ÇµÛÜÒ ÞßÉ¬ÐçØÚÒ¾ËÂÇÚÐçË ËÂÇµÉïÐçÒÒÊôÙÛ×ÐçËÂÉ³èàÞßÉÞßÊÌÍìÓÛÜÒ ÔÜÍÂÉÉ³è ÐçØÚèàËÂÇµÉïÐçèÚè«ÍÂÉÒÒ
ÉIPôÎ Û²ÍÂÉ³Ò ^ÒÉ³É " É³ÙËÂÛÜÊØ \ é é `´é êÔ æ¹ÉãØµÊ¦æ æ·ÍÂÛñËÉÏÔ×ÊÌÍòIÉ P«ÐÞÀÎÚáÜÉ
int i = 3;
int & j = foo (i );
// j refers to expired object
std :: cout < < j < < "\n" ; // undefined behavior
ËÂÇµÉÓÍÂÉ³Ô×ÉÍÂÉØÚÙÉ ÍÂÉÔ×ÉÍÂÒòËÊÀÐØóÉGP¥ÎÚÛñÍÂÉ³è Ê _ É³ÙÄË¬íÚÐØµè ËÂÇµÉÓÍÂÉ³ÒÖµáñËÂÛñØÚå _ ÉÇ ÐXUôÛÜÊÌÍÈÊÔ ËÂÇµÉÏÎµÍÂÊåÌÍÂÐÞ ÛÜÒ
ÖµØµèÚIÉ Y ØµÉ³è¤é j
Reference Guideline:
S¼ÇµÉ³ØµÉ UÉÍ ìÊÖóÙÄÍÂÉ¬ÐçËÂÉâÐØóÐçáÜÛ×ÐÒ¹Ô×ÊÌÍ·ÐØóÊ _ ÉÙË¬í«É³ØµÒÖ«ÍÂÉÏËÂÇÚÐçËòËÇÚÉÏÊ _ ÉÙË
èµÊôÉ³ÒÈØµÊÌË·ÉGP¥ÎÚÛñÍÂÉ
_ É³Ô×ÊÌÍÂÉÏËÂÇµÉâÐçáÜÛ×ÐÒ´é
ÛÜØµå«é
ÆÈÇµÉòÙ³ÊÞÀÎÚÛÜáÜÉÍ ÖµÒÖÚÐáÜá²ì ØÚÊÌËÛÜÙ³É³ÒU¥ÛÜÊáÜÐçËÂÛÜÊØµÒ ÊÔËÂÇµÉ
É³Ô×ÉÄÍÂÉ³ØµÙ³É
ÖµÛÜèµÉ³áñÛÜØµÉ¹ÐØµèÀÛÜÒÒÖµÉ³ÒÐÓæÈÐÑÍÂØ
4.2.4 More user-defined operators
ÉË! ÒÃåÉÄË_ ÐÙ ]àËÂÊÓËÂÇµÉÈÐèµèµÛñËÂÛÜÊØ¿ÐÒÒîÛÜåØµÞßÉ³Ø Ë ÊÌÎÉÍäÐçËÂÊÌÍ
Rational numbers: addition assignment.
Ô×ÊÌÍëÊÖ«ÍãØµÉæ ÒîËÍÂÖµÙË
é êØ ÊÌÍÂèµÉÍÏËÂÊ Y P ÊÖµÍãÔªÐÛÜáñÉ³èÝÐçËîËÂÉ³ÞÀÎµËÏÔÜÍÂÊÞ ËÇÚÉ_ É³åÛñØÚØµÛÜØµå ÊÔ
rational
ËÂÇµÛÜÒ¹ÒÉ³ÙÄËÂÛÜÊØ¤í«æ¹ÉãØµÉ³É³èóËÊ ÐèµèË æ¹ÊÙ´ÇÚÐçÍÂÐÙËÂÉÍÒòÊØµáñìé
Ò·ÛñØóËÂÇµÉãÎµÍÂÉ UôÛÜÊÖµÒ·Ô×ÖµØµÙËÂÛÜÊØ
íÚËÂÇµÉâÔ×ÊÌÍÂÞ®Ðá Î ÐçÍäÐçÞßÉËÂÉÍ Þ¿ÖµÒîË_ÉâÎ ÐÒÒÉ³è ÐçÒ Ð
increment
ÍÂÉ³Ô×ÉÄÍÂÉ³ØµÙ³ÉÌíÐØµèËÂ
Ê _ ÉâÙ³ÊÞÀÎÚáÜÛÜÐØ Ë æ·ÛñËÂÇ®ËÂÇµÉãÖµÒÖÚÐá¾ÒÉ³Þ®ÐçØ ËÂÛÜÙÒÈÊÔ í«æ¹ÉãaÐáÜÒÊ ÍÂÉËÖµÍØËÂÇµÉãÍÂÉÒÖµáñË
+=
ÐÒòÐ ÍÉ³Ô×ÉÍÂÉ³ØµÙ³É
// POST : b has been added to a ; return value is the new value of a
rational & operator += ( rational & a , rational b)
{
a .n = a. n * b.d + a. d * b.n ;
a .d *= b .d;
return a ;
}
Æ ÇµÉÕÊÌËÂÇµÉÍ ÐçÍÛñËÂÇµÞßÉËÂÛÜÙ ÐçÒÒÛÜåØµÞßÉ³Ø¯Ë¤ÊÌÎ ÉÍäÐçËÊÌÍÂÒ ÐçÍÉÕÒÛñÞßÛÜá×ÐçÍ¬í¦ÐçØÚèëæ¹ÉÃèµÊØ Ë¡áÜÛÜÒîË¾ËÂÇµÉ³Þ ÇÚÉÄÍÂÉÉIP È
ÎÚáÜÛÜÙ³Û²ËÂáñìé¡Æ ÊåÉËÂÇµÉÍ æ Û²ËÂÇ¿ËÇÚÉÈÐçÍÛñËÂÇµÞßÉËÂÛÜÙÐØµèÍÂÉá×ÐçËÂÛÜÊØÚÐáôÊÌÎ ÉÍäÐçËÊÌÍÂÒ èÚÛñÒÙ³ÖµÒÒÉ³è ÛÜØ " ÉÙËÂÛÜÊØ é é í
æ¹ÉÏØÚÊ¦æ ÇÚKÐ UÉâÐàÖÚÒîÉ³Ô×Öµá ÒÉËòÊÔ ÊÌÎ ÉÄÍäÐçËÂÛÜÊØµÒïÊØÍÂÐçËÂÛÜÊØÚÐá Ø ÖÚÞ_ ÉÍÒ³é
\ \
ÉËÖµÒ áÜÊôÊ ] ÐçË
ÍÊåÌÍäÐÞ
ÊØµÙ³É ÞßÊÌÍÂÉÌí·æ·ÛñËÂÇ ËÇÚÉ
Rational numbers: input and output.
Ô×ÖµØµÙËÂÛÜÊØ®ØÚÐÞßÉ
ÍÂÉÄÎ áÜÐÙ³É³è _ôì
ÐØµèÀËÂÇµÉ ×Ô ÖµØµÙËÂÛÜÊØßÙ³ÐáÜá
ÍÂÉÎÚá×ÐÙ³Éè_ ì
add
operator+
add (r, s)
é " ËÂÛÜáÜá»íôæ¹ÉãÙ³ÐØóÒîÎ ÊÌËÈÎ ÊÌËÂÉØ ËÂÛ×Ðçá ÛÜÞÀÎµÍÂÊ UÉÞßÉ³Ø ËÂÒ ÜÛ ØµÒîËÂÉ¬ÐçèÊÔ æòÍÂÛñËÂÛÜØµå
r + s
std :: cout < < " Sum is " < < t.n < < "/" < < t.d < < "\n" ;
ÛÜØ áÜÛÜØµÉ íµæ¹É è ÍÂÐçËÂÇµÉÍÈæ·ÍÂÛñËÉ
std :: cout < < " Sum is " < < t < < "\ n";
µÖ ÒîËÈáÜÛV]É æ¹ÉâÐÑÍÂÉÓèÚÊÛñØÚå ÛñËÈÔ×ÊÌÍòÔ×ÖµØÚèÚÐÞßÉØ ËäÐá Ë ì Î É³Ò³é ÔÜËÂÉÄÍ·ÐçáÜáíôæ¹É æÈÐØ¯ËÈËÊ¿ËÂÇµÛÜØ ]®ÊÔ¡Ð ÍÂÐçËÂÛÜÊØÚÐá
ØôÖµÞ_ ÉÍòÐÒòÐ¿ÒîÛÜØµåáÜÉUÌÐáñÖÚÉÏÔÜÍÂÊÞ ËÂÇµÉãÒîÉË ÐçØÚèØÚÊÌËòÐÒ¹Ë æ¹Ê UÐáÜÖµÉ³ÒÈÔÜÍÂÊÞ ËÂÇµÉâÒÉË é
[ ÍÂÊÞ æ ÇÚÐçËòæ¹ÉëÇÚXÐ UÉëèµÊØÚÉëÐ _ ÊUÉÌíìÊÖ Ù¬ÐØ åÖµÉ³ÒÒ·ËÇ ÐÑË ÐáÜá¾æ¹ÉëÇÚÐXUÉâËÂÊßèµÊÀÛÜÒ·ËÊßÊ UÉÍáÜÊÐè
ËÂÇµÉ ÊÖµËîÎ Ö«ËÊÌÎ ÉÍäÐÑËÂÊÌÍ
éêØ èÚÛñÒÙ³ÖµÒÒÛÜØµå ËÂÇµÉÊÖ«ËÎÚÖ«Ë¿ÊÌÎ ÉÍäÐçËÊÌÍ ÛÜØ " É³ÙËÂÛÜÊØ \ é é K æ¹ÉÇÚÐXUÉ
<<
ÐçÍÂåÖµÉ³è ËÂÇÚÐçËòËÂÇµÉëÊÖ«ËÎÚÖ«Ë ÒîËÍÉ¬ÐÞðÎ ÐÒîÒÉ³è ËÂÊßÐØµè ÍÂÉËÂÖ«ÍÂØµÉ³è _ ìËÂÇµÉëÊÖ«ËÎÚÖ«Ë ÊÌÎ ÉÄÍäÐçËÂÊÌÍ·ÞÖµÒîË_ É
ÐØóá UÌÐáñÖÚÉíÚÒîÛÜØµÙ³ÉÏËÇÚÉãÊÖ«ËÎÚÖµËòÊÌÎ ÉÍÂÐçËÂÊÌÍÈÞßÊôèµVÛ YÚÉ³ÒïËÂÇµÉÏÒîËÍÂÉ¬ÐçÞ é XÐ UôÛÜØµåÍÂÉ³Ô×ÉÄÍÂÉ³ØµÙ³ÉâË ìôÎÉ³Ò ÐçËòÊÖ«Í
èµÛÜÒîÎ ÊÒÐáíµËÂÇµÛÜÒ Ù¬ÐØ É¬ÐçÒÛÜáñì _ ÉëèµÊØµÉ Èæ¹ÉàÒîÛÜÞÀÎÚáñì®ÎÐçÒÒ ÐØµè ÍÂÉËÂÖ«ÍÂØ ËÂÇµÉàÊÖ«ËÎÚÖ«Ë ÒîËÍÉ¬ÐÞ ^»æ·ÇµÊÒÉ
Ë ìôÎ ÉÏÛÜÒ
`¹ÐÒòÐ ÍÂÉÔ×ÉÍÂÉ³ØµÙ³É
std::ostream
// POST : a has been written to o
std :: ostream & operator < < ( std :: ostream & o , rational r)
{
return o < < r.n < < " /" < < r .d;
}
Æ ÇµÉÍÉ ÛÜÒ ØµÊ ÍÂÉ¬ÐÒÊØ ËÂÊ ÒîËÂÊÌÎÇÚÉÄÍÂÉ ®Ô×ÊÌÍ ËÂÇµÉ ÛñØµÎÚÖ«Ë¬í æ¹Éßæ¹ÊÖµáÜè ÛñØ ËÂÇµÉ®ÒÂÐÞßÉßÔªÐçÒÇµÛÜÊØáÜÛ ]É
È
ËÂÊ ÍÂÉÎÚá×ÐÙ³É¿ËÂÇµÉ¿Ë æ¹ÊóÛñØµÎÚÖ«ËÏÒîËäÐçËÉ³ÞßÉ³Ø ËÒ
ÐØµè
_ôì ËÂÇµÉ
std::cin >> r.n;
std::cin >> r.d;
ÒÛÜØµåáÜÉÓÒîËÂÐçËÂÉ³ÞßÉ³Ø¯Ë
std :: cin > > r;
^ºÐØµèóËÂÇµÉëÒÐÞßÉâÔ×ÊÌÍ ËÇÚÉâÛÜØ«Î Ö«Ë·ÊÔ `Äé ·åÐÛÜØ¤íÚæ¹ÉâØÚÉÉ³è ËÂÊ Î ÐÒÒÓÐØµèóÍÂÉÄËÂÖ«ÍÂØ ËÂÇµÉëÛÜØ«ÎÚÖµËòÒîËÍÉ¬ÐÞ
^ÊÔÃË ì Î É
` ÐÒÏÐ®ÍÉ³sÔ×ÉÍÂÉ³ØµÙ³Éé êØ ÐèµèÚÛ²ËÂÛÜÊØ¤íæ¹É¿ÞÖµÒîË Î ÐÒÒÓËÂÇµÉàÍäÐçËÂÛñÊØ Ðçá ØôÖµÞ_ ÉÍ
Ó
ËÂÇÚÐçË¹æ¹ÉãæÈstd::istream
ÐçØ Ë¹ËÂÊÍÉ¬Ðè ÐÒòÐàÍÂÉ³Ô×ÉÍÂÉØÚÙÉÌíÒÛñØÚÙÉÏËÂÇµÉÏÛÜØ«ÎÚÖµËÈÊÌÎ ÉÄÍäÐçËÂÊÌÍÈÇÚÐÒ¹ËÂÊ ÞßÊ èÚÛñÔÜìßÛñËÂÒ ÌU ÐçáÜÖµÉÌé
ÆÈÇµÉ ÊÌÎ ÉÍäÐçËÊÌÍ µY ÍÂÒîË¿ÍÂÉ¬ÐçèÚÒ¿ËÂÇµÉØ ÖÚÞßÉÄÍäÐçËÂÊÌÍ¿ÔÜÍÂÊÞ ËÂÇµÉÒîËîÍÂÉ¬ÐÞ íÕÔ×ÊáÜáÜÊ¦æ¹É³è_ ì Ð ÒÉÎ ÐçÍäÐÑËÂÛÜØµå
ÙäÇÚÐçÍäÐÙËÉÍ¬íÐçØÚè ÚY ØÚÐáÜá²ì ËÇÚÉãèµÉ³ØµÊÞßÛÜØÚÐçËÂÊÌÍ¬éÆÈÇ ÖÚÒí æ¹ÉëÙ¬ÐçØ ÍÉ¬Ðè Ð¿ÍÂÐçËÂÛÜÊØÚÐá¤ØôÖµÞ_ ÉÍ·ÛÜØ ÊØµÉâåÊ
_ôì É³Ø¯ËÂÉÍÂÛñØÚåÔ×ÊÌÍ ÉI«
P ÐÞÀÎÚáÜÉ
é
1/2
// POST : r has been read from i
// PRE : i starts with a rational number of the form " n /d"
std :: istream & operator > > ( std :: istream & i , rational & r)
{
char c ; // separating character , e .g . ’/ ’
return i > > r.n > > c > > r.d ;
}
ê ØÙ³ÊØ¯ËÍäÐÒËïËÂÊ
í¥ËÂÇµÛÜØµåÒïÙ¬ÐØ åÊ æ·ÍÂÊØµå«íôÉé å«éñíôÛÜÔ¾ËÇÚÉ ÖµÒÉÄÍ·É³Ø¯ËÂÉÍÒ¹ËÇÚÉÓÙäÇÚÐçÍäÐÙÄËÂÉÍ
operator<<
ÒÉXW ÖµÉ³ØµÙ³É æ ÇµÉ³Ø®ÎÚÍÊÞÀÎÚËÉ³è®Ô×ÊÌÍ ÐâÍäÐÑËÂÛÜÊØÚÐáØ ÖµÞb_ ÉÄÍ¬é áÜÒÊ«íôæ¹É ÎµÍÂÊ _ Ð _Úáñì èÚÊØ ËïæÈÐØ ËïËÂÊ
A/B
ÐÙ³ÙÉÎµË
ÐÒÏÐßÍÂÐçËÂÛÜÊØÚÐá¡ØôÖµÞb_ÉÍÏÐÒÓÊÖ«ÍãÛñØµÎÚÖ«ËÓÊÌÎ ÉÍÂÐçËÂÊÌÍÓèµÊôÉ³ÒéàÆÈÇµÉÍÉÐÑÍÂÉ ÞßÉ³ÙäÇ ÐçØÚÛñÒÞßÒòËÂÊ
èµÉ¬Ðá æ Û²3.4
ËÂÇÒÖµÙäÇ ÛÜÒÒîÖÚÉÒ³í _ Ö«ËÈæ¹ÉÏæ¹Ê
Ø ËòèµÛÜÒîÙ³ÖµÒÒÈËÇÚÉÞðÇµÉÍÂÉé
\ ÄÉ ËâÖµÒâÙÊØÚÙáÜÖµèÚÉ ËÂÇµÛÜÒÏÒÉ³ÙËÛÜÊØ æ Û²ËÂÇ Ð _ É¬ÐçÖµËÛVYÚÉ³è UÉÍÒÛÜÊØÝÊÔ ÍÂÊåÌÍäÐçÞ éS¼Ç ÐÑËâÞ®Ð ]ÉÒ
ËÂÇµÛÜÒUÉÍÂÒÛÜÊØ®É UÉØ ØµÛÜÙÉÍïÛÜÒÃËÂÇµÉ ÔªÐÙË ËÇ ÐÑËÛÜØÀËÇÚÉ
Ô×ÖµØµÙËÂÛÜÊØ¤íËÂÇµÉ ØµÉæ Ë ì Î É ÛÜÒÕÖÚÒîÉ³è®ÉIPµÐÙÄËÂáñì
main
áÜÛV]ÉÏÐØ ÐçËÂÊÞßÛÜÙ àÔ×ÖµØµèÚÐÞßÉ³Ø¯ËäÐáË ìôÎ ÉÏÒÖµÙäÇ ÐÒ
é
êØÝËÂÇµÉÒÎ Û²ÍÂÛñËÏÊÔ " É³ÙËÛÜÊØ é é ÊØ ÞßÊôèµÖµá×Ðçint
ÍÂÛ³ÐçËÂÛÜÊØ¤í æ¹ÉÐÙËÖ ÐçáÜáñì ÒîÎÚáÜÛ²ËÏËÇÚÉ¿ÎµÍÂÊåÌÍäÐÞ ÛÜØ¯ËÂÊ
ËÂÇÚÐçË ÙÊØ ËäÐÛñØÚÒòËÂÇµÉàèÚIÉ Y ØµÛñËÛÜÊØóÊÔ ËÂÇµÉàÒîËÍÂÖµÙÄË
í¤ÐçáÜÊØµå
ËÂÇ«ÍÂÉ³Z
É YÚáÜÉÒ ·ÐbY áñÉ
rational.h
rational
æ Û²ËÂÇ èµÉ³Ù³áÜÐçÍäÐçËÂÛñÊØÚÒ ÊÔËÂÇµÉòÊ UÉÍáÜÊÐèµÉ³èÀÊÌÎ ÉÄÍäÐçËÂÊÌÍÂÒ ô
Ð Y áñÉ
ËÂÇÚÐçËÕÙ³ÊØ ËÂÐÛÜØµÒÃËÇÚÉòèµÉ YÚØÚÛ rational.C
ËÂÛÜÊØµÒ¹ÊÔ ËÂÇµÉ³ÒîÉâÊÌÎ ÉÍÂÐçËÂÊÌÍÂÒ YÚØÚÐáÜáñìí¥
Ð YÚáÜÉ
ËÂÇÚÐçËÈÙ³ÊØ¯ËäÐÛÜØµÒ¹ËÂÇµÉâÞ®ÐÛñØ®ÎµÍÂÊåÌÍÂÐÞ é
userational2.C
ïË ËÂÇµÉ ÒÐÞßÉÃËÛÜÞßÉÌí¦æ¹ÉÕÎÚÖ«Ë ÊÖ«Í ØµÉæ Ë ì Î É
ÐØµèëËÂÇµÉ ÊÌÎÉÍäÐçËÂÛñÊØÚÒ¾ÊØàÛñË ÛÜØ ËÂÊòØÚÐÞßÉ³ÒîÎ ÐÙ³É
ÛÜØóÊÌÍÂèµÉÍ·ËÂÊÀXÐ UÊÛÜèóÎ ÊÒîÒVÛ _ÚáÜÉÏØ ÐçÞßÉÏÙá×Ðrational
ÒÇµÉ³Ò³é P«ÉÍÂÙÛÜÒÉ
ÐÒ ]ôÒ·ìÊÖóËÂÊßÐÙÄËÂÖÚÐáÜáñì®ÛÜØ¯ËÂÉ³åÌÍäÐÑËÂÉ
ifm
??
ËÂÇµÉâØµÉæ ÍäÐçËÛÜÊØÚÐá ØôÖµ
Þ _ ÉÍÈË ì Î ÉâÛñØ ËÂÊ¿ËÂÇµÉâÞ®ÐçËÂÇ áÜVÛ _µÍäÐçÍîì ËÇ ÐÑËòìÊÖ ÇÚXÐ UÉ _ÚÖµÛÜáñËÈÛÜØ P«ÉÍÙ³ÛÜÒÉ í
ÒÊ¿ËÂÇÚÐçË ÍÂÊåÌÍäÐçÞ _ É³áÜÊ¦æ Ù¬ÐØ _ ÉãÙÊÞÀÎ ÛñáÜÉ³è®ÖµÒÛÜØµåËÂÇµÛÜÒ¹áÜÛ _ÚÍÂÐçÍìé
// Program : userational2.C
// Add two rational numbers .
# include < iostream >
# include " rational .C "
int main ()
{
// input
std :: cout < < " Rational number r :\ n";
rational r;
std :: cin > > r;
std :: cout < < " Rational number s :\ n";
rational s;
std :: cin > > s;
// computation and output
std :: cout < < " Sum is " < < r + s < < " .\ n" ;
return 0;
}
Program 33: ¢¤£³Á ³£Ä©¯î ©¯ Å
// Program : rational .h
// Define a type for rational numbers , and declare
// operations on it .
# include < iostream >
M"@.6#9"K(P
#*.*6N7
*C7
A*
"
BIE$:#*
/
%M$&SB
)2 Q*C$&K
A'(%6#(
K*
(6#
"
%*TF/
$
5*E$7
I
.*
operator+
%*
/ %**'"
7
-*
/
M.P0J
.#*
*
r + s
/1J
.*!#6%"
. :2
,)$& 7
%*
J
#A"
*T
%I
+.*
\ namespace ifm {
// the new type rational
struct rational {
int n ;
int d ; // INV : d != 0
};
// POST : return value is the sum of a and b
rational operator + ( rational a , rational b );
// POST : a has been written to o
std :: ostream & operator < < ( std :: ostream & o , rational a );
// POST : a has been read from i
// PRE : i starts with a rational number of the form " n /d"
std :: istream & operator > > ( std :: istream & i , rational & a );
}
Program 34: ¢¤£³Á £©¯î ©¯×Å×
// Program : rational .C
// Define a type rational and operations on it
// the new type rational
struct rational {
int n;
int d ; // INV : d != 0
};
// POST : return value is the sum of a and b
rational operator + ( rational a , rational b )
{
rational result ;
result . n = a.n * b .d + a.d * b.n;
result . d = a.d * b .d;
return result ;
}
// POST : b has been added to a ; return value is the new value of a
rational & operator += ( rational & a , rational b ) {
a.n = a .n * b.d + a.d * b. n;
a.d *= b.d;
return a;
\ }
// POST : a has been written to o
std :: ostream & operator < < ( std :: ostream & o , rational a )
{
return o < < a.n < < "/" < < a.d ;
}
// POST : a has been read from i
// PRE : i starts with a rational number of the form " n/d "
std :: istream & operator > > ( std :: istream & i , rational & a)
{
char c ; // separating character , e.g . ’/ ’
return i > > a.n > > c > > a. d;
}
Program 35: ¢¤£³ÄÁ £©¯ ©¯ªÅ
·ÉÍÂÉãÛÜÒÈÐØóÉIP«ÐÞÀÎÚáÜÉ ÍÂÖµØóÊÔ¡ËÇÚÉÓÎµÍÂÊåÌÍäÐÞ é
Rational number r:
1/2
Rational number s:
1/3
Sum is 5/6.
4.2.5 Const-types
ÉËÏÖÚÒÓÙ³ÊÞßÉZ_ ÐÙ ] ËÂÊ®ËÂÇµÉ¿ÐèµèµÛñËÂÛÜÊØ ÊÌÎ ÉÄÍäÐçËÂÊÌÍÓÔ×ÊÌÍÓÍäÐçËÛÜÊØÚÐá ØôÖµÞ_ ÉÍÂÒ ÔÜÍÊÞ ÍÂÊåÌÍäÐçÞ é á ËÂÇµÊÖµåÇ ËÇÚÛñÒàÊÌÎ ÉÍäÐçËÊÌÍ èÚÊ É³Ò
ÛñØ ËÂÉ³Øµè ËÂÊ Ù´ÇÚÐØµåÉßËÂÇµÉLUÐáÜÖµÉ³Ò ÊÔòÛ²ËÂÒ Ù¬ÐáÜáÎ ÐçÍÂÐÞßÉËÂÉÍÒ³í ËÂÇµÉ

É Ù³ÛÜÉ³ØµÙìÔªÐØÚÐçËÂÛÜÙ¹ÛÜØìÊÖÀÞßÛÜåÇ Ë ÒîÖÚååÉÒîËÕËÊâÒîÎ ÉÉ³èßÖ«ÎÀËÂÇµÛÜÒ ÊÌÎ ÉÍäÐÑËÂÊÌÍ _ ì¿ÖÚÒîÛÜØµåãÍÂÉ³Ô×ÉÄÍÂÉ³ØµÙ³É·Ë ìôÎÉ³Ò
ÐØ¯ìôæÈÐ³ì
// POST : return value is the sum of a and b
rational operator + ( rational & a , rational & b )
{
rational result ;
result .n = a .n * b.d + a.d * b. n;
result .d = a .d * b.d ;
return result ;
}
êØµèµÉ³É³è¤íËÂÇµÛÜÒ UÉÍÂÒîÛÜÊØ ÛÜÒ¿Ù³ÊÌÍÍÂÉÙËÀÐçØÚè Î ÊÌËÂÉ³Ø¯ËÂÛ×ÐáÜá²ìÝÔªÐÒîËÉÍ ËÂÇÚÐØ ËÂÇµÉ ÎµÍÂÉ UôÛÜÊÖµÒ ÊØÚÉí ÒÛÜØµÙ³É®ËÂÇµÉ
ÛÜØµÛñËÂÛÜÐáÜÛ¬ÐÑËÂÛÜÊØâÊÔÐÏÔ×ÊÌÍÂÞ®Ðçá¥ÎÐÑÍäÐÞßÉËÂÉÄÍ ÛÜÒÕèµÊØµÉ _ôì¿Ù³ÊÌÎôìôÛÜØµå ÖµÒîË
ÐçèÚè«ÍÂÉÒÒ³íÍäÐçËÂÇµÉÍÃËÂÇÚÐØ

UÌÐçáÜÖµÉ³Ò ÐÒ¡ÛÜØëËÂÇµÉÞßÉ³
Þ _ ÉÍ æ ÛñÒÉ Ù³ÊÌÎôìãËÂÇÚÐçË ËäÐ ]É³Ò ÎÚá×ÐÙ³ÉÃÖµØµèÚÉÄÍ ËÂÇµÉÙ¬ÐçáÜá _ ì UÌÐáñÖÚÉÕÒÉ³Þ®ÐØ¯ËÂÛÜ¸Ó
Ù³Ò é
int
UÉØ ÛñÔ¡ËÂÇµÉÏÒÂXÐ U¥ÛÜØµå ÛñÒòÒÞ®ÐçáÜá ÛÜØ ËÇÚÛñÒòIÉ P«ÐÞÀÎÚáÜÉÌíôìÊÖ Ù³ÐØóÛÜÞ®ÐåÛñØÚÉòËÂÇÚÐçË·ÞßÉÞb_ ÉÄÍæ·ÛÜÒÉ Ù³ÊÌÎ ì
Ù¬ÐØ _ ÉàÎµÍÂÉËË ì GÉ P¥Î É³ØµÒÛ UÉ¿ÛñØ½ÒîËîÍÂÖµÙËÂÒ ËÂÇÚÐçËÓÐçÍÉ¿ÞßÊÌÍÉëÉ³á×Ð _ÊÌÍäÐçËÂÉâËÇ ÐçØ
¤ÛÜØ Ù³ÊØ ËÍÂÐÒîË¬í
rational
\ Ù¬Ðáñá _ ì®ÍÂÉ³Ô×ÉÍÂÉØÚÙÉëÛÜÒÈÔªÐÒîËòÔ×ÊÌÍ
Ë ì Î É³Ò³íµÉ UÉØóËÂÇµÉãÞßÊÒîËÈÙ³ÊÞÀÎÚáÜÛñÙ¬ÐçËÂÉ³è®ÊØµÉ³Ò³é
¯
©
»

·ØµÔ×ÊÌÍËÂÖµØÚÐçËÂÉ³á²ìíËÂÇµÉóÙ¬ÐáñáÈÎ ÐçÍäÐçÞßÉËÂÉÍÂÒ¿Þ¿ÖÚÒË _ Éóá UÌÐáñÖÚÉÒ ÖµØµèÚÉÄÍÀÙ¬Ðáñá_ôì ÍÂÉ³Ô×ÉÍÉ³ØµÙ³ÉÌíÈÒÊÝæ¹É
Ù¬Ð
Ø Ëïæ·ÍÂÛñËÉ·ËÂÇµÉ IÉ P¥ÎµÍÂÉÒÒÛÜÊØ
í¥Ô×ÊÌÍÈÉIP«ÐÞÀÎÚáÜÉÌí¯É UÉ³ØóÛÜÔ
ÐçÍÂÉUÐçÍÂÛ×Ð _ áñÉ³Ò ÊÔ¤Ë ì Î É
^»æ·Ç ì `Äé " ËÛÜáÜáíËÂÇµaÉ¿+ÔªÐÒîbËÂÉÄ+Í UcÉÍÒÛÜÊØÝÞßÛÜåÇ¯Ë æ¹ÊÌÍ ] ÛÜØ½Êa,
ÖµÍâb,
ÐçÎµÎÚcáÜÛÜÙ¬ÐÑËÂÛÜÊØ ÛñËÏèµÊôÉ³ÒÏÒÊ ÛÜØ
rational
ÍÂÊåÌÍäÐÞ í Ô×ÊÌÍ¹IÉ PµÐçÞÀÎ áñÉÌí ÒîÛÜØµÙ³É ÛÜØßËÂÇµÛÜÒÎµÍÂÊåÌÍÂÐÞ íæ¹É Ù¬ÐáÜá
æ·ÛñËÂÇßVá UÐáÜÖµÉ·ÊÌÎÉÍäÐØµèµÒ
operator+
ÊØµáñìé
ØµÉáÜÉ³ÒÒÏÊ _ UôÛÜÊÖµb
Ò ^ÐØµèÝÞÖµÙäÇ ÞßÊÌÍÉàè ÐçØÚåÉÄÍÂÊÖµÒ ` ÎµÍÂÊ _ÚáÜÉÞ ÍÂÉ³Þ®ÐÛÜØµÒ³íËÇÚÊÖµåÇ ÓÛÜØ½Î ÐÒîÒÛÜØµå
ËÂÇµÉ Î ÐçÍäÐÞßÉËÉÍÂÒ®ÐÒ®ÍÂÉ³Ô×ÉÍÉ³ØµÙ³É³Ò³í æ¹É
ËÂÇµÉ ÊÌÎ ÉÍäÐçËÊÌÍßËÂÊ ÙäÇÚÐØµåÉ ËÂÇµÉ UÐáÜÖµÉ³Ò®ÊÔãÛñËÂÒ®Ù¬ÐáÜá
Î ÐçÍäÐÞßÉÄËÂÉÍÂÒ ÛñØ ËÂÇµ
É YµÍÂÒîËïÎÚá×ÐçÙ³ÉÌí¥É UÉØÛÜ©¯Ô »ËÂªÇÚ¸ ÐçË¹ÇÚÐçÎµÎ É³ØµÒïÖµØµÛÜØ ËÉ³Ø ËÂÛñÊØ ÐçáÜáñìé¡êØÔ×ÖµØµÙËÂÛÜÊØµÒËÂÇÚÐçËÈÐÑÍÂÉ
á×ÐçÍåÉÍ ËÂÇÚÐØ ËÂÇµÉ Ð _ Ê UÉ
íÛñË Ù³ÐØ É¬ÐÒÛÜá²ìóÇÚÐçÎµÎ É³Ø ËÂÇÚÐçË æ¹ÉàÞßÊ èµÛÜÔÜì ÒÊÞßÉëÊÔ ËÂÇµÉàÙ¬ÐáÜá
operator+
Î ÐçÍäÐÞßÉÄËÂÉÍÂÒ¹ÒÛÜÞÀÎÚáñì _ôì ÞßÛÜÒîËäÐ ]ÉÌé
O ÊÌË Þ®Ð ]ôÛÜØµå ÒÖµÙäÇ ÞßÛñÒîËäÐ ]É³ÒòÛÜÒ ËÂÇµÉâÎµÍÂÛÜÞßÉÏÍÂÉ³ÒîÎÊØÚÒîVÛ _ÚÛÜáÜÛñË ì ÊÔ ËÂÇµÉëÎÚÍÊåÌÍäÐÞßÞßÉÍ¬íµÊÔ Ù³ÊÖ«ÍÂÒÉí
_ÚÖ«Ë·ËÂÇµÉâÙáÜÉ UÉÍ ÎµÍÂÊåÌÍÂÐÞßÞßÉÍ¹Ù¬ÐáñáÜÒÈËÇÚÉãÎÚÍÊåÌÍäÐÞßÞßÛÜØµåàá×ÐØµåÖÚÐåÉÓÔ×ÊÌÍ ÇµÉ³áñÎ æ·ÇµÉ³ØµÉ UÉÍ Î ÊÒîÒVÛ _ÚáÜÉÌé
êØ ËÇÚÛñÒïÒîÎÚÛñÍÂÛñË¬í¯ËÂÇµÉÏÐ _ Ê U
É îÉ Ù³ÛÜÉØÚÙÄì Y P¿Ô×ÊÌÍ
ÛñÒ¹Ð _ Ðè ÞßÊ UÉí¥ÒÛÜØµÙ³ÉÓÛñË¹ÛÜØ¯ËÍÂÊ èÚÖµÙ³ÉÒ
operator+
Ð ØÚÉÄæ ÎÊÒÒVÛ _ÚáÜÉÓÒÊÖ«ÍÂÙ³ÉãÊÔ ÉÍîÍÂÊÌÍÂÒ³é
êÔËÂÇµÛÜÒÓÒÊÖµØµèµÒÏËÂÊ ÊÐ _ ÒËÍäÐÙËÏÔ×ÊÌÍãìÊÖ¾í¾ÇµÉÍÂÉ¿ÛÜÒãÐØ IÉ P«ÐÞÀÎÚáÜÉëæ ÇµÉÍÉÛñËÏÛÜÒÏÒÛÜÞÀÎÚáñì æ·ÍÂÊØµå®ËÂÊ
ÞßÊ UÉ¹ËÊãÙ¬ÐçáÜá _ ì »ÍÂÉ³Ô×ÉÍÂÉØÚÙÉ·ÒÉ³Þ®ÐçØ ËÂÛÜÙÒ ËÂÇµÉòÙ³ÊÞÀÎÚÛÜáñÉÍ ÇÚÐÒÃØÚÊÏÙäÇÚÐØµÙ³ÉòËÂÊÏèµÉËÂÉ³ÙÄËÕËÂÇµÛÜÒ ÉÍîÍÂÊÌÍÒîÛÜØµÙ³É
ÛñË·ÛÜÒòÎÚÖ«ÍÂÉ³áñì ÒÉ³Þ®ÐçØ ËÂÛÜÙ³Ðáé ïÊØµÒÛÜèµÉÍ·ËÇÚÉãÖµØ ÐÑÍìóÒÖ _µËÍäÐçÙËÂÛÜÊØóÊÌÎÉÍäÐçËÂÊÌÍ·Ô×ÊÌÍ ËÂÇµÉãË ì Î É
rational
ÔÜÍÂÊÞ " É³ÙËÂÛñÊØ é é é
// POST : return value is -a
rational operator - ( rational a)
{
a .n = - a .n;
return a ;
}
ïÇ ÐçØÚåÛñØÚåàËÂÇµÛÜÒ¹ËÂÊ
rational operator - ( rational & a)
{
a .n = - a .n;
return a ;
}
ÚÇ ÐÒ¿Ðóè«ÍäÐÒîËÂÛñÙ ^ºÐØµè ÖµØµèµÉ³ÒÛñÍÉ³è `âÙ³ÊØµÒÉXWôÖµÉ³ØµÙ³É ÀËÂÇµÉßÉIP¥ÎµÍÂÉ³ÒîÒÛÜÊØ
æ ÛÜáñáÒîËÂÛñáÜáÇÚÐXUÉ ËÇÚÉßÒÐÞßÉ
-a
UÐáÜÖµÉ ÐÒ _ É³Ô×ÊÌÍÂÉÌí _ÚÖµË ÛñËÕæ·ÛÜáÜáµÇÚÐXUÉ·ËÂÇµÉ ÐèµèµÛñËÂÛñÊØ Ðçá«É É³ÙÄË¹ÊÔ Ù´ÇÚÐØµåÛÜØµåÏËÂÇµÉUÐáÜÖµÉ ÊÔ é SÝÉ ÇÚÐXUÉ
a
ÐÙ³ÙÛÜèµÉ³Ø ËäÐçáÜáñ
ì ÙÍÉ¬ÐçËÂÉ³èóÐ¿Ù³ÊÞÀÎÚáÜÉÄËÂÉ³áñìßèµÛ ÉÍÂÉ³Ø¯Ë·ÊÌÎÉÍäÐçËÂÊÌÍ³é
ÒÏÞ®ÐØ ì ÊÌËÂÇµÉÍÏÇµÛÜå
Ç ºáñÉ UÉ³á ÎµÍÂÊåÌÍäÐÞßÞßÛÜØµåá×ÐØµåÖÚÐåÉ³Ò³í Ê ÉÍÂÒëÐßÞßÉ³ÙäÇÚÐØµÛÜÒÞ ËÇ ÐÑË #
ÛÜÔÈÎÚÍÊÌÎ ÉÍÂáñì ÖµÒÉè # ÐçáÜáÜÊ¦æ ÒàËÂÇµÉ ÙÊÞÀÎ ÛñáÜÉÍëËÂÊ½èµÉËÂÉ³ÙË ÖµØÚèµÉ³ÒîÛñÍÂÉ³è ÙäÇÚÐØµåÉ³ÒÊ
Ô UÐáÜÖµÉ³ÒÐÒ ÛÜØ ËÂÇµÉ
ÎµÍÂÉ UôÛÜÊÖµÒÈGÉ PµÐÞÀÎÚáÜÉÌé Æ¹ÇµÉÓÛÜèµÉ¬ÐàÛÜÒïËÂÊ
ËÂÇÚÐçËòÐàÙ³ÉÍËäÐçÛÜØ UÐáÜÖµÉ æ ÛÜáñáØµÊÌË _ ÉÓÙ´ÇÚÐØµåÉ³è¤íµÐØµè
¨
ËÂÇµÉ³Ø¿áÜÉË ËÇÚÉïÙÊÞÀÎ ÛñáÜÉÍ¡Ù´ÇµÉ³Ù ] æ ÇµÉËÇÚÉÄ¢¤Í £³æ¹
É ]»É³ ÉÎ ÊÖµÍ ÎµÍÂÊÞßÛñÒÉÌé êØàËÂÇµÉïÙ¬ÐçáÜá _ ì »ÍÂÉ³Ô×ÉÍÂÉØÚÙÉ UÉÄÍÂÒÛÜÊØ
ÊÔ¡ËÇÚÉÓÖµØÚÐçÍì®ÒÖ _µËÍäÐçÙËÂÛÜÊØ ÊÌÎ ÉÍäÐÑËÂÊÌÍ¬í¥ËÂÇµ
É ^ÔªÐáÜÒîÉ `ÎÚÍÊÞßÛÜÒÉ Ù¬ÐØ _ ÉÏåVÛ UÉ³ØóÐçÒòÔ×ÊáÜáñÊÑæ·Ò³íôÖµÒÛÜØµå ËÂÇµÉ
]Éì æ¹ÊÌÍÂè
é
const
rational operator - ( const rational & a)
{
\ a .n = - a .n ; // error : a was promised to be constant
return a ;
}
ê ØÀÙ³ÊÞÀÎÚÛÜáñÛÜØµå·ËÂÇµÛÜÒUÌÐçÍÛ×ÐØ ËÃÊÔËÇÚÉÈÊÌÎ ÉÍÂÐçËÂÊÌÍ¬íËÂÇµÉÈÙ³ÊÞÀÎÚÛÜáÜÉÍ æ·ÛÜáÜá¥ÛÜÒîÒÖµÉ ÐØÀÉÄÍÍÂÊÌÍÕÞßÉ³ÒÒÂÐåÉíÎÊÛÜØ Ë
ÛÜØµå ÊÖ«ËÈËÂÇµÉãÞßÛñÒîËäÐ ]ÉÌéSÝÉãÙ¬ÐØóËÂÇµÉ³Ø YaPÛñË_ôì É³ÛñËÂÇµÉÍòåÊÛÜØµå_ÐçÙ ] ËÂÊÙ¬ÐáÜá _ôì UÐáÜÖµÉãÒîÉ³Þ®ÐØ ËÛÜÙ³Ò³í
ÊÌ
Í _ôì ÛÜØ ËÍÊôèµÖµÙ³ÛÜØµå Ð ÍÉ³ÒÖµáñ
Ë UÌÐçÍÛ×Ð _ÚáÜÉÓáÜÛV]ÉÓÛÜØ
Ð _ ÊUÉÌé
operator+
[ ÍÂÊÞ ËÂÇµÉ®ÒîËÍÂÛÜÙÄËÂáñìÝÔ×ÖµØµÙËÂÛñÊØ ÐçáÎ ÊÛÜØ¯ËàÊÔRU¥ÛÜÉæâí ËÂÇµÛÜÒëÎµÍÂÊÞßÛÜÒÉÞßÉ³ÙäÇ ÐçØÚÛñÒÞ ÛÜÒàÒÖ«Î ÉÄÍ QÚÖÚÊÖµÒ³í
ÐØµè ËÂÇµÉÍÂÉ ÐçÍÂÉÀÎµÍÂÊåÌÍäÐÞßÞßÛñØÚå á×ÐØµåÖÚÐåÉ³ÒëÛñØ ÖµÒÉ®ËÂÇÚÐçË èµÊ
Ø ËàÇÚXÐ UÉ Û²
Ë ^ ÖµÒÉ³è ËÂÊ _ É®ÒÖµÙ´ÇÐ
á×ÐØµåÖÚÐåÉÌíÖµØ¯ËÂÛÜá¥ËÂÇµÉ
]ÉÄìôæ¹ÊÌÍÂè æÈÐçÒÐèµèµÉ³èÛÜØ íÞßÊÌËÂVÛ UÐçËÂÉb
è _ ìÛ²ËÂÒ ÒÖµÙ³Ù³É³ÒÒÛÜ
Ø `Äé
const
áÜÒîÊ«í«ØÚÊ _Êôè«ì®Ô×ÊÌÍÂÙ³É³Ò·ÖÚÒÈËÂÊ Þ®Ð ]ÉÏÖµÒÉãÊÔ¡ËÂÇµÉÏÎÚÍÊÞßÛÜÒÉ ÞßÉ³ÙäÇÚÐØµÛÜÒÞ é Ö«ËÈËÂÇµÉãæ·ÇÚÊáñÉÓÎÊÛÜØ ËÈÊÔ
ÇµÛÜåÇ ºáÜIÉ UÉ³á¯ÎµÍÂÊåÌÍÂÐÞßÞßÛÜØµåïá×ÐØµåÖÚÐåÉ³Ò¾ÛÜÒ ËÂÊ·Þ®Ð ]ÉÃËÇÚÉÕÎÚÍÊåÌÍäÐÞßÞßÉ!Í Ò áÜÛÜÔ×ÉÕÉ¬ÐÒÛÜÉÄÍ ÌËÂÇµÉ Ù³ÊÞÀÎÚÛÜáÜÉÄÍ¤ÛÜÒ
ÊÖ«ÍïÔÜÍÂÛÜÉØÚè®ÐçØÚè®Ù¬ÐçØßÇÚÉáñÎ ÖÚÒÕËÂÊàXÐ UÊÛÜèßÞ®ÐØ¯ìËÂÛñÞßÉ ºÙ³ÊØµÒÖµÞßÛÜØµå ÉÍîÍÂÊÌÍÂÒ³éÕÆÈÇµÉ
ÞßÉ³ÙäÇÚÐØµÛÜÒÞ
const
ÛÜÒÏáÜVÛ ]ÉÐ ÙäÇµÉ³Ù ]ÝèµÛÜåÛ²Ë _ôì ÎµÍÂÊ UôÛÜèµÛÜØµåÐçèÚèµÛñËÛÜÊØÚÐá ÍÂÉ³èµÖµØµèÚÐØ ËãèÚÐçËäÐ ^»ËÂÇµÉ
]Éì æ¹ÊÌÍÂè `Äí¡æ¹É
Þ®Ð ]ÉßÒÖ«ÍÂÉßËÂÇÚÐçË¿ÛÜØµÙ³ÊØµÒÛñÒîËÂÉ³ØµÙ³ÛÜÉÒàÛÜØ ËÇÚÉßæ·ÇÚÊáñÉÀèÚÐçËäÐóÒÉË ^»ËÂÇµÉÀÎµÍÂÊåÌÍäÐçÞ const
`âÐçÍÂÉ ÐÖ«ËÂÊÞ®ÐçËÛÜÙ¬ÐáÜá²ì
èµÉËÂÉ³ÙÄËÂÉ³è¤é
êÔ ÛÜÒ·ÐçØ ì®Ë ìôÎ ÉÌí«ËÂÇµÉ³Ø
Definition.
const
ÛÜÒâËÇÚÉ
Ë ì Î É ^Ù³ÊØµÒîË Ë ìôÎ ÉßÔ×ÊÌÍëÒÇµÊÌÍËG`ãÊÔ í ÐØµè ÛñËÂÒîÉ³áÜÔ ÛÜÒãËÂÇµÉ

¯
©

F
éÃÆÈÇµÉïÙ³ÊØµÒîË WôÖÚÐáñÛVYÚÉ³èZUÉÍÂÒÛñÊØÊÔ ÇÚÐÒ ÉIPµÐçÙËÂáñìëËÇÚÉïÒÐÞßÉ UÐáÜÖµÉ ÍäÐØµåÉ¹ÐØµè¿Ô×ÖÚØµÙËFµÛÜ³Ê£¦ØÚ ÐJ áÜ°ÛñË ì Á
Ð J¦Ò ¢ é Æ¹ÇµÉÀÊØµáñìÝèµÛ ÉÍÂÉ³ØµÙ³ÉßÛÜÒãËÂÇÚÐçË ÐØ ÉIPôÎÚÍÉ³ÒÒÛÜÊØ ÊÔ¹Ù³ÊØµÒîË»Ë ìôÎ ÉßÛÜÒâØÚÊÌËàÐáÜáÜÊ¦æ¹É³èÝËÂÊ ÙäÇÚÐØµåÉ
ÛñËÂÒ UÌÐçáÜÖµÉ ^ÛÜØ ÊÌËÇÚÉÄÍâæ¹ÊÌÍÂèµÒ³í Û²ËëÛÜÒãÙ³ÊØµÒîËäÐØ¯ËG` ËÂÇµÛÜÒãÛÜÒâÊÖ«ÍâÎµÍÂÊÞßÛñÒÉÌí ÐçØÚèÝËÇÚÉÙ³ÊÞÀÎÚÛÜáÜÉÍÏÙäÇµÉ³Ù ]¥Ò
æ ÇµÉËÇÚÉÄÍòæ¹
É ]É³ÉÎ ËÂÇÚÐçË¹ÎµÍÂÊÞßÛÜÒîÉÌé
êÔ æ¹ÉÓæ·ÍÂÛñËÉÏÔ×ÊÌÍ·IÉ PµÐçÞÀÎ áñÉ
const int n = 5;
n = 6;
ËÂÇµÉ ÙÊÞÀÎ ÛñáÜÉÍ æ ÛñáÜá ÛÜÒÒÖµÉÐØ ÉÍÍÊÌÍëÞßÉ³ÒÒÐåÉÙÊØÚÙÉÍÂØµÛÜØµå ÂË ÇµÉÐÒÒÛñåØÚÞßÉØ Ë
í¡ÒîÛÜØµÙ³É ÇÚÐÒ
n = 6
n
ËÂÇµÉÏÙ³ÊØµÒîËË ì Î É
é
const int
ÕÐáÜÖµÉ³Ò¹ÊÔ Ù³ÊØµÒîË»Ë ìôÎ ÉÏÞÖµÒîËòÐá²æÈÐ¬ìôÒR_ ÉâÛÜØµÛñËÛ×ÐáÜÛÉ³è¤é"S Í ÛñËÂÛÜØµå
const int n ; // error : uninitialized constant
ÛÜÒÈÛÜáÜáÜÉåÐá^ºÐØµè Þ®Ð ]É³ÒÈØÚÊ¿ÒÉ³ØµÒÉí ÒÛñØÚÙÉãæ¹ÉÏÙ¬ÐØóØµÉ UÉÍ ÐÒÒÛñåØóÐ UÌÐáñÖÚÉÓËÂÊ
n
á×ÐçËÉÍG`Äé
4.2.6 What exactly is constant?
ÉËÈÖÚÒ¹Ù³ÊØµÒÛñèÚÉÄÍ¹ÒÊÞßÉ áVUÐáÜÖµÉ ÊÔ¾Ë ì Î É
éµêÔ¤ËÂÇµÉÓÖÚØµèµÉÍÂá²ì¥ÛÜØµåëË ìôÎ É ÜÛ ÒïØµÊÌËòÐâÍÂÉ³Ô×ÉÍÉ³ØµÙ³É
const
Ë ìôÎ ÉíôËÂÇµÉ³Ø®ËÂÇµÉ Vá UÐáÜÖµÉ ÛÜÒ¹ÐÒîÒÊôÙ³ÛÜÐçËÂÉ³èßæ Û²ËÂÇ®ÐâÙ³ÊØµÒîËäÐçØ Ë é [ ÊÌÍ¹ÉIPµÐçÞÀÎ áñÉÌ¯í ËÇÚÉ èµÉ³Ùá×ÐçÍäÐçËÛÜÊØ
¥ ±¬
*
/
/4,
0G:$/
&*
22
24
A07
#6%"@"
A&7O
A BI
DU M/
A
7
5D 292*SB@ BI
/
O
.*
7
%+
-
O"
(7
2* -"
#, 2&7
(Q"K( %G%,S
&*
%< J
-2*
.
8*SB$&B>J
A= <
\ ÚÎ ÍÊÞßÛÜÒÉ³ÒÓËÂÇÚÐçËÏËÂÇµÉ UÌÐçáÜÖµÉ ÊÔËÂÇµÉÊ _ É³ÙË _ É³ÇµÛÜØµè ËÂÇµÉ UÐçÍÂÛÜÐ Ú_ áÜÉ æ ÛÜáÜá ØµÊÌË
_ ÉàÞßÊ èÚÛ Y Éè¾éRS ÉàÞ®Ð¬ì ^ºÐÙ³ÙÛÜèµÉ³Ø ËäÐçáÜáñìa` ËÍìËÂÊßÙäÇÚÉ³ÐçËÏÐçÍÂÊÖµØµè ÂË ÇµÛÜÒ·ÎµÍÂÊÞßÛÜÒÉ _ôì µ
Ö ÒÛñØÚnå®ÐçØÚÊÌËÇÚÉÄÍ
ØÚÐÞßÉÓÔ×ÊÌÍòËÇÚÉÏÊ _ ÉÙË¬í _ÚÖµË¹ËÂÇµÉãÙ³ÊÞÀÎÚÛÜáÜÉÍ æ·ÛÜáÜá Ù¬ÐÑËÂÙäÇóÖµÒ
const int n = 5
const int n = 5;
int i & = n ; // error : const - qualification is discarded
i = 6;
SÝÉ·Ù¬ÐØµØµÊÌËïÖµÒÉ ÐØ®ÉIPôÎÚÍÉ³ÒÒÛÜÊØ®ÊÔ Ë ìôÎ É
ÂË ÊàÛñØÚÛ²ËÂÛ×ÐáÜÛ³É^ÊÌÍ¹ÐÒîÒÛÜåØÀËÂÊ `ÐçØ ÉGP¥ÎµÍÂÉ³ÒÒîÛÜÊØ®ÊÔ
const
Ë ìôÎ É íÒÛÜØµÙ³ÉâËÇ ÐÑË·æ¹ÊÖµáÜè ÙÍÂÉ³ÐçËÂÉ Ð ÞßÊ èÚÛ YÐ _ áñÉÓÐ áÜÛ×ÐÒÈÔ×ÊÌÍÏÐØ Ê _ ÉÙË ËÂÇÚÐçËòæÈÐÒ·ÎµÍÂÊÞßÛÜÒÉèËÂÊ
_ ÉÏÙ³ÊØµÒîËäÐØ¯Ë¬é
ØßËÂÇµÉ ÊÌËÇÚÉÄÍïÇÚÐØµè¾í ÐØßáVUÐáÜÖµÉ ^ºÐÙËÂÖÚÐáñáñìíÐØ ì ÉIP¥ÎµÍÂÉÒÒÛÜÊØ `ÕÊÔË ìôÎÉ
ÛñÒÕËÂÇµÉ ÐáñÛ×ÐÒ
const
ÊÔ ÐçØ Ê _ ÉÙË¬í _ÚÖµËÏËÂÇÚÐçËÓÊ _ É³ÙËãÛÜÒ
ØµÉ³Ù³É³ÒîÒÂÐçÍÂÛÜá²ì Ù³ÊØµÒîËäÐØ¯ËâÛñËÒÉ³áÜÔºéàÆ¹ÇµÉÙÊØÚÒË W Ö ÐçáÜÛVYÚÙ¬ÐçËÂÛñÊØ

ÛÜØßËÂÇµÛÜÒ Ù³ÐÒÉÓÛÜÒ ÞßÉÍÉ³áñìßÐâÎµÍÂÊÞßÛñÒÉòËÂÇÚÐçËËÂÇµÉ Ê _ É³ÙË Ò UÐáÜÖµÉ æ ÛñáÜá ØµÊÌË _É ÞßÊôèµÛVYÚÉ³è
ËÂÇµÉ
º¥£³ÚÁ
ÐáÜÛÜÐÒïÛÜØ W ÖÚÉÒîËÂÛÜÊØ¤é·
ÉÍÂÉÏÛÜÒòÐØóÉIP«ÐÞÀÎÚáÜÉ ËÂÇÚÐçËòÛÜáÜáÜÖµÒîËîÍäÐçËÂÉ³ÒÕËÂÇµÛÜÒÈÎÊÛÜØ Ë¬é
int n = 5;
const int & i = n ;
int & j = n ;
i = 6;
j = 6;
//
//
//
//
i becomes a non - modifiable alias of n
j becomes a modifiable alias of n
error : n is modified through const - reference
ok : n receives value 6
·ÉÍÉÌíïæ¹ÉóèµÊ Ú
Ø ÊÌË ÇÚÐKUÉ Ð Ù³ÊØµÒîËäÐØ¯Ë
í _ÚÖµËÀÐ Ù³ÊØµÒîËäÐçØ Ë
^ØÚÐÞßÉ³áñì `´é Ø
i
¥ Ù¬ÐáÜ±¬áñÉ³ èÐ
¬¢¤£³¬³
ÉIPôÎÚÍÉ³ÒÒÛÜÊØÊÔ Ë ìôÎ É
ÛÜÒÈÐáÜÒÊ
é
const
±¤ £³¶ç³£³³ ±Ä
4.2.7 Const-references
[ ÛñÍÒîËÃÊÔÐáñáíËÂÇµÉ Ë ìôÎ ÉØ ÐçÞßÉ
ÛÜÒ Î ÐçÍÉ³Ø ËÂÇµÉ³ÒîÛ³É³è ÐÒ
^ `ÄíÛé ÉÌé æ¹É¹åÉË Ù³ÊØµÒîËäÐçØ Ë
const
const
UÐáÜÖµÉ³ÒëÊÔ ÍÂÉÔ×ÉÍÂÉ³ØµÙ³ÉßË ìôÎ Éé ïÊØµÒîËÍÉ³Ô×ÉÍÂÉ³ØµÙ³ÉÒÐÑÍÂÉ UÉÍìÝÖÚÒîÉ³Ô×ÖµáïÐØµè ÊÔÜËÂÉ³Ø ÐçÎµÎ É¬ÐçÍâÛÜØÝÍÂÉ¬Ðçá ºáÜÛñÔ×É
Ù³Ê èÚÉé ÉÄËÖµÒ Ù³ÊÞßL
É _ ÐÙ ] ËÂÊ ÊÖ«ÍÔªÐçÒîËÂÉb
Í UÉÍÂÒÛñÊØÊÔ
Ô×ÊÌÍÍÂÐçËÂÛÜÊØÚÐáÕØôÖµÞb_ÉÍÂÒ³é ê»ËÂÒ
operator+
îÒÂÐÔ×É UÉÍÂÒÛñÊØóÛÜÒïËÂÇµÛÜÒ
// POST : return value is the sum of a and b
rational operator + ( const rational & a , const rational & b ) {
rational result ;
result .n = a .n * b.d + a.d * b. n;
result .d = a .d * b.d ;
return result ;
}
Æ¹ÇÚÉàÔªÐçÙËÓËÂÇÚÐçË ËÂÇµÛÜÒ Ù³ÊÞÀÎÚÛÜáñÉ³Ò·Ù³ÊØ YµÍÂÞßÒ·ËÇ ÐÑË æ¹É ÐçÍÂÉ ØµÊÌËÓÙäÇ ÐçØÚåÛñØÚåÀËÇÚÉZUÐáÜÖµÉ³Ò ÊÔ ËÂÇµÉàÔ×ÊÌÍÂÞ®Ðá
Î ÐçÍäÐÞßÉÄËÂÉÍÂÒ ÊÌÍ æ·ÛñËÂÇµÛÜØ ËÇÚÉ_ Êôè«ìÀÊÔ ËÂÇµÛÜÒÕÔ×ÖµØÚÙÄËÂÛÜÊØ¤é Ö«Ë ËÂÇµÉÍÂÉ æÈÐÒ ÐØµÊÌËÂÇµÉÍ ÎµÍÂÊ _ÚáÜÉ³Þ ËÂÇÚÐçË
æ¹É ÐçÎµÎ ÐçÍÂÉ³Ø¯ËÂaáñìèÚÛñbèÚØ Ë ÒÊáVUÉëìÉÄËòÎ ÐÒÒîÛÜØµåÀÎÐÑÍäÐÞßÉËÂÉÄÍÂÒ_ôìÍÂÉ³Ô×ÉÍÂÉØÚÙÉ¿ÍÂÉ WôÖµÛñÍÂÉ³Ò á UÌÐáñÖÚÉÒÓÐÒ Ù¬ÐáÜá
Î ÐçÍäÐÞßÉÄËÂÉÍÂÒ³íÐçØÚè ËÂÇµÛÜÒ ÒÉIUÉÍÂÉ³áñì ÍÂÉÒîËÍÂÛÜÙÄËÂÒ ËÂÇµÉòÐÑÎÚÎÚáÜÛñÙ¬Ð _ÚÛÜáÜÛñË ìÏÊÔµËÂÇµÉòÊÌÎ ÉÄÍäÐçËÂÊÌÍ¬é [ ÊÌÍËÂÖµØÚÐçËÂÉ³áñìíÌËÇÚÛñÒ
ÛÜÒÃÐ ØµÊØ ÛÜÒÒÖµÉ Ù³ÊØµÒîË ÍÉ³Ô×ÉÍÂÉ³ØµÙ³É
Ò ^ÛÜØ¿ÎÐÑÍËÂÛÜÙ³Öµá×ÐÑÍ¬íÌÔ×ÊÌÍÂÞ®Ðá ÎÐÑÍäÐÞßÉËÂÉÄÍÂÒ ÊÔÙ³ÊØµÒîË ÍÂÉ³Ô×ÉÄÍÂÉ³ØµÙ³É »Ë ì Î É`
Ù¬ÐØ _ ÉãÛñØÚÛ²ËÂÛ×ÐáÜÛ ³É³è®ÔÜÍÂÊÞ Í UÌÐçáÜÖµÉ³Ò·ÐçÒÈæ¹É³áñáé ÆÈÇµÛÜÒÈÞßÉ¬ÐØµÒïËÂÇÚÐçË¹æ¹ÉâÙ³ÐØæ·ÍÛñËÂÉ
const int & i = 3;
\ É³ÇµÛÜØµèãËÇÚÉÃÒÙ³ÉØÚÉÒ³íçËÂÇµÉÃÙ³ÊÞÀÎÚÛÜáÜÉÍÙÍÂÉ¬ÐçËÉ³Ò
Ð ËÂÉ³ÞÀÎÊÌÍäÐçÍì Ê _ É³ÙË¤ËÂÇÚÐçË¾ÇµÊáÜèµÒËÂÇµÉUÐáÜÖµÉ íçÐØµèãËÂÇµÉ
ï
Ðèµè«ÍÂÉ³ÒÒÊÔËÂÇµÛÜÒÕËÂÉÞÀÎ ÊÌÍäÐçÍì¿Ê _ É³ÙË ÛñÒ ÖµÒÉ³èßËÂÊëÛÜØµÛñËÛ×ÐáÜÛÉ ËÂÇµÉ Ù³ÊØµÒîË»ÍÂÉ³Ô×ÉÍÂÉØÚÙÉ éÃÆ¹ÇµÉ 3Ù³ÊÞÀÎÚÛÜáÜÉÍ
Þ®Ð ]É³ÒàÒîÖµÍÉ®ËÇ ÐÑËàËÂÇµÉÀËÂÉ³ÞÀÎ ÊÌÍäÐÑÍìÝÊ _ É³ÙË èµÊôÉ³ÒàØµÊÌË IÉ PôÎÚÛñÍÂÉ _ É³Ô×ÊÌÍÂÉßËÂÇµÉßÙ³ÊØµÒîiË ÍÂÉ³Ô×ÉÄÍÂÉ³ØµÙ³É ËÇ ÐÑË
ÍÂÉ³Ô×ÉÄÍÂÒ·ËÂÊ¿ÛñË ^ÒÉ³ÉÏËÂÇµÉ É³Ô×ÉÍÂÉØÚÙÉ ÖµÛÜèµÉ³áÜÛñØÚÉ ÊØ ÐåÉ \ `´é
ÆÈÇµÉ¿ÒÂÐÞßÉëÇÚÐçÎµÎ ÉØÚÒ æ·ÇÚÉØÝÐßÔ×ÊÌÍÂÞ®Ðá Ô×ÖµØµÙËÂÛñÊØ Î ÐçÍäÐçÞßÉËÂÉÍ ÊÔÃÙ³ÊØµÒîË »ÍÂÉ³Ô×ÉÍÉ³ØµÙ³É Ë ì Î ÉÀÛÜÒ ÛÜØµÛ ËÂÛ×ÐçáÜÛ³É³è ÔÜÍÂÊÞðÐØÍ UÐáÜÖµÉÌé
Î ÐçÍäÐÞßÉÄËÂÉÍãÊÔ¹Ë ì Î É
ÛñÒâËÂÇµÉÍÂÉ³Ô×ÊÌÍÉÀËÂÇµÉßÐáÜá ÛÜØ ºÊØµÉ¿èÚIÉ U¥ÛÜÙ³É ÒÖµÛñËäÐ _ÚáÜÉÔ×ÊÌÍàÉ UÉÍì
const
ÎÚÖ«ÍÎ ÊÒÉ
ËÂÇµÉÙ¬ÐáÜá Î ÐçÍäÐÞßÉËÉÍâÛñÒëÐØ Vá UÐáÜÖµÉÌí¾ËÂÇµÉ ÛÜØµÛñËÛ×ÐáÜÛ ³ÐçËÂÛÜÊØ ÛÜÒ UÉÍì É Ù³ÛÜÉØ L
Ë ^»ÊØÚá²ì ÛñËÒ

¶
Ðèµè«ÍÂÉ³ÒÒ ØµÉ³É³èµÒËÂÊ _É Ù³ÊÌÎÚÛÜÉ³è `ÄíÓÐçØÚè ÊÌËÂÇµÉÍæ ÛñÒÉÌí æ¹É ÉÒÒÉ³Ø¯ËÂÛ×ÐáÜá²ì ÔªÐçáÜ
á _ÐçÙ ] ËÂÊÙ³ÐáÜá _ôì UÌÐçáÜÖµÉ
ÒÉ³Þ®ÐçØ ËÂÛÜÙÒ³é
É³ÒîÎÚÛñËÂÉ ËÂÇµÛÜÒ³í¹ËÂÇµÉÍÂÉ ÐçÍÉ ÒËÂÛÜáÜá ÒîÛñËÂÖÚÐçËÂÛÜÊØµÒ æ·ÇµÉÍÂÉ ÛÜÒÀÎµÍÂÉ³Ô×ÉÍäÐ _ áñÉ Ê UÉÍ
ÐçÒ Ð
Î ÐçÍäÐÞßÉÄËÂÉÍ Ë ìôÎ Éé êÔ ÛñÒÕÐÓÔ×ÖµØµè ÐçÞßÉ³Ø ËäÐçáôË ìôÎ ÉÈÊÌÍ ÐÓÒîËÍÖÚÙÄËÕæ·ÛñËÂÇ¿ÒÞ®ÐáÜá¥ÞßÉÞßÊÌÍconst
ìâÍÂÉ WôÖµÛñÍÂÉ³ÞßÉØ ËÂÒ³í
ÛñËïèµÊ É³Ò¹ØÚÊÌËïÎ Ð³ìßÊ óËÊ¿ÞßÊ UÉ ËÂÊ
í¥ÒîÛÜØµÙ³É ËÂÇµÉ ÒÂXÐ U¥ÛÜØµå ÛÜØ®ÇÚÐØµèÚáñÛÜØµåëVá UÐáÜÖµÉ·Î ÐçÍäÐÞßÉÄËÂÉÍÂÒ
const
ÛÜÒ¹ÒÊÒÞ®ÐáÜ"
á ^ÊÌÍÈÉ UÉ³ØóØµÊØµIÉ P«ÛñÒîËÂÉ³Ø¯GË `ËÂÇÚÐçËÈÛñËïæ¹Ê
Ø Ë¹Ù³ÊÞÀÎ ÉØÚÒÐçËÂÉ Ô×ÊÌÍÈËÂÇµÉ ^ÒîáÜÛÜåÇ Ëáñaì ` ÞßÊÌÍÂÉÓÙ³ÊÒîËÂáñì
ÐÙ³ÙÉ³ÒÒ ËÂÊÀËÂÇµÉëÔ×ÊÌÍÞ®Ðá¡Ô×ÖµØµÙËÂÛñÊØ Î ÐçÍÂÐÞßÉËÂÉÍòÛÜØ ËÂÇµÉëÔ×ÖµØµÙËÂÛÜÊØ _Êôè«ìé·êØµèµÉ³É³è¤íÙ³ÐáÜ"
á _ ìÍÂÉÔ×ÉÍÂÉ³ØµÙ³É
ÐèµèµÒïÊØµÉ ÛÜØµèÚÛ²ÍÂÉ³ÙËÂÛñÊØ ¾ËÂÊ áÜÊôÊ ] Ö«Î ËÂÇµ
É UÐáÜÖµÉ ÊÔ¾ÐëÔ×ÊÌÍÂÞ®Ðá Ô×ÖÚØµÙËÛÜÊØßÎÐÑÍäÐÞßÉËÂÉÄÍïÖµØÚèµÉÍ¹Ù³ÐáÜá _ôì ÍÂÉ³Ô×ÉÄÍÂÉ³ØµÙ³ÉïÒÉ³Þ®ÐçØ ËÂÛÜÙÒ³íçæ¹É ÇÚKÐ UÉÕËÂÊ áÜÊôÊ ]ÏÖ«Î ÛñËÂÒ Ðèµè«ÍÂÉ³ÒÒ ÐçØÚèëËÇÚÉØ¿áñÊôÊ ]ÏÖ«Î¿ËÇÚÉÐÙËÂÖÚÐá UÐáÜÖµÉ
ÐçËòËÂÇÚÐçË Ðèµè«ÍÂÉ³ÒîÒ³é òØÚèµÉÍ·Ù¬Ð£¦áÜá _ôì UÐáÜÖµÉâÒîÉ³Þ®ÐØ ËÛÜÙ³Ò³í«ËÂÇµÉàÐèµè«ÍÂÉ³ÒîÒ ÊÔ ËÂÇµÉ UÐáÜÖµÉëÛÜÒ îÇÚÐçÍÂè«æ ÛñÍÉ³
è ^ºÐØµè ÍÂÉ³Ô×ÉÍÂÒòËÂÊ ÒÊÞßÉÓÊ _ É³ÙËòÊØËÂÇµÉãÙ³ÐáÜá ÒîËäÐçÙ ]í ÒîÉ³É " É³ÙÄËÂÛÜÊØ é \ é \ `Äé
áÜÒÊ«íÛ²ËÕÛñÒÃÊÔÜËÉ³Ø Ù³ÊØ UÉ³ØµÛÜÉ³Ø¯ËÕËÂÊÏÖµÒÉïËÂÇµÉòÔ×ÊÌÍÞ®ÐáôÎ ÐçÍäÐÞßÉÄËÂÉÍ ÐÒÕÐ áÜÊôÙ³Ðá UÐçÍÂÛ×Ð _ÚáÜÉ¹ÐçØÚèÞßÊôèµÛÜÔÜì
ÛñËÂR
Ò UÐáÜÖµb
É ^ÒÉ³É
Ð _ Ê UÉ ` µÔ×ÊÌÍòËÇ ÐÑË¬í«ÛñËÂÒïË ìôÎ ÉãÞÖµÒîËÈØµÊÌË _ ÉãÐ¿Ù³ÊØµÒîË Ë ìôÎ Éé
operator-
4.2.8 Const-types as return types.
ïÊØÚÒË Ë ìôÎÉ³Ò Þ®Ð³ìóÐáñÒÊ®ÐçÎµÎ É¬ÐçÍ ÐÒ ÍÂÉËÖµÍØ Ë ì Î É³Ò ÊÔ Ô×ÖµØÚÙÄËÂÛÜÊØµÒ³í ÖµÒîËÓáÜÛ ]ÉëÐØ¯ì ÊÌËÂÇµÉÍ Ë ìôÎÉ³Ò³é êØ
ËÂÇÚÐçËÈÙ¬ÐÒîÉÌíÚËÇÚÉ
ÎµÍÂÊÞßÛÜÒÉ³ÒïËÂÇÚÐçË¹ËÂÇµÉÏÔ×ÖµØÚÙÄËÂÛÜÊØóÙ¬ÐçáÜá¤ÉIP¥ÎµÍÂÉ³ÒîÒÛÜÊØóÛñËÒÉ³áÜÔ¾ÛÜÒÈÙ³ÊØµÒîËäÐØ¯Ë¬é
const
êÔËÂÇµÉ¹ÍÂÉËÂÖ«ÍÂØ Ë ì Î ÉòÛñÒÃØµÊÌËÐÓÍÉ³Ô×ÉÍÂÉ³ØµÙ³ÉòË ì Î ÉÌíËÂÇµÉÈÔ×ÖÚØµÙËÛÜÊØÀÙ¬Ðáñá«IÉ P¥ÎµÍÂÉ³ÒîÒÛÜÊØÀÛÜÒÕÐØ Í UÌÐáñÖÚÉÈÐØµè
ÇµÉ³ØµÙ³ÉÏØÚÊÌËÈÞßÊ èÚÛ YÐ _ áñÉ ÐØ¯ìôæÈÐ³ìéÃêØËÂÇµÛÜÒïÙ¬ÐÒîÉÌí«ËÂÇµÉ
]Éì æ¹ÊÌÍÂèóÛÜÒ¹áÜÉ³åÐçá _ Ö«ËÈÇÚÐÒ¹ØµÊ É É³ÙË¬é
const
ïÊØÚÒË Ë ìôÎÉ³ÒÈËÇÚÉÄÍÂÉ³Ô×ÊÌÍÂÉãÊØÚá²ì Þ®Ð ]ÉãÐ èµÛ ÉÄÍÂÉ³ØµÙ³ÉÏÛÜÔ¡ËÂÇµÉÏÔ×ÖµØÚÙÄËÂÛÜÊØÍÂÉÄËÂÖ«ÍÂØµÒ·ÐàÍÂÉ³Ô×ÉÍÉ³ØµÙ³ÉÌé
O ÊÌËÂÉËÇ ÐÑËÀÛñË ÛÜÒ
åÉ³ØµÉÍäÐáñáñì UÌÐáñÛÜè ËÂÊÝÍÂÉÎÚá×ÐçÙ³ÉÍÂÉËÖµÍØ Ë ì Î É _ ì
·æ·ÇµÛÜáÜÉ
ËÂÇµÛÜÒ¿ÒÂÐÔ×É³á²ì æ¹ÊÌÍ ]ôÒ Ô×ÊÌ ÍËÂ ÇµÉÔ×ÊÌÍÂÞ®Ðá Î ÐçÍÂÐÞßÉËÂÉÍ Ë ìôÎÉ³Ò³íïÛñË¿Ù¬ÐØ Ô×ÊÌÍËÂÇµÉ ÍÂÉÄËÂÖ«const
ÍÂØ Ë ìôÎ É ÍÂÉ³ÒÖµáñËÛÜØ
ÒîìôØ ËäÐÙÄËÂÛÜÙ¬ÐáñáñìßÙ³ÊÌÍÍÉ³Ù
Ë _ÚÖ«ËÈÒÉ³Þ®ÐØ¯ËÂÛÜÙ¬ÐçáÜáñì æòÍÂÊØµåÙÊôèµÉÌé
ÒßÐØ GÉ PµÐÞÀÎÚáÜÉÌíÃáÜÉË ÒÍÉÎÚá×ÐÙ³É
_ ì
ÐçÒ ËÂÇµÉ ÍÂÉÄËÂÖ«ÍÂØ Ë ìôÎ ÉóÊÔ
rational
const rational&
operator+
const rational & operator + ( const rational & a , const rational & b ) {
rational result ;
result .n = a .n * b.d + a.d * b. n;
result .d = a .d * b.d ;
return result ;
}
êØÝÉIP¥É³Ù³Ö«ËÂÛÜØµå®ËÂÇµÉ
ÒîËäÐçËÉ³ÞßÉ³Ø Ë³íËÂÇµÉ¿ÍÂÉÄËÂÖ«ÍÂØ UÐáÜÖµÉ »^ ÛÜØ ËÇÚÛñÒÏÙ³ÐÒÉÀÐ®Ù³ÊØµÒîË»ÍÂÉ³Ô×ÉÍÉ³ØµÙ³É`ÓËÊ
_ É Î ÐÒîÒÉ³è ËÂÊóËÂÇµÉßreturn
Ù¬ÐáÜáñÉÍëÊÔ ËÂÇµÉßÔ×ÖµØµÙËÂÛÜÊØ ÛñÒëÛÜØµÛñËÂÛ×ÐçáÜÛ³É³è æ ñÛ ËÇÝËÂÇµÉÀÉIP¥ÎµÍÂÉÒÒÛÜÊØ
é O Ê¦æ
result
\ ÍÂÉ³Ù³ÐáÜá¹ËÂÇÚÐçË ËÂÇµÉóÛñØÚÛ²ËÂÛ×ÐáÜÛ¬ÐçËÂÛÜÊØ ÊÔÓÐ ^Ù³ÊØµÒîË` ÍÂÉ³Ô×ÉÄÍÂÉ³ØµÙ³É ÔÜÍÂÊÞ ÐçØ áVUÌÐçáÜÖµÉÒÛÜÞÀÎÚáñì Þ®Ð ]É³Ò ÛñË ÐØ
ÐáÜÛÜÐÒïÊÔ¡ËÂÇµÉÏáVUÐáÜÖµÉÌé Ö«ËòËÂÇµÉÏáVUÐáÜÖµÉÓÛÜØ W ÖÚÉÒîËÂÛÜÊØ ^ØÚÐÞßÉ³áñì
`¹ÛÜÒòÐàáÜÊôÙ³Ðá UÌÐçÍÛ×Ð _ÚáÜÉ æ ÇµÊÒÉ
result
ÞßÉ³ÞßÊÌÍìÛÜÒÏÔÜÍÂÉ³É³è ÐçØÚè½æ ÇµÊÒÉ¿Ðèµè«ÍÂÉ³ÒÒ _ É³Ù³ÊÞßÉ³ÒÓÛÜØ UÌÐçáÜÛÜè æ ÇµÉ³Ø½ËÇÚÉ¿Ô×ÖµØµÙËÂÛÜÊØ½Ù¬ÐáÜá ËÂÉÄÍÂÞßÛÜØÚÐçËÂÉ³Ò
^ÒÉÉ " É³ÙËÛÜÊØ \ é é ÐØµè " É³ÙËÂÛñÊØ é \ é `Äé Æ¹ÇÚÉÈÙ³ÊØµÒÉXW ÖµÉ³ØµÙ³É ÛÜÒ ËÂÇÚÐçËÃËÂÇµÉÈÍÉËÂÖ«ÍÂØµÉ³è ÍÂÉÔ×ÉÍÂÉ³ØµÙ³É·æ·ÛÜáÜá
_ ÉÏËÂÇµÉàÐáÜÛ×ÐçÒÈÊÔ ÐçØ IÉ P¥ÎÚÛñÍÂÉè Ê _ É³ÙË³íÐØµè ÖµÒÛÜØµå¿ËÂÇµÛÜÒòÍÂÉÔ×ÉÍÂÉ³ØµÙ³ÉëÍÂÉÒÖµáñËÂÒ·ÛñØ ÖÚØµèµÉ YÚØµÉ³è _ É³ÇÚÐXU¥ÛñÊÌÍ
ÊÔ¡ËÇÚÉÏÎµÍÂÊåÌÍäÐÞ é
ÍîÍÂÊÌÍÂÒ áÜVÛ ]É ËÂÇµÛÜÒ®ÐçÍÂÉ UÉÍì¼ÇÚÐçÍÂè¼ËÂÊ Y Øµè ^ºÐØµè æ¹É Ù¬ÐØµØµÊÌË®ÍÂÉ³áÜÛÜÐ _Úáñì Ù³ÊÖµØ¯ËÊØ Ù³ÊÞÀÎÚÛÜáÜÉÍ
æÈÐçÍÂØµÛÜØµåÒëÇµÉÍÉ `Äí ÒÛÜØµÙ³ÉÀËÇÚÉÀÎµÍÂÊåÌÍÂÐÞ
æ¹ÊÌÍ ]ÝÐÒàÛÜØ ËÉ³ØµèÚÉè¾í Ô×ÊÌÍ GÉ PµÐÞÀÎÚáÜÉÛÜÔïËÂÇµÉßÞßÉ³ÞßÊÌÍì
ËÂÇÚÐçËàæÈÐÒ ÐçÒÒÊôÙÛ×ÐçËÂÉ³è ËÂÊ ËÂÇµÉ®IÉ P¥ÎÚÛñÍÉ³è ¨½
Ê _ © J É³ÙË ÛÜÒàØµÊÌË ÛÜÞßÞßÉèÚÛÜÐçËÂÉ³áñì ÍÉ³ÖµÒÉ³è¤é Ö«Ë ÊØ ÐçØÚÊÌËÇÚÉÄÍ
ÎÚá×ÐçËÂÔ×ÊÌÍÞ í ËÂÇµÉÓÎÚÍÊåÌÍäÐÞ Þ®Ð¬ì _ É³ÇÚXÐ UÉâèµÛ ÉÄÍÂÉ³Ø Ëáñì®ÊÌÍ·IÉ UÉ³Ø ÙÍäÐÒîÇ¾é
4.2.9 When to use const?
S¼ÇµÉ³ØµÉ UÉÍ·ìÊÖ ËÂÇµÛÜØ ]®Ð _ ÊÖ«ËïËÂÇµÉãÐçÎµÎµÍÂÊÌÎµÍÂÛ×ÐçËÂÉòË ì Î ÉÏÊÔ ÐZUÐçÍÂÛ×Ð _ÚáÜÉí«ÐàÔ×ÊÌÍÂÞ®ÐáÔ×ÖµØµÙËÂÛÜÊØ Î ÐçÍäÐÞ
ÉËÂÉÄÍ¬í¤ÊÌÍâÐßÔ×ÖµØµÙËÂÛÜÊØ Ò ÍÂÉËÂÖ«ÍÂØ UÌÐáñÖÚÉí ÛñËÏÛÜÒ åÊôÊ è ÎµÍäÐÙÄËÂÛÜÙ³ÉàËÂÊßËÂÇµÛÜØ ] Ð _ÊÖµËÓÙÊØÚÒË W Ö ÐçáÜÛVYÚÙ¬ÐçËÂÛñÊØ
ÐçËãËÂÇµÉ ÒÂÐÞßÉ ËÂÛÜÞßÉé ÔÜËÂÉÄÍàÐáÜá»í¾ìÊÖ ÒÇµÊÖµáÜè ]ôØµÊÑæ æ ÇÚÐçËÏìÊÖ æÈÐØ ËÏËÂÊóèµÊæ·ÛñËÂÇ½ËÇÚÉ UÌÐçÍÛ×Ð _ÚáÜÉÌí
Î ÐçÍäÐÞßÉÄËÂÉÍ¬í¯ÊÌÍ ÍÂÉËÖµÍ
Ø UÐáÜÖµZ
É ^»ÛÜÔìÊÖ®èÚÊØ Ë¬í ËÂÇµÛÜÒÃÎ ÐçÍäÐåÌÍäÐÑÎ ÇßÛñÒ É UÉ³Ø ÞßÊÌÍÂÉ·ÛÜÞÀÎÊÌÍËäÐØ¯ËG`ÄíÌÒÊàìÊÖ
ÐáÜÒîb
Ê ]ôØÚÊ¦æ æ ÇµÉËÂÇµÉÍÈËÂÇµÉÓÎÚÍÊåÌÍäÐÞ ØµÉ³É³èµÒòËÂÊÙäÇ ÐçØÚåÉÏÛñËÂÒ UÐáÜÖµÉÏÐçË·ÒîÊÞßÉÓÎÊÛÜØ Ë¬é
ÆÈÇµÉ _ÐçÒÛÜÙ ÍÂÖµáÜÉÓËÂÊ Ô×ÊáÜáñÊÑæ ÛñÒÈËÂÇµÛÜÒ
·ÒîÉâÙ³ÊØµÒîË
Ë ì Î É³Òòæ·ÇµÉ³ØµÉ UÉÍÈËÂÇµÛÜÒòÛñÒ¹ÎÊÒÒÛV_ÚáÜÉÏÐØµèóÞ®Ð ]ÉÒòÐ èµÛ ÉÄÍÂÉ³ØµÙ³ÉéÚê»Ë
ÐáñæÈÐ³ìôÒÈÞ®Ð ]É³Ò·Ð èµÛ ÉÄÍÂÉ³ØµÙ³ÉÏÛÜØÙ³ÊØµØµÉ³ÙËÛÜÊØæ Û²ËÂÇ ÍÉ³Ô×ÉÍÂÉ³ØµÙ³ÉãË ì Î É³Ò¬é
Const Guideline:
−
êØµèÚÉÉ³è¤íÈÛ²ËÀÛÜÒÞßÊÌÍÂÉ ËÇ ÐçØ ËÂÇµÉÎµÍÂÊÞßÛÜÒÉ ÊÔ Ù³ÊØµÒîËäÐØ¯Ë UÐáÜÖµÉËÂÇÚÐçËèµÛÜÒîËÂÛñØÚåÖµÛÜÒîÇÚÉÒ¿ËÂÇµÉË ì Î É
ÔÜÍÂÊÞ æ·ÇµÛÜáÜÉ æ¹ÉßØµÉ³É³è áVUÐáÜÖµÉ³ÒâËÂÊ ÛÜØµÛñËÛ×ÐáÜÛÉ ÐØµè ÐÒÒÛñåØ ËÊ Ê _ É³ÙËÂÒàÊÔ¹Ë ìôÎ É
const
ÛñÒ·ÎµÍÂÉ³Ô×ÉÍÂÐ _ÚáÜÉâËÂÊ
íÍ UÐáÜÖµÉ³Ò·ÒÖ Ù³É Ô×ÊÌÍ
éS ÉàÇÚÐKUÉ ÐçáÜÒÊ®ÐçÍÂåÖµÉ³è ËÂÇÚÐçË
const
const
ÛñØÞ®ÐØ¯ì®ÒÛñËÂÖÚÐçËÂÛñÊØÚÒí¥ÒÛÜÞÀÎÚáñìßÔ×ÊÌÍòÉ Ù³ÛÜÉ³ØµÙì ÍÂÉ¬ÐÒîÊØÚÒé ÊÖóÙ¬ÐØµØµÊÌËÈÛÜåØµÊÌÍÂÉÓËÂÇµÉ³ÒÉâÔªÐçÙËÂÒ³íµÉ UÉ³Ø
ÛÜÔ¾ìÊÖóèµÊØ ËÈÙ¬ÐçÍÂÉâÐ _ ÊÖ«Ë¹ËÂÇµÉÏÎµÍÂÊÞßÛÜÒîÉ ÞßÉ³ÙäÇ ÐçØÚÛñÒÞ ÊÌËÂÇµÉÍæ ÛñÒÉÌé
êÔ ÛÜÒÓØµÊÌËÏÐÀÍÂÉ³Ô×ÉÍÉ³ØµÙ³É¿Ë ì Î ÉÌí ËÂÇµÉ³Ø½ËÇÚÉ W ÖµÉ³ÒîËÂÛÜÊØ æ ÇµÉËÇÚÉÄÍ
Þ®Ð ]ÉÒÓÐ®èµÛ ÉÍÂÉ³ØµÙ³É
const
ÔÜÍÂÊÞ ÇÚÐÒßÖµÒÖÚÐáÜá²ì ØµÊÌËßÒÖµÙäÇ ÐÝÙ³áñÉ¬ÐçÍ®ÐØµÒîæ¹ÉÄÍ¬íÈæ·ÛñËÂÇ ÊØµÉ IÉ P«ÙÉÎµËÂÛÜÊØ ½ÛñØ ÍÉËÂÖ«ÍÂØ Ë ì Î É³Ò®ÊÔ
]Éìôæ¹ÊÌÍè ÍÂÉ³ÐáÜáñì½Þ®Ð ]É³ÒãØÚÊ èµÛ ÉÍÂÉ³ØµÙ³É
Ô×ÖµØµÙËÂÛÜÊØµÒâËÂÇÚÐçËâèµÊ ØÚÊÌËãÍÂÉËÂÖ«ÍÂØ ÍÂÉÔ×ÉÍÂÉ³ØµÙ³É³ÒíÃËÂÇµÉ
const
ÐØµèóÒÇµÊÖµáÜè ËÂÇµÉÍÂÉ³Ô×ÊÌÍÂÉ _ ÉÏÊÞßÛñËËÂÉ³è¤é
êØ ËÇÚÉ®ÒÂÐçÞßÉÀÒîÎÚÛñÍÂÛñË³í ËÂÇµÉ
]Éìôæ¹ÊÌÍèÛÜÒëË ìôÎÚÛÜÙ³ÐáÜáñì½ÊÞßÛñËËÂÉè Ô×ÊÌÍ Ô×ÊÌÍÂÞ®Ðá Ô×ÖµØµÙËÂÛÜÊØ Î Ð ÍäÐÞßÉÄËÂÉÍÂÒ ËÂÇÚÐçËãÐçÍÉ¿ØµÊÌË ÍÂÉÔ×ÉÍÂÉ³const
ØµÙ³É³Òé êØ ËÂÇµÛÜÒ ÒÛñËÂÖÚÐçËÛÜÊØ¤í
ÛÜÒ
ÍÂÉèÚÖµØµèÚÐØ Ë³íËÂÇµÊÖÚåÇ ÛñÔ

const
ÐÔ×ÊÌÍÂÞ®Ðçá Î ÐçÍÂÐÞßÉËÂÉÍâÛÜÒâÊÔïÙÊØÚÒË Ë ìôÎÉÌí æ¹É ÎµÍÂÊÞßÛÜÒÉ¿ØµÊÌËâËÂÊ ÖµÒÉËÂÇµÉÀÔ×ÊÌÍÂÞ®Ðá Î ÐçÍäÐÞßÉËÉÍëÐçÒàÐ
ÞßÊôèµVÛ Y Ð _ÚáÜÉáÜÊôÙ³Ðá UÌÐÑÍÂÛ×Ð _ÚáÜÉÌé Ö«Ë¿ËÂÇµÛÜÒàÎµÍÂÊÞßÛÜÒÉßÛÜÒ¿ØµÉ³ÛñËÂÇµÉÍ¿ØµÉ³Ù³ÉÒÒÂÐçÍì ËÂÊ ÎÚÍÉ UÉ³Ø ËÀÐÙ³Ù³ÛñèÚÉØ ËäÐá
ÞßÊôèµVÛ YÚÙ¬ÐÑËÂÛÜÊØ ÊÔ ËÂÇµÉëÙ¬ÐáÜá¤Î ÐçÍäÐÞßÉÄËÂÉÍ ^Ù¬ÐáÜá _ôì UÌÐáñÖÚÉâÐáñÍÂÉ³Ðè«ì ËäÐ ]É³Ò Ù¬ÐçÍÂÉâÊÔ ËÂÇµÛÜÒ `ÄíÚØµÊÌÍ èµÊôÉ³Ò Û²Ë
ÛÜØ QÚÖµÉ³ØµÙ³É ËÂÇµÉÓÊÖµËÒÛÜèµ
É _ É³ÇÚXÐ U¥ÛÜÊÌÍïÊÔ¾ËÂÇµÉÓÔ×ÖÚØµÙËÛÜÊØ ÛñØÐØ¯ìßæÈÐ³ìéÕêØ ÔªÐÙË¬í«ÛñÔ¾ìÊÖ æ·ÍÂÛñËÉ Ô×ÖµØÚÙÄËÂÛÜÊØµÒ
Ô×ÊÌÍ®Ð áñVÛ _µÍäÐçÍì ^ÒÉÉ " É³ÙËÂÛÜÊØ é é `ÄíÕìÊÖ _ÉËËÂÉÍ ÍÂÉ³ÔÜÍäÐÛñØ¼ÔÜÍÊÞ ÒÖµÙäÇ¼ÙÊØÚÒË Ë ìôÎÉóÖµÒÂÐåÉÌíÈÐÒÀÛ²Ë
ÖµØµØÚÉÙ³É³ÒÒÂÐÑÍÂÛÜáñìÍÂÉ³ÒîËÍÛÜÙËÂÒ ìÊÖ ÛÜÔ ìÊÖ áÜÐçËÂÉÍïèµÉ³ÙÛÜèµÉ ËÂÊ Ù´ÇÚÐØµåÉ ËÂÇµÉ Ô×ÖµØµÙËÂÛÜÊØ®èµÉ YÚØµÛñËÂÛÜÊØ¤íìÊÖÐÑÍÂÉ
Ù³ÊÞßÞßÛñËîËÂÉ³èËÊëËÂÇµÉ Ù³ÊØµÒîË »Ë ì Î É Î ÐçÍäÐÞßÉÄËÂÉ
Í ^»É UÉ³Ø ÛÜÔ ËÂÇµÛÜÒÕËÂÖ«ÍÂØµÒïÊÖ«Ë ËÂZ
Ê _É ÛÜÞÀÎµÍäÐÙËÂÛñÙ¬Ðá `ÄíÖÚØµáÜÉÒÒ
ìÊÖ ÐáñÒÊ ÙäÇÚÐØµåÉÏËÇÚÉÏÇµÉ¬ÐèµÉ
Í YÚáÜÉÏËÇ ÐÑËòÙÊØ ËäÐÛñØÚÒïËÂÇµÉÏÔ×ÖµØÚÙÄËÂÛÜÊØóèµÉ³Ù³áÜÐçÍäÐçËÂÛñÊØ¾é
\ Üá ÒÊ«íÚØµÊÌË Ðáñá UÐçÍÂÛÜÐ _ÚáÜÉ³Ò¹ËÂÇÚÐçË Ù³ÊÖµáÜè _ ÉëèµÉ³Ù³á×ÐÑÍÂÉ³è
ÛñØ Ð¿ÎµÍÂÊåÌÍäÐçÞ ÐçÍÉâË ìôÎÚÛÜÙ¬Ðáñáñì®èµÊØÚÉ
const
ÒÊ«íÕÒÛÜÞÀÎÚáñì _ ÉÙ¬ÐÖµÒÉÛ²ËÞ®Ð ]É³Ò^ÊÌÍÀÐÑÎÚÎÉ¬ÐçÍÂÒ ËÂÊ Þ®Ð ]É`âØµÊ èµÛ ÉÄÍÂÉ³ØµÙ³ÉÛñØ ËÂÇµÉ Ù³ÊØ ËÂÉGP¥ËÀÊÔÈËÂÇµÉ
èµÉ³Ù³á×ÐÑÍäÐçËÂÛÜÊØ¤é ·ÒïÐØ®ÉIPµÐçÞÀÎ áñÉÌí ÙÊØÚÒîÛÜèµÉÍïáÜÛÜØµÉ ÛÜØ ÍÂÊåÌÍÂÐÞ \ ÛñË
ÎÊÒÒÛV_ÚáÜÉÈËÂÊàèµÉ³Ù³á×ÐçÍÉ·ËÂÇµÉ
Þ®Ð ]ÉÈÐÏèµÛ ÉÍÂÉ³ØµÙ³ÉÌí¯ÒÛÜØµÙ³É
UÐçÍÂÛ×Ð _ÚáÜÉ Ð
Ò _É³ÛÜØµåãÊÔÙ³ÊØµÒîËË ì Î É
í_ÚÖµËÃÛñËÕèµÊôÉ³ÒîØ ËÃ
ËÂÇµÛÜR
Ò UÐçÍÂÛ×Ð t_ÚáÜÉÓÊ Ù³Ù³Ö«ÍÂÒ·ÊØµÙ³ÉÏÊØµáñì ÐÔÜËÂconst
ÉÍîæÈÐçÍÂèµrational
Ò³íµÐØµèËÂÇµÛÜÒ¹Ê Ù³Ù³Ö«ÍÍÂÉ³ØµÙ³ÉãÛÜÒÖµÒîËÈËÂÇ«ÍÂÉ³ÉãáÜÛÜØµÉ³ÒR_ É³áñÊÑæâé
[ ÊÌÍïÙ³ÊØµÙÍÂÉËÉ³ØµÉ³ÒÒ³í¥áñÉË ÖµÒ ÒîËÛñÎÚÖÚáÜÐçËÂÉïËÂÇÚÐçËï
Ð UÌÐÑÍÂÛ×Ð _ÚáÜÉ¹ËÂÇÚÐçË ÛñÒÞßÉ¬ÐØ¯ËÕËÊëÇÚÐKUÉ·Ù³ÊØµÒîËäÐØ¯Ë UÐáÜÖµÉ
ÒÇµÊÖµáÜèèµÉ YÚØµÛñËÂÉáñì®åÉËòÙ³ÊØµÒîË Ë ì Î ÉãÛñÔ ÛñËÂÒÈÒÙ³ÊÌÎÉãÒîÎ ÐØµÒòÞßÊÌÍÂÉ ËÂÇÚÐØ áÜÛÜØµÉ³Ò¹ÊÔ Ù³Ê èµÉÌé
4.2.10 Goals
Ë¹ËÂÇµÛÜÒ¹Î ÊÛÜØ¯Ë¬íôìÊÖóÒîÇÚÊÖµáÜè é³é³é
¹
Dispositional.
`ßÖµØÚèµÉÍÒîËäÐØµèËÂÇµÉãÐáÜÛ×ÐçÒïÙ³ÊØµÙ³ÉÎµË_ ÉÇÚÛñØÚè ÍÂÉ³Ô×ÉÍÉ³ØµÙ³ÉâË ôì Î ÉÒ·ÐØµèËÇÚÉ ÉÔ×ÉÍÂÉ³ØµÙ³É ÖµÛÜèµÉ³áÜÛÜØµÉ ` ÖµØÚèµÉÍÒîËäÐØµè ËÂÇµÉÈèµÛ ÉÍÉ³ØµÙ³É
_ ÉË æ¹É³É³Ø
Ð Øµè
ÒîÉ³Þ®ÐØ ËÛÜÙ³Ò Ô×ÊÌÍ
\ ß
´
±
¯
©
»

K
J
D
ô
¯
©

´
±
¯
©
»

K
ß
J

¶
³

£
³

Ô×ÖÚØµÙËÛÜÊØÎ ÐçÍäÐÞßÉÄËÂÉÍÂÒ
`ßÖµØÚèµÉÍÒîËäÐØµèóÙ³ÊØµÒîË Ë ìôÎ ÉÒ ÐØµè ËÇÚÉ ¹ÊØµÒîË
Operational.
ÖÚÛñèÚÉáÜÛÜØµÉÌé
êØÎ ÐçÍËÂÛñÙ³Öµá×ÐçÍ¬í¯ìÊÖ ÒÇµÊÖÚáñè _ ÉãÐ _ÚáÜÉ ËÂÊéé¬é
`®ÒËäÐçËÂÉ¹ÉIP«ÐÙËÃÎÚÍÉ ÐØµè ÎÊÒîËÂÙ³ÊØµèµÛñËÂÛÜÊØµÒ Ô×ÊÌÍÕÔ×ÖµØµÙËÂÛÜÊØµÒÃÛÜØ UÊáVUôÛÜØµå Ô×ÊÌÍÂÞ®Ðá¥Î ÐçÍäÐçÞßÉËÂÉÍ Ë ìôÎÉ³Ò
^
ÌÊ ÍÈÍÂÉËÂÖ«ÍÂØ Ë ì Î É³ÒòÊÔ¡ÍÂÉ³Ô×ÉÄÍÂÉ³ØµÙ³ÉàÐØµè ÊÌÍÈÙ³ÊØµÒîË»Ë ì Î É ^ \ `ßæòÍÂÛñËÂÉÓÔ×ÖµØµÙËÂÛÜÊØµÒïËÂÇÚÐçËòÞßÊôèµÛÜÔÜì ^»ÒÊÞßÉÓÊÔ ` ËÂÇµÉ³ÛñÍÈÙ¬Ðáñá¤Î ÐçÍÂÐÞßÉËÂÉÍÒ
^ ` YÚØµè Òîì¥Ø¯ËäÐÙËÂÛñÙ¬Ðá ÐØµè ÒÉ³Þ®ÐØ¯ËÂÛÜÙ¬ÐçáÉÍÍÊÌÍÂÒòÛñØ®ÎµÍÂÊåÌÍÂÐÞßÒËÂÇÚÐçËÈÐçÍÉÏèµÖµÉ ËÂÊ ÛÜÞÀÎµÍÂÊÌÎ ÉÍÇÚÐØ
èµáÜÛÜØµå ÊÔ ÍÂÉÔ×ÉÍÂÉ³ØµÙ³ÉâË ì Î É³Ò ^
` YÚØµè Òîì¥Ø¯ËäÐÙËÂÛñÙ¬Ðá ÐØµè ÒÉ³Þ®ÐØ¯ËÂÛÜÙ¬ÐçáÉÍÍÊÌÍÂÒòÛñØ®ÎµÍÂÊåÌÍÂÐÞßÒËÂÇÚÐçËÈÐçÍÉÏèµÖµÉ ËÂÊ ÛÜÞÀÎµÍÂÊÌÎ ÉÍÇÚÐØ
èµáÜÛÜØµå ÊÔ Ù³ÊØµÒîË Ë ìôÎ ÉÒ ^ ` YÚØµè ËÂÇµÉèµÉ³Ù³áÜÐçÍäÐçËÂÛñÊØÚÒ·ÛÜØ ÐßåVÛ UÉØ½ÎµÍÂÊåÌÍäÐÞ æ·ÇµÊÒÉàË ì Î É³ÒÓÒÇµÊÖµáÜè _É¿ÙÊØÚÒË ÐÙÙ³ÊÌÍÂèµÛÜØµå
ËÂÊ¿ËÂÇµÉ ïÊØµÒîË ÖµÛÜèµÉ³áÜÛÜØµÉÌé
4.2.11 Exercises
Exercise 115 Ó¤ Fµ³£Àºò¶°§¸¹»ÁÏ¶Ñ©¨ J Â¶¹¶ ±¬î¤
T foo (S i)
{
return ++ i;
}
¸ º
¹
³³°ÁÝ ®Â¶ º J¦¢ ¬ int int& ©F const int& Ó©F
³³°ÁÝ Â¶ º
¦J ¢ ¬ int const int int& © F const int& ÅLHç¥» Fµ ¬ F ³£³¤ ¶ ±¬î¤NMôÅ
©KM
° F º ó±¨ ¬» ©¯¤Â¶
© F
¶ç£À¸È¥î±¬Ýºß£¬»Áë¶ ±³î Fµ »î
»¿IJ ¤î ©µ±¬±Ä©¯°EJ Dô©¯NF ©FÝ ¦¢¾ª ©° ÚJ £®©¤¸Ó³£ÌÅ
\ \
M
¨ ÁÝºÝ±¨ ¬» ©¯¤â¶ FÝ° ©KM
F ºÝ±¨ ¬» ©¯¤ Â¶
©F
¶£
¸È¥î±³ßºÏ£¬»Á¹¶Ä ±¬ Fµ » ã©¯ ³¨ ©¤î±Ä©¯°EJ Dô©¯NF ¨ ´© »Áº©¯
¶Ä ±¬¼±´©¯»À©¯¸ © J à©D¥à¸Ó¬» GFµ GF ôD ©¯ À©F½ Ä±¬ ¦¢¾×©» J £®©¤Ä¸Ï£ÌÅ
±M
£ ©¯»â±Ä¨½ ¬° ©¯î¤ ¶F » M
¢F»Áã¶ ±³î foo Å
Á D¥ ¢¤£±³» ¢ ±F¤¿¶ç£ º ±Ä£¬£³
^
Ì^
Ì^
^
Exercise 116 Ý£¬» ©â¶ ±³î º©¯·¸ ©¬¢¾¿º Dô©¯ ¬®Â¶¿¸Ï int Dô©£¬©« ¦ª¬çÅ
[ ÊÌÍ·ÉGPµÐÞÀÎÚáÜÉÌí
`
\ `
int a = 5;
int b = 6;
// here comes your function call
std :: cout < < a < < "\n" ; // outputs 6
std :: cout < < b < < "\n" ; // outputs 5
ØµÊÌÍÂÞ®ÐçáÜÛ³É³Ò
¸ © ¤ ©D¥¼©½¶ ±¬î º©¯
© £©¯ ©¯ë ¨ ³ ³£ ÅÅ
£©¤»¶ç£¬¨óà¹»¤ º ¿ à£³î¢¤£¬³¤î©¯ » È
¸ ¥î±¬ ¨½³£©¯£ ©F Fµ³ ¨ » ©¯ £
©£³£³¬×©¯ D¥¬ J®¢¤£¬»¨ ©F º Fµ ¨» ©¯£ ¢ » D¥Å £«©¨¢¾ª
Exercise 117
21
−14
£¬¨½©¯ IF
−3
.
2
Õ³£³ ©£À¸Óó ©¯£Ä©¯D¥³£¦¤ Â¶¿º¥¹¶ ±¬î
// POST : r is normalized
void normalize ( rational & r );
// POST : return value is the normalization of r
rational normalize ( const rational & r );
Â ¨¢¾ª³¨½³¤â
Hint: JÚ ¨½©J
Â¶ º¨ Ó©F ©£îÁ¡ ¸ÈJ JÚ ©Dô ¬± ç³» ã ¥D ³£Àº
º³£ Å
¸ ©¤Óó º ¶ ±¬î gcd ¶Ä³£ ¨
±¬ Å ¹¨ F IF ¶^£Ï¢\ ©Ì£^ ©¨ \ `
¬³£¦ Â¶¿ ÑJ ¢ int H¸ FµÌ³º¥»¿¨½ F ±´©¯î ª Ì ô M¥Å
Exercise 118 È£³ DNFµ © Fµ
Â¶¿ºÈ¶»ª¸¹»Áã¶ ±¬î Å
// POST : return value indicates whether the linear equation
//
a * x + b = 0 has a real solution x ; if true is
//
returned , the value s satisfies a * s + b = 0
bool solve ( double a , double b , double & s );
\ ¬JÚ£·¶Ä ±¬° © ¢¤£Á£©¨ð¶£®©¯ §´©¯³ ºÏ¢©»£¦ (a, b) ¶Ä£³¨
º ³
{(2, 1), (0, 2), (0, 0), (3, −4)}.
^
\ `
Ä±¤ µF ³£ º¾¶»ª¸¹»Á¹¢¤£³ÁÌ£©¨óÓ©Fâ Fµ³¤ ¶IJ º µF Ä±¬ª©£Ä©¯î¤ HÂ¶RDô©£¬
©« ¦ª¬ë £ ¶ £¬¨ ©¯¢©£Ä©¨ ¬³£¦ M° ¸ò¥±¬LJÚÝ± F £³¢¾ª©µ±Ä¿©¿ J¦¢
K Jºë±Ä¤³ ND¥£¦î
const Å
Exercise 119
©KM ¹£³Á£Ä©¨
H Ó©Á«
M ¹£³Á£Ä©¨
H Ó©Á«
± M ¹£³Á£Ä©¨
FKM ¹£³Á£Ä©¨
M ¹£³Á£Ä©¨
M
M
H Ó©Á« H Ó©Á«
H Ó©Á«
KM
M
M
^
`
» F ©¯»Ã ¨î© ô¬ Hî ¶ © J M ° ºÓ¶°§¸¹°Áß¢¤£³Á£©¨ à©F ¦¢¾ª©° ¸ÈJ
º¬ ©£³ß¨ ³î© ¬çÅ »¾¢¤£Á£©¨ ©£³®ºÓ¶»ª¸¹»Á ¸Ó¿¶ ±¬î Fµ »î¤©F
¤ J F ³£À° º³°£ main ¶ ±¬î¤çÅ
Exercise 120
int foo ( int & i ) {
return i += 2;
}
const int & bar ( int & i ) {
return i += 2;
}
©KM
M
int main ()
{
const int i = 5;
int & j = foo ( i );
}
int main ()
{
int i = 5;
const int & j = foo ( i );
}
\ ±M
int main ()
{
int i = 5;
const int & j = bar ( foo ( i ));
}
FKM int main ()
{
int i = 5;
const int & j = foo ( bar ( i ));
}
M
int main ()
{
int i = 5;
const int j = bar (++ i );
}
4.2.12 Challenges
Ù³Ê
Þ ³ î©F«©£ F ¬£Ä©£KJ ©¯ß±¤î©°¤â©¿ JÑ¢ ¶£à±¨¿¢¤°Áà¸¹º
Å
±¨¢¾ª ¨½ ³£ó¸È³£³ º º£³´©¯Ó©F º»¨ ©Á» ©£KJó¢©£¦©£³
double ¿©¯ ¦J ¢ std::complex<double> H J ÄI F #include <complex> » £IFµ³£
ÀÁ«¬ïº¥»â ¦J ¢ Å Â £ Fµ³£ ÀÁ«³·©ó© ±¨¿¢¾§ ® ¨½ ³£ ¸¹º£³´©¯Ú¢©£¦ r ©Fß»¨ ©Á» ©£KJ
¢©£¦ i RJÚ ±Ä© º ¦¢¤£³³³î
Exercise 121 ÎÚáÜÉIP½ØôÖµÞ_ ÉÍÂÒ
std :: complex < double >(r ,i ); // r and i are of type double
Ïº£¬¸È»
±Ä¨¢¾ªë ¨½ ³£¦ò¸Ó£ à©¯ ¦¢ Ä±¬IF«Å »¥ºò³î©F«©£ F ³¢ ³£©¯£¦ HÄ©£¬»º
¨ ³î± ¹£³¬×©¯î ©¯ M ©
F¨ ©¯º³¨½©¯±Ä©¯¥¶Ä ±¬¤ H std::sqrt std::abs std::pow ÅÅÌÅ M
©£³©D ©×©« ¦§çÅ Õó³¢ ³£Ä©¯£¦ ©¯²çó¸Ó£ » ¨ ÚGF ¦¢¤£³¬î¤ ¸È³£ ó³¢ ³£©F »
Â¶ J¦¢ std::complex<double> ©
Fº º³£ Â¶ J¦¢ double Å ¶ ± £¦ bJ
±Ä© ©¯² »¥¢¤Ï©
FÝñ¢¤Ó±¨¿¢¾ § ¨½ ³£¦çÅ
³

³
£
à

µ
©
¬
±
¾©¯Ã±³©¯»ª³Á« »¨¿¢¾§¨ ³¤ïºï¶ç»ª¸È°Á ¶Ä ±¬ß¶£ DÌ»Áó±³¡ ¬±
¡©¯î¤® D¥³£Àº±¨¿¢¾§ ¨ ³³£¦
// POST : return value is the number of distinct ( complex ) solutions
//
of the cubic equation ax ^3 + bx ^2 + cx + d = 0. If there
//
are infinitely many solutions ( a=b= c=d =0) , the return
//
value is -1. Otherwise , the return value is a number n
//
from {0 ,1 ,2 ,3} , and the solutions are written to s1 ,.. , sn
int solve_cubic_equation ( std :: complex < double > a ,
std :: complex < double > b ,
std :: complex < double > c ,
std :: complex < double > d ,
\ std :: complex < double >& s1 ,
std :: complex < double >& s2 ,
std :: complex < double >& s3 );
¬£ ©¿¢¤£³ÁÌ£©¨ º©¯ò¬³ºJ £ò¶Ä ±¬ Å £ µ©¨¿¢¾§ JÚ ¨½©J Ä¡ ¦³»º
¤ £³¬ £¬ IF KJ º ©« ³ Dôß¶ ±³î »¤ ax3 + bx2 + cx + d = 0 ©F ±¬±
¸È¬º³£ßº ¦¢¤£³³³î »FµÄIF XDô©¯¡©¯³ HÄ©¬¢«¢¤£³ ¯°¨½©¯¬ J M ¹ÐçÍÂ³èÚ£¥ÐØµÅ Ê Ò¹Ô×ÊÌÍÂÞ¿Öµá×Ð
Fóº Ä±¬ç©£KJ ºÄ£KJ Fµ³£ º ôXJ¯¸Ó£ F
Å
Hint:
\ 4.3 Classes
¬ò¨ ¬» JÚ º¥»à» ±³§°Á
¸ ¸Ó ¨ °Á«ò³¨ »¨¿¢ °Á
·£¬³ò¨½ ¶à¸Ó XD¥³£ß¸ »J £ÀÄ±¬î
ë³X D¥ ¨½»ºÈ¶ç£ JÚ£ ¸È ¢¤£³±³î
»»
¨»º ³ »
H M
ª©µ±
¥ Ä±¬î °¤£³ F ±Ä¬º ± ±Äî¢¾ÝÂ¶ ±¬ª©¯¬ ©¯ ©ð ³¤ Â¶ º
Õ
³£¬¡±¬·±Ä ±î¢¾¶Ä£³¨
Ä±¬î Å ¥Å ¸È»» ªÄ©£¬ ©« F«©¯î©³ ±Ä©¬¢¾ª©¯
©¯ © F»³»Á ³¥°Á¶çÄ©¯ £³Â¶ ±¬×©¯³¬ÑÅ Õ¥à¶ ´©¯£³Ý¨ © ô¬½ JÑ¢ ½»¨¿¢¾§
¨½³¤î©¯î¤ ¨ £³½ç©ä¶ç ©F ¯ ¦ªÅ
¸¹»» £¦³¿§´©£¬ ±¬×©¯³³ ¶çÄ©¯ £³
K J ¶çÄ©¯£ ¶£½£©¯î ©¯ ¨½ ³£¦ ©F º³ ¸Ï ±¨¢¾ª¬ ±³ª©¯³ç¬ °
±Ä ±¬ ¸È»º £©
Fµ¨ ¨ ³³£ÀÁ¥³ ³£©¯ Å
4.3.1 Encapsulation
êØ½ËÇÚÉàÎµÍÂÉ UôÛÜÊÖµÒ Ë æ¹Ê ÒÉÙËÂÛÜÊØµÒ³íæ¹É ÇÚÐXUÉ¿èµÉ YÚØµÉ³è Ð®ØµÉæ ÒËÍÂÖµÙË Ë ìôÎ É
æ·ÇµÊÒÉZUÌÐçáÜÖµÉ
rational
ÍäÐØµåÉ¹ÞßÊôèµÉ³áñÒ ËÂÇµÉÈÞ®ÐÑËÂÇµÉ³Þ®ÐçËÂÛÜÙ³Ðá¯Ë ì Î É ^»ËÂÇµÉ¹ÒÉËÕÊÔ ÍäÐÑËÂÛÜÊØÚÐá¥Ø ÖÚÞ_ ÉÍÒ `ÄíÐØµè æ¹É¹ÇÚÐXUÉòÒÇµÊ¦æ Ø
ÇµÊÑæ ÛñË¹Ù¬ÐØ _ ÉÓXÉ WôÖµÛñÎµÎ É³è®æ Û²ËÂÇ®ÒÊÞßÉ ÖµÒÉ³Ô×Öµá¤Ô×ÖÚØµÙËÛÜÊØÚÐáÜÛñË ì ^ºÐçÍÂÛñËÇÚÞßÉÄËÂÛÜÙòÐØµè ÍÂÉ³áÜÐçËÂÛÜÊØÚÐáÚÊÌÎ ÉÍäÐ ËÂÊÌÍÂÒíÚÛñØµÎÚÖ«ËòÐØµèÊÖ«ËÎÚÖ«GË `Äé
Æ ÊÞßÊÌËVÛ UÌÐÑËÂÉëËÂÇµÉ¿ËîÍäÐØµÒÛñËÂÛñÊØ½ÔÜÍÂÊÞ ÒîËÍÖÚÙÄËÂÒÏËÊÙ³á×ÐçÒÒÉ³ÒãÛÜØ½ËÂÇµÛÜÒÏÒÉ³ÙÄËÂÛÜÊØ ^ÐØµèÝÛÜØ½Î ÐçÍîËÂÛÜÙ³Öµá×ÐçÍ
ËÂÇµÉÀÐçÒîÎ É³ÙËëÊÔ É³ØµÙ¬ÐçÎÚÒÖµá×ÐçËÛÜÊØ `Äí¤áÜÉËëÖµÒãÒîËäÐçÍËâÊ æ·ÛñËÂÇÝÐ®ËÂÇµÊÖµåÇ ËëGÉ P¥Î ÉÍÛÜÞßÉ³Ø Ë³é " Ö«ÎµÎ ÊÒÉ ìÊÖ
ÇÚXÐ UÉÀÎ Ö«ËàËÂÇµÉßÒîËîÍÂÖµÙË
ÐØµèÐáñáÕËÂÇµÉÀÔ×ÖµØµÙËÂÛÜÊØÚÐáñÛñË ì½ËÂÇÚÐçËâæ¹ÉßÇ XÐ UÉßèµÉ UÉ³áñÊÌÎ É³è ÛÜØ ËÂÊóÐ
ØµÛÜÙ³É®áÜVÛ _µÍäÐÑÍìé êØ P«ÉÍrational
Ù³ÛÜÒÉ
ìÊÖÇÚXÐ UÉ ÐÙËÂÖÚÐáÜá²ìÝèµÊØµÉÀËÂÇµÛÜÒ³í Ô×ÊÌÍ ËÂÇµ
É UÉÄÍì _ÐçÒÛÜL
Ù UÉÍÂÒîÛÜÊØÊÔ
??
ËÂÇµÉ Ë ìôÎÉ
ÔÜÍÊÞ ÍÊåÌÍäÐÞ ÐçØÚè ÍÂÊåÌÍäÐÞ é O Ê¦æ ìÊÖ ÇÚXÐ UÉßÒÊáÜè ËÂÇµÉÀáÜÛ _ÚÍÂÐçÍì
ËÂÊ Ð¿Ù³ÖµÒîËÊrational
ÞßÉÍ «áÜÉË Ò¹Ù¬ÐáÜá¤ÛñË ÕÆ ^
`´é ÕÆ ÛñÒòÛÜØµÛñËÛ×ÐáÜáñìÇÚÐçÎµÎôì®æ Û²ËÂÇ
¯
©

¯
©

Á
Â ±Å
ËÂÇµÉ¿Ô×ÖÚØµÙËÛÜÊØÚÐáÜÛñË ìËÇ ÐÑËÏËÇÚÉ¿áÜVÛ _µÍäÐÑÍìóÎµÍÂÊ U¥ÛÜèµÉ³Ò³í¾ÐØµè½ÒîËäÐçÍËÒÏæ¹ÊÌÍ ]ôÛÜØµåßæ ÛñËÂÇ ÛñË¬é Ö«ËÓËÂÇµÉ³ØÝÒîÊÞßÉ
ÖµØ«Î áñÉ¬ÐÒÂÐØ¯ËÈÛÜÒÒÖµÉ³ÒòÙ³ÊÞßÉÓÖµÎ¾é
" ÊÞßÉ Ù³Ê èµÉßèÚÉIUÉ³áÜÊÌÎ É³è ÐÑË
ÃÆ ØµÉ³É³èµÒëËÂÊ ÛÜØµÛñËÂÛÜÐáÜÛ³É Ð
Issue 1: Initialization is cumbersome.
ØµÉæ UÐçÍÂÛ×Ð _ÚáÜÉ æ Û²ËÂÇ ËÂµÇ É ÍÂÐçËÂÛÜÊØÚÐá¡ØôÖµÞb_ÉÍ
¤Ô×ÊÌÍÓËÂÇµÛÜÒ³íËÂÇµÉàÎµÍÂÊåÌÍäÐÞßÞßÉÍ·ÛÜØ Ù´ÇÚÐçÍåÉ¿Þ¿ÖµÒîË
r
1/2
æ·ÍÛñËÂÉ
rational r ; // default - initialization of r
r.n = 1;
// assignment to data member
r.d = 2;
// assignment to data member
Æ ÇÚÉÏèµÉ³Ù³á×ÐÑÍäÐçËÂÛÜÊØ
¹
èµÉ³ÔªÐÖµáñËÛÜØµÛñËÂÛ×ÐçáÜÛ³É³Ò í_ Ö«Ë¹ËÂÇµÉãÐÙËÂÖÚÐá UÐáÜÖµÉÏÊÔ Þ¿ÖµÒîËR_ ÉÓÎµÍÂÊ rational r
U¥ÛñèÚÉèÀËÂÇ«ÍÂÊÖµåÇ
ÐçÒÒÛÜåØµÞßÉ³Ø¯ËÂÒ á×ÐçËÂÉÍ³é ÃÆËÂÉ³áÜáµrìÊÖÀËÂÇÚÐçËÃËÂÇµÉì æ¹ÊÖµáÜè¿ÎµÍÂÉ³Ô×ÉrÍïËÂÊãÛÜØµÛñËÂÛ×ÐçáÜÛ³É
ÔÜÍÂÊÞ ËÂÇµÉëØôÖµÞßÉÍä¸Ó
ÐÑËÂ ÊÌÍòÐØµè èÚÉØÚÊÞßÛñØ ÐÑËÂÊÌÍÈÛÜØóÊØµÉëåÊ«í ÐØµèóìÊÖ ÍÉ¬ÐáÜÛÉÏËÂÇÚÐçËòËÂÇµÉì Ç ÐXUÉàÐ Î ÊÛÜØ¯rË
\ µÇ ÉÍÂÉÌé êØµèÚÉÉ³è¤íÕÛÜÔ¹ËÂÇµÉÀÎµÍÂÊåÌÍäÐÞßÞßÉÄÍàÐçË ÕÆ Ô×ÊÌÍÂåÉËÂÒ ÊØµÉ®ÊÔÈËÇÚÉ ÐÒîÒÛÜåØµÞßÉ³Ø ËÒ³í ÇÚÐÒ ÖµØµèµÉ YÚØµÉ³è UÐáÜÖµÉb^ºÐØµè ìÊÖóåÉÄËòËÂÊ¿ÇÚÐØµèµáÜÉ ËÂÇµÉ
_ÚÖµåÍÉÎ ÊÌÍËÂÒ `ÄéÃêÔ¡ËÂÇµÉÏÒîËÍÂÖµÙËòÛÜÒ¹á×ÐçÍÂåÉÄÍ ^»Ù³rÊØµÒÛÜèµÉÍ¹ËÂÇµÉ
IÉ P«ÐÞÀÎÚáÜÉ ÊÔ
ÊØ Î ÐåÉ \ `Äí¥ËÂÇµÉãÎµÍÂÊ _ÚáÜÉÞ ÛÜÒ·ÐçÞÀÎ áñÛVYÚÉ³è¤é
rational_vector_3
Ø ì®áÜÉåÐá UÌÐçáÜÖµÉÏÊÔ¤ËÂÇµÉÓË ì Î É
Þ¿ÖµÒîËÈÇ XÐ UÉ
Issue 2: Invariants cannot be guaranteed.
Ð ØµÊØ ÉÍÂÊ èÚÉØÚÊÞßÛñØ ÐÑËÂÊÌÍ¬é ÊÖ ÇÚÐXUÉ®ÒîËÂñÛ ÎÚÖµá×ÐçËÂÉè ËÇÚÛñÒÐçÒÐçØÛÜØ UÌÐçÍÛ×ÐØ Ëërational
ÛÜØ ÍÂÊåÌÍäÐçÞ í _Ú«Ö Ë
ËÂÇµÉÍÂÉÏÛÜÒÈØµÊæÈÐ³ì ÊÔ É³ØµÔ×ÊÌÍÂÙ³ÛÜØµå ËÂÇµÛÜÒÈÛÜØ UÐçÍÂÛ×ÐçØ Ë¬é ê»ËòÛÜÒ¹Î ÊÒÒÛV_ÚáÜÉÓÔ×ÊÌÍ·ÐØ ìÊØµÉÏËÂÊ¿æòÍÂÛñËÂÉ
rational r ;
r.n = 1;
r.d = 0;
ÐØµèßËÂÇôÖµÒ UôÛÜÊá×ÐçËÉÈËÂÇµÉ
ÊÔ ËÂÇµÉ·Ë ìôÎÉÌí ËÂÇµÉ Ù³ÊÌÍîÍÂÉ³ÙËÂØµÉ³ÒîÒïÊÔËÂÇµÉ ÛÜØ ËÂÉÄÍÂØÚÐáÚÍÉÎµÍÂÉ³ÒÉ³Ø¯ËäÐçËÂÛñÊØ¾é
° ¤Á£¬» J
ÊÖ ÞßÛñåÇ Ë ÐÑÍÂåÖµÉëËÂÇÚÐçË ÛñË æ¹ÊÖµáÜè _É WôÖµÛñËÂÉëÒËÂÖ«Î Ûñè ËÊ®æòÍÂÛñËÂÉ
í¤ÐØµè ÉIUÉ³Ø½ËÂÇµÉëÎµÍÂÊ r.d = 0
åÌÍäÐÞßÞßÉÄÍ¹ÐçË ÃÆ Ù¬Ð
Ø Ë _ ÉÓËÂÇÚÐçËòÒîËÂÖ«ÎÚÛÜè¤é Ö«ËòÛÜØ ÃÆ ÒÈÐçÎµÎÚáÜÛÜÙ¬ÐÑËÂÛÜÊØ¤í ËÂÇµÉ
UÌÐçáÜÖµÉ³Ò·ÊÔ ÍÂÐçËÂÛÜÊØÚÐá
ØôÖµÞ_ ÉÍÂÒ®ÐÑÍÂÛÜÒÉ ÔÜÍÂÊÞ ÙÊÞÀÎ áñÛÜÙ¬ÐçËÂÉè Ù³ÊÞÀÎÚÖµËÂÐçËÂÛÜÊØµÒ¿ÒÊÞßÉæ ÇµÉÍÉ É³áÜÒÉ ÛÜØ ËÂÇµÉ ÎµÍÂÊåÌÍäÐÞ ·ËÇÚÉÒÉ
Ù³ÊÞÀÎÚÖ«ËäÐçËÂÛñÊØÚÒ Þ®Ð¬ìâÍÂÉ³ÒÖµáñËÕÛÜØÀÐ ³ÉÍÊâèµÉ³ØµÊÞßÛÜØÚÐçËÂÊÌÍ ÒîÛÜÞÀÎÚáñì _ ìÞßÛñÒîËäÐ ]ÉÌíÐçØÚè ÛÜØÀÐáñáÜÊÑæ·ÛÜØµåUÐáÜÖµÉ
ËÊ _ É ÐçÒÒÛÜåØµÉ³èÀËÂÊ
íËÂÇµÉ ÞßÛÜÒîËäÐ ]ÉÈÔ×Ö«ÍËÇÚÉÄÍ ÎµÍÂÊÌÎ ÐåÐçËÂÉ³ÒÃÛÜØµÒîËÂÉ³ÐèÀÊÔ _ É³ÛñØÚåÏæ ÛñËÇÚè«ÍäÐ³æ·Ø ÔÜÍÂÊÞ
0
Ù³ÛñÍÙ³Öµá×ÐçËÂÛÜÊØ ^ºÐåÐçÛÜØ¤íôìr.d
ÊÖóåÉËÈËÂÊ ÇÚÐØµèµáÜÉ ËÂÇµ
É _ Öµå¿ÍÂÉÎ ÊÌÍËÒ `Äé
ÊÖ ËÇÚÛñaØ ]âÐ _ ÊÖ«Ë ÇÚÊ¦æ _ÊÌËÂÇàÛÜÒÒÖµÉ³Ò ÙÊÖÚáñZ
è _ É¹Ðèµè«ÍÂÉ³ÒÒÉè¿ÛñØ ËÇÚÉØÚGÉ P¥Ë ÍÂÉáÜÉ¬ÐÒÉïÊÔ¥ËÇÚÉ ÍÂÐçËÂÛÜÊØÚÐá
ØôÖµ
Þ _ ÉÍÂÒ®áÜÛ _ÚÍÂÐçÍìíòÐØµè ìÊÖ Ù³ÊÞßÉ Ö«Î æ·ÛñËÂÇ ËÂÇµÉ½Ô×ÊáÜáñÊÑæ·ÛÜØµå ÒÊáÜÖ«ËÂÛÜÊØ Ò ÐØµÊÌËÂÇµÉÍ®ÎÚÛÜÉ³Ù³É ÊÔ
Ô×ÖµØµÙËÂÛÜÊØÚÐáÜÛ²Ë ì ÊØÝËÇÚÉ¿Ë ìôÎ É
í¡ìÊÖ èÚIÉ Y ØµÉ Ð Ô×ÖµØµÙËÂÛÜÊØ ËÇ ÐÑËëÙÍÂÉ³ÐçËÂÉ³Òà
Ð UÌÐçáÜÖµÉÊÔ Ë ìôÎÉ
rational
ÔÜÍÂÊÞ Ë æ¹Ê UÐáÜÖµÉ³ÒÈÊÔ Ë ìôÎ É
é
rational
int
// PRE : d != 0
// POST : return value is n/d
rational create_rational ( int n , int d ) {
// somehow check here that d != 0
rational result ;
result .n = n ;
result .d = d ;
return result ;
}
ÊÖóËÂÇµÉ³Ø Ðè U¥ÛñÒÉ ÃÆ ËÂÊßÖµÒÉãËÂÇµÛÜÒ·Ô×ÖµØµÙËÂÛÜÊØóæ·ÇµÉ³ØµÉ UÉÍ ËÂÇµÉì æÈÐØ ËòËÂÊÀÛÜØµÛñËÛ×ÐáÜÛÉ ÊÌÍ ÐÒÒîÛÜåØóËÂÊ
ÐàÍäÐçËÂÛÜÊØÚÐáØôÖµÞb_ÉÍ¬é [ ÊÌÍ·ÉIP«ÐÞÀÎÚáÜÉÌí
rational r = create_rational (1 , 2);
æ¹ÊÖµáÜèÛñØÚÛ²ËÂÛ×ÐáÜÛ³É æ ÛñËÂÇ
ÛÜØÊØµÉÏåÊ«íµÐØµèóÐçËÈËÂÇµÉÓÒÂÐÞßÉ ËÂÛÜÞßÉ Þ®Ð ]ÉÓÒÖ«ÍÂÉÓËÂÇÚÐçË¹ËÂÇµÉÏèÚÉØÚÊÞßÛ r
1/2
ØÚÐçËÂÊÌÍÈÛÜÒÈØµÊØ ÉÍÂÊ«é
" ÖµÙ´ÇëÐ¹ÙÄÍÂÉ¬ÐçËÂÛñÊØâÔ×ÖµØµÙËÂÛÜÊØÏÙ³ÉÍËÂÐÛÜØµáñì Þ®Ð ]ÉÒ¾ÒîÉ³ØµÒÉÕÔ×ÊÌÍ¡ÒîËÍÖÚÙÄËÂÒ¡ÛñØãåÉ³ØµÉÍÂÐáí_ Ö«Ë ËÂÇµÉ Ë æ¹ÊòÛÜÒîÒÖµÉ³Ò
Ð _ Ê UÉ èµÊ
Ø ËÏÍÂÉ¬ÐçáÜáñì åÊóÐ³æÈÐ³ìéßÆ¹ÇÚÉ¿ÍÂÉ¬ÐÒîÊØ ÛÜÒãËÂÇÚÐçËÏËÂÇµÛÜÒÏÒÂÐÔ×ÉÀÙÄÍÂÉ¬ÐçËÂÛñÊØÝÙ¬ÐØ _ ÉÙ³ÛñÍÂÙÖÚÞUÉ³Ø¯ËÂÉ³è
_ôì ØµÊÌËÓÖµÒÛÜØµå ÛñË¬éëêØ½ÔªÐÙË¬í ìÊÖ«ÍãÐè U¥ÛÜÙÉ¿ÞßÛñåÇ Ë ØµÊÌËÏÇÚKÐ UÉàÍÂÉ³ÐÙäÇÚÉè ËÂÇµÉ¿ÎµÍÂÊåÌÍÂÐÞßÞßÉÍ ÐÑË ÃÆÏí
ÐØµèßÉ UÉ³Ø ÛÜÔ ÛñËÕèµÛÜè¤í ËÂÇµÉÈÎÚÍÊåÌÍäÐÞßÞßÉÍÃÞßÛÜåÇ¯
Ë _É·ËÂÊôÊâá×Ð ì ^ÊÌÍ ËÂÊ ÊëÒîËÂÖ _a_ ÊÌÍÂØ ` ËÂÊëÔ×ÊáÜáÜÊ¦æ¼ÛñË¬é ê»ËÛÜÒ
ËÂÇµÉÍÂÉÔ×ÊÌÍÂÉÓÒîËÂÛÜáÜáÚÎ ÊÒÒÛ _ áñÉòËÂÊàæ·ÍÛñËÂÉ
ÐØµè®Ô×ÊÌÍÂåÉËòÐ _ÊÖ«ËïèÚÐçËäÐâÞßÉ³
Þ _ ÉÍÐÒÒîÛÜåØµÞßÉ³Ø Ë¬í
rational r;
\ ÐØµè ÛñËïÛÜÒïÒîËÛÜáÜáÎ ÊÒîÒÛV_ÚáÜÉ·ËÂÊ¿ÐÒîÒÛÜåØ ËÂÊ
é
É³ÇµÛÜØµè ËÇÚÛñÒ áÜÛÜÉ³ÒïÛñØ ÔªÐÙÄËÈÐëÞÖµÙäÇ áÜÐçÍÂåÉÍïÎµÍÂÊ _ÚáÜÉ³Þ í
0
r.d
ÐÒ¹ìÊÖ èµÛÜÒÙ³ÊUÉÍòØÚÉGP¥Ë¬é
ÜÔ ËÂÉÍóÇ XÐ UôÛÜØµå ÖµÒÉ³è ËÂÇµÉ ÍäÐçËÛÜÊØÚÐá
Issue 3: The internal representation cannot be changed.
ØôÖµÞ_ ÉÍÂÒïáñÛV_µÍäÐçÍì Ô×ÊÌÍ·ÒÊÞßÉ ËÛÜÞßÉÌí Ã
Æ ÐçÎµÎµÍÂÊÐÙäÇµÉ³ÒïìÊÖ æ·ÛñËÂÇ â
Ð ÍÂÉXW ÖÚÉ ÒîË·Ô×ÊÌÍòÐZUÉÍÂÒîÛÜÊØ æ·ÛñËÂÇ Ð
á×ÐçÍåÉÍTUÐáÜÖµÉÍäÐçØÚåÉíÌÒÛÜØµÙ³ÉËÇÚÉÄìàÇÚÐKUÉÊ _ ÒîÉÍ UÉ³è ËÂÇÚÐçË ÛÜØ¯ËÂÉÍÂÞßÉèÚÛÜÐçËÂÉ UÌÐçáÜÖµÉ³Ò ÒÊÞßÉËÛÜÞßÉ³Ò¤Ê UÉÄÍ QÚÊÑæâé
ÊÖÍÂÉ³Ù³ÐáÜáôËÂÇµÉ¹Ë ìôÎ É
ÔÜÍÂÊÞ ÐåÉ \ ÐØµèÍÂÉ³ÐáÜÛ³É ËÇ ÐÑËÕÊØµÉïËÂÇµÛÜØµå ìÊÖ Ù³ÊÖµáÜè
extended_int
É¬ÐÒîÛÜáñì èµÊãÛñÒÃËÂÊÏÙäÇÚÐØµåÉÈËÂÇµÉ¹Ë ìôÎ ÉÈÊÔØ ÖÚÞßÉÄÍäÐçËÂÊÌÍÕÐØµè èµÉ³ØµÊÞßÛÜØÚÐçËÊÌÍ ÔÜÍÂÊÞ
ËÂÊ
ÐØµè ÒîËÂÊÌÍÂÉ ËÂÇµÉ ÒÛñåØ ÊÔ ËÂÇµÉ ÍäÐÑËÂÛÜÊØÚÐáÚØôÖµb
Þ _ÉÍïÒÉÎ ÐçÍÂÐçËÂÉ³áñìßÐÒ¹ÐâèÚÐçËäÐâÞßÉÞbint
_ ÉÄÍ ÊÔ Ëunsigned
ìôÎ É
é int
Ô×ÊÌÍ
bool
IÉ P«ÐÞÀÎÚáÜÉ áÜVÛ ]ÉÓËÂÇµÛÜÒ
struct rational {
unsigned int n ;
// absolute value of numerator
unsigned int d ;
// absolute value of denominator
bool is_negative; // sign of the rational number
};
»ê ËòÛÜÒ·ÐçáÜÒÊ¿ØÚÊÌËÈËÂÊ ÊÇÚÐçÍÂè ËÂÊÍÂÉÄæ·ÍÂÛñËÉÓËÇÚÉÏáÜÛV_µÍäÐÑÍì YÚáÜÉ³Ò
ÐØµè
ËÂÊÍÉ QÚÉ³ÙË
rational.h
rational.C
ËÂÇµÛÜÒ¹ÙäÇÚÐØµåÉâÛÜØ ÍÂÉÎµÍÂÉ³ÒÉØ ËäÐçËÛÜÊØ¤é
Ö«Ë ÒÇµÊÌÍËÂá²ìÀÐÔÜËÉÍ ìÊÖßÇÚÐXUÉ ÒÇµÛñÎµÎ É³èÀËÂÇµÉòØÚÉÄæ UÉÍÒÛÜÊØ®ÊÔìÊÖ«ÍïáÜÛV_µÍäÐÑÍì ËÂÊ ÃÆ ^»ìÊÖ®ÇÚÐXUÉ
É UÉ³ØÛÜØµÙ³áÜÖµèµÉ³è ËÂÇµÉ®ÒÂÐÔ×É ÙÍÂÉ³ÐçËÂÛÜÊØ Ô×ÖÚØµÙËÛÜÊØ
ÔÜÍÂÊÞ Ð _ ÊUÉ ÛÜØ ËÂÇµÉ®ÇÚÊÌÎÉ®ËÊ
create_rational
ÍÂÉ³ÒîÊVá UÉëÛÜÒÒîÖÚÉÒ ÐØµè \ Ð _ Ê UÉ `ÄíµìÊÖ ÍÂÉ³Ù³É³Û UÉ ÐØ ÐØµåÌÍì Î ÇµÊØµÉëÙ¬ÐçáÜá¡ÔÜÍÊÞ ËÂÇµÉâÞ®ÐØÚÐåÉÞßÉ³Ø ËÈÊÔ
ÃÆ ËÂÇµÉòÎµÍÂÊåÌÍäÐÞßÞßÉÄÍ ÍÂÉÎ ÊÌÍËÒËÂÇÚÐçËÐáñËÂÇµÊÖµåÇ ËÂÇµÉ ÐÑÎÚÎÚáÜÛñÙ¬ÐçËÂÛÜÊØ¿Ù³Ê èÚÉòÒîËÂÛÜáñáÚÙÊÞÀÎ ÛñáÜÉ³Ò æ ÛñËÇ ËÂÇµÉ
ØµÉæ UÉÍÂÒÛñÊØóÊÔ¡ËÂÇµÉãáÜVÛ _µÍäÐçÍìí
æ¹ÊÌÍ ]¥Ò·ÐØ ìôÞßÊÌÍÂÉ º¥»Á
ÔÜËÂÉÍ®ËäÐ ]¥ÛÜØµå Ð W ÖÚÛñÙ ] áÜÊôÊ ] ÐçËßËÂÇµÉ ÐçÎµÎÚáÜÛÜÙ¬ÐçËÛÜÊØ Ù³Ê èÚÉíÈìÊÖ¼ÒÖµèµèÚÉØÚá²ì ÍÂÉ¬ÐáñÛ ³Éæ·Ç ÐÑËÀËÂÇµÉ
ÎµÍÂÊ _ÚáÜÉ³Þ ÛñÒ ËÂÇµÉÓÙ³ÊôèµÉ ÛÜÒïÙ³áñÖµËîËÂÉÍÂÉ³è®Ö«Î æ ÛñËÂÇ®IÉ PôÎµÍÂÉ³ÒÒÛñÊØÚÒ¹ÊÔ ËÂÇµÉÓÔ×ÊÌÍÂÞ
ÐØµè
í«ÐÒ
.n
.d
ÛÜØ
rational r ;
r.n = 1;
r.d = 2;
ñá ÍÉ¬Ðè«ì®ËÂÇµÛÜÒÈÎ ÐçÍîËÂÛÜÙ³Öµá×ÐçÍ¹ÎÚÛÜÉ³ÙÉâÊÔ Ù³Ê èÚÉãèµÊôÉ³ÒòØÚÊÌËòæ¹ÊÌÍ ]ÐØ ìôÞßÊÌÍÂÉ ÕÐ¿ÍäÐçËÛÜÊØÚÐá¤ØôÖµÞb_ÉÍ·ÛñÒ·ØµÊÑæ
ÍÂÉÎµÍÂÉÒÉ³Ø ËÉ³è _ôì
èÚÐçËäÐÀÞßÉÞb_ ÉÄÍÂÒ³í _ÚÖ«ËãËÇÚÉ^Êáñè ` ÐÑÎÚÎÚáÜÛñÙ¬ÐçËÂÛÜÊØ ÙÊôèµÉÊ _ U¥ÛÜÊÖµÒá²ì èµÊôÉ³ÒÏØµÊÌË
Ä ÞßÉ³Þ_ ÉÍ¿ÊÔ Ë ìôÎ É
ÛÜØµÛñËÂÛÜÐáÜÛ ³ÉÀËÇÚÉ ^ØµºÉ¥æ
£³`à
é O Ê¦æðìÊÖ ÍÉ³åÌÍÂÉËßØµÊÌË¿ËÂÊÝÇÚÐXUÉÎµÍÂÊU¥ÛÜèµÉ³è ËÂÇµÉ
bool
Ô×ÖµØµÙËÂÛÜÊØÛÜØ ËÂÇµÉ
YµÍÂÒîËÈÎ áÜÐÙ³É «ÛÜØµèµÉ³É³è¤í«ËÂÇµÉÏÒîËäÐçËÂÉÞßÉ³Ø Ë
create_rational
rational r = create_rational (1 , 2);
æ¹ÊÖµáÜè ÒËÂÛÜáÜá¥æ¹ÊÌÍ ]íôÐçÒÒÖµÞßÛÜØµå ËÂÇÚÐçË ìÊÖÀÇÚÐKUÉòÙ³ÊÌÍÍÂÉÙËÂáñìÐèÚÐçÎµËÂÉ³èËÂÇµÉòèµÉ YÚØµÛñËÂÛÜÊØ¿ÊÔËÂÇµÉÈÔ×ÖµØÚÙÄËÂÛÜÊØ
ËÊ èµÉ¬Ðçá æ ÛñËÂÇ ËÂÇµÉàØµÉæ ÍÂÉÄÎÚÍÉ³ÒÉ³Ø¯ËäÐçËÂÛÜÊØ¤é Ö«Ë ËÂÇµÉàÎÚÍÊ _ áñÉ³Þ ÛÜÒÓÞ¿ÖÚÙäÇ ÞßÊÌÍÂÉ
create_rational
ÔªÐçÍ »ÍÂÉ¬ÐÙäÇµÛÜØµåàÐØµèÀÞ®ÐçØÚÛñÔ×É³ÒîËÂÒÃÛñËÂÒÉáÜÔÛÜØ É¬ÐÙäÇ®ÐØµèÀÉ UÉÍîìÀÊôÙÙ³Ö«ÍÍÂÉ³ØµÙ³É ÊÔ
ÊÌÍ
ÛÜØ ËÂÇµÉ
.n
.d
ÐçÎµÎÚáÜÛÜÙ¬ÐÑËÂÛÜÊØßÙ³Ê èÚÉí«ÒÛÜØµÙ³É ËÂÇµÉÓè ÐÑËäÐëÞßÉ³Þ_ ÉÍÂÒ ÇÚÐXUÉÓÙäÇ ÐçØÚåÉè ËÂÇµÉ³ÛñÍ¹ÞßÉ³ÐØµÛÜØµå ^»ËÂÇµÉì®ÞßÛÜåÇ¯Ë É UÉ³Ø
ÇÚXÐ UÉëÙäÇ ÐçØÚåÉèóËÂÇµÉ³ÛñÍòØ ÐçÞßÉ³Ò ` ÃÛÜØóáÜÉÄËËÂÛÜØµå ÃÆ ÐÙ³Ù³ÉÒÒ ØôÖµÞßÉÍÂÐçËÂÊÌÍòÐØµè èµÉ³ØµÊÞßÛÜØÚÐçËÂÊÌÍïËÂÇ«ÍÂÊÖµåÇ
èÚÐçËäÐïÞßÉ³
Þ _ ÉÍÂÒ ËÂÇÚÐçË¡ÐçÍÉÒîÎ ÉÙ³VÛ YÚÙÃËÊ Ð¹Ù³ÉÍËäÐçÛÜØãÍÂÉÄÎÚÍÉ³ÒÉ³Ø¯ËäÐçËÂÛÜÊØ¤íÑìÊÖàÐçÍÂÉÕØÚÊ¦æ Ù³ÊÞßÞßÛñËîËÂÉ³è ËÂÊòËÇ ÐÑË
ÍÂÉÎµÍÂÉÒÉ³Ø ËÂÐçËÂÛÜÊØ¤í ÐØµè ìÊÖ Ù¬Ð
Ø ËëÙ´ÇÚÐØµåÉÀÛ²Ëâæ Û²ËÂÇµÊÖµËâÐÒ ]ôÛÜØµå ÃÆ ËÂÊ ÙäÇÚÐØµåÉÀÛñËÂÒâÐçÎµÎ áñÛÜÙ¬ÐçËÂÛñÊØ
Ù³Ê èÚÉãÐÒ¹æ¹É³áÜ
á ^»æ·ÇÚÛñÙ´Ç ËÂÇµÉì®æ ÛÜáñáÍÂÉ³Ô×ÖµÒÉÌíÚÊÔ Ù³ÊÖ«ÍÂÒîÉ `Äé
\ S¼ÇÚÉØ ËÂÇµÉ
ÃÆ
Þ ÐØÚÐåÉ³ÞßÉ³Ø¯Ë·ÍÂÉ³ÐáÜÛ³ÉÒÏËÂÇÚÐçË ËÇÚÉ¿ØµÉæ ÍäÐçËÂÛñÊØ Ðçá ØôÖµÞ_ ÉÍÂÒ æ·ÛñËÂÇ ÉGP¥ËÂÉ³ØµèµÉ³è
®
UÐáÜÖµÉ ÍäÐØµåÉÓÐçÍÂÉ ÖµÒÉáÜÉ³ÒÒ¹Ô×ÊÌÍïËÂÇµÉ³Þ í¯ËÂÇµÉìÀËÂÉÍÞßÛÜØÚÐçËÂÉòËÇÚÉ ÙÊØ ËÍäÐçÙËïæ ÛñËÇÀìÊÖ¾é ÛÜÒÂÐÑÎÚÎÊÛÜØ ËÂÉè®ÐÒ
ìÊÖÐçÍÂÉÌíÌìÊÖ¿ÒîËÂÛÜáñá¯ÍÂÉ³ÐáÜÛ ³ÉÕËÂÇÚÐçË æ ÇÚÐçË ìÊÖØµÉ³É³è ËÂÊÓXÐ UÊÛÜè ÒÖµÙäÇËÍÊÖa_ÚáÜÉÒ ÛÜØàËÂÇµÉ Ô×Ö«ËÂÖ«ÍÂÉ¹ÛÜÒ
³ ÔÜ±´Í©¬Ê¢ Þ
Ð¿ÞßÉ³ÙäÇ ÐçØÚÛñÒÞ ËÂÇÚÐçË
ËÂÇµÉâÐçÙËÂÖÚÐá¤ÍÂÉÎµÍÂÉ³ÒÉØ ËäÐçËÛÜÊØóÊÔ ËÂÇµÉÏË ì Î É
ËÂÇµÉ
ª©¯ÙÖÚÒ ËÂÊÞßÉÍ¬íïÐçØÚè¼ÐçËËÂÇµÉÒÂÐçÞß
¥ É Fµ¬ËÂ ÛñÞßÉ Ê ÉÍÂÒ ËÂÇµÉ Ù³ÖµÒîËÂÊÞßÉÍ
rational
£³î¢¤£³¬ç³¤î©¯î »
Fµî¢
Fµ³¤
æÈÐ³ì¥ÒÈÊÔ æ¹ÊÌÍ ]ôÛÜØµå¿æ·ÛñËÂÇ®ÍäÐçËÂÛÜÊØÚÐá Ø Öµb
Þ _ ÉÄÍÂÒ³é
êØ í É³ØµÙ¬ÐÑÎ ÒîÖÚáÜÐçËÂÛÜÊØÝÛñÒëXÐ UÌÐçÛÜá×Ð _ÚáÜÉ ËÂÇ«ÍÂÊÖµåÇ ËÂÇµÉÀÖµÒÉÀÊÔ¹Ù³á×ÐÒîÒÉ³Ò³í ÐØµèÝæ¹ÉÀÒËäÐçÍZ
Ë _ ìÝIÉ P ÎÚá×ÐÛÜØµÛÜØµåàÇµÊ¦æ ËÊ ÇµÛÜèµÉ ËÂÇµÉÏÍÂÉÄÎÚÍÉ³ÒÉ³Ø¯ËäÐçËÂÛÜÊØÊÔ ÐàË ì Î ÉÏÔÜÍÂÊÞ ËÇÚÉãÙ³ÖµÒîËÂÊÞßÉÍ³é
4.3.2 Public and private
·ÉÍÉãÛÜÒòÐàÎÚÍÉ³áÜÛÜÞßÛÜØÚÐçÍîì
UÉÍÂÒÛñÊØóÊÔ
class
struct rational
ËÂÇÚÐçË¹ËäÐ ]ÉÒ·Ù¬ÐçÍÉãÊÔ èÚÐçËÂÐ ÇµÛÜèµÛÜØµå«é
class rational {
private :
int n;
int d ; // INV : d != 0
};
êØ ÂË ÇµÉ ÒÂÐçÞßÉÝæÈÐ¬ì ÐçÒ Ð
íàÐ
ÐååÌÍÂÉ³åÐçËÉ³Ò ÒÉ UÉÍÂÐá èÚÛ ÉÍÂÉØ Ë Ë ìôÎ É³Ò ÛÜØ¯ËÂÊ Ð
struct
class
ØµÉæ Ë ì Î ÉÌíR_ÚÖµËËÂÇµÉ
]Éì æ¹ÊÌÍÂè ÛÜØµèµÛÜÙ¬ÐçËÂÉÒËÂÇÚÐçË
Þ®Ð³ì ÊôÙ³Ù³Ö«Í¬íïÍÂÉ³Ðá µ
©

±
³

³
£
¬

¬
£

¬
±

class
Û³ÉèËÂÇ«ÍÂÊÖµåÇ ËÂÇµ
É ]Éìôæ¹ÊÌÍèÚÒ
ÐØµè
é
public
private
ðèÚÐçËÂÐ ÞßÉ³
Þ _ ÉÍ ÛñÒ
ÛÜÔ·ÐØµè ÊØµáñì ÛÜÔòÛñËÂÒ¿èÚÉÙ³á×ÐçÍäÐÑËÂÛÜÊØ ÐçÎµÎ É¬ÐçÍÂÒ¿ÒÊÞßÉæ·ÇµÉÍÂÉ ÐÔÜËÂÉÍ Ð
± ØÚÊ
ÒÎ É³Ù³VÛ YÚÉÍ³í ÐØµè ¢¤æ·¾ÛñËÂ ¦Ç½
ÒîÎ É³ÙÛVYÚÉÍâÛñØ _ ÉË æ¹É³ÉØ¾é ê»ËâÛñÒ
ÊÌËÂÇµÉÍæ ÛñÒÉÌé
¤
¢
¬
£

ô
D
¯
©

public:
private:
êØ Î ÐçÍËÛÜÙ³Öµá×ÐçÍ¬íÛÜÔÕËÂÇµÉ Ù³á×ÐçÒÒãèµÉ YÚØµÛñËÂÛñÊØ ^»ÒÉ³É " ÉÙËÂÛÜÊØ é é _ É³áñÊÑæ Ô×ÊÌÍãËÂÇµÉàÎÚÍÉ³Ù³ÛÜÒÉ¿ÞßÉ¬ÐØµÛÜØµå®ÊÔ
ËÂÇµÛÜÒâËÉÍÂÞ `ãÙ³ÊØ¯ËäÐÛÜØµÒëØµÊ
ÒîÎ É³Ù³Û Y ÉÄÍ¬í ÐáÜá èÚÐçËÂÐÞßÉ³
Þ _ ÉÍÂÒâÐçÍÂÉßÎµÍÂVÛ UÐçËÂÉ _ ì èÚÉÔªÐÖµáñË¬é êØ
public:
Ù³ÊØ¯ËÍäÐÒîË³í Ð ÒîËÍÖÚÙÄË·ÛÜÒÈÐ Ù³á×ÐÒîÒÈæ·ÇÚÉÄÍÂÉëÐáñá è ÐÑËäÐàÞßÉ³b
Þ _ÉÍÂÒ¹ÐçÍÂÉÏÎÚaÖ _ÚáÜÛñ
Ù _ôì èÚÉÔªÐÖµáñË¬é êÔ¾ÐâèÚÐçËäÐãÞßÉb
Þ _ ÉÄÍÛÜÒÕÎÚÍVÛ UÌÐÑËÂÉÌíôÛ²ËïÙ¬ÐØµØµÊÌË _ ÉÓÐçÙ³Ù³É³ÒÒîÉ³è _ ìßÙ³ÖµÒîËÂÊÞßÉÄÍÂÒËÂÇ«ÍÂÊÖµåÇÀËÂÇµÉ ÞßÉ³
Þ _ ÉÍ
ÐÙ³ÙÉ³ÒÒâÊÌÎÉÍäÐçËÂÊÌÍ³é êÔ ÐßèÚÐçËäÐßÞßÉ³
Þ _ ÉÍ ÛÜÒÓÎÚÖ _ áñÛÜÙÌíËÂÇµÉÍÂÉÐçÍÂÉ ØµÊÒîÖÚÙäÇ ÍÉ³ÒîËÍÂÛñÙËÂÛÜÊØµÒ³é ·ØµèµÉÍÏÊÖ«Í
Ð _ Ê UÉÏèÚIÉ Y ØµÛñËÛÜÊØ ÊÔ
íµËÇÚÉÏÔ×ÊáÜáÜÊ¦æ ÛñØÚåàæ·ÛÜáÜáËÂÇµÉÍÉ³Ô×ÊÌÍÂÉâØµÊÌËòÙ³ÊÞÀÎÚÛÜáÜÉ
class rational
rational r ;
r.n = 1;
// error : n is private
r.d = 2;
// error : d is private
int i = r. n ; // error : n is private
ê Ø ÎÐÑÍËÂÛÜÙ³Öµá×ÐÑÍ¬íÑËÇÚÉ¹ÐçÒÒÛÜåØµÞßÉ³Ø¯Ë
_ É³Ù³ÊÞßÉÒ ÛÜÞÀÎ ÊÒÒîÛV_ÚáÜÉ^°æ ÇµÛÜÙäÇàÛÜÒ åÊôÊ è `Äí _ÚÖ«ËÃÐçË Ð ^°ËÂÊôÊ `
r.d = 0
ÇµÛÜåÇ ÎµÍÂÛÜÙ³É ¿ØµÊÑæ ìÊÖ«Í Ù³ÖµÒîËÂÊÞßÉÍàÙ³ÐØµØÚÊÌËëèÚÊ ÐçØ ìôËÇÚÛñØÚåóæ·ÛñËÂÇ Ð ÍÂÐçËÂÛÜÊØÚÐáÕØôÖµÞb_ÉÍ¬í ÐØµè É UÉ³Ø
ìÊÖ®Ù¬ÐØµØµÊÌË ÛÜÞÀÎÚáÜÉÞßÉ³Ø Ë
íôÒÐ¬ìíôÐÒÕìÊÖ®ÖµÒÉ³èßËÂÊâèÚÊàÛ²ËÛÜØ ÍÂÊåÌÍäÐçÞ éS¼ÇÚÐçËæ¹É ÐÑÍÂÉ
operator+
ÒîËÂÛñáÜáµá×ÐÙ ]¥ÛÜØµåãÛÜÒÕÒîÊÞßÉÈæÈÐ³ì
ÊÔ¤ÐÙ³Ù³É³ÒîÒÛÜØµåâËÂÇµÉ·ÉØÚÙ³ÐçÎÚÒÖµá×ÐçËÂÉ³èÍÂÉÎµÍÂÉ³ÒÉØ ËäÐçËÛÜÊØ¤é ÆÈÇµÛÜÒÕÔ×ÖÚØµÙËÛÜÊØÚÐáÜÛñË ì
ÛÜÒ¹ÎµÍÂÊ UôÛÜèµÉ³è _ ìóÐ ÒÉ³ÙÊØÚèÙ¬ÐçËÂÉ³åÊÌÍîì ÊÔ Ù³á×ÐÒÒÈÞßÉ³
Þ _ ÉÍÂÒí¥ØÚÐÞßÉ³áñì
é
¨½³¨½ ³£ ¶Ä ±¬¤
2
*
*
2%,
R
A #$&.*6
2
,#7
7%*
-
J
M "
-7#"
N"KB? 7
N /C.*
.*
--
>/
% ,0
$
%*
9 2A.)
K$&.:
K
#6%"
N$&
7
(&
%*
%
A
7
N%+
J
K( K*SB
\ 4.3.3 Member functions
ÉËÃÖÚÒÃØµÊÑæ¼ÐçèÚèËÂÇµÉÈÞßÛñÒÒÛÜØµå Ô×ÖµØµÙËÂÛÜÊØÚÐáÜÛ²Ë ìãËÂÊ
ËÂÇ«ÍÂÊÖµåÇ ÞßÉ³Þ_ ÉÍ Ô×ÖÚØµÙËÛÜÊØµÒ³é
class rational
ê»Ëâæ¹ÊÖµáÜè ÒÉ³É³Þ ØÚÐçËÖµÍÂÐáÃËÊóÒîËäÐçÍîËâæ Û²ËÂÇÝÒÂÐÔ×É ÙÍÂÉ¬ÐçËÛÜÊØ¤í"_ÚÖ«ËëÒÛÜØµÙ³ÉËÂÇµÉÍÂÉßÐçÍÉ ÒîÎ ÉÙ³ÛVYÚÙ ÞßÉ³Þ_ ÉÍ
Ô×ÖµØµÙËÂÛÜÊØµÒàÍÂÉ³ÒÉÄÍ UÉ³è Ô×ÊÌÍ ËÂÇµÛÜÒàÎÚÖµÍîÎ ÊÒÉÌíÃáÜÉË ÖµÒYÚÍÒîË¿ÒÇµÊÑæ Ë æ¹ÊîåÉ³ØµÉÍäÐá½ÞßÉ³Þ_ ÉÍâÔ×ÖµØÚÙÄËÂÛÜÊØµÒ
ËÂÇÚÐçËåÌÍäÐØ¯Ë ÒÂÐçÔ×ÉóÐÙ³ÙÉ³ÒÒËÊ½ËÂÇµÉØ ÖµÞßÉÍäÐçËÂÊÌÍÐØµè èµÉ³ØµÊÞßÛÜØÚÐçËÊÌÍàÊÔ Ð ÍäÐçËÂÛÜÊØÚÐáïØ ÖµÞb_ ÉÄÍ ^°æ¹É áÜá
èµÛÜÒÙ³ÖµÒb
Ò _ É³áÜÊ¦æ æ ÇÚÐçË
ÐØµè
ÞßÉ¬ÐØ ÇµÉÍÂÉ ÐØµè ÛÜÔòìÊÖ æ¹ÊØÚèµÉÍæ Ç¯ì æ¹É Ù¬ÐØ ÖµÒÉ
ÐØµè
_ ÉÔ×ÊÌÍÂÉÀËÂÇµÉì *this
ÐÑÍÂÉßèµÉ³Ù³á×ÐçÍÉ³èconst
ËÇÚÛñÒëÛÜÒàÐÒÎ É³Ù³Û×ÐçáÔ×É¬ÐçËÖµÍÉßÊÔÈÙá×ÐÒÒëÒîÙ³ÊÌÎ ÉÌí IÉ PôÎ áÜÐÛÜØµÉ³è ÛÜØ
n
d
" É³ÙËÛÜÊØ é é `Äé
class rational {
public :
// POST : return value is the numerator of * this
int numerator () const
{
return n;
}
// POST : return value is the denominator of * this
int denominator () const
{
return d;
}
private :
int n;
int d ; // INV : d != 0
};
êÔ
r
ÛñÒ·Ð UÌÐÑÍÂÛ×Ð _ÚáÜÉÓÊÔ¡Ë ì Î É
rational
int n = r. numerator ();
int d = r. denominator ();
íµÔ×ÊÌÍ·ÉGPµÐÞÀÎÚáÜÉÌí ËÂÇµÉãÙ³ÖµÒîËÊÞßÉÍÈÙ¬ÐØËÂÇµÉ³Ø æ·ÍÂÛñËÉ
// get numerator of r
// get denominator of r
ÖµÒÛÜØµåãËÇÚÉòÞßÉ³Þ_ ÉÍÃÐÙ³ÙÉ³ÒÒ ÊÌÎ ÉÄÍäÐçËÂÊÌÍÐÒÔ×ÊÌÍèÚÐçËäÐÓÞßÉ³Þb_ÉÍÂÒ³é ÆÈÇµÉ·ÙÖÚÒËÂÊÞßÉÍÕÙ¬ÐØßÙ¬ÐáñáµËÇÚÉÒÉòË æ¹Ê
Ô×ÖµØµÙËÂÛÜÊØµÒ³íÒîÛÜØµÙ³ÉëËÂÇµÉì ÐçÍÂÉàèµÉ³Ùá×ÐçÍÂÉ³è
é Ù³ÙÉ³ÒÒÓÒîÎ É³ÙÛVYÚÉÍÂÒ ÇÚÐXUÉ ËÇÚÉàÒÐÞßÉëÞßÉ¬ÐØµÛÜØµåÀÔ×ÊÌÍ
public
ÞßÉ³
Þ _ ÉÍ¹Ô×ÖµØµÙËÂÛÜÊØµÒòÐÒÈÔ×ÊÌÍ èÚÐçËäÐ ÞßÉ³Þ_ ÉÍÂÒ ÐàÎµÍÂÛVUÐçËÂÉÏÞßÉ³Þ_ ÉÍ¹Ô×ÖµØµÙËÂÛÜÊØóÙ³ÐØµØÚÊÌË _ ÉãÙ¬ÐáÜáñÉ³è _ ì
ËÂÇµÉÃÙ³ÖµÒîËÂÊÞßÉÍ¬é ÆÈÇµÛÜÒ ]ôÛÜØµèëÊÔ¥ÐÙÙ³É³ÒÒ ËÊÈËÂÇµÉÃÍÂÉÎµÍÂÉ³ÒÉØ ËäÐçËÛÜÊØâÛÜÒ QÚIÉ P«Û _ áñÉÌíçÒîÛÜØµÙ³ÉÃËÇÚÉÃÙ³ÊÌÍÍÉ³ÒîÎ ÊØµèµÛÜØµå
ÞßÉ³
Þ _ ÉÍ Ô×ÖµØµÙËÂÛñÊØÚÒ Ù¬ÐçØ É¬ÐÒîÛÜáñZ
ì _ ÉÈÐèÚÐçÎµËÂÉèËÂÊÏÐ ØµÉæ ÍÂÉÎµÍÂÉ³ÒîÉ³Ø ËäÐÑËÂÛÜÊØ Û²Ë ÛÜÒÃÐáÜÒÊÓÒÂÐÔ×Éí ÒÛñØÚÙÉÈÛ²Ë ÛÜÒ
ØµÊÌËïÎ ÊÒÒÛ _ áñÉòËÂÊ ÙäÇÚÐØµåÉ ËÂÇµ
É UÌÐáñÖÚÉÒïÊÔ¾ËÂÇµÉ èÚÐçËäÐãÞßÉ³b
Þ _ÉÍÂÒÕËÇµÍÊÖÚåÇßËÂÇµÉ Ô×ÖµØµÙËÂÛÜÊØµÒ
ÐØµè
é ·Ò ÐåÉ³ØµÉÍäÐáÕÍÂÖµáÜÉÀÊÔïËÂÇôÖµ
Þ _¾í ÐáÜáÃèÚÐçËäÐÞßÉ³
Þ _ ÉÍÒâÊÔÈÐÙ³á×ÐçÒÒënumerator
ÒÇµÊÖµáÜè _ É
denominator
ÎµÍÂVÛ UÐçËÂÉ ^»ÊÌËÂÇµÉÍæ ÛñÒÉÌí ìÊÖ É³ØµÙ³ÊÖ«ÍäÐåÉ ËÇÚÉßÙÖÚÒËÂÊÞßÉÍâËÊ ÐÙ³Ù³É³ÒîÒ ËÇÚÉ è ÐÑËäÐÞßÉ³
Þ _ ÉÍÂÒí¡æ Û²ËÂÇÝËÂÇµÉ
Öµåáñì®Ù³ÊØµÒXÉ W ÖµÉ³ØµÙ³É³Ò ÞßÉ³Ø¯ËÂÛÜÊØµÉ³è ÛÜØ êÒÒîÖÚÉ Ð _ Ê UKÉ `Äé
êØëÊÌÍÂèµÉÍ¡ËÂÊ·Ù¬ÐçáÜáôÐ¹ÞßÉ³Þ_ ÉÍ¾Ô×ÖµØµÙËÂÛÜÊØ¤í¦æ¹ÉØµÉ³É³è ÐØàÉGP¥ÎµÍÂÉ³Ò ÒÛÜÊØ ÊÔ ËÂÇµÉëÙ³áÜÐÒÒ Ë ìôÎ ÉàÔ×ÊÌÍ æ·ÇµÛÜÙäÇóæ¹É
ËÂÇµÉàÔ×ÖµØµÙËÂÛÜÊØ¤íÐçØÚè ËÂÇµÛÜÒ·ÉIP¥ÎµÍÂÉ³ÒîÒÛÜÊØ ^ºÐçÎµÎ É¬ÐçÍÛÜØµå
µ
©

±
³

_ É³Ô×ÊÌÍÉÏËÂÇµÉ ` ÛÜÒÈÐØ
æ·ÇµÊÒ
É UÌÐáñÖÚÉÓÞ®Ð³ì®ÊÌÍÈÞ®Ð³ìßØµÊÌ
Ë _ÉãÞßÊ èÚÛ Y Éè_ ì
ËÂÇµÉÏÔ×ÖÚØµÙËÛÜÊ.ØóÙ¬Ðáñáé »¨¿¢¾±³Ó±Ä©¯°©£îÁ ¨ ³¤
The implicit call argument and *this.
\ S¼ÛñËÂÇµÛÜØóÉ¬ÐçÙ´Ç ÞßÉÞb_ ÉÄÍòÔ×ÖµØµÙËÂÛñÊØ¾í«ËÂÇµÉãáVUÌÐçáÜÖµÉ
ÍÂÉ³Ô×ÉÍÂÒòËÂÊ ËÂÇµÛÜÒÈÛÜÞÀÎÚáÜÛÜÙ³Û²Ë Ù¬Ðáñá ÐçÍåÖÚÞßÉØ Ë
*this
ÐØµèÀÉIôP ÎÚá×ÐÛÜØµÒ ËÂÇµÉ·ÐÑÎÚÎÉ¬ÐçÍäÐØµÙ³ÉÈÊÔ
ÛÜØ¿ËÂÇµÉÈÎ ÊÒËÂÙ³ÊØµµè ÛñËÂÛÜÊØµÒ ÊÔËÂÇµÉ¹Ë æ¹ÊâÞßÉ³Þ_ ÉÍ Ô×ÖµØµÙËÂÛñÊØÚÒ
*this
Ð _ ÊU ÉÌé ê»ËèµÊ É³Ò ØÚÊÌË ÉG¥P ÎÚá×ÐÛÜØ æ·Ç ì ÐØÐÒîËÂÉÍÛÜÒ ] ÐçÎµÎ É³ÐçÍÂÒ ÛÜØ
í_ Ö«Ëàæ¹Éßæ ÛñáÜá åÉËàËÂÊ ËÂÇµÛÜÒ
*this
á×ÐçËÉÍ¬é
]Éìôæ¹ÊÌÍÂè ÐçÔÜËÂÉÍâËÇÚÉ¿Ô×ÊÌÍÂÞ®ÐáÃÐçÍåÖÚÞßÉØ ËÓáÜÛÜÒËãÊÔ Ð ÞßÉ³Þ_ ÉÍ
Ô×ÖµØµÙËÂÛÜÊØÍÉ³Ô×ÉÍÂÒÈËÂÊËÂÇµÉÏÛÜÞÀÎÚáÜÛÜÙ³Û²ËïÐçÍÂåÖµÞßÉ³Ø¯Ë
ÐçØÚè ËÂÇµÉÍÂÉ³Ô×ÊÌÍÂÉÏÎµÍÂÊÞßÛÜÒÉ³ÒïËÂÇÚÐçË¹ËÂÇµÉÏÞßÉ³Þ_ ÉÍ
*this
Ô×ÖµØµÙËÂÛÜÊØ®Ù¬ÐáñáèµÊôÉ³ÒïØµÊÌËïÙäÇÚÐØµåÉ ËÂÇµÉÌU ÐçáÜÖµÉ »^ ÍÂÉÎµÍÂÉ³ÒîÉ³Ø ËÂÉè ô_ ì ËÂÇµÉÌU ÐáñÖÚÉÒïÊÔ¤ËÇÚÉ èÚÐçËÂÐâÞßÉ³Þ_ ÉÍÂÒ `
ÊÔ
éS ÉãÙ¬ÐçáÜá¤ÒÖµÙ´Ç Ð ÞßÉ³Þ_ ÉÍïÔ×ÖµØµÙËÂÛñÊØ Ð
é
Const member functions.
const
±Ä¤³·¨½³¨½ ³£ ¶ ±¬î
*this
Member function call.
Æ¹ÇÚÉÏåÉ³ØµÉÍäÐçá¡ÒîìôØ ËäÐPÊÔ Ð¿ÞßÉÞb_ ÉÄÍ¹Ô×ÖÚØµÙËÛÜÊØóÙ¬Ðáñá ÛÜÒ
é ^ í¤éñé²éñí `
·ÉÍÉÌí ÛñÒ ÐØàÉGP¥ÎµÍÂÉ³ÒÒîÛÜÊØàÊÔ«Ð¹Ù³á×ÐÒÒ¾Ë ìôÎ ÉÕÔ×ÊÌÍ æ ÇµÛÜÙäÇàÐÈÞßÉ³Þ_ ÉÍ¤Ô×ÖµØµÙËÂÛÜÊØàÙ¬ÐçáÜáÜÉ³è ÛÜÒãèÚÉÙ³á×ÐçÍÂÉè¾í íñé³éé³í ÐçÍÂÉËÇÚÉ Ù¬ÐáÜáÕÐçÍÂåÖµÞßÉ³Ø¯ËÂÒ³í ÐçØÚè
ÛñÒãËÂÇµÉ ÞßÉÞb_ ÉÄÍ
.
ÐÙ³ÙÉ³ÒÒ®ÊÌÎ ÉÍäÐÑËÂÊÌÍ¬é êØ¼ÞßÊÒîËÀÙ³ÐÒÉ³Ò³í ÛÜÒ®ÐØ áVUÐáÜÖµÉóÊÔ ËÂÇµÉ Ù³á×ÐçÒÒÀË ìôÎ ÉÌí¹Ë ìôÎÚÛÜÙ¬ÐçáÜáñì Ð
UÐçÍÂÛ×Ð _ÚáÜÉé
S Û²ËÂÇµÛÜØ ËÂÇµÉ_ Ê è«ì Ê Ô¤ÐâÞßÉÞb_ ÉÄÍÕÔ×ÖµØµÙËÂÛñÊØ ÊÔ¤Ð
Ù³á×ÐçÒÒ³í«ÐØ¯ìßÞßÉ³Þ_ ÉÍ^èÚÐçËäÐãÞßÉÞb_ ÉÄÍ ÊÔ¤ÞßÉ³Þ_ ÉÍ Ô×ÖµØµÙËÂÛñÊØ Ã
` ÊÔ¤ËÇÚÉ Ò ÐÞßÉ Ù³á×ÐçÒÒïÙ¬ÐØ_ ÉÓÐÙ³ÙfÉ³ÒÒÉ³è
æ Û²ËÂÇµÊÖµË ÐßÎµÍÂÉIYaP
¤ÛÜØ ËÂÇµÛÜÒÓÙ¬ÐÒîÉÌí æ¹É¿ÛñÞÀÎ áñÛÜÙ³ÛñËÂá²ì ÐÙ³Ù³É³ÒîÒãÛñËÓÔ×ÊÌÍ
éëêØ½ÊÖ«ÍÏÉIµP ÐçÞÀÎ áñÉÌí
.
*this
ËÂÇµÉ®ÉIP¥ÎµÍÂÉÒÒÛÜÊØ
ÛÜØ ËÂÇµÉßÍÂÉËÂÖ«ÍÂØ ÒîËäÐçËÉ³ÞßÉ³Ø ËëÊÔÈËÂÇµÉ®ÞßÉ³Þ_ ÉÍàÔ×ÖµØµÙËÂÛñÊØ
ÍÂÉ³Ô×ÉÍÂÒ ËÂÊ
n
numerator
ËÂÇµÉ èÚÐçËäÐÏÞßÉ³Þb_ÉÍ ÊÔ
éÃÆÈÇµÉ Ù¬ÐçáÜá
ËÂÇµÉÍÉ³Ô×ÊÌÍÂÉ èµÊôÉ³Ò æ·ÇÚÐçËæ¹É ÉIôP Î É³ÙË ÃÛ²Ë
ÍÂÉËÖµÍØÚÒ¹ËÂÇµÉâØ ÖµÞßÉÍänÐçËÂÊÌÍ¹*this
ÊÔ¾ËÂÇµÉãÍäÐÑËÂÛÜÊØÚÐáØ r.numerator()
ÖÚÞ_ ÉÍ é
S¼ÛñËÂÇµÛÜØ ÞßÉÞb_ ÉÄÍÓÔ×ÖÚØµÙËÛÜÊØµÒ³í æ¹É Ù¬ÐØÝÐáÜÒÊ ÐÙ³Ù³rÉÒÒâÞßÉ³Þ_ ÉÍÒ Ô×ÊÌÍëÊÌËÂÇµÉÍãÉI¥P ÎµÍÂÉÒÒÛÜÊØµÒâÊÔÃËÂÇµÉ
ÒÂÐÞßÉÏÙ³á×ÐçÒÒòË ìôÎÉãËÂÇ«ÍÂÊÖµåÇóËÂÇµÉâÞßÉÞb_ ÉÄÍ·ÐçÙ³Ù³É³ÒÒ·ÊÌÎ ÉÍäÐçËÊÌÍ ^ áñÛV] ÉâÐ¿Ù³ÖµÒîËÂÊÞßÉÍÈæ¹ÊÖµáÜèóèµÊÀÛñË Ä` é áÜá
ÐÙ³ÙÉ³ÒÒÉ³ÒÓËÂÊßÙ³áÜÐÒÒ ÞßÉ³Þ_ ÉÍÂÒÈæ ÛñËÇÚÛñØ ÞßÉ³Þb
_ ÉÍ Ô×ÖµØµÙËÂÛÜÊØµÒ ÊÔ ËÂÇµÉàÒÂÐÞßÉâÙ³á×ÐÒÒ ÐçÍÂÉ
í

£
¬

¬
£
î

³
±

F
ÍÂÉ³åÐÑÍÂèµáÜÉ³ÒÒ ÊÔòæ·ÇµÉËÂÇµÉÍ ËÂÇµÉ®ÞßÉ³Þb
_ ÉÍàÛÜØôW ÖµÉ³ÒîËÂÛñÊØÛÜÒàÎÚÖ _ áñÛÜÙÀÊÌÍ¿ÎÚÍÛVÌU ÐÑËÂÉÌé Æ¹ÇµÉ
ÐØµè
ÒîÎ ÉÙ³ÛVÚY ÉÍÂÒòÐçÍÂÉÏÊØµáñì®ÍÂÉ³áÜÉ U ÐØ¯Ë·Ô×ÊÌÍòËÂÇµÉãÙ³ÖµÒîËÊÞßÉÍ¬íÚ_ Ö«Ë·ØµÊÌËòÔ×ÊÌÍ ÞßÉ³Þ_ ÉÍ¹public:
Ô×ÖµØµÙËÂÛÜÊØµÒÈÊÔ
Access to members within member functions.
private:
ËÂÇµÉÏÙ³á×ÐÒÒÈÛñËÂÒîÉ³áÜÔºé
)É³
Þ _ ÉÍòÔ×ÖÚØµÙËÛÜÊØµÒ·ÐçÍÉãÒÊÞßÉÄËÂÛÜÞßÉ³ÒïÐçáÜÒÊ ÍÂÉ³Ô×ÉÍÍÉ³è ËÂÊ ÐçÒ
¨ ³º F¯
ÊÔ¡ËÂÇµÉÏÙ³á×ÐÒÒé
ê ØóËÂÇµÉâÒîÎÚÛñÍÛñËÈÊÔ " É³ÙËÂÛÜÊØ é é íµÛñËÈæ¹ÊÖµáÜè _ÉâÖµÒÉ³Ô×Öµá¡ËÊ
ÒÊÖ«ÍÂÙ³ÉòÊÖ«ËËÂÇµÉÈÞßÉ³Þb_ÉÍÃÔ×ÖÚØµÙËÛÜÊØÀèµÉ YÚØµÛñËÂÛÜÊØµÒ³íÛÜØ ÊÌÍÂèµÉÍËÂÊâÐáÜáÜÊ¦æ ÒÉÎ ÐçÍÂÐçËÂÉ·ÙÊÞÀÎ Ûñá×ÐçËÂÛÜÊØ¤é¾Æ¹ÇÚÛñÒ
æ¹ÊÌÍ ]ôÒëáÜÛV]É¿Ô×ÊÌÍàÊÌÍÂèµÛÜØÚÐçÍì½Ô×ÖµØµÙËÂÛÜÊØµÒ³í¡ÉIP«Ù³ÉÄÎÚËâËÂÇÚÐçËëÛÜØ Ð ÞßÉ³
Þ _ ÉÍãÔ×ÖÚØµÙËÛÜÊØ èµÉ YÚØÚÛ²ËÂÛÜÊØ ÊÖ«ËÂÒîÛÜèµÉ
ÊÔ ËÂÇµÉóÙ³áÜÐÒÒ èµÉ YÚØµÛñËÂÛñÊØ¾íÃËÂÇµÉÔ×ÖµØµÙËÂÛÜÊØ ØÚÐÞßÉ ÞÖµÒîb
Ë _ É W ÖÚÐáÜVÛ YÚÉ³è æ ÛñËÂÇ ËÂÇµÉóÙ³á×ÐÒîÒ ØÚÐÞßÉÌé êØ
ËÂÇµÉ ÇµÉ¬ÐèµÉÍ Y áñÉ
æ¹É æ¹ÊÖµáÜè ËÂÇµÉ³Ø æ·ÍÂÛñËÉàÊØÚá²ì ËÂÇµÉ¿èµÉ³Ùá×ÐçÍäÐçËÛÜÊØµZ
Ò ^ºÐÒÓÖµÒÖÚÐá æ·ÛñËÂÇµÛÜØ
rational.h
ØÚÐÞßÉ³ÒîÎ ÐÙÉ
`
Member functions and modularization.
ifm
class rational {
public :
// POST : return value is the numerator of * this
\ \
int numerator () const ;
// POST : return value is the denominator of * this
int denominator () const ;
private :
int n;
int d ; // INV : d != 0
};
Æ¹ÇÚÉ Þ®ÐçËÙ´ÇµÛÜØµåßèµÉ YÚØµÛñËÂÛÜÊØµÒ·æ¹ÊÖµáÜè ËÂÇµÉ³Ø ÐçÎµÎ É¬ÐçÍÏÛÜØ ËÂÇµÉ Ò ÊÖ«ÍÂÙ³É¿Ù³ÊôèµÉ YÚáÜÉ
^ºÐçåÐÛÜØ
rational.C
æ Û²ËÂÇµÛÜØ ØÚÐÞßÉÒîÎ ÐÙ³É
íÐØµèÐÔÜËÂÉÍ·ÛÜØµÙ³áÜÖµèµÛÜØµå
` ÐÒòÔ×ÊáñáÜÊÑæ·Ò³é
¹
ifm
rational.h
int rational :: numerator () const
{
return n ;
}
int rational :: denominator () const
{
return d ;
}
4.3.4 Constructors
Ù³ÊØµÒîËÍÂÖµÙËÂÊÌÍ ÛÜÒ¡ÐòÒîÎ ÉÙ³Û×ÐáÞßÉ³Þ_ ÉÍ¾Ô×ÖµØµÙËÂÛÜÊØâËÇ ÐÑË¡ÎµÍÂÊU¥ÛÜèµÉ³Ò¾ÒÂÐÔ×ÉÛÜØµÛñËÂÛ×ÐáñÛ¬ÐçËÂÛñÊØ ÊÔ¥Ù³á×ÐçÒÒ UÐáÜÖµÉ³Òé
Æ¹ÇÚÉÓØÚÐÞßÉ ÊÔ¡ÐëÙ³ÊØµÒîËÍÖÚÙÄËÂÊÌÍÈÙ³ÊÛÜØµÙ³ÛÜèµÉ³Òæ·ÛñËÂÇßËÂÇµÉÓØÚÐÞßÉ ÊÔ¤ËÂÇµÉÏÙá×ÐÒÒ³í¥ÐØµè#¼ËÇÚÛñÒ èµÛÜÒîËÂÛÜØµåÖµÛÜÒÇµÉ³Ò
Ù³ÊØµÒîËÍÖÚÙÄËÂÊÌÍÂÒÈÔÜÍÂÊÞðÊÌËÇÚÉÄÍòÔ×ÖµØµÙËÂÛñÊØÚÒ # ÛñË¹èµÊôÉ³ÒÈØµÊÌË·ÇÚXÐ UÉâÐàÍÂÉËÖµÍØ Ë ìôÎ Éí ÐçØÚè Ù³ÊØµÒÉXW ÖÚÉØ ËÂáñì ØµÊ
ÍÂÉËÖµÍb
Ø UÐáÜÖµÉÌé Ù³áÜÐÒÒ ÖµÒÖÚÐáÜáñìâÇÚÐÒ ÒÉ UÉÍäÐçá«Ù³ÊØµÒîËÍÂÖµÙËÂÊÌÍÒ³í¯ÐØµè¿ËÂÇµÉ¹Ù³ÊÞÀÎÚÛÜáÜÉÄ"
Í YÚåÖ«ÍÂÉ³Ò ÊÖ«Ë æ ÇµÛÜÙäÇ
ÊØµÉ·ÛÜÒÕÞßÉ¬ÐØ¯ËÕÛñØ®ÐÏåVÛ UÉØßÙ³ÊØ ËÂGÉ P¥
Ë ^»ÖÚÒîÛÜØµåãËÂÇµÉòÍÖÚáñÉ³ÒÊÔ Ê UÉÍÂáÜÊÐçèÚÛñØÚåÓÍÂÉ³ÒîÊáÜÖ«ËÂÛÜÊØ¤íÒÉ³É·ËÂÇµÉ ÓÉÄËäÐÛÜáÜÒ
ÊÔ " ÉÙËÂÛÜÊØ é `Äé
ÆÈÇµÉãÒîì¥Ø¯ËäÐ P ÊÔ Ð Ù³ÊØµÒîËÍÂÖµÙËÊÌÍ·èµÉ YÚØµÛñËÂÛÜÊØ Ô×ÊÌÍ Ð Ù³á×ÐçÒÒ ÛÜÒòÐÒÈÔ×ÊáÜáÜÊ¦æ Ò³é
^
·ÉÍÉÌí
í í¤éñéñé²í I `
^ `´í¾é²éñé²í ^ íñé´é¬é³í
³£
^ í²éÄé¬é³í G
`Äí¤é²éñéñí
`
ÐçÍÂÉÓËÂÇµÉâÔ×ÊÌÍÂÞ®Ðçá¤ÐçÍÂåÖµÞßÉ³Ø ËÒïÊÔ ËÂÇµÉãÙÊØÚÒËÍÂÖµÙËÂÊÌÍ¬éÕêØËÂÇµÉ
^
`
ÉÍÒ³í¥ÐØµè » »©¯
ÐÑÍÂÉãèÚÐçËÂÐ¿ÞßÉ³
Þ _
íñé³é³éí
ÐçÍÂÉÏÉIP¥ÎµÍÂÉ³ÒîÒÛÜÊØµÒ
Ê Ô·Ë ìôÎÉ³Òæ·ÇÚÊÒîÉ UÐáÜÖµÉ³ÒÙ¬ÐØ _ É Ù³ÊØ UÉÄÍËÂÉ³è ËÂÊ ËÂÇµÉ ÍÂÉ³ÒÎ É³ÙËÂÛ UÉèÚÐçËäÐ ÞßÉ³Þ_ ÉÍ Ë ìôÎ ÉÒ³é Æ¹ÇµÉ³ÒÉ
UÐáÜÖµÉ³Ò ÐÑÍÂÉòÖµÒÉèÀËÂÊâÛÜØµÛñËÛ×ÐáÜÛÉËÂÇµÉ·èÚÐçËäÐÓÞßÉ³Þ_ ÉÍÂÒí _ É³Ô×ÊÌÍÂÉ
ÛñÒÕÉIP¥É³Ù³Ö«ËÂÉ³è¤í«ÐçØÚè ÛÜØËÂÇµÉòÊÌÍÂèµÉÍ
ÛÜØëæ·ÇÚÛñÙ´ÇãËÂÇµÉ ÞßÉ³Þ_ ÉÍÂÒ¡ÐçÍÂÉÕèÚÉÙ³á×ÐçÍÂÉèàÛÜØëËÂÇµÉ Ù³á×ÐçÒÒ³é êØàÊÌËÂÇµÉÍ¡æ¹ÊÌÍÂèµÒ³íËÂÇµÉ ÊÌÍèÚÉÄÍ ÛÜØëËÂÇµÉ ÛñØÚÛ²ËÂÛ×ÐáÜÛ³ÉÍ
ÛÜÒëÛñåØÚÊÌÍÉ³è¤T
í _ÚÖµËëÛñËëÛÜÒâåÊôÊôè ÎµÍäÐÙÄËÂÛÜÙ³É¿ËÂÊ ÖµÒÉÀËÂÇµÉßèµÉ³Ù³á×ÐçÍÂÐçËÂÛÜÊØÝÊÌÍèÚÉÄÍ ÇÚÉÄÍÂÉßÐÒëæ¹É³áñáéêÔÈÐèÚÐçËÂÐ
\ ÞßÉ³Þ_ É Í·ÛÜÒ ØµÊÌËÓáÜÛÜÒîËÂÉè ÛñØ ËÂÇµÉàÛÜØµÛñËÛ×ÐáÜÛÉÍ¬íÚÛñË ÛÜÒ èµÉ³ÔªÐÖµáñË ºÛÜØµÛñËÂÛÜÐáÜÛ³Éè¾é·êØ ËÇÚÉàÙÊØÚÒËÍÂÖµÙËÂÊÌÍ
_ Ê èµì
í¥æ¹ÉãÙ¬ÐçØ ÒîËÂÛÜáÜáÒÉË ÊÌÍòÙäÇÚÐØµåÉÏËÂÇµÉ UÌÐáñÖÚÉÒòÊÔ¡ÒÊÞßÉÓÊÔ ËÂÇµÉãèÚÐçËÂÐ ÞßÉÞb_ ÉÄÍÂÒ³é
[ ÊÌÍ ËÂÇµÉâË ìôÎ É
í ÇµÉÍÂÉàÛÜÒ·Ð Ù³ÊØµÒîËÍÂÖµÙËÊÌÍ ËÂÇÚÐçË ÛñØÚÛ²ËÂÛ×ÐáÜÛ³É³Ò¹ÐÍäÐÑËÂÛÜÊØÚÐá¤ØôÖµÞ_ ÉÍ ÔÜÍÊÞ
rational
Ë æ¹Ê ÛÜØ¯ËÂÉ³åÉÍÒ³é
// PRE : d != 0
// POST : * this is initialized with numerator / denominator
rational ( int numerator , int denominator)
: n ( numerator) , d ( denominator)
{
// somehow check that d != 0
}
Æ ÊÖÚÒîÉÏËÂÇµÛÜÒÈÙ³ÊØµÒîËÍÖÚÙÄËÂÊÌÍ·ÛñØóÐ UÌÐçÍÛ×Ð _ÚáÜÉÓèµÉ³Ù³á×ÐçÍÂÐçËÂÛÜÊØ¤íôæ¹ÉÓæ¹ÊÖµáÜèÔ×ÊÌÍòÉIPµÐçÞÀÎ áñÉ æ·ÍÂÛñËÉ
rational r (1 ,2); // initializes r with value 1/2
êØóåÉ³ØµÉÍäÐçáí¥ËÂÇµÉâèµÉ³Ù³áÜÐçÍäÐçËÂÛñÊØ
^ í¾é²éñé²í
`
èµÉ YÚØµÉ³Ò ÐUÐçÍÂÛÜÐ _ÚáÜÉ ÊÔôË ì Î É ÐØµè ÐçË¡ËÂÇµÉ ÒÂÐÞßÉ ËÂÛÜÞßÉÃÛÜØµÛñËÂÛÜÐáÜÛ³ÉÒ ÛñË"_ ìëÙ¬ÐçáÜáÜÛÜØµå¹ËÂÇµÉïÐçÎµÎµÍÂÊÌÎµÍÂÛ×ÐçËÂÉ
Ù³ÊØµÒîËÍÖÚÙÄËÂÊÌÍòæ·ÛñËÂÇ Ù¬ÐáÜá¤ÐçÍåÖÚÞßÉØ ËÂÒ
í²é³é³é³í
é
ÆÈÇµÉãÙ³ÊØµÒîËÍÂÖµÙËÂÊÌÍòÙ¬ÐØóÐáÜÒÊ_ ÉÏÙ¬ÐáÜáÜÉèÉIPôÎ áñÛÜÙ³ÛñËÂá²ì®ÐÒÈÛÜØ
rational r = rational (1 , 2);
Æ¹ÇÚÛñÒ ÛÜØµÛñËÂÛ×ÐáñÛ ³É³Ò ØµÊÌË µè ÛñÍÂÉ³ÙÄËÂáñìàÔÜÍÂÊÞ Ë æ¹ÊÏÛÜØ ËÉ³åÉÍÂÒ³í _ÚÖ«ËÃÔÜÍÂÊÞ ÐØÉIP¥ÎµÍÂÉ³ÒîÒÛÜÊØ Ê ÔµË ìôÎ É
ËÂÇÚÐçË®ÛÜÒßÙÊØÚÒËÍÂÖµrÙËÂÉ³è _ôì ËÂÇµÉ ÉIPôÎÚáÜÛÜÙ³ÛñËÀÙ³ÊØµÒîËÍÂÖµÙËÂÊÌÍ®Ù³ÐáÜá
^»·
æ ÇÚÛñÙ´Ç¼ÛÜrational
ÒßÊÔÓË ìôÎ É
rational(1,2)
`Äé
rational
4.3.5 Default constructor
©¯î Ô×ÊÌÍ ËÂÇµÉ ]ôÛÜØµè ÊÔ
êØ " É³ÙÄËÂÛÜÊØ é é íòæ¹É ÇÚÐXUÉ ÛÜØ¯ËÍÂÊôèµÖµÙ³É³è¼ËÂÇµÉ ËÉÍÂÞ
µ
F

ç
¶

¯
©

ÛÜØµÛñËÂÛÜÐáÜÛ ¬ÐÑËÂÛÜÊØ ËÂÇÚÐçË¹ËäÐ ]É³ÒÈÎÚá×ÐçÙ³ÉãÛñØèµÉ³Ù³á×ÐÑÍäÐçËÂÛÜÊØµÒ¹áÜÛV]É
rational r ;
[ ÊÌÍÈÔ×ÖÚØµèÚÐÞßÉ³Ø¯ËäÐáË ì Î É³Ò³íµèµÉ³ÔªÐÖµáñË ÛÜØµÛñËÂÛ×ÐçáÜÛ¬ÐçËÛÜÊØáñÉ¬ÐXUÉ³ÒÈËÇÚÉUÐáÜÖµÉÓÛÜØ W ÖµÉ³ÒîËÂÛÜÊØ ÖµØÚèµÉ YÚØµÉ³è¤í _ÚÖ«Ë
×Ô ÊÌÍÈÙ³á×ÐÒîÒ Ë ì Î É³Ò³í¥ËÇÚÉ
ÛÜÒ¹ÐÖ«ËÂÊÞ®ÐçËÛÜÙ¬ÐáÜá²ì Ù¬ÐáÜáÜÉè®ËÊ¿ÛñØÚÛ²ËÂÛ×ÐáÜÛ³ÉÈËÇÚÉUÐáÜÖµÉÌéÃêÔ
µ
F

Ñ
¶

¬
£
¡

¬
±

£
ÎµÍÂÉ³ÒÉØ Ë¬íËÇÚÉÕèµÉ³ÔªÐÖµáñË¾Ù³ÊØµÒîËÍÂÖµÙËÊÌÍ ÛÜÒ¤ËÂÇµÉÖµØµÛ WôÖµÉÃÙ³ÊØµÒîËÍÂÖµÙËÊÌÍ¡æ·ÛñËÂÇëÐçØàÉ³ÞÀÎµË ìÓÔ×ÊÌÍÞ®Ðá¯ÐçÍÂåÖµÞßÉ³Ø Ë
áÜÛÜÒË¬é
ÕìÎµÍÂÊ U¥ÛÜèµÛÜØµå½Ð½èµÉ³ÔªÐÖµáñËÙ³ÊØµÒîËÍÖÚÙÄËÂÊÌÍ¬íæ¹É Ù¬ÐØ ËÂÇ ÖÚÒÞ®Ð ]É ÒÖ«ÍÂÉËÂÇÚÐçËÙ³á×ÐÒîÒË ìôÎ É UÌÐçáÜÖµÉ³Ò
ÐçÍÂÉ
ÎµÍÂÊÌÎ ÉÄÍÂáñì ÛÜØµÛñËÂÛ×ÐáñÛ ³É³è¤éïêØ Ù¬ÐçÒÉàÊÔ ËÂÇµÉëÙ³á×ÐçÒÒ
^ÊÌÍ ÐçØ ì ÐÑÍÂÛñËÂÇµÞßÉËÂÛñÙÓË ì Î É `Äí
¯
©

¸
©
J

rational
èµÉ³ÔªÐÖµáñË ÛÜØµÛñËÂÛ×ÐçáÜÛ ¬ÐçËÛÜÊØæ Û²ËÂÇ UÐáÜÖµÉ
ÒÉ³É³ÞßÒ ËÂ
Ê _É½ËÂÇµÉ Ù³ÐØµÊØÚÛñÙ¬ÐáÏÙ´ÇµÊÛÜÙÉÌíÓÐØµè ÇµÉÍÂÉ ÛñÒ ËÂÇµÉ
Ù³ÊÌÍÍÉ³ÒîÎ ÊØµèµÛÜØµåèµÉ³ÔªÐçÖÚá²Ë·Ù³ÊØµÒîËîÍÂÖµÙËÂÊÌÍ¬0é
\ // POST : * this is initialized with 0
rational ()
: n (0) , d (1)
{}
êØÔªÐÙÄË¬í¥æ¹É
ÎµÍÂÊU¥ÛÜèµÉÏÐàèµÉ³ÔªÐÖµáñË¹Ù³ÊØµÒîËÍÖÚÙÄËÂÊÌÍÈÛÜÔ æ¹ÉÓæÈÐçØ Ë¹ËÂÇµÉÓÙ³ÊÞÀÎÚÛÜáÜÉÍÕËÂÊ ÐçÙ³Ù³ÉÎµË¹ËÂÇµÉ
¨
µè É³Ù³á×ÐÑÍäÐçËÂÛÜÊØ
éëÆ¹ÇÚÛñÒÏÞ®Ð ]É³Ò Ù³á×ÐÒîÒÓË ì Î É³ÒÓÒÂÐÔ×ÉÍÏËÂÇÚÐØ Ô×ÖµØÚèÚÐÞßÉØ ËäÐá¡Ë ì Î É³Ò³í¤ÒÛñØÚÙÉ¿Û²Ë
rational r
ÛÜÒÈØµÊÌËÈÎÊÒÒÛV_ÚáÜÉ ËÂÊÙ³ÛñÍÂÙ³ÖµÞUÉ³Ø ËòÐ Ù³ÊØµÒîËÍÂÖµÙËÊÌÍ·Ù¬ÐçáÜá¤ÛÜØèµÉ³Ù³áÜÐçÍÂÛÜØµåÐUÐçÍÂÛ×Ð _ áñÉÌé
ÆÈÇµÉÕÙ¬ÐçÍÂÉ³Ô×ÖµáÍÂÉ¬ÐçèÚÉÄÍ æ ÛñáÜáÌØµÊÌËÂÛÜÙ³É ËÂÇÚÐçË¤ËÂÇµÉÍÉÞ¿ÖÚÒË _ÉÐØãÉIP«ÙÉÎµËÂÛÜÊØâËÂÊÈËÂÇµÛÜÒ ÍÂÖµáÜÉ ÍÂÊåÌÍäÐÞ \
ÛÜØ " É³ÙËÛÜÊØ é Ù³ÊØ¯ËäÐÛÜØµÒ ËÂÇµÉ èµÉ³Ùá×ÐçÍäÐçËÛÜÊØàÒîËäÐçËÉ³ÞßÉ³Ø Ë
Ðá²ËÂÇµÊÖÚåÇàÛñØ ËÇ ÐÑË ÎµÍÂÊåÌÍäÐÞ í
rational r;
ËÂÇµÉÕË ì Î É
ÛÜÒ ÐÈÒËÍÂÖµÙË æ·ÛñËÂÇµÊÖ«Ë¾ÐØ ìãÙ³ÊØµÒîËÍÂÖµÙËÊÌÍÂÒ³é Æ¹ÇµÛÜÒ ÛÜÒ¾ÛÜØàÔªÐÙÄË ËÂÇµÉÊØµáñìÏÉIP¥Ù³ÉÎµËÂÛÜÊØ
Ô×ÊÌÍ¿ÐóÙ³á×ÐçÒrational
Òëæ ÛñËÇÚÊÖ«ËëÐØ ì ÙÊØÚÒËÍÂÖµÙËÂÊÌÍÂÒí ËÇÚÉßèµÉ³ÔªÐçÖÚá²ËàÙ³ÊØµÒîËÍÂÖµÙËÊÌÍ ÛÜÒëÛÜÞÀÎÚáÜÛÜÙÛñËÂáñìóÎµÍÂÊ U¥ÛñèÚÉè _ ì
ËÂÇµÉ®Ù³ÊÞÀÎÚÛÜáÜÉÍ³í ÐØµè ÛñË ÒîÛÜÞÀÎÚáñìÝèµÉ³ÔªÐÖµáñË ºÛÜØµÛñËÂÛÜÐáÜÛ ³ÉÒÏËÂÇµÉ®èÚÐçËäÐóÞßÉ³
Þ _ ÉÍÒ ÛÜÔòÐóèÚÐçËäÐ ÞßÉ³
Þ _ ÉÍâÛÜÒ
ÊÔ Ù³á×ÐÒîÒ·Ë ìôÎ ÉÌí ËÂÇµÛÜÒ·ÛÜØ ËÂÖ«ÍÂØ Ù¬ÐçáÜáÜÒòËÇÚÉàèµÉ³ÔªÐçÖÚá²Ë Ù³ÊØµÒîËÍÖÚÙÄËÂÊÌÍÓÊÔ ËÂÇµÉàÙ³ÊÌÍÍÂÉÒîÎ ÊØµèµÛÜØµåÀÙ³á×ÐÒîÒ³é Æ¹ÇÚÛñÒ
IÉ P¥Ù³ÉÎµËÂÛÜÊØÝÇÚÐ
Ò _ É³ÉØÝÞ®ÐèµÉàÒÊ ËÇ ÐÑËâÒîËîÍÂÖµÙËÂb
Ò ^»æ ÇµÛÜÙäÇ ÇÚÐÒÏÛÜØµÇµÉÍÂÛñËÉ³è½ÔÜÍÊÞ ÛñËÒÓÎµÍÂÉÙ³Ö«ÍÂÒÊÌÍ
`RYµËÈÛÜØ ËÊËÂÇµÉÏÙ³á×ÐÒÒÈÙ³ÊØµÙ³ÉÎµËòÊ
Ô àé
4.3.6 User-defined conversions
ïÊØÚÒËÍÂÖµÙËÂÊÌÍÂÒÕæ ÛñËÂÇ ÊØÚÉ ÐçÍÂåÖµÞßÉ³Ø ËÃÎÚá×³Ð ìÀÐÏÒîÎ É³ÙÛ×Ðá ÍÊáÜÉ ËÂÇµÉìÀÐçÍÂÉ
é

£
µ
F

ô
D
³

¦
£

[ ÊÌÍòËÇÚÉÏÙ³á×ÐçÒÒ
í«ËÂÇµÉâÙÊØÚÒËÍÂÖµÙ ËÂÊÌÍ
rational
// POST : * this is initialized with value i
rational ( int i)
: n ( i ) , d (1)
{}
ÛÜÒïÐãÖÚÒîÉÍºèµÉ YÚØµÉ³è Ù³ÊØ UÉÄÍÂÒÛÜÊØ®ÔÜÍÂÊÞ
ËÂÊ
é·ØµèµÉÍ ËÂÇµÛÜÒÕÙ³ÊØµÒîËÍÂÖµÙËÂÊÌÍ³í
_É³Ù³ÊÞßÉ³Ò
int
Ð Ë ìôÎ Éóæ·ÇµÊÒÉ UÐáÜÖµÉ³ÒßÙ¬ÐØ _ É Ù³ÊØ int
UÉÍËÂÉè rational
ËÂÊ
«é ÆÈÇµÛÜÒ Ô×ÊÌÍ®ÉIPµÐÞÀÎÚáÜÉ ÞßÉ¬ÐØµÒ ËÂÇÚÐçË
rational
æ¹ÉÙ³ÐØ ÎµÍÂÊ UôÛÜèµÉÐ Ù¬Ðáñá·ÐçÍåÖÚÞßÉØ Ë¿ÊÔ·Ë ìôÎ É
æ ÇµÉ³ØµÉ UÉÍßÐ Ô×ÊÌÍÂÞ®ÐçáÈÔ×ÖµØµÙËÂÛñÊØ ÐçÍÂåÖµÞßÉ³Ø¯ËÊÔ
int
Ë ìôÎ É
ÛñÒëIÉ P¥ÎÉ³ÙËÂÉ³è ÃÛÜØÝËÂÇµÉÀÛÜÞÀÎÚáÜÛñÙ³ÛñËÓÙ³ÊØ UÉÍÂÒÛÜÊØ ËÂÇÚÐçËãËäÐ ]É³ÒëÎÚá×ÐçÙ³ÉÌí¡ËÇÚÉ Ù³ÊØ UÉÍîËÂÛÜØµå
Ù³ÊØµÒîËÍrational
ÖÚÙÄËÂÊÌÍÓÛÜÒ Ù¬ÐçáÜáÜÉ³è¤
é S¼ÛñËÂÇ ÖµÒÉÍ ºèµÉ YÚØµÉ³è½ÙÊØ UÉÍÂÒîÛÜÊØµÒ³íæ¹ÉàåÊ _ ÉìÊØµè ËÇÚÉâÒÉËÏÊÔ
F«©ËÂÇµ£ FÉ
ËÂÇÚÐçË ÐçÍÉ _ÚÖÚÛñáñË ºÛñØ ^áÜVÛ ]ÉÏËÂÇµÉëÊØµÉëÔÜÍÂÊÞ
ËÂÊ
`´í _ÚÖ«Ë ÛÜØ Ù³ÊØ¯ËÍäÐ³ÒîîË·©ËÂÊÀ
îÉ³¤Ò¹ ÛÜØµÙ³ÊÞÀÎÚáÜÉËÉ ` ÒîËäÐØµèÚÐçÍÂè Ù³ÊØ UÉÍÂÒÛñÊØ ÍÖÚáñÉ³int
^±ÒÊ ÞßDôÉÄ³ËÂ£¦ÛÜÞß
Ò ÒîËÂÛ²Î Öµá×double
ÐÑËÂÉ³è _ ìËÂÇµÉ ÒîËäÐØµèÚÐçÍÂè¤íµæ¹É
Þ®Ð ]É ËÂÇµÉÓÍÂÖµáÜÉ³ÒÈÊÖµÍÒÉ³Vá UÉ³Òé
ÆÈÇµÉÍÉ ÐçÍÂÉóÞßÉ¬ÐçØÚÛñØÚåÔ×ÖµáÈÖµÒÉÍ èÚIÉ Y ØµÉ³è¼Ù³ÊØ UÉÍÂÒÛÜÊØµÒ ËÇ ÐÑËßÙ¬Ð
Ø Ë _ ÉóÍÂÉ¬ÐçáÜÛ ³É³è _ ì Ù³ÊØµÒîËÍÖÚÙ ËÂÊÌÍÂÒé [ ÊÌÍ IÉ PµÐÞÀÎÚáÜÉí ÛÜÔ æ¹ÉÀæÈÐØ¯Ë ÐÙ³ÊØ UÉÍÂÒÛñÊØ ÔÜÍÂÊÞ
ËÂÊ
í æ¹ÉÀÙ¬Ð
Ø ËàÐèµè Ð
rational
double
Ù³ÊÌÍÍÉ³ÒîÎ ÊØµèµÛÜØµåóÙ³ÊØµÒîËÍÖÚÙÄËÂÊÌÍëËÂÊóËÂÇµÉ Ë ì Î É
í ÒÛñØÚÙÉ
ÛÜÒãØÚÊÌËàÐÙ³á×ÐÒîÒâË ìôÎ ÉÌé UÉ³Ø
double
double
Ù³ÊØ UÉÍÂÒÛÜÊØµÒÏËÂÊÒÊÞßÉ¿Ù³á×ÐçÒÒÏË ì Î É ÞßÛÜåÇ ËÓØµÊÌË _ ÉÎ ÊÒîÒVÛ _ÚáÜÉ ÛÜØ ËÂÇµÛÜÒÓæÈÐ³ì ÏÛÜÔ ÛÜÒâØµÊÌË îÊÖ«Í Ë ìôÎ É ^ _ÚÖ«Ë·Ù³ÊÞßÉÒòÔÜÍÂÊÞ ÐáÜÛ _ÚÍÂÐçÍìí¥ÒÂÐ³ì `ÄíÚæ¹ÉãÙ¬ÐØµØµÊÌË ÒÛñÞÀÎ á²ì®Ðèµè ÐÙÊØÚÒËÍÂÖµÙËÂÊÌÍòËÂÊ éêØÒÖµÙ´Ç
ÒÛñËÖ ÐÑËÂÛÜÊØµÒ³íæ¹É¹ÒÛÜÞÀÎÚáñìãËÂÉ³áÜá
Ë ìôÎÉòÇµÊ¦æ ÛñËÂÒ UÐáÜÖµÉ³Ò ÒÇµÊÖµáÜè _ É¹Ù³ÊØ UÉÍËÂÉè ËÂÊÓËÂÇµÉÈËäÐÑÍÂåÉË Ë ìôÎ Éé
£ ËÊ
Æ¹ÇÚÉÈÙ³ÊØ UÉÍÂÒÛÜÊØßÔÜÍÊÞ
í¯Ô×ÊÌÍGÉ PµÐÞÀÎÚáÜÉÌíÙÊÖÚáñè _ ÉÈèÚÊØµÉ¹ËÂÇ«ÍÂÊÖµåÇÀÐÓÞßÉ³
Þ _ ÉÍ
rational double
Ô×ÖµØµÙËÂÛÜÊØ Ø ÐçÞßÉ³è
áñVÛ ]É ËÂÇµÛÜÒ³é
operator double
// POST : return value is double - approximation of * this
operator double ()
\ {
return double (n )/ d;
}
êØ å É³ØµÉÍäÐá»íËÂÇµÉóÞßÉ³Þ_ ÉÍÔ×ÖµØÚÙÄËÂÛÜÊØ
Ç ÐçÒÀÛÜÞÀÎÚáÜÛÜÙ³Û²ËëÍÂÉËÂÖ«ÍÂØ Ë ìôÎÉ ÐØµè ÛÜØµèÚÖµÙ³ÉÒ
Ð ÖµÒÉÄÍºèµÉ YÚØµÉ³è Ù³ÊØ UÉÍÂÒÛÜÊØÝËÊ ËÂÇµÉË ì operator
Î É ËÂÇÚÐçã
Ë ÛÜÒãÐÖ«ËÂÊÞ®ÐçËÂÛñÙ¬ÐáÜáñì ÛÜØ UÊ ]³É èÝæ Çµ³É ØµÉ UÉÍãËÂÇµÛÜÒÏÛÜÒ
ØµÉ³Ù³É³ÒîÒÂÐçÍìé
4.3.7 Member operators
áÜáÚÔ×ÖµØµÙËÂÛÜÊØÚÐáÜÛ²Ë ì ÊÔÍÂÐçËÂÛÜÊØÚÐáÚØôÖµÞ_ ÉÍÂÒÃËÂÇÚÐçËÕæ¹É·Ç ÐXUÉ ÎµÍÂÉ UôÛÜÊÖµÒáñì ÎµÍÂÊU¥ÛÜèµÉ³èßËÂÇ«ÍÂÊÖµåÇ îåáÜÊ _ Ðá
Ô×ÖµØµÙËÂÛÜÊØµ
Ò ^
í
íñé³éé ` Þ¿ÖµÒîË ØµÊ¦æ _ É ÍÉ³Ù³ÊØµÒÛÜèµÉÍÂÉè¾íôÒÛÜØµÙ³É èµÛñÍÂÉÙËÂáñìÀÐÙ³Ù³É³ÒîÒ ÛÜØµåÀËÂÇµÉëèÚÐçËäoperator+
ÐÞßÉ³b
Þ _ÉÍÂÒÈoperator+=
ÛÜÒ ØµÊ®áÜÊØµåÉÍ·Î ÊÒîÒÛV_ÚáÜÉÌé·êØÚÒËÂÉ¬Ðè¤í æ¹Éëæ·ÛÜáÜá¾ÖµÒÉëËÂÇµÉëÞßÉ³Þ_ ÉÍòÔ×ÖµØÚÙÄËÂÛÜÊØµÒ
ÐØµè
Ô×ÊÌÍòØµÊØ ºÞßÊôèµÛÜÔÜìôÛÜØµåàÐÙ³ÙÉ³ÒÒòËÊËÂÇµÉ ÍÂÉÎµÍÂÉ³ÒîÉ³Ø ËäÐÑËÂÛÜÊØ¤íµÐØµèóÐ Ù³ÊØ
numerator
ÒîËÍÖÚÙÄËÂÊÌÍ·Ô×ÊÌÍòÍÂÉËÂÖ«denominator
ÍÂØµÛÜØµåÐ ÍÉ³ÒÖµáñË¬
é èµèµÛñËÂÛÜÊØ Ô×ÊÌÍ IÉ P«ÐÞÀÎÚáÜÉ ËÂÇµÉ³Øóæ¹ÊÌÍ ]ôÒòáñÛV]ÉÏËÇÚÛñÒ ^ÐØµè _ É³Ù³ÊÞßÉÒ
Ð _ Û²ËòáñÉ³ØµåÌËÂÇ ì `
// POST : return value is the sum of a and b
rational operator + ( rational a , rational b)
{
int rn = a. numerator () * b. denominator () +
a. denominator () * b. numerator ();
int rd = a. denominator () * b. denominator ();
return rational ( rn , rd );
}
Ö«ËóÖµØµèÚÉÄÍóÐÙÙ³É³ÒÒÍÉ³ÒîËÍÂÛñÙËÂÛÜÊØµÒ³í·ËÇÚÉÄÍÂÉÝÐçÍÉ
Ò ÊÞßÉ ËÂÇµÛÜØµåÒ®ËÂÇÚÐçË æ¹É½Ù¬ÐØµØµÊÌËèµÊ ÎÚÍÊÌÎ ÉÍÂáñì
ËÂÇ«ÍÂÊÖµåÇóåáÜÊ _ Ðá¤Ô×ÖµØµÙËÂÛÜÊØµÒ³é Ò ÐØ ÉGPµÐÞÀÎÚáÜÉÌí«ÙÊØÚÒîÛÜèµÉÍ
é¹ÆÈÇµÛÜÒòÊÌÎ ÉÍÂÐçËÂÊÌÍòØÚÉÉ³èµÒòËÂÊ
ÙäÇÚÐØµåÉ®ËÇÚÉ UÐáÜÖµÉßÊÔòÐ ÍÂÐçËÂÛÜÊØÚÐáÕØôÖµÞb_ÉÍ¬í_ÚÖ«ËàËÂÇµÉÍÂÉ®ÛÜoperator+=
Ò ØµÊ ÒÎ É³Ù³ÛVYÚÙ®ÞßÉ³Þ_ ÉÍâÔ×ÖÚØµÙËÛÜÊØ ËÇ ÐÑË
ÐáÜáñÊÑæ·Ò¿ÖµÒ ËÊÝèµÊ ËÂÇµÛÜÒ³é S ÉóÙ¬ÐØ ÊØµáñì ÒÛÜÞ¿Öµá×ÐçËÂÉÀËÇÚÉÙ´ÇÚÐØµåÉ ËÂÇ«ÍÂÊÖµåÇËÂÇµÉ ÐèµèµÛñËÂÛÜÊØÐØµè ÐØ
ÐÒÒîÛÜåØµÞßÉ³Ø Ë¬íôáÜÛV]É ËÂÇµÛÜÒ³é
// POST : b has been added to a ; return value is the new value of a
rational & operator += ( rational & a , rational b)
{
return a = a + b ;
}
Æ¹ÇÚÛñÒÓæ¹ÊÌÍ ]¥Ò_ÚÖµËÏÛÜÒÓÛÜØµÉ Ù³ÛÜÉ³Ø¯Ëb^Ù³ÊØµÒÛÜèµÉÍÏá×ÐçÍåÉÍÏÒîËÍÂÖµÙËÂÒ `Äí¤ÒÛÜØµÙ³Éàæ¹ÉbYµÍÂÒËãÙÊØÚÒËÍÂÖµÙËãÐØ ÛñØ ËÂÉÍ ÞßÉ³èµÛ×ÐçËÉòÍÂÉ³ÒîÖÚá²Ë
æ ÇµÛÜÙäÇ ÛñÒ ÒÖ _ÚÒÉXWôÖµÉ³Ø¯ËÂáñì®Ù³ÊÌÎÚÛÜÉè _ ÐÙ ]ßÛÜØ¯ËÂÊ éÕêØ ÔªÐÙË³í
æÈÐÒ
a + b
a
operator+=
èµÉ³ÒÛÜåØµÉ³è ËÂÊÀXÐ UÊÛÜè ÉIPµÐÙÄËÂáñì®ËÂÇµÛÜÒ¹èµÉËÂÊÖ«ÍòËÇ ÐÑËÈæ¹ÉãØµÉ³É³èËÂÊ¿ËäÐ ]ÉÏØµÊÑæâé
_ ÉËîËÂÉÍ æÈÐ³ì ËÂÊßåÊÀÛÜÒ·ËÊ ÍÂÉ¬ÐáñÛ³É
ÐÒ Ð ÎÚÖa_ÚáÜÛñÙâÞßÉ³Þ_ ÉÍÈÔ×ÖµØµÙËÂÛÜÊØ ^ºÐ
operator+
`´í ÇÚXÐ U¥ÛÜØµåÓÊØµáñìàÊØµÉ¹Ô×ÊÌÍÂÞ®Ðá«ÐÑÍÂåÖµÞßÉ³Ø ËR^»Ô×ÊÌÍ `ÄíÐçØÚè
ËäÐ ]¥ÛñØÚå ËÂÇµÉïÍÂÊáÜÉïÊÔ ¨ ³é ¨ ÆÈ ³Çµ³ÛÜ£ Ò
b
*this
a
áÜ³Ê ¢ Ê ³]¥£ÄÒÈ©¯ÐÒÈ£ Ô×ÊáÜáÜÊ¦æ Ò³é
// POST : b has been added to * this ; return value is
//
the new value of * this
\ rational & operator += ( rational b)
{
n = n * b.d + d * b. n;
d *= b.d ;
return * this ;
}
S¼ÛñËÂÇµÛÜØÏËÂÇµÛÜÒ ÞßÉ³Þ_ ÉÍ¤Ô×ÖµØµÙËÂÛñÊØ¾íÑËÂÇµÉÍÂÉÕÛÜÒ ØµÊ·ÎµÍÂÊ _ÚáÜÉ³Þ ÛÜØ ÐçÙ³Ù³É³ÒÒîÛÜØµå·ËÂÇµÉèÚÐçËäÐ¹ÞßÉ³Þ_ ÉÍÂÒ¤èµÛñÍÂÉÙËÂáñìí
ÒÛÜØµÙ³ÉÃËÂÇµÉ ÐÙÙ³É³ÒÒ ÍÉ³ÒîËÍÂÛñÙËÂÛÜÊØµÒ¤èµÊ ØµÊÌË ÐçÎµÎÚáñìÓËÂÊ ÞßÉÞb_ ÉÄÍ¤Ô×ÖÚØµÙËÛÜÊØµÒ³é Æ¹ÇµÛÜÒ"UÉÍÒÛÜÊØëÊÔ
ÛÜÒÓÐÒÓÉ Ù³ÛÜÉ³Ø¯ËãÐÒ ËÂÇµÉ¿ÊØµÉàÎµÍÂÉ ôU ÛÜÊÖµÒáñìóÖµÒÉèÝÔ×ÊÌÍ
í¤ÐçØÚè ÛñËÓÙ¬ÐçØ ÛÜØ operator+
ËÂÖ«ÍÂØ É UÉ³Ø
struct rational
ÒÉÍ U ÉëÐÒ·Ð_ ÐÒÛñÒÈÔ×ÊÌÍ Ð¿ÞßÊÌÍÉÏÒîÖÚÙÙ³ÛÜØµÙËòÛÜÞÀÎÚáÜÉ³ÞßÉ³Ø¯ËäÐçËÂÛÜÊØÊÔ
operator+
// POST : return value is the sum of a and b
rational operator + ( rational a , rational b)
{
return a += b;
}
ÊÖÞßÛÜåÇ ËÃÐÑÍÂåÖµÉòËÂÇÚÐçËÃÉ UÉ³Ø
operator+
Prefer nonmember operators over member operators.
ÒÇµÊÖµáÜè_ É³Ù³ÊÞßÉ ÐàÞßÉ³Þ_ ÉÍÔ×ÖµØÚÙÄËÂÛÜÊØ ÊÔ
íµÐØµè ÛÜØµèÚÉÉ³è¤íôËÂÇµÛÜÒ æ¹ÊÖµáÜèß
ÎµÍÂÊ _ Ð _Úáñì
class rational
ÐáÜáñÊÑæ ÐÒîáÜÛÜåÇ Ëáñì ÞßÊÌÍÂÉ¿É Ù³ÛñÉ³Ø ËàÛñÞÀÎ áñÉ³ÞßÉ³Ø ËÂÐçËÂÛÜÊØ¤é¿ÆÈÇµÉÍÉÀÛÜÒâÊØµÉÛÜÞÀÎ ÊÌÍËäÐçØ Ë ÍÂÉ¬ÐçÒÊØ ËÂÊ ]É³ÉÎ
ËÂÇµÛÜÒ¹ÊÌÎ ÉÍÂÐçËÂÊÌÍòåáÜÊ _ Ðáí ËÇÚÊÖµåÇ¤íµÐØµèËÂÇµÛÜÒ¹ÇÚÐÒ¹ËÂÊ èµÊ¿æ ÛñËÂÇ ÖµÒÉÍèÚÉIY ØµÉ³èÙ³ÊØ UÉÍÒÛÜÊØµÒ³é
XÐ U¥ÛÜØµå ËÂÇµÉãÙÊØ UÉÍÂÒîÛÜÊØ ÔÜÍÂÊÞ
ËÂÊ
ËÇ ÐÑËÈæ¹ÉÏåÉËÈËÂÇ«ÍÂÊÖµåÇËÂÇµÉÏÙ³ÊØµÒîËÍÂÖµÙËÂÊÌÍ
int
rational
// POST : * this is initialized with value i
rational ( int i );
æ¹ÉâÙ³ÐØ Ô×ÊÌÍ ÉIP«ÐÞÀÎÚáÜÉ æ·ÍÂÛ²ËÂÉÓÉIP¥ÎµÍÂÉ³ÒîÒÛÜÊØµÒ áÜÛV]É
ÊÌÍ
íÚæ·ÇµÉÍÂÉ ÛÜÒòÊÔ¡Ë ì Î É
é
êØßÙ³ÊÞÀÎÚÛÜáÜÛñØÚå ËÇÚÛñÒ³íËÂÇµÉ·Ù³ÊÞÀÎÚÛÜáÜÉÄÍÐçÖµËÊÞ®ÐçËÂÛÜÙ¬ÐçráÜáñìÏ+ÛÜØµ2ÒÉÍËÒ 2Ðã+ÙÊrØ UÉÍËÛÜØµåârÙ³ÊØµÒîËÍÖÚÙÄËÂÊÌÍ rational
Ù¬ÐçáÜáé O Ê¦æëí
ÇÚXÐ U¥ÛÜØµå
ÐçÒàÐ ÞßÉÞb_ ÉÄÍÏæ¹ÊÖµáÜè½ÍÂÉ³ÞßÊUÉ¿ËÂÇµÉÒÉ³Ù³ÊØµè Î ÊÒÒÛ _ ÛñáÜÛñË ì ÊÔæòÍÂÛñËÂÛÜØµå
é
operator+
2 + r
S¼Ç ì É!Ë Ò YµÍÂÒîË ÒÉÉëæ ÇÚÐçË ÇÚÐçÎµÎ ÉØÚÒÈæ ÇµÉ³Ø
ÛÜÒ·Ù³ÊÞÀÎÚÛÜáÜÉ³è¤éïêÔ
ÛÜÒ Ð¿ÞßÉ³Þb_ÉÍ
r + 2
operator+
Ô×ÖµØµÙËÂÛÜÊØ¤í¥ËÂÇµÉ³Ø
îÞßÉ¬ÐØµÒ r + 2
r. operator + (2)
ê Ø ÙÊÞÀÎ ÛñáÜÛÜØµå¿ËÇÚÛñÒ³íËÂÇµÉ Ù³ÊÞÀÎÚÛÜáÜÉÍ·ÛÜØµÒÉÍËÂÒ ÂË ÇµÉàÙ³ÊØ UÉÍÂÒÛñÊØ ÔÜÍÂÊÞ ÂË ÇµÉ Ù¬ÐáÜá ÐçÍÂåÖµÞßÉ³Ø Ë·Ë ì Î É
ËÂÊëËÂÇµÉ Ô×ÊÌÍÞ®ÐáÐÑÍÂåÖµÞßÉ³Ø ËÃË ìôÎ É
ÊÔ
í«ÐØµè®É UÉÍ ì ËÂÇµÛÜØµåâæ¹ÊÌÍ ]ôÒ¹ÐÒ ÉIPôÎ É³ÙËÂint
Éè¾é
rational operator+
íµÇµÊÑæ¹IÉ UÉÍ¬íµæ¹ÊÖÚáñèÞßÉ¬ÐØ
2 + r
2. operator + ( r )
æ µÇ ÛÜÙäÇóÞ®Ð ]É³ÒòØµÊÀÒîÉ³ØµÒÉâæ ÇÚÐçËÒÊôÉ UÉÄÍ¬éïêÔ æ¹ÉÏæ·ÍÂÛñËÉâÐ_ ÛñØ ÐÑÍì ÊÌÎ ÉÍäÐçËÊÌÍ ÐÒ·Ð¿ÞßÉÞb_ ÉÄÍòÔ×ÖµØµÙËÂÛñÊØ¾í
ËÂÇµÉ³Ø ËÇÚÉ YµÍÂÒîËãÙ¬Ðáñá ÐçÍÂåÖµÞßÉ³Ø Ë
_ É¿ÊÔÕËÂÇµÉ ÍÂÉ³ÒîÎ ÉÙËÂÛVUÉ¿Ù³á×ÐÒîÒÏË ìôÎÉÌé¿êÞÀÎ áñÛÜÙ³ÛñË·Ù³ÊØ UÉÍÒÛÜÊØµÒ
¨

èµÊØµÊÌËÈæ¹ÊÌÍ ]®ÇµÉÍÂÉ ËÂÇµÉì ÊØµáñì ÐèÚÐçÎµË¹Ù¬Ðáñá¡ÐÑÍÂåÖµÞßÉ³Ø ËÂÒ ËÊÔ×ÊÌÍÂÞ®Ðçá ÐçÍÂåÖµÞßÉ³Ø¯ËïË ì Î É³ÒÈÊÔ Ù³ÊØµÙÍÉËÂÉ
Ô×ÖµØµÙËÂÛÜÊØµÒ³í _ÚÖ«Ë ËÂÇµÉì Ù¬ÐØµØµÊÌ
Ë _ ÉÉGP¥Î É³ÙÄËÂÉ³è ËÂÊ NYÚØµè
ËÂÇµÉ Ù³á×ÐçÒÒ æ ÇµÊÒÉ
ÇÚÐÒ ËÂÊ_ É
operator+
ÐçÎµÎÚáÜÛÜÉ³è¤é
\ 4.3.8 Nested types
Æ¹ÇÚÉÄÍÂÉ®ÛÜÒ ÐËÂÇµÛñÍÂè Ù¬ÐçËÂÉåÊÌÍì ÊÔÈÙ³á×ÐÒÒàÞßÉÞb_ ÉÄÍÂÒ³í ÐØµè ËÂÇµÉ³ÒÉ ÐçÍÂÉ
éÝÆ Ê ÞßÊÌËÂÛVUÐçËÂÉ
ËÂÇµÉ³ÒÉí áÜÉÄËòÖµÒòÙ³ÊÞßÉ_ÐçÙ ] ËÂÊ êÒÒÖµÉ Ð _ ÊUÉÌí«ËÂÇµÉâÊØµÉãÙÊØÚÙÉÍÂØµÛÜØµå ËÂ¬Çµ³ÉÏGÛÜØ F ËÉ ÍÂJ¦ØÚ¢ Ð¬á ÍÂÉÎµÍÂÉ³ÒîÉ³Ø ËäÐÑËÂÛÜÊØ
ÊÔ ÍäÐçËÂÛñÊØ ÐçáòØ ÖÚ
Þ _ ÉÍÒ³é êÔ ìÊÖ ËÂÇµÛÜØ ] Ð _ÊÖµË Ù³ÊØµÒÉXW ÖÚÉØ ËÂáñì ÇÚÛñèÚÛñØÚå ËÂÇµÉóÍÂÉÎµÍÂÉ³ÒîÉ³Ø ËäÐÑËÂÛÜÊØ ÊÔÓÐ
ÍäÐçËÛÜÊØÚÐáØ ÖÚ
Þ _ ÉÍ¹ÔÜÍÊÞ ËÂÇµÉÏÙÖÚÒËÂÊÞßÉÍ¬íôËÇÚÉØ ìÊÖ ÎÚÍÊ _Ð _ á²ì®ÐáÜÒÊàæÈÐØ¯ËïËÂÊ ÇµÛÜèµÉ ËÂÇµÉÏØ ÖµÞßÉÍäÐçËÂÊÌÍ
ÐØµè èµÉ³ØµÊÞßÛÜØÚÐçËÂÊÌÍ¤Ë ìôÎ ÉÌé ·Ò ÛÜØµèµÛÜÙ¬ÐçËÂÉèàÛÜØàËÂÇµÉ IÉ PµÐçÞÀÎ áñÉÌíçËÂÇµÉ³ÒîÉ Ë ì Î É³Ò ÞßÛÜåÇ¯Ë ÛñØ ËÂÉÍØ ÐçáÜáñìÏÙäÇ ÐçØÚåÉí
_ÚÖ«ËâÛÜØÝËÇÚÉ¿ÞßÉ³
Þ _ ÉÍÏÔ×ÖµØµÙËÂÛÜÊØµÒ
ÐçØÚè
í¡ìÊÖ ÒËÂÛÜáÜá ÎµÍÂÊÞßÛÜÒÉàËÂÊ ÍÂÉËÂÖ«ÍÂØ
numerator
denominator
UÐáÜÖµÉ³Ò³é
int
_ ÉËËÉÍ¹ÒÊáÜÖ«ËÂÛÜÊØÀæ¹ÊÖµáÜè _ É ËÂÊëÎµÍÂÊÞßÛñÒÉ ÊØµáñìßÐâË ì Î É æ ÛñËÇ ÙÉÍËäÐÛñØ ÎµÍÂÊÌÎ ÉÄÍËÂÛÜÉ³Òí _ôì®ÒÐ¬ìôÛÜØµå
Ô×ÊÌÍ IÉ P«ÐÞÀÎÚáÜÉ ËÂÇÚÐçË ËÂÇµÉÝÔ×ÖµØÚÙÄËÂÛÜÊØµÒ
ÐØµè
ÍÂÉËÂÖ«ÍÂØ ÐØ ÛÜØ ËÂÉåÌÍäÐáÓË ì Î É
numerator
denominator
^ " É³ÙÄËÂÛÜÊØ \ é \ é `Äé Æ¹ÇµÉ³ØßìÊÖßÙ¬ÐØ®ÛÜØ ËÉÍÂØÚÐáÜáñì ÙäÇÚÐØµåÉ ÔÜÍÂÊÞ ÊØµÉ ÛÜØ ËÉ³åÌÍäÐáµË ìôÎ ÉÈËÂÊàÐâèµÛ ÉÍÂÉ³Ø¯Ë ÊØµÉ
æ Û²ËÂÇµÊÖµËÈÐçØÚØµÊ¦ì¥ÛÜØµå ËÂÇµÉãÙÖÚÒËÂÊÞßÉÍ¬éÃÆ É³ÙäÇµØµÛÜÙ¬ÐáñáñìíôËÂÇµÛÜÒ¹Ù¬ÐØ _ ÉãèµÊØµÉãÐÒòÔ×ÊáñáÜÊÑæ·Ò³é
class rational {
public :
// nested type for numerator and denominator
typedef int rat_int ;
...
// realize all functionality in terms of rat_int
// instead of int , e.g .
rational ( rat_int numerator , rat_int denominator ); // constructor
rat_int numerator () const ;
// numerator
...
private :
rat_int n;
rat_int d ; // INV : d != 0
};
êØóÙ³ÖµÒîËÊÞßÉÍÈÙ³ÊôèµÉÌí«ËÂÇµÛÜÒ¹Ù¬ÐØ _ ÉâÖµÒÉ³èÔ×ÊÌÍ ÉIP«ÐÞÀÎÚáÜÉ áÜÛV]É ËÂÇµÛÜÒ³é
typedef rational :: rat_int rat_int ;
rational r (1 ,2);
rat_int numerator = r. numerator ();
rat_int denominator = r. denominator ();
// 1
// 2
SÝÉòÐáñÍÉ¬Ðè«ì ÒÉ³ÉÈÊØµÉòÊÔÚËÂÇµÉÈÎµÍÂÊÌÎ ÉÄÍËÂÛÜÉ³Ò ËÂÇÚÐçË ËÂÇµÉÈØµÉ³ÒîËÂÉ³èË ì Î É
ÞÖµÒîËÃÇÚÐXUÉ
rational::rat_int
ÛÜØ ÊÌÍÂèµÉÍÓÔ×ÊÌÍ ËÂÇµÛÜÒÈËÂÊßæ¹ÊÌÍ
] é [ ÊÌÍÓÉGPµÐÞÀÎÚáÜÉÌíaUÐáÜÖµÉ³Ò ÊÔ Ë ìôÎ É
ÞÖµÒîË_ ÉàÙ³ÊØ UÉÍËÂÛV_ÚáÜÉãËÂÊßÛñË¬é êÔ
ìÊÖóÇÚÐXU ÉâÒÉÄË Ö«Î É U ÉÍîìôËÂÇµÛÜØµå¿Ù³áÜÉ¬ÐØµáñìí¥ìÊÖÙ¬ÐØóØµÊ¦æ Ô×ÊÌÍòÉIP«int
ÐÞÀÎÚáÜÉ ÍÂÉÎÚá×ÐÙ³É ËÂÇµÉãáÜÛÜØµÉ
typedef int rat_int ;
_ôì®ËÂÇµÉãáÜÛÜØµÉ³Ò
typedef ifm :: integer rat_int ;
ÐØµèËÂÇ ÖµÒòÛÜÞßÞßÉèÚÛÜÐçËÂÉ³áñì¿åÉËòÉIP«ÐÙËÈÍäÐçËÂÛÜÊØÚÐá Ø ÖµÞb_ ÉÄÍÂÒïæ ÛñËÂÇµÊÖ«ËÈÐØ¯ìÊUÉÍ QÚÊ¦æ ÛñÒÒÖµÉ³Ò³é
\ ÛÜØ¯ËÍÂÊôèµÖµÙ³É³ÒÈÐëØµÉæ ØÚÐÞßÉ Ô×ÊÌÍòÐØ ÉIP«ÛñÒîËÂÛÜØµåàË ìôÎ É

¦
J
¢
G

µ
F

¶
µ
F

¬
±
×

£
¯
©

ÛÜØ¯ËÂÊóÛñËÂÒãÒÙ³ÊÌÎ Ééóê»ËàèµÊôÉÒ
ÛÜØ ËÍÊôèµÖµÙ³ÉÀÐØµÉæ Ë ìôÎ ÉÌéêØ ÔªÐÙË¬í ËÇÚÉ ØÚÉÄæ ØÚÐÞßÉ Ù³ÐØ _ ÉÀÖµÒÉ³è

ÒîìôØÚÊØ¯ì¥ÞßÊÖµÒáñì æ ÛñËÇ ËÇÚÉ ÊáÜè ØÚÐÞßÉëÛÜØ½ÐáÜá Ù³ÊØ ËÂÉGP¥ËÂÒ³éâêØ½ËÂÇµÉ Ð _Ê UÉ Ù³Ê èÚÉí¤æ¹ÉàÒîÉ³É¿ËÇÚÛñÒ Ë æ·ÛÜÙ³É
æ Û²ËÂÇµÛÜØ ËÂÇµÉëÙá×ÐÒÒ
íÚËÂÇµÉâË ìôÎ É³èµÉ³Ô èµÉ³Ù³áÜÐçÍäÐçËÂÛñÊØÛÜØ¯ËÍÂÊôèµÖµÙ³É³Ò ÐØµÉ³ÒîËÂÉè Ë ì Î É
íÐ
rational
rat_int
ØµÉæ ØÚÐÞßÉãÔ×ÊÌÍ ËÂÇµÉëË ì Î É
éÓêØ ËÇÚÉâÙ³ÖµÒîËÂÊÞßÉÍ ÙÊôèµÉÌíËÂÇµÉàÙ³áÜÐÒÒ Ò ØÚÉÒîËÂÉ³è Ë ìôÎ É ^»ËÇ ÐÑË Ù¬ÐØ _ É
ÐÙ³ÙÉ³ÒÒÉ³èßÖµÒÛñØÚå ËÂÇµÉÈÒÙ³ÊÌÎ É¹int
ÊÌÎ ÉÄÍäÐçËÂÊÌÍ¬íÛÜÔµËÂÇµÉ¹ØµÉ³ÒîËÂÉ³èË ì Î É¹èµÉ³Ù³á×ÐçÍÂÐçËÂÛÜÊØ ÛÜÒ ÎÚÖ _ áñÛÜÙ`¡ÍÂÉ³ÙÉ³ÛVUÉ³ÒÕÐ ØÚÉÄæ
^ÒÇµÊÌÍËÉGÍ `ïØ ÐçÞßÉÌé
êØßÍÂÉ¬Ðá áÜÛÜÔ×É ÙÊôèµÉÌí ËÇÚÉÄÍÂÉ ÐçÍÂÉ·ØÚÉÒîËÂÉ³èßË ìôÎ ÉÒïÊÔ ØÚÉÒîËÂÉ³èßË ìôÎ ÉÒ ÊÔ ØµÉ³ÒîËÂÉ³èßË ìôÎÉ³Ò³íñé¬éé³í¥ÐØµè
Ë ìôÎ ÉØ ÐçÞßÉ³ÒÈËÉ³ØµèóËÂÊ åÉ
Ë UÉÍìáÜÊØµåèµÖµÉãËÂÊËÂÇµÛÜÒ³é Æ¹ÇµÉãË ìôÎÉ³èµÉ³Ô ÞßÉ³ÙäÇÚÐØµÛÜÒÞ ÐáÜáÜÊ¦æ Ò¹ÖµÒòËÊ ]É³ÉÎ
ÊÖ«Í·ÙÊôèµÉÏÍÉ¬ÐèÚÐ _ÚáÜÉÌé
Typedef declarations.
4.3.9 Class definitions
SÝÉàØµÊ¦æ ÇÚÐXUÉàÒ É³É³Ø ËÂÇµÉëÞ®Ð ÊÌÍ ÛÜØµåÌÍÂÉèÚÛñÉ³Ø ËÂÒòÊÔÃÐÀÙ³á×ÐÒÒé [ ÊÌÍÂÞ®ÐáÜáñìíµÐÀÙá×ÐÒÒ èµÉ YÚØµÛñËÂÛñÊØóÇÚÐÒ ËÂÇµÉ
Ô×ÊÌÍÂÞ
− class
é²éñé
− æ ÇµÉÍÉ ÜÛ Ò¿ÐØ ÛÜèµÉ³Ø ËÛVYÚÉÍ¬é½ÆÈÇµÉ®ÒÉXW ÖÚÉØÚÙÉÊÔ ÒëÞ®Ð³ì _É ÉÞÀÎÚË ìé&ÕÐçÙ´Ç ÛñÒÀÐØ ©µ±Ä±¬³ ¢ ±³ ³£ ^ public: ÊÌÍ private: `´íïÊÌÍßÐ ¨½³¨½ ³£ Fµ±¬×©£©¯ é ÞßÉ³
Þ _ ÉÍ¡èµÉ³Ù³á×ÐçÍÂÐçËÂÛÜÊØëÛñÒ Ð·èµÉ³Ù³áÜÐçÍäÐçËÂÛñÊØëÒîËäÐçËÉ³ÞßÉ³Ø Ë¡ËÂÇÚÐçË Ë ì Î ÛñÙ¬ÐáÜáñìÏèµÉ³Ù³áÜÐçÍÂÉ³Ò ÐòÞßÉ³Þb_ÉÍ Ô×ÖµØµÙËÂÛñÊØ¾í
Ð èÚÐçËÂÐ ÞßÉ³
Þ _ ÉÍ³í¾ÊÌÍëÐ ØµÉ³ÒîËÉ³è Ë ì Î ÉÌé ïÊáÜáÜÉ³ÙÄËÂÛVUÉ³áñìíËÇÚÉÒÉßÐçÍÂÉÙ¬ÐáÜáñÉ³è
ÊÔËÂÇµÉÀÙá×ÐÒÒ³í
¨

½
¨
³

¦
£

ÐØµè ËÂÇµÉ³ÛñÍ ØÚÐÞßÉÒ·Þ¿ÖÚÒ
Ë _ ÉàÛÜèµÉ³Ø¯ËÂVÛ YÚÉÍÂÒé Ùá×ÐÒÒ èµÉ YÚØµÛñËÂÛÜÊØÛÜØ ËÍÊôèµÖµÙ³É³Ò Ð ØµÉæ Ë ìôÎ ÉÌíÐØµè ËÇÚÛñÒ
Ë ìôÎ ÉÏÛÜÒÈÙ¬ÐáÜáñÉ³èóÐ
íÚÐÒòÊÌÎµÎ ÊÒÉ³èËÊ Ð Ô×ÖµØÚèÚÐÞßÉØ ËäÐáË ì Î ÉÌé
±¬×©¯³ J¦¢
ÞßÉ³b
Þ _ÉÍ Ô×ÖµØµÙËÂÛÜÊØ èµÉ YÚØµÛñËÂÛÜÊØ®ÛÜÒ¹ÐàèµÉ³Ù³á×ÐÑÍäÐçËÂÛÜÊØ®ÐçÒ¹æ¹ÉáÜáí _ÚÖ«ËÈÛÜÔ ËÂÇµÉÓÙ³á×ÐÒÒïèµÉ YÚØµÛñËÂÛñÊØ èµÊ É³Ò
ØµÊÌË¿Ù³ÊØ ËäÐçÛÜØ ËÂÇµÉ èµÉ YÚØÚÛ²ËÂÛÜÊØ ÊÔ Ð ÞßÉ³
Þ _ ÉÍ Ô×ÖµØµÙËÂÛÜÊØ¤í ËÂÇµÛÜÒ Ô×ÖÚØµÙËÛÜÊØÞ¿ÖµÒîË¿Ç XÐ UÉ Ð Þ®ÐÑËÂÙäÇÚÛñØÚå
"
èµÉ YÚØµÛñËÂÛÜÊØ ÒÊÞßÉÄæ ÇµÉÍÂÉëÉáÜÒÉ ^ÒÉÉ É³ÙËÂÛÜÊØ é é `Äé áñá ÞßÉ³
Þ _ ÉÍòÔ×ÖµØÚÙÄËÂÛÜÊØ èÚIÉ Y ØµÛñËÛÜÊØµÒÈËÂÊåÉÄËÂÇµÉÍ
Ô×ÊÌÍÂÞ ËÂÇµÉ
é
±³ª©¯³ »¨¿¢¾§¨ ³¤î©¯
Ø ìÞßÉ³Þ_ ÉÍÈèµÉ³Ù³á×ÐçÍÂÐçËÂÛÜÊØóÊÔ Ð¿Ù³á×ÐÒîÒ ÛÜÒ·ÒÐÛÜèóËÂÊÀÇÚÐXUÉ
éòê»ËÂÒòèÚÉÙ³á×ÐçÍ
¬
±
×

¯
©
³

±
³

ÐçËÂÛ UÉÍÂÉ³åÛñÊØÝÛÜÒÓËÂÇµÉÀÙ³á×ÐÒîÒâèµÉ YÚØµÛñËÂÛÜÊØ¤é¹á×ÐçÒÒâÒÙÊÌÎ É èµÛ ÉÍÂÒâÔÜÍÊÞ áÜÊ Ù¬Ðá ÒÙ³ÊÌÎ É ^ " É³ÙËÂÛñÊØ \ é é `
ÛÜØ ÊØµÉ ÐÒîÎ ÉÙË¬éÓÆ¹ÇÚÉëÎÊÌËÂÉ³Ø ËÛ×Ðá ÒîÙ³ÊÌÎ ÉàÊÔÃÐÀÞßÉ³Þ_ ÉÍòèÚÉÙ³á×ÐçÍäÐÑËÂÛÜÊØ ÛÜÒ ØµÊÌËÓÊØµáñì ËÂÇµÉëÎÐÑÍË ÊÔ ËÂÇµÉ
Ù³á×ÐçÒÒ èµÉ YÚØµÛñËÂÛÜÊØ N_É³áÜÊ¦æ ãËÇÚÉ·èÚÉÙ³á×ÐçÍäÐÑËÂÛÜÊØ¤í _ÚÖ«Ë Û²ËïÒîÎ ÐØµÒËÂÇµÉ
Ù³á×ÐçÒÒ èµÉ YÚØµÛñËÂÛÜÊØ¤í
ËÂÇµÉ
È
¸

F
Ô×ÊÌÍÂÞ®ÐáÕÐçÍÂåÖµÞßÉ³Ø¯ËëáÜÛñÒîËÂÒëÐçØÚè _ Ê èµÛÜÉ³ÒëÊÔÈÐáÜáÃÞßÉ³
Þ _ ÉÍãÔ×ÖµØÚÙÄËÂÛÜÊØ èµÉ YÚØÚÛ²ËÂÛÜÊØµÒ³é êØ ÒîÇÚÊÌÍîË¬í ÐÙ³á×ÐçÒÒ
ÞßÉ³
Þ _ ÉÍ¹Ù¬ÐçØ _ ÉÏÖÚÒîÉ³
è îÉ UÉÍì æ ÇµÉÍÂÉ ÀÛÜØ ËÇÚÉÏÙ³á×ÐçÒÒ³é
êÔ¾Ë æ¹Ê Ù³á×ÐçÒÒïèµÉ YÚØÚÛ²ËÂÛÜÊØµÒ Ô×ÊÌÍÂÞ èµÛÜÒ ÊÛÜØ¯ËïèµÉ³Ù³á×ÐÑÍäÐçËÂVÛ UÉòÍÂÉ³åÛÜÊØµÒ³í¯ËÂÇµÉÍÂÉÓÛÜÒïØµÊàÎÚÍÊ _ áñÉ³Þ ÛÜØ ÖµÒÛÜØµå
ËÂÇµÉÏÒÂÐÞßÉÓØÚÐÞßÉÓÔ×ÊÌÍ·ÞßÉ³
Þ _ ÉÍÒïÊT
Ô _ ÊÌËÂÇ Ù³á×ÐÒÒîÉ³Ò³é
Class Scope.
\ 4.3.10 Random numbers
SÝÉ·ØÚÊ¦æ Ç ÐXUÉÓÐáñáÚËÂÇµÉòÞßÉ¬ÐØµÒÕËÂÊãÎ Ö«Ë ËÊåÉËÂÇµÉÍïÐãÙ³ÊÞÀÎÚáÜÉËÂÉòÐØµèßÖµÒÉ³Ô×ÖµáÛÜÞÀÎÚáÜÉ³ÞßÉØ ËäÐçËÛÜÊØàÊÔËÂÇµÉ
Ë ìôÎ É
ÐÒòÐ¿Ùá×ÐÒÒòÛÜØ a_ Ö«ËòÒÛÜØµÙ³ÉÓæ¹ÉâÇÚÐXUÉëÐá²ÍÂÉ¬Ðè«ì®ÒÉ³ÉØ ÞßÊÒîËÈÊÔ ËÂÇµÉãØµÉ³ÙÉ³ÒÒÂÐçÍîì
Ù³Ê èÚÉÛÜrational
Ø " É³ÙËÂÛÜÊØ é ÐØµè ÛÜØëËÂÇµÛÜÒ ÒÉÙËÂÛÜÊØ¤íæ¹É áñÉ¬ÐXUÉËÂÇµÉïÔ×ÖµáÜáôÙ³áÜÐÒÒ ÐÒ P¥ÉÍÂÙ³ÛÜÒîÉ \ \ ÐØµèàÙ³ÊØ ËÂÛñØôÖµÉ
ÇµÉÍÂÉÏæ ÛñËÂÇ Ð ÔÜÍÂÉ³ÒÇ Ù³áÜÐÒÒ¹ËÂÇÚÐçË·ÇÚÐÒòÐ áÜÛñËËÂáñÉ ÞßÊÌÍÂÉÓÉ³Ø ËÉÍËäÐÛñØÚÞßÉØ Ë¹ÛÜØóÒËÂÊÌÍÂÉÌé
ÃáÜÐ¬ìôÛÜØµåàåÐÞßÉ³ÒïÊØ®ËÂÇµÉÓÙ³ÊÞÀÎÚÖµËÉÍ æ¹ÊÖµáÜè _ É ÎµÍÂÉËË ìL_ ÊÌÍÛÜØµåëæ ÛñËÂÇµÊÖ«ËïÒÊÞßÉ ÖµØ«ÎµÍÂÉ³èµÛÜÙËÂÐ _ÚÛÜá ÛñË ì ÃÐàÙäÇµÉ³ÒÒ¹ÎÚÍÊåÌÍäÐÞ ÒÇµÊÖµáÜè ØÚÊÌËòÐá²æÈÐ¬ìôÒïÙ³ÊÞßÉ Ö«Îæ·ÛñËÂÇ®ËÂÇµÉÏÒÐÞßÉ ÊáÜè ÞßÊ UÉ³Ò¹ÛÜØ®ÍÂÉ¬ÐçÙËÂÛÜÊØ®ËÂÊ
ìÊÖ«Í ÒÂÐÞßÉãÊáÜè ÞßÊ UÉ³Ò³íÐçØÚè ÛñØ½ÐçØ ÐÙËÂÛÜÊØ åÐçÞßÉÌíÚËÇÚÉâÉ³ØµÉ³ÞßÛÜÉ³ÒÈÒÇµÊÖµáÜè ØÚÊÌË ÐáñæÈÐ³ì¥ÒÈÎ ÊÌÎ Ö«Î ÐÑË
ËÂÇµÉâÒÐÞßÉÏËÛÜÞßÉÏÐØµè áÜÊôÙ¬ÐÑËÂÛÜÊØ¤éêØ ÊÌÍèÚÉÄÍ·ËÂÊßÐçÙ´ÇµÛÜIÉ UÉëÖµØµÎµÍÂÉèÚÛñÙËäÐ _ÚÛÜáÜÛ²Ë ìí¯ËÂÇµÉãÎµÍÂÊåÌÍäÐçÞ Ë ìôÎÚÛÜÙ³ÐáÜáñì
ÖµÒÉ³Ò·Ð
é ÆÈÇµÛÜÒ¹ËÂÉÍÂÞ ÛÜÒÈÞßÛÜÒáÜÉ³ÐèµÛÜØµå«í¯ËÂÇµÊÖµåÇ¤í«ÒÛÜØµÙ³ÉÏËÂÇµÉãØ ÖÚ
Þ _ ÉÍÒ

µ
F

¨
³
³

£
«
Á
³

£
¯
©

£
ÐçÍÂÉÓÛÜØ®ÍÂÉ¬ÐçáÜÛñË ìßåÉ³ØµÉÍäÐÑËÂÉ³èóÐçÙ³Ù³ÊÌÍÂèµÛÜØµåàËÂÊ¿ÒÊÞß
É Y P«É³è ÍÂÖµáÜÉÌí¥ÛÜØ ÒÖµÙäÇóÐëæÈÐ³ìßËÂÇÚÐçËïËÂÇµÉì
ËÂÊ
_ ÉÓÍäÐØµèµÊÞ é Ö«Ë·Ô×ÊÌÍòÞ®ÐØ ìßÎÚÖ«ÍÎ ÊÒîÉ³Ò ^»ÛÜØµÙ³áÜÖµèµÛÜØµåàåÐÞßÉ³Ò `Äí ËÂÇµÛÜÒ¹ÛÜÒòÙÊÞÀÎ áñÉËÂÉ³áñìÀÒÖ Ù³©¬ÛÜ¢«É¢ Ø ´Ë¬©íµ£ ÐØµè
æ¹ÉÏÙ¬ÐáÜá¤ÒÖµÙäÇ ØôÖµ
Þ _ ÉÍÂÒ
é
¢¾³ Fµ£©
Fµ¨
ÒîÛÜÞÀÎÚáÜÉ ÐØµè æ ÛÜèµÉ³áñì ºÖµÒÉ³èßËÂÉÙ´ÇµØµÛ W ÖÚÉ ÊÔ¾åÉËËÛÜØµå Ð ÒîÉXWôÖµÉ³ØµÙ³É
ÊÔ ÎÚÒÉÖÚèµÊÌÍäÐçØÚèµÊÞ ØôÖµÞ_ ÉÍÂÒ¾ÛÜÒ ËÂÇµÉ
é ÛVUÉ³ØàÐ
í
° Ä©£ëÐç±ÄØÚè ÁÐ £¬ ³¤ ©¯Ú¨½¬ºí áÜF ÉËëÖÚÒâÙ³ÊØµÒ¨ÛÜèµÉÍë
²ËÂ¢¾ÇµÉß
î³Ò£ É WôaÖµÉ³ØµÙ³É
í Ð
ÐØ
¬ ÊcÔ¡ØÚÐçËÂÖ«ÍäÐá¤¨½
èµmÉ YÚØµÉ³è _ ì ËÂÇµÉÓÍÂÖµÄáÜI É F x0
ØôÖµ ÞF
_ ÉÍÂÒ¹
Linear congruential generators.
x1 , x2 , . . .
xi = (axi−1 + c)
ÞßÊôè
m,
i > 0.
Ò Þ®ÐçáÜáÉIPµÐÞÀÎÚáÜÉÃÛÜÒ¾ËÂÇµÉÎÚÒÉ³ÖµèµÊÌÍäÐØµèµÊÞ ØôÖµÞ_ ÉÍ åÉ³ØµÉÍäÐçËÂÊÌÍ
íÌèµÉ YÚØµÉ³è _ôìâËÇÚÉÔ×ÊáÜáÜÊ¦æ ÛÜØµå
knuth8
Î ÐçÍäÐÞßÉÄËÂÉÍÂÒ³é
a = 137,
Æ¹ÇÚÉÏÒÉXW ÖµÉ³ØµÙ³É
c = 187,
í
x1 , x2 , . . .
í
í
m = 28 = 256,
x0 = 0.
ÊÔ¡ØôÖµÞ_ ÉÍÂÒïËÂÇÚÐçËÈæ¹ÉãåÉÄË·ÔÜÍÂÊÞ ËÂÇµÛÜÒ¹ÛÜÒ
í
í
í
í
í
í
í
í
í
í
í
í
187 206 249 252 151 138 149 120 243 198 177 116 207 130 77
í í
í
í
í í
í í
í í
í í
í í
í
í
í
í
240 43 190 105 236 7 122 5 104 99 182 33 100 63 114 189 224 155
í
í
í
í
í
í í
í
í
í í
í í í
í í
í
174 217 220 119 106 117 88 211 166 145 84 175 98 45 208 11 158
í
í
í í
í í í
í í í í í
í
í
í
í
í
í
73 204 231 90 229 72 67 150 1 68 31 82 157 192 123 142 185 188
í í í í
í
í
í í
í í í
í
í
í í
í
í í
87 74 85 56 179 134 113 52 143 66 13 176 235 126 41 172 199 58
í í í
í
í í
í í
í
í í
í
í
í í í í í
197 40 35 118 225 36 255 50 125 160 91 110 153 156 55 42 53 24
í
í í í
í í
í
í
í í í
í
í í
í í í í
í
147 102 81 20 111 34 237 144 203 94 9 140 167 26 165 8 3 86 193
í
í í í
í í í
í
í í í í
í
í í í
í í í
4 223 18 93 128 59 78 121 124 23 10 21 248 115 70 49 244 79 2
í
í
í í
í
í
í
í
í
í
í í
í
í
í
í í
205 112 171 62 233 108 135 250 133 232 227 54 161 228 191 242 61
í í í í í
í
í
í
í í í í
í í
í
í í
í
96 27 46 89 92 247 234 245 216 83 38 17 212 47 226 173 80 139
í
í í
í
í
í
í
í í
í
í
í
í í í
í í
30 201 76 103 218 101 200 195 22 129 196 159 210 29 64 251 14
í í
í
í
í
í í í
í
í í
í
í í
í
í
í í
57 60 215 202 213 184 51 6 241 180 15 194 141 48 107 254 169 44
í
í í
í
í
í í
í
í
í
í í
í
í í í
í
71 186 69 168 163 246 97 164 127 178 253 32 219 238 25 28 183
í
í
í í
í
í
í
í
í
í í í
í
í í í
í
170 181 152 19 230 209 148 239 162 109 16 75 222 137 12 39 154
í
í
í
í í
í í
í
í í
í é¬é³é
37 136 131 214 65 132 95 146 221 0 187
\
[ ÍÂÊÞðÇµÉÍÉãÊØ¤í¥ËÂÇµÉÏÒÉXWôÖµÉ³ØµÙ³ÉÏÍÂÉÎ É³ÐçËÂÒÈÛñËÂÒÉ³áñÔ ^ÛñØåÉ³ØµÉÍäÐçáí¥ËÂÇµÉ Î ÉÍÂÛÜÊ èÙ¬ÐçØóØµÉ UÉÍ_ ÉãáñÊØÚåÉÄÍ
ËÂÇÚÐØ
´` é Ö«ËÏÖµØ ËÂÛñá ËÂÇµÛÜÒ Î ÊÛÜØ Ë³í¾Û²ËÏÐçÎµÎ É¬ÐçÍÂÒÓËÂÊ_ É ÎÚÍÉËË ì ÍäÐØµèµÊÞ ^ºÐáñËÂÇµÊÖµåÇ½Ð Ù³áñÊÒÉÍÏáÜÊôÊ ]
m
ÍÂÉ U É³ÐáÜÒïËÂÇÚÐçËÈÛñË¹ÛÜÒ¹ØµÊÌË¹ÍäÐØµèµÊÞ ÐçËòÐáÜá èÚÊ ìÊÖóèµÛÜÒÙÊ UÉÍòÐàÒîËÍÛV]¥ÛÜØµå ÒÛÜåØ ÊÔ¡ØµÊØ«ÍäÐØµèµÊÞßØµÉ³ÒÒ `Äé
êØÊÌÍÂèµÉÍÈËÂÊ Þ®Ð ] É ËÂÇµÉÓÞ®ÐåØµÛñËÂÖµèµÉ·ÊÔ¤ËÂÇµÉÏÍÂÐØµèÚÊÞ Ø ÖµÞb_ ÉÄÍÂÒïÛÜØµèµÉÎ É³ØµèµÉ³Ø ËÈÔÜÍÂÊÞ ËÂÇµÉÓÞßÊôè
ÖµáÜÖµÒ³í«ÛñËòÛÜÒ¹Ù³ÊÞßÞßÊØ®ÎÚÍÂÐÙËÂÛÜÙÉÓËÊ
ËÇÚÉÏØôÖµÞ_ ÉÍÂÒÈÒÊ¿ËÂÇÚÐçË¹ËÂÇµÉìóÐÑÍÂÉÏÍÉ¬Ðá¤ØôÖµÞ_ ÉÍÂÒÈÛÜØ

¬
£
½
¨
¯
©

é
ËÂÇµÉÏÛÜØ ËÂÉÄÍ ÌU Ðçá
[0, 1)
ÍÊåÌÍäÐÞ _ É³áñÊÑæ¼Ù³ÊØ¯ËäÐÛÜØµÒ ËÂÇµÉ·èµÉ Ú
Y ØµÛñËÂÛÜÊØ¿ÊÔ ÐÓÙ³á×ÐÒîÒ
ÛÜØÀØÚÐÞßÉ³ÒîÎ ÐÙÉ
Ô×ÊÌÍåÉ³Ø
random
ifm
ÉÍäÐÑËÂÛÜØµåãØµÊÌÍÂÞ®ÐçáÜÛ³É³èàÎÚÒÉ³ÖµèµÊÌÍäÐØµèµÊÞ ØôÖµÞb
_ ÉÍÂÒÃÐÙ³Ù³ÊÌÍèÚÛñØÚåÓËÂÊÏËÂÇµÉòáÜÛñØÚÉ³ÐçÍÃÙ³ÊØµåÌÍÂÖµÉ³Ø ËÂÛÜÐá«ÞßÉËÂÇµÊôè¤é
Æ¹ÇÚÉÄÍÂÉßÛÜÒ ÐóÙ³ÊØµÒîËÍÖÚÙÄËÂÊÌÍàËÂÇÚÐçË ÐáñáÜÊÑæ·ÒãËÂÇµÉßÙ³ÖµÒîËÊÞßÉÍëËÂÊ ÎµÍÂÊ¥U ÛÜèµÉ ËÂÇµÉÀÎ ÐçÍäÐçÞßÉËÂÉÍÂÒ
í
a, c, m, x0
ÐØµè½Ð ÞßÉ³Þ_ ÉÍ Ô×ÖµØµÙËÂÛñÊØ
ËÂÊ®åÉË ËÂÇµÉëÍÂÉ³ÒÎ É³ÙËÂÛ U É¿ØµÉIôP ËÓÉ³áÜÉ³ÞßÉØ Ë ÛñØ ËÇÚÉà
ÒîÉXôW ÖµÉ³ØµÙ³É
ÊÔ¡ËÇÚÉ
xi
é
operator()
// Prog : random .h
// define a class for pseudorandom numbers .
namespace ifm {
// class random : definition
class random {
public :
// POST : * this is initialized with the linear congruential
//
random number generator
//
x_i = ( a * x_ {i -1} + c ) mod m
//
with seed x0 .
random ( unsigned int a , unsigned int c ,
unsigned int m , unsigned int x0 );
// POST : return value is the next pseudorandom number
//
in the sequence of the x_i , divided by m
double operator ()();
private :
const unsigned int a_ ; // multiplier
const unsigned int c_ ; // offset
const unsigned int m_ ; // modulus
unsigned int xi_ ;
// current sequence element
};
} // end namespace ifm
Program 36: ¢¤£³ÄÁ £©Fµ¨ Å×
ÆÈÇµÉÓÔ×ÖµØµÙËÂÛÜÊØ
ÇÚÐÒïØµÊ¿ÐçÍÂåÖµÞßÉ³Ø¯ËÂÒÛÜØ ÊÖ«ÍïÙ¬ÐÒîÉb^°ËÂÇÚÐçË! ÒÕæ Ç ìÀÛñËÂÒïèµÉ³Ù³áÜÐçÍäÐçËÂÛñÊØÀÛÜÒ
í æ operator()
ÇµÛÜÙäÇ ÐèµÞßÛñËËÂÉ³èµáñì®áñÊôÊ ]¥Ò·Ð _ÚÛñË Ô×ÖµØµØ ì `Äí ÐØµè ÛñË ÊUÉÍÂáñÊÐèµÒ ËÂÇµÉ
operator()()
îÉ Ù³á×±ÄÐç©¯Ò°Ò
í«ÒÉ³ÉãÆ Ð _ÚáÜÉ ÛÜØßËÂÇµÉ ·ÎµÎ É³ØµèµÛ Pé êØ åÉ³ØµÉÍÂÐáí¥ÛÜÔ
ÛÜÒ¹ÐØ ÉIPôÎÚÍÉ³ÒÒÛÜÊØ ¶Ê Ô¾ Ò±¬ÊÞß
£Ä©¯æ Çµ£ ÛÜÙäÇóÇÚÐÒ¹ËÂÇµÉÏÞßÉ³Þb_ÉÍ¹Ô×ÖµØÚÙÄËÂÛÜÊØ
Ë ³ìô¢ Î ³ÉÓ
operator
\ ^`I^ I
íñé²éñé²í Ù¬ÐØ _ÉãÖµÒÉ³è
^ íñé²éñéñí G `
ËÂÇµÉ³Ø
áñÛV]ÉÏÐ¿Ô×ÖµØµÙËÂÛÜÊØ
`
ËÂÇµÉâÉGP¥ÎµÍÂÉ³ÒÒîÛÜÊØ
ÛÜÒ¤ÉXW ÖÚÛ UÌÐáñÉ³Ø Ë ËÂÊòÐïÙ¬ÐáÜáÊÔ ËÂÇµÉ ÞßÉ³Þ_ ÉÍÔ×ÖµØµÙËÂÛñÊØ
æ Û²ËÂÇÏÐçÍÂåÖµÞßÉ³Ø¯ËÂÒ
í²é³é³é³í G
Ô×ÊÌÍÈËÂÇµÉâÉIPôÎÚÍÉ³ÒÒÛÜÊØ
éSÝÉÏæ ÛÜáÜáÒÉ³ÉâÒîÖÚÙäÇ Ù¬ÐçáÜoperator()
áÜÒ¹ÛÜØ ÍÂÊåÌÍäÐçÞ ÐØµè ÍÂÊåÌÍäÐÞ
_É³áÜÊ¦æëé
·ÉÍÂÉ ÛñÒÀËÂÇµÉóÛÜÞÀÎÚáÜÉ³ÞßÉ³Ø¯ËäÐçËÂÛñÊØ ÊÔ ËÂÇµÉ Ù³áÜÐÒÒ
ÛÜØ æ ÇµÛÜÙäÇ æ¹ÉóÒÉÉ ÇµÊ¦æ
random
operator()
Ö«Î èÚÐçËÂÉ³ÒÀËÇÚÉ UÐáÜÖµÉóÊÔÓËÂÇµÉ èÚÐçËäÐ ÞßÉÞb_ ÉÄÍ
ËÂÊ _ ÉóËÂÇµÉóÍÂÉ³ÒîÎÉ³ÙËÂÛVUÉ ØµÉIPôË®É³áÜÉ³Þß
ÉØ ËßÛÜØ ËÂÇµÉ
x_i
ÒXÉ W ÖµÉ³ØµÙ³ÉëÊÔ¡ËÇÚÉ é
xi
// Prog : random .C
// implement a class for pseudorandom numbers .
# include < IFM / random .h >
namespace ifm {
// class random : implementation
random :: random ( unsigned int a , unsigned int c ,
unsigned int m , unsigned int x0 )
: a_ ( a ) , c_ (c ) , m_ (m ) , xi_ ( x0 )
{}
double random :: operator ()()
{
// update xi acording to formula ,...
xi_ = ( a_ * xi_ + c_ ) % m_ ;
// ... normalize it to [0 ,1) , and return it
return double ( xi_ ) / m_ ;
}
} // end namespace ifm
Program 37: ¢¤£Á £©Fµ¨ Å
)óÐØ ì½Ù³ÊÞßÞßÊØµáñìÖÚÒîÉ³è ÍäÐØµèµÊÞ ØôÖµÞ_ ÉÍãåÉ³ØµÉÍäÐçËÂÊÌÍÒâÐçÍÉÊ _µËäÐÛñØÚÉè ÛÜØ½ÉIPµÐÙÄËÂáñì ËÂÇµÛÜÒÓæÈÐ¬ìé
[ ÊÌÍ ÉIP«ÐÞÀÎÚáÜÉÌíÌËÂÇµÉ¹æ¹É³áÜá ]¥ØµÊ¦æ Ø åÉØÚÉÄÍäÐçËÂÊÌÍ
ÍÂÉËÖµÍØÚÒÃÎÚÒÉ³ÖµèµÊÌÍäÐØµèµÊÞ Ø ÖµÞb_ ÉÄÍÂÒÕÛñØ
a = 25214903917,
[0, 1)
drand48
ÐÙ³ÙÊÌÍÂèµÛÜØµå¿ËÂÊ¿ËÂÇµÉÏÎ ÐçÍÂÐÞßÉËÂÉÍÒ
c = 11,
m = 248 ,
ÐØµè ÐÒîÉ³É³è ÙäÇÚÊÒîÉ³Ø _ ì ËÇÚÉâÙ³ÖµÒîËÂÊÞßÉÍ¬é ê»Ë ÛÜÒòÙ³áÜÉ¬ÐçÍòËÂÇÚÐçË·æ¹ÉãØµÉ³É³è Ðá×ÐÑÍÂåÉâÞßÊ èÚÖµáÜÖµÒïËÂÊÀÊ _µËäÐçÛÜØ
Ð ÖÚÒîÉ³Ô×Öµá¡åÉ³ØµÉÍÂÐçËÂÊÌÍ¬í«ÒÛñØÚÙÉ
ÛñÒ·ÐØ Ö«ÎÚÎÉÍ_ÊÖÚØµè Ô×ÊÌÍòËÂÇµÉÏØôÖµÞ_ ÉÍÈÊÔ èµÛ ÉÍÂÉ³Ø¯Ë·Ø ÖÚÞ_ ÉÍÒ¹ËÇ ÐÑË
-
A%+
.*
"@*
6Q7
(
JQ#*
random
Q$&BK
:
22
m
H%*
6L.1
6
A$&BLO6I
,$
unsigned
int
(7.#* %Q 8{0, BI. . . , 232 −1} *
/
double
J
#"5
*
-."
! (. 7 6 *
+(7*
6
7
8"L/
%(2!
*
6
%(7
\\
æ ÉÏÙ¬ÐØÎ ÊÒîÒÛV_Úáñì®åÉËòÔÜÍÂÊÞ ËÂÇµÉãåÉØÚÉÄÍäÐçËÂÊÌÍ¬éÕÆ¹ÇÚÛñÒÈÞßÉ³ÐØµÒ¹ËÇ ÑÐ Ë
¹
ÔÜÍÂÊÞðÐ _ ÊUÉãÛÜÒ¹ÍäÐçËÇÚÉÄÍ·Ð
knuth8
ËÂÊ¦ì åÉ³ØµÉÍäÐçËÂÊÌÍ³é
·ÉÄÍÂÉòÛñÒÕÐÓåÐÞßÉËÇ ÐÑËÃìÊÖ Ù³ÊÖµáÜè¿ÎÚá×Ð³ìàæ·ÛñËÂÇ
ì ÊÖµÍÕÔÜÍÂÛÜÉØÚè¿æ ÇµÛÜáÜÉ
The game of choosing numbers.
æÈÐÛñËÛÜØµåÔ×ÊÌÍ ÐèµÉ³á×Ð³ìÉ³è ËÍäÐçÛÜØ¤éÃ
ÐÙäÇ ÊÔ ìÊÖ ÛñØÚèµÉÎÉ³ØµèÚÉØ ËÂáñì®æòÍÂÛñËÂÉ³ÒÈèµÊÑæ·Ø ÐØ ÜÛ Ø¯ËÂÉ³åÉÍ_ ÉË æ¹É³É³Ø
ÐçØÚè é Æ¹ÇÚÉØóËÂÇµÉâØôÖµÞ_ ÉÍÂÒÈÐçÍÉâÙ³ÊÞÀÎ ÐçÍÂÉè¾éÕêÔ ËÂÇµÉìÐçÍÂÉãÉXWôÖÚÐá»íµËÇÚÉÏåÐÞßÉÓÛÜÒ·Ð¿è«ÍäÐ³æëéêÔ ËÂÇµÉ
ôØ ÖµÞ_ É6ÍÂÒ èµÛ ÉÍ_ôì¿ÊØµÉÌí¯ËÇÚÉïÎÚá×Ð³ìÉÍÃæ ÛñËÇ¿ËÂÇµÉÈÒÞ®ÐáÜáñÉÍ ØôÖµÞ_ ÉÍ åÉËÒ [ \ ÔÜÍÂÊÞ ËÂÇµÉÈÊØÚÉ¹æ Û²ËÂÇ
ËÂÇµÉÏá×ÐçÍÂåÉÍÈØ ÖÚÞ_ ÉÍ³éÕêÔ¾ËÂÇµÉÏË æ¹ÊØôÖµÞb_ÉÍÂÒ¹èµÛ ÉÍ_ôì®Ë æ¹Ê¿ÊÌÍ·ÞßÊÌÍÂÉí¥ËÂÇµÉÏÎÚá×Ð³ìÉÍÈæ ÛñËÇ ËÂÇµÉÏá×ÐçÍÂåÉÄÍ
ØôÖµ
Þ _ ÉÍãåÉËÂÒ [ ÔÜÍÂÊÞ ËÂÇµÉÊØµÉ¿æ ÛñËÂÇ ËÂÇµÉ ÒÞ®ÐçáÜáÜÉÍÓØ ÖÚÞ_ ÉÍ³é ÊÖÝÙ¬ÐØ½ÍÂÉÎ É¬ÐÑËâËÂÇµÛÜÒÏÖµØ ËÂÛÜá
ËÂÇµÉÓËÍäÐÛÜØÐçÍÍÂÛ UÉ³Ò ^»ÊÌÍ·ÖµØ ËÛÜá ÊØµÉãÊÔ ìÊÖ ÍÂÖµØÚÒòÊÖ«ËòÊÔ Ù¬ÐÒÇ¤íÚÐØµèÇµÊÌÎ É³Ô×ÖµáÜá²ì®ÛñË! ÒïìÊÖ«Í ÔÜÍÛÜÉ³Øµè `Äé
êÔ ìÊÖ ËÂÇµÛÜØ ] Ð _ ÊÖ«ËëÇµÊÑæ ËÂÊÎÚá×Ð³ì ËÂÇµÛÜÒÏåÐÞßÉÌí Û²!Ë ÒÏØµÊÌËëÊ _ UôÛÜÊÖµÒãæ·Ç ÐÑËãËÂÊèÚÊ«é ØµÉ ËÂÇµÛÜØµå
Ê _ UôÛÜÊÖµÒ³í ËÂÇµÊÖµåÇ ìÊÖ½ÒîÇÚÊÖµáÜè ØµÊÌË æòÍÂÛñËÂÉâèÚÊ¦æ ØóËÇÚÉàÒÐÞßÉâØ ÖÚ
Þ _ ÉÍ ÛÜØ É UÉÍì ÍÊÖÚØµè¤íÒîÛÜØµÙ³É
ËÂÇµÉ³Ø ìÊÖ«ÍÔÜÍÂÛÜÉ³ØµL
è W ÖÚÛñÙ ]ôáñì¿áÜÉ¬ÐçÍØÚÒ ËÂÊãIÉ P¥ÎÚáÜÊÛ²ËÃËÇÚÛñ
Ò _ ì¿æòÍÂÛñËÂÛñØÚåÓèµÊ¦æ ØßÐÓØôÖµ
Þ _ ÉÍ ËÂÇÚÐç
Ë _ É¬ÐçËÂÒ ìÊÖ«Í
ØôÖµ
Þ _ ÉÍ ^ _ ì èµÉ³ÒîÛÜåØóÊÔ¡ËÇÚÉÏåÐÞßÉÌí¥ËÇÚÛñÒÈÛñÒ·Ðá²æÈÐ¬ìôÒïÎ ÊÒÒVÛ _ÚáÜKÉ `Äé
ÊÖ ÒÇµÊÖÚáñè ËÂÇµÉÍÂÉ³Ô×ÊÌÍÂÉ ÐçèÚè ÒîÊÞßÉ ÖµØ«ÎÚÍÉ³èµÛÜÙËäÐ _ÚÛÜáñÛñË ì ËÂÊìÊÖ«Í ÙäÇµÊÛÜÙ³É³Òé
ÊÖ Ù³ÊÖµáÜè¤íÓÔ×ÊÌÍ
IÉ P«ÐÞÀÎÚáÜÉÌíÒÉÙÍÂÉËÂá²ì ÍÂÊáÜá ÐÀèµÛÜÙ³É ÛÜØ É UÉÍîì ÍÊÖÚØµè½ÐØµè æòÍÂÛñËÂÉàèµÊ¦æ Ø ËÂÇµÉ ØôÖµb
Þ _ÉÍ ËÂÇÚÐçË ÛñËÓÒîÇÚÊ¦æ Òé
Ö«Ë P«ÉÄÍÂÙ³ÛÜÒÉ \ ÍÉ UÉ¬ÐáÜÒÈËÂÇÚÐçË¹ìÊÖ«Í·ÔÜÍÂÛñÉ³ØµèóÙ¬ÐØ IÉ PôÎÚáÜÊÛñËïËÂÇµÛÜÒÈÐÒÈæ¹É³áÜáé
ÊÖ ÞÖµÒîËÈÒÊÞßÉ³ÇµÊ¦æ Y ØµÉËÖÚØµÉÏìÊÖµÍòÍäÐØµèµÊÞðÙäÇµÊÛÜÙ³ÉÒ³í _ÚÖ«Ë ÇµÊ¦æ êØ ÊÌÍèÚÉÄÍ·ËÂÊÀGÉ P¥Î ÉÍÛÜÞßÉ³Ø Ë
æ Û²ËÂÇèµÛ ÉÍÂÉ³Ø¯Ë èµÛÜÒîËîÍÂVÛ _ÚÖ«ËÂÛÜÊØµÒ³í ìÊÖ èµÉ³Ù³ÛÜèµÉ ËÂÊ èµÉ YÚØµÉÐçØÚèÛÜÞÀÎÚáÜÉ³ÞßÉ³Ø¯Ë Ð Ùá×ÐÒÒ
ËÂÇÚÐçË ÍÊáÜáÜÒòËÇÚÉàèµÛÜÙÉàÛÜØ ÒÖµÙ´Ç½Ð æÈÐ³ìËÂÇÚÐçË·ËÂÇµÉëÎÚÍÊ _Ð _ ÛñáÜÛñË ì Ô×ÊÌÍÓØôÖµb
Þ _ÉÍ ËÂÊßÙ³ÊÞßloaded_dice
ÉëÖ«Î ÛÜÒ XÉ W Ö Ðçá
i
ËÂÊ Ð ÎµÍÂÉ³ÒîÎ ÉÙ³VÛ YÚÉ³è UÐáÜÖµÉ
^ºÐ½ÔªÐÛ²ÍÀèµÛÜÙ³É Ç ÐçÒ
Ô×ÊÌÍßÐáÜá
`Äé Æ¹ÇµÉ³Ø ìÊÖ
pi = 1/6
. . , 6}
Ù³ÊÖµáÜè áÜÉÄËàèµÛ ÉÍÉ³Ø ËàáñÊÐèµÉ³pè i èµÛÜÙ³É³ÒâÎÚá×Ð³ìÝÐåÐÛÜØµÒîËã
É¬ÐÙäÇ ÊÌËÂÇµÉÍ¬í ÐØµè iÛÜØÝËÂ{1,
ÇµÛÜÒã. æÈ
Ð¬ì èµÛÜÒÙ³Ê UÉÍ ÒÖµÛñË Ð _ÚáÜÉÏÎÚÍÊ _Ð _ ÛñáÜÛñËÂÛÜÉÒ¹ËÊÀÖµÒÉ ÐåÐÛÜØµÒîËÈìÊÖ«Í ÔÜÍÂÛÜÉØÚè ^»æ·ÇµÊÀÛÜ
Ò _ ìËÂÇµÉâæÈÐ³ìØÚÊÌË ÒîËÂÖµè«ì¥ÛÜØµå ÙÊÞÀÎ Ö«ËÂÉÄÍ
ÒÙ³ÛñÉ³ØµÙ³É `Äé
ÍÊåÌÍäÐÞ ÒÇµÊÑæ·Ò Ð ÒÖµÛñËäÐ _ÚáÜÉÏÙ³á×ÐÒîÒ·èµÉ YÚØµÛñËÂÛÜÊØ ^»ËÂÇÚÐçËòÛÜØóËÂÖ«ÍÂØÍÉ³áÜÛÜÉ³ÒòÊØËÂÇµÉâÙá×ÐÒÒ
ÔÜÍÂÊÞ Ð _ Ê UÉÌí æ ÛñËÇ¼ËÂÇµÉ½ØÚÊÌÍÞ®ÐáÜÛ ¬ÐÑËÂÛÜÊØ ËÂÊ ËÂÇµÉ½ÛÜØ¯ËÂÉÍ UÐá
`Äé SÝÉ½æ·ÛÜáÜá åÉÄËËÂÊ ËÂÇµrandom
É½Ù³áÜÐÒÒ
ÛÜÞÀÎÚáÜÉ³ÞßÉØ ËäÐçËÛÜÊØ ^ºÐØµè ËÂÇµÉãÞßÉ¬ÐçØÚÛñØÚå ÊÔ¾ËÂÇµÉãèÚÐçËäÐ ÞßÉ³
Þ _ ÉÍÂ[0,
Ò `Õ1)
ÛÜØ ÍÂÊåÌÍäÐÞ _ É³áÜÊ¦æëé
1
# include < IFM / random .h >
// Prog : loaded_dice.h
// define a class for rolling a loaded dice .
namespace ifm {
// class loaded_dice : definition
class loaded_dice {
public :
// PRE : p1 + p2 + p3 + p4 + p5 <= 1
// POST : * this is initialized to choose the number
//
i in {1 ,... ,6} with probability pi , according
//
to the provided random number generator; here ,
//
p6 = 1 - p1 - p2 - p3 - p4 - p5
loaded_dice ( double p1 , double p2 , double p3 , double p4 ,
double p5 , ifm :: random & generator);
// POST :
//
//
unsigned
\ return value is the outcome of rolling a loaded
dice , according to the probability distribution
induced by p1 ,... , p6
int operator ()();
private :
// p_upto_i is p1 + ... + pi
const double p_upto_1 ;
const double p_upto_2 ;
const double p_upto_3 ;
const double p_upto_4 ;
const double p_upto_5 ;
// the generator ( we store an alias in order to allow
// several instances to share the same generator)
ifm :: random & g;
};
} // end namespace ifm
Program 38: ¢¤£Á ª© FµIF F±ÅÜ
Æ Ê ÛÜØµÛñËÂÛ×ÐçáÜÛ³ÉàËÂÇµÉßáÜÊÐèµÉ³è èµÛÜÙ³Éí æ¹ÉßÇÚÐXUÉÀËÂÊ ÎµÍÂÊU¥ÛÜèµÉÀËÂÇµÉÀÎµÍÂÊ _ Ð _ÚÛÜáÜÛñËÛÜÉ³Ò
^
p 5 p6 =
P5
`´í¤ÐØµè½ËÂÇµÉàÍäÐØµèµÊÞ ØôÖµÞ_ ÉÍÓåÉ³ØµÉÍÂÐçËÂÊÌÍÓËÂÇÚÐçËÓÛÜÒ_ ÉÛÜØµå ÖµÒÉè½ËÂÊ pÐ1Ù,ËÂ.ÖÚ.Ð.áñ,áñì
ÍÂÊáñá ËÂÇµÉ
1 − i=1 pi
ËÂÊâÍÂÉ¬ÐçáÜÛ³ÉÈËÂÇµÉ·Ô×ÖµØµÙËÂÛÜÊØÚÐáÜÛ²Ë ì ÊÌÔÍÂÊáÜáÜÛÜØµåÓËÂÇµÉ èµÛÜÙ³ÉòÊØÚÙÉÌé
èµÛÜÙ³ÉÌé ·åÐÛÜØ¤í¯æ¹ÉòÊ UÉÄÍÂáÜÊÐè
operator()
·Ê¦æ èÚÊÈæ¹ÉÕÛñÞÀÎ áñÉ³ÞßÉ³Ø Ë ËÂÇµÛÜÒ¾Ô×ÖÚØµÙËÛÜÊØÚÐáÜÛñË ì S ÉÕÎ ÐçÍËÛñËÂÛÜÊØÓËÂÇµÉÕÛñØ ËÂÉÍ UÌÐá
ÛÜØ ËÂÊ ÍÂÛÜåÇ Ë ºÊÌÎ É³Ø
[0, 1)
6
ÛÜØ¯ËÂÉÍ UÐáÜÒ³í¥æ·ÇµÉÍÂÉÏÛÜØ ËÂÉÄÍ UÌÐçá ÇÚÐÒÈáÜÉ³ØµåÌËÂÇ
i
p1
p2
pi
p3
p1 + p 2
0
p4
x
p1 + p 2 + p 3
p5
p6
1
Æ ÇµÉ³Ø æ¹É è«ÍäÐ³æ ÐëØôÖµÞ_ ÉÍ ÐçËïÍäÐØµèµÊÞ ÔÜÍÂÊÞ
È
í ÖÚÒîÛÜØµåàÊÖµÍ¹åÉØÚÉÄÍäÐçËÂÊÌÍ¬éÃêÔ¤ËÂÇµÉ ØôÖµÞ_ ÉÍ
x
[0, 1)
ËÂÇÚÐçË¡æ¹É åÉË æ¹ÉÍÉËÍÂÖµáñìÏÍäÐØµèµÊÞ íÑËÇÚÉØ¿Û²Ë æ¹ÊÖµáÜèàÉ³Øµè Ö«Î ÛÜØàÛÜØ ËÉÍ UÐá æ Û²ËÂÇëÎµÍÂÊ _ Ð _ÚÛÜáÜÛñË ì ÉIP«ÐÙËÂá²ì
é ·ØµèµÉÍ ËÂÇµÉòÐÒÒÖµÞÀÎµËÂÛÜÊØàËÂÇÚÐçËÃÊÖ«Í Î ÒîÉ³ÖµèÚÊÌÍÂÐØµèÚÊÞ Ø ÖÚÞ_ ÉÍÒ_ É³ÇÚiÐXUÉ·áÜÛV]É ÍÂÐØµèÚÊÞ Ø ÖÚÞ_ ÉÍÒ
pi
ÛÜØÐ¿ÒÖµÛñËÂÐ _ÚáÜÉ æÈÐ³ìíµæ¹ÉÓËÂÇµÉÍÂÉÔ×ÊÌÍÂÉãèµÉ³Ùá×ÐçÍÂÉ ÐÒ¹ËÂÇµÉãÊÖ«ËÂÙÊÞßÉÏÊÔ¾ÍÂÊáÜáÜÛñØÚåëËÇÚÉÏèµÛÜÙ³ÉÏÛÜÔ ÐçØÚè ÊØµáñì®ÛÜÔ
i
É³ØµèµÒòÖ«Î ÛñØóÛÜØ ËÉÍ UÐá éÕÆ¹ÇÚÛñÒÈÛñÒÈËÂÇµÉÏÙ¬ÐÒÉã
ÛÜÔ ÐçØÚè ÊØµáñì ÛÜÔ
x
i
p1 + . . . + pi−1
x < p1 + . . . + pi .
ÆÈÇµÛÜÒâÉIP¥ÎÚá×ÐÛñØÚÒãËÂÇµÉÀèÚÐçËäÐÞßÉ³Þ_ ÉÍÒ
íñé³éé³í
^°æ¹ÉÀèµÊØ ËàØµÉ³É³è
p_upto_1
p_upto_5
^
`òÐØµè
^
` `ÄéÈÆ¹ÇÚÉâÙ³ÊØµÒîËÍÂÖµÙËÂÊÌÍ ÛÜØ ÍÊåÌÍäÐÞ ÒÛÜÞÀÎÚáñì®ÒÉËÒ ËÂÇµÉ³ÒÉëp_upto_0
ÞßÉ³Þ_ ÉÍÒ
=0
p_upto_6 = 1
ÔÜÍÂÊÞ ËÂÇµÉòèÚÐçËäÐ ÎµÍÂÊ U¥ÛÜèµÉ³è¤í ÐØµè ËÂÇµÉ·ÛÜÞÀÎÚáÜÉÞßÉ³Ø ËäÐÑËÂÛÜÊØëÊÔ
ÖµÒÉ³ÒÃËÂÇµÉ³Þ ÛñØÀÉIPµÐçÙËÂáñì ËÂÇµÉ
æÈÐ³ì®ËÂÇÚÐçË¹æÈÐÒòÉØ U¥ÛñÒÛÜÊØµÉ³è _ ì ËÇÚÉÓÎµÍÂÉ UôÛÜÊÖµÒòXÉ W ÖÚÐçËÂÛÜÊØ¤é operator()
\
// Prog : loaded_dice.C
// implement a class for rolling a loaded dice .
# include < IFM / loaded_dice.h >
namespace ifm {
// class loaded_dice : implementation
loaded_dice:: loaded_dice
( double p1 , double p2 , double p3 , double p4 , double p5 ,
ifm :: random & generator)
: p_upto_1 ( p1 ) ,
p_upto_2 ( p_upto_1 + p2 ) ,
p_upto_3 ( p_upto_2 + p3 ) ,
p_upto_4 ( p_upto_3 + p4 ) ,
p_upto_5 ( p_upto_4 + p5 ) ,
g ( generator)
{}
unsigned int loaded_dice :: operator ()()
{
double x = g ();
if ( x <= p_upto_1 ) return 1;
if ( x <= p_upto_2 ) return 2;
if ( x <= p_upto_3 ) return 3;
if ( x <= p_upto_4 ) return 4;
if ( x <= p_upto_5 ) return 5;
return 6;
}
} // end namespace ifm
Program 39: ¢¤£³Á §© FµGF Fî±çÅ
O Ê¦æ ìÊÖÀÙ¬ÐØÀÙÊÞÀÎÐÑÍÂÉ Ë æ¹ÊãèµÛ ÉÍÂÉ³Ø¯ËáÜÊÐçèÚÉèèµÛÜÙ³ÉÒÃËÂÊ
YÚØµèÀÊÖ«ËÃæ·ÇµÛÜÙäÇ ÊØµÉ¹ÛÜÒ_ÉËËÂÉÍÃÛÜØ¿ËÂÇµÉ
åÐÞßÉ ÊÔ¾ÙäÇµÊôÊÒÛÜØµå ØôÖµÞ_ ÉÍÂÒé ÍÂÊåÌÍÂÐÞ
èµÊ É³Ò ËÂÇµÛÜÒ³í«ÐçÒÒÖµÞßÛÜØµåëËÂÇÚÐçË ìÊÖÐçÍÂÉÓÖµÒÛÜØµå¿ÐëáÜÊÐèµÉ³è
èµÛÜÙ³ÉëËÂÇÚÐçËòÎµÍÂÉ³Ô×ÉÍÂÒÏá×ÐçÍåÉÍ Ø ÖÚÞ_ ÉÍÒ³í ÐØµè ìÊÖ«ÍÓÔÜÍÂÛÜÉ³Øµè ÖÚÒîÉ³ÒÓÐÀáÜÊÐèµÉ³è èµÛÜÙ³ÉâËÇ ÐÑË ÒîËäÐ³ì¥Ò ÞßÊÌÍÉëÛÜØ
ËÂÇµÉâÞßÛñèÚèµáÜÉéê»ËòËÂÖ«ÍÂØµÒ ÊÖ«ËòËÂÇÚÐçË·ÛÜØËÂÇµÛÜÒòÒÉËËÛÜØµå«íµìÊÖóæ·ÛÜØóÛÜØ ËÂÇµÉëáÜÊØµåÍÂÖµØ¤ía_ Ö«Ë·ØÚÊÌËô_ ìÞÖµÙäÇ
^ [
ÊØ ÐKU ÉÄÍäÐåÉ Î ÉÍÏÍÂÊÖµØÚè Ä` é ¥P ÉÍÂÙ³ÛÜÒîÉ \ Ù´ÇÚÐáñáÜÉ³ØµåÉ³Ò ìÊÖ ËÂÊÚY Øµè½ËÂÇµÉ_ É³ÒîËÏáÜÊÐèµÉ³è
0.12
èµÛÜÙ³ÉÓËÂÇÚÐçË¹ìÊÖóÙ³ÊÖµáÜèÎÊÒÒÛVÚ_ áñì®ÖµÒÉÏÛÜØËÂÇµÛÜÒ¹åÐÞßÉé
// Prog : choosing_numbers.C
// let your loaded dice play against your friend ’s dice
// in the game of choosing numbers .
# include < iostream >
# include < IFM / loaded_dice.h >
\
// POST : return value is the payoff to you ( possibly negative ) ,
//
given the numbers of you and your friend
int your_payoff ( unsigned int you , unsigned int your_friend)
{
if ( you == your_friend) return 0;
// draw
if ( you < your_friend ) {
if ( you + 1 == your_friend ) return 2; // you win 2
return -1;
// you lose 1
} // now we have your_friend < you
if ( your_friend + 1 == you ) return -2; // you lose 2
return 1;
// you win 1
}
int main () {
// the random number generator; let us use the generator
// ANSIC instead of the toy generator knuth8 ; m = 2^31;
ifm :: random ansic (1103515245u , 12345 u , 2147483648u , 12345 u );
// your strategy may be to prefer larger numbers and use
// the distribution (1/21 , 2/21 , 3/21 , 4/21 , 5/21 , 6/21)
double p = 1.0/21.0;
ifm :: loaded_dice you (p , 2*p , 3* p , 4* p , 5* p , ansic );
// your friend ’s strategy may be to stay more in the middle
// and use the distribution (1/12 , 2/12 , 3/12 , 3/12 , 2/12 , 1/12)
double q = 1.0/12.0;
ifm :: loaded_dice your_friend (q , 2* q , 3* q , 3* q , 2* q , ansic );
// now simulate 1 million rounds ( the train may be very late ...)
int your_total_payoff = 0;
for ( unsigned int round = 0; round < 1000000; round ++) {
your_total_payoff += your_payoff ( you () , your_friend ());
}
// output the result :
std :: cout < < " Your total payoff is "
< < your_total_payoff < < "\n ";
return 0;
}
Program 40: ¢¤£Á
±¬»Á ¨½ ³£¦çÅ
\ 4.3.11 Details
" ÊÞßÉËÛÜÞßÉ³Ò³íÌæ¹É·æÈÐçØ ËËÊëåÌÍäÐØ¯Ë ØµÊØµÞßÉ³Þb_ÉÍÃÔ×ÖÚØµÙËÛÜÊØµÒ ÐÙ³ÙÉ³ÒÒ ËÂÊëËÂÇµÉòÛÜØ ËÂÉÄÍ
ØÚÐá¥ÍÂÉÎµÍÂÉÒÉ³Ø ËÂÐçËÂÛÜÊØ¿ÊÔÐ Ùá×ÐÒÒ³é ÆÃìôÎÚÛÜÙ¬ÐáôÔ×ÖÚØµÙËÛÜÊØµÒ Ô×ÊÌÍÃæ·ÇµÛÜÙäÇ¿ËÂÇµÛÜÒ Þ®Ð ]É³Ò ÒÉ³ØµÒÉòÐçÍÂÉïËÂÇµÉïÛÜØ¾ÐØµè
ÊÖ«ËÎÚÖ«Ë ÊÌÎ ÉÍäÐÑËÂÊÌÍÂÒ
ÐçØÚè
éÏêØÚèµÉ³Éè¾í æ·ÍÂÛñËÛÜØµåÀÊÖ«Ë ÊÌÍ ÍÂÉ¬ÐèµÛÜØµåßÛÜØ¯ËÂÊÀËÂÇµÉ
ÛÜØ¯ËÂÉÍÂØÚÐáÃÍÂÉÎµÍÂÉ³ÒÉØ Ëäoperator<<
ÐçËÛÜÊØ ÊÔÜËÂÉ³Ø ÍÉXWôÖµÛñoperator>>
ÍÂÉÒàÒÊÞßÉ ]ôØÚÊ¦æ áñÉ³èµåÉÀÊÔÈËÇÚÛñÒâÍÂÉÎµÍÂÉ³ÒîÉ³Ø ËäÐÑËÂÛÜÊØ ËÂÇÚÐçËàåÊ É³Ò
_ ÉìÊØµèæ·ÇÚÐçËÈÊÌËÂÇµÉÍ·Ô×ÖµØµÙËÂÛñÊØÚÒÈØµÉ³É³è¤é
SÝÉ Ù³ÐØµØÚÊÌË¡ÍÂÉ¬ÐÒîÊØ Ð _ á²ìãæòÍÂÛñËÂÉ
ÐçØÚè
ÐÒ ÞßÉb
Þ _ ÉÄÍÂ
Ò ^»æ Ç¯ìâØµÊÌË `Äí _ÚÖ«Ë
operator<<
operator>>
æ¹É¹Ù¬ÐØÞ®Ð ]É ËÂÇµÉ³ÒÉ¹Ô×ÖµØµÙËÂÛÜÊØµÒ
ÊÔµËÂÇµÉÈÙá×ÐÒÒ³é ·ÒÕÐ ÔÜÍÂÛñÉ³Øµè¾í¯Ð Ô×ÖµØµÙËÂÛÜÊØ¿ÇÚÐÒ ÖµØµÍÉ³ÒîËÍÂÛñÙËÂÉ³è
ÉÍÂF¯Ò é ê»ËïÛÜÒïÙ³áñÉ¬ÐçÍ ËÂÇÚÐçËËÂÇµÉ
ÐÙ³ÙÉ³ÒÒïËÂÊàËÂÇµÉ·ÎµÍÂVÛ UÐçËÂÉ Ù³áÜÐÒÒïÞßÉ³¶
Þ £¬_ ³
Þ¿ÖÚÒËïèµÉ³Ù³á×ÐçÍÉÓÐâÔ×ÖµØÚÙÄËÂÛÜÊØßËÂÊ
¬
±
ª

¯
©

_ É¿ÛñËÂÒÓÔÜÍÂÛÜÉØÚè¤í¾ÐØµè ØµÊÌËÏËÂÇµÉ ÊÌËÂÇµÉÍãæÈÐ³ì½ÐçÍÂÊÖµØµè¤í ÒÛÜØµÙ³É¿Ûñ!Ë ÒÓËÇÚÉ¿Ù³á×ÐçÒÒ ËÂÇÚÐçËÏÇ ÐçÒÏËÂÊ®ÎµÍÂÊÌËÂÉ³ÙÄËâÛñËÒ
ÎµÍÂVÛ UÐÙìí¤ÐØµè ØµÊÌË ËÂÇµÉàÔ×ÖÚØµÙËÛÜÊØ¤é [ ÊÌÍÂÞ®ÐçáÜáñìí Ð
ÛÜÒÓÐÀÞßÉ³
Þ _ ÉÍ èµÉ³Ùá×ÐçÍäÐçËÛÜÊØ

¶
¬
£
î

F
µ
F
Ä

¬
±
×

£
¯
©

ÊÔ¡ËÇÚÉãÔ×ÊÌÍÂÞ
Friend functions.
friend
− Æ¹ÇÚÛñÒÈèµÉ³Ùá×ÐçÍäÐçËÛÜÊØ®Þ®Ð ]É³Ò ËÂÇµÉÓÍÂÉ³ÒîÎ ÉÙËÂÛVUÉÓÔ×ÖµØµÙËÂÛÜÊØÐ ÔÜÍÛÜÉ³ØµèÊÔ¾ËÂÇµÉÏÙá×ÐÒÒÈÐØµèåÌÍÂÐØ ËÂÒÈÐÙÙ³É³ÒÒ¹ËÂÊ
ÐáÜá¤èÚÐçËäÐàÞßÉ³
Þ _ ÉÍÒ³íôæ ÇµÉËÇÚÉÄÍòËÂÇµÉì ÐçÍÉãÎÚÖ _ áñÛÜÙ ÊÌÍ·ÎµÍÂÛVUÐçËÂÉé [ ÊÌÍÈËÂÇµÉãÙ³áÜÐÒÒ
íµæ¹ÉÏÙ³ÊÖµáÜè
rational
ÍÂÉæòÍÂÛñËÂÉÓËÂÇµÉ
ÒîÉ³ÙËÂÛÜÊØóÐÒ·Ô×ÊáÜáñÊÑæ·ÒÈËÂÊ èÚÉÙ³á×ÐçÍÂÉãÛÜØÐØµè ÊÖµËîÎ Ö«ËòÊÌÎ ÉÍäÐÑËÂÊÌÍÂÒòËÊ _ ÉâÔÜÍÛÜÉ³ØµèµÒ
private
ÊÔ¡ËÇÚÉãÙ³á×ÐÒîÒ³é
class rational {
private :
friend std :: ostream & operator < < ( std :: ostream & o , rational r );
friend std :: istream & operator > > ( std :: istream & i , rational & r );
int n ;
int d ; // INV : d != 0
};
êØÀËÂÇµÉòèµÉ YÚØµÛñËÂÛñÊØ ÊÔËÂÇµÉ³ÒîÉ·ÊÌÎ ÉÍÂÐçËÂÊÌÍÂÒ³íæ¹ÉòÙ¬ÐØÀËÂÇµÉ³ØßÐçÙ³Ù³É³ÒÒÕËÂÇµÉòØ ÖÚÞßÉÄÍäÐçËÂÊÌÍÕÐØµèÀèµÉ³ØµÊÞßÛÜØÚÐ ËÂÊÌÍ ËÂÇ«ÍÂÊÖµåÇ
ÐØµè
ÐÒ æ¹ÉÈÖµÒÉ³è ËÂÊÏèµÊãÛ²Ë ÛÜØ " É³ÙÄËÂÛÜÊØ é \ é é êÔÚÎ ÊÒÒÛ _ áñÉÌíÔÜÍÂÛÜÉ³ØµèèÚÉÙ³á×ÐçÍäÐÑËÂÛÜÊØµÒ
ÒÇµÊÖµáÜL
è _ É·KÐ U.n
ÊÛñèÚÉè¾í Ò.d
ÛÜØµÙ³ÉÈËÂÇµÉìÀÙ³ÊÞÀÎµÍÂÊÞßÛñÒÉïÉ³ØµÙ¬ÐçÎÚÒÖµá×ÐçËÛÜÊØ _ÚÖ«Ë ÒÊÞßÉÄËÂÛÜÞßÉ³Ò³íÌËÇÚÉÄìÀÐçÍÉ ÖµÒÉ³Ô×Öµá
ÛÜØ ÊÌÍÂèµÉÍ·ËÂÊ¿ÒÂXÐ UÉâÖµØµØµÉ³Ù³É³ÒîÒÂÐçÍìÞßÉ³Þb_ÉÍïÔ×ÖµØµÙËÂÛÜÊØµÒ³é
4.3.12 Goals
Ë¹ËÂÇµÛÜÒ¹Î ÊÛÜØ¯Ë¬íôìÊÖóÒîÇÚÊÖµáÜè é³é³é
¹
Dispositional.
`L_ ÉãÐ _ÚáÜÉÏËÊÉIPôÎÚá×ÐÛÜØ ËÂÇµÉÏÎÚÖ«ÍÎ ÊÒÉÏÊÔ Ð Ù³á×ÐÒîÒòÛÜØ \ `ßÖµØÚèµÉÍÒîËäÐØµèËÂÇµÉãØÚÉÄæ Òì¥Ø ËÂÐÙËÂÛÜÙ³Ðá¾ÐØµèóÒÉ³Þ®ÐØ¯ËÂÛÜÙ¬ÐçáËÂÉÍÂÞßÒòÐÒÒÊ Ù³Û×ÐçËÉ³è æ·ÛñËÂÇ ÛÜØÎ ÐçÍËÂÛñÙ³Öµá×ÐçÍ
í
íµÐçØÚè
µ© ±±Ä¬³¢ ± ³£¦ ½
¨ ³¨½ ³£ ¶ ±¬î¤
±¤³£¬¡±¬£¦
ÐØµè
ÛÜØèµÉËäÐçÛÜáé
`ßÖµØÚèµÉÍÒîËäÐØµèËÂÇµÉÏÙ³á×ÐÒîÒÉ³Ò
ifm::random
ifm::loaded_dice
Ù³áÜÐÒÒÉ³Òí
Operational.
\ êØÎ ÐçÍËÂÛñÙ³Öµá×ÐçÍ¬í¯ìÊÖ ÒÇµÊÖÚáñè _ ÉãÐ _ÚáÜÉ ËÂÊéé¬é
` YÚØµè Òîì¥Ø¯ËäÐÙËÂÛñÙ¬ÐáÐçØÚè ÒÉ³Þ®ÐçØ ËÂÛÜÙ³ÐáÉÍîÍÂÊÌÍÂÒ¹ÛÜØ ÐëåÛVUÉ³Ø Ù³á×ÐÒîÒïèÚÉIY ØµÛñËÛÜÊØ®ÐØµè ÛñÞÀÎ áñÉ³ÞßÉ³Ø ËÂÐ ^
ËÂÛñÊØ \ `®èµÉ³ÒîÙÍÂÛV_ É
UÐáÜÖµÉÏÍÂÐØµåÉâÐØµè Ô×ÖÚØµÙËÛÜÊØÚÐáÜÛñË ìßÊÔ ÐàË ìôÎ ÉÏåÛVUÉ³Ø _ôì Ð¿Ùá×ÐÒÒÈèµÉ YÚØÚÛ²ËÂÛÜÊØ ^
` ÐçèÚè Ô×ÖµØÚÙÄËÂÛÜÊØÚÐáÜÛñË ìÀËÊÀÐ åÛVUÉ³ØóÙ³áÜÐÒÒ¹ËÂÇ«ÍÂÊÖµåÇÞßÉ³Þ_ ÉÍ¹Ô×ÖµØµÙËÂÛÜÊØµÒ
^
`ßæòÍÂÛñËÂÉÓÒÛñÞÀÎ áñÉ Ù³á×ÐÒîÒÉ³Ò·ÊØ ìÊÖµÍòÊ¦æ Ø ^
^
`ßæ¹ÊÌÍ ] æ·ÛñËÂÇ ÐØµè ÐçÍÂåÖµÉãÐ _ ÊÖ«Ë¹ÎÚÒÉ³ÖµèµÊÌÍäÐØµèµÊÞ
Ø ÖµÞb_ ÉÄÍÂÒ³é
4.3.13 Exercises
Exercise 122 È£³ DNFµ®©â¶Ä»¾»¨¿¢¾§³¨½³¤î©¯î
Â¶ë£©¯ ©¯ ¨½ ³£¦ß©¯ © ±¬×©¯³ J¦¢ ©F
¬³ÃîÅßÕÏ JÑ¢â³ F® £©¯»¤©£¬º¥¨½¬î± ³¢ ³£Ä©¯£¦ Hî° ±¬"F»Á ° ·©F FµÄ±³£³³¨½³¤
©F¿ºâ ©£¬»º¥¨ ¬±â©¯³§Á ¨ ³¤º M £³¬×©¯î ©¯¾³¢ ³£©¯£¦ ©¯ò¸Ó¬»¤©¯ò° È©F®ñ¢¤Õ©F
ç³£ GFµ GF ± D¥£¦î¤LH²¶£³¨ int ©F® doubleM¥Å à© » Dô©£¬ ©¤ » ³ Fß F
º©¯ÓºÀ»¤£¬ ©¯£³î¢¤£³³³¤î©¯î¼ £¬¨½©¯ G F H ©¯² Ú³£±³ MôÅ £ ©¯»Ãº
¶ ±¬î ©¯ J J ¢¤£³ DÌNFµ Fµ±³ FµÏ¸È¬º£ã»Õ³ Fß Ó£³Ä©¯ G F X JÀ¨ ³¨ ³³£Õ¶ ±¬î¤
£ KJ ¨½³¨½ ³£à¶ ±¬î¤çÅ ½±¬ª©¯ ³ F ©¯²çÝ© D¥ ©Ý ¬³G F ¨½³£©¯£Ý©
F
Fµ³ ¨ » ©¯£ J¦¢ ©µ±¬¥X D¥ ¨ £³
¯ ¬ J
©
F º³£³ F ³ © ±^ D¥ `I£¦^ î `
¶ ±¬î ¶£³¨ Dô©¯ ³ Â¶¿º¥¿ J¦¢ Å
Exercise 123
©Ý±³ª©¯³
KJ
³¸¹£¬»ëºÏ³£¬ ±³ Tribool º©¯JÚ ©D¥ FµXD¥¬ª³¢ GFÀ»
©KM ¨ © »Áº F«©¯î©¨ ³¨ ³³£¦Ó¢¤£¬ Dô©¯
M © F F°Á ±Ä£¬£³¬¢ F°Á ©µ±Ä±¬³ò¶ ±³î¤
Ú³£³±³»
°¤
± M © F F°Á © µ© ±±³³Ã¶Ä ±¬ is_bool() const º©¯Õ£³¬£¬¤ã£¬ Ï ¶ë©F¤EJß ¶ãº
Dô©¯ ¿ ò
¸¹ Ï©F
FKM © F F°Á³£ GFµ IFÝ±Ä D¥³£¦¤ò¶Ä£³¨
©Fóóº JÑ¢ bool Å
^
`
Exercise 124
©KM
° F ©¯» ³£¬£³£¦ë» ºï¶ç»ª¸¹»Áë¢¤£³ÄÁ£©¨½Å º³¨ ©F Fµ¬±³£¬ ³ º ï¶Ä ±¬
©¯ J Â¶óº J¦¢ Clock À KJ ¢¤£³ DÌNF°Á ¢¤£³ ä©F ¢³±F»î¤ë¶£ º ^ ¨½ `G³^ ¨½ \ ³` £
¶Ä ±¬¤çÅ
\
# include < iostream >
class Clock {
Clock ( unsigned int h , unsigned int m , unsigned int s );
void tick ();
void time ( unsigned int h , unsigned int m ,
unsigned int s );
private :
unsigned int h_ ;
unsigned int m_ ;
unsigned int s_ ;
};
Clock :: Clock ( unsigned int & h ,
unsigned int & m ,
unsigned int & s )
: h_ ( h ) , m_ ( m ) , s_ ( s )
{}
void Clock :: tick ()
{
h_ += ( m_ += ( s_ + = 1 ) / 6 0 ) / 6 0 ;
h_ %= 24; m_ %= 60; s_ %= 60;
}
void Clock :: time ( unsigned int & h ,
unsigned int & m ,
unsigned int & s )
{
h = h_ ;
m = m_ ;
s = s_ ;
}
int main () {
Clock c1 (23 , 59 , 58);
tick ();
unsigned int h ;
unsigned int m ;
unsigned int s ;
time (h , m , s );
std :: cout < < h < < " : " < < m < < " : " < < s < < " \ n " ;
return 0;
}
_ ` êÞÀÎÚáÜÉ³ÞßÉØ Ë¹ÐØ ÊÖ«ËÎÚÖµËòÊÌÎ ÉÍÂÐçËÂÊÌÍÈÔ×ÊÌÍòËÇÚÉãÙ³á×ÐÒîÒ
Exercise 125 Ý£¬» ©ß¢¤£³Á£©¨
Clock
é
^
`
random_triangle.C ½Ä»¨ª©¯®º ¶ç»ª¸¹»Á £©Fµ¨
¢¤£³Ì±Ä¬³àÁ£©¬¢¾¥±Ä©¯» JµÅ ¤NFµ£ © ÚIFó£¬©Áª t ©FÝ±³Ìß©¼©£Ì ³£©£KJ D¥£¦ Â¶ t
©¯ © î©£¦»ÁÀ¢ °¤îÅ Â ´©µ±¬ ³î¢ ã±¬Ì ©¯ © ¢ °¤ ºÀ¨ F¬¢ »¤â ¬¸Ó³ º
±³ £¬£³³¤¢ °¤Ï©
F © Hî ¶£¬¨EJM £©Fµ¨ó J ¬ª±¬GF D¥³£¦ Â¶ t Å
ó»¨ ×©¯î ©¯À´©µ±¬ ³î¢F£©¸ÈóºÝ±³ £¬£³³¤·¢ °¤ë»¤ ©
»Fµ¸ Å
º
¸¹»
Fµ¸ ¥ ±³ ifm::wio Fµ GF » <IFM/window> ¶ç£ ÁÌ£©¬¢¾¥î±´©¯Óñ¢¤ à©F ±³Ì
º½£¬©Áª½¸¹»º D¥£¦î±³ (0, 0) (512,
0) ©F (256, 512) Å º½£©Fµ¨ ¨½ ³£
Á«³ £©¯£ ansic ¶£³¨ ¹£³Á£©¨
Å Ï ³ Á» ·ºÏ¢¤£Á£©¨ ³ F £³Ä© F» ©óGFë¶ç£
\
º £©Fµ¨ ¨½ ³£Á«³ £©¯£ ©F º ¨ ³ ³£ Â¶à°¨×©¯î ³¢¾à ¢³£º¶£¬¨ Å ^ £ `
¬³°Á ¢¤ £¢ ¬ ¹ª¬ º »¨ ×©¯î ¬£ ¶ £ «© ³î¢¾çÅ
Exercise 126 Ó¤ Fµ³£ º Á« ³£©¯£ ansic çIF½» È£³ÄÁ£©¨
Å » ±Äº ¨ F
m = 231 Àº½»¤³£¬ ©¯à±¨¿¢¤î©¯¤ Â¶ º½Á¥³ ³£©¯£ ¸¹°ë±³£¦î©°¤EJ D¥ £ ¸ ¶
¬º ©£³ GF £³î¢¤£³¬ç³¤ unsigned int Dô©¯¡¬çÅ
¬ ¢¤ º¥ º½ ¡³ ± Â¶
¢¾³ Fµ£©Fµ¨ ¨ ³ ³£¦ ±Ä¨¢¤IF K J º ãÁ¥³ ³£©¯£ ±£¬£±¬ ©F ±° ±³NFµ³ã¸¹º^ » º`
¨½©¯º¨½©¯î±´©¯ Fµ Å ¦¢¾×©» º¥ »
»
F © ª© FµG F F î±Ä º©¯ ³´©¯º ºó¶ç©°£ F± ° ºÁ¥©¨½ Â¶¼±¬Ì^ ° Á `
¨ ³³£¦ÑÅ HÕ¥àÀ©óº£KJ Ú³£±³çÅ M
Exercise 127
4.3.14 Challenges
©¯ »ãº ³¬³ïª© FµGF Fî±¶£·¢¾×©J °ÁßºâÁô©¨ Â¶¿±¬ÌÄ»ÁÀ ¨½ ³£¦
D¥Ý»ºF³£¬ ¬
± F¼£KJ ©¬¢«¢¤£³ ¯»¨ ©¯ º F£¬ ¬î ¬¢ ³£¬»¨½³
î©¯» J à£ ç¨ ¬¸ ±¨¿
¤
¢

®îÅ H Hint: »
£ Fµ³£ F © Äî©« ¬§®ºÄ£³¬î±´©¯ï¨ Fµ¬
´©£³±¬ ¶£ º ³£¬¨
³£³ îÄ¨ Á¥©¨½¬ ß£ F°£³±¬º JÁ« º ±£¬£¬¢ F°Á ±³©¬¢
³£ ° http://www.inf.ethz.ch/personal/gaertner/cv/lecturenotes/ra.pdf Å ±
JÚ © D¥ã¶£¬¨ ×©¯I F º¢¤£¥ ¦§¨ ©¯ © ³£³ îÄ¨ Á¥©¨ J ±Ä© D¥ ëÄ»Á
^ ¶ £`
µ©¨¿¢¾§ º¸Ó »¤£º¶ç©µ± http://banach.lse.ac.uk/form.html
Exercise 128
\
Appendix A
C++ Operators
Description
ÒÙÊÌÎ É
ÒÖ _ÚÒÙÍÛñÎµË
Ô×ÖµØµÙËÂÛÜÊØÙ¬ÐáÜá
Ù³ÊØµÒîËîÍÂÖµÙËÂÛÜÊØ
ÞßÉ³
Þ _ ÉÍÈÐçÙ³Ù³É³ÒÒ
ÞßÉ³
Þ _ ÉÍÈÐçÙ³Ù³É³ÒÒ
Î ÊÒË ºÛÜØµÙÍÉ³ÞßÉ³Ø Ë
Î ÊÒË ºèµÉ³ÙÍÉ³ÞßÉ³Ø Ë
è«ì¥ØÚÐÞßÛñÙ Ù¬ÐÒîË
ÒîËÂÐçËÂÛÜÙÓÙ¬ÐÒË
ÍÂÉÛÜØ ËÂÉÄÍÎµÍÂÉËòÙ¬ÐÒîË
Ù³ÊØµÒîËòÙ¬ÐÒË
Ë ìôÎÉãÛÜèµÉ³Ø¯ËÂVÛ YÚÙ¬ÐçËÛÜÊØ
ÎµÍÂÉ ÛÜØµÙÍÂÉ³ÞßÉØ Ë
ÎµÍÂÉ èµÉ³ÙÍÂÉ³ÞßÉØ Ë
èµÉÍÂÉÔ×ÉÍÂÉ³ØµÙ³É
Ðèµè«ÍÂÉ³ÒîÒ
_ÚÛñË æ ÛñÒÉÓÙ³ÊÞÀÎÚáÜÉ³ÞßÉ³Ø¯Ë
áÜÊåÛñÙ¬Ðá ØµÊÌË
ÒÛñåØ
ÒÛñåØ
ÒÛ ³É³ÊÔ
ØµÉæ
èµÉ³áÜÉÄËÂÉ
Ù¬ÐçÒîË
ÞßÉ³
Þ _ ÉÍïÎÊÛÜØ ËÂÉÄÍ
ÞßÉ³
Þ _ ÉÍïÎÊÛÜØ ËÂÉÄÍ
Operator
::
[]
()
()
.
->
++
-dynamic_cast<>
static_cast<>
reinterpret_cast<>
const_cast<>
typeid
++
-*
&
~
!
+
sizeof
new
delete
(
)
->*
.*
é³é¬é
\ Arity
\
Prec.
!
\
\
\
\
\
\
Assoc.
ÍÂÛÜåÇ¯Ë
áÜÉ³ÔÜË
áÜÉ³ÔÜË
ÍÂÛÜåÇ¯Ë
áÜÉ³ÔÜË
áÜÉ³ÔÜË
áÜÉ³ÔÜË
áÜÉ³ÔÜË
ÍÂÛñåÇ Ë
ÍÂÛÜåÇ¯Ë
ÍÂÛÜåÇ Ë
ÍÂÛÜåÇ¯Ë
ÍÂÛñåÇ Ë
ÍÂÛÜåÇ¯Ë
ÍÂÛÜåÇ¯Ë
ÍÂÛÜåÇ¯Ë
ÍÂÛÜåÇ¯Ë
ÍÂÛÜåÇ¯Ë
ÍÂÛÜåÇ¯Ë
ÍÂÛÜåÇ¯Ë
ÍÂÛÜåÇ¯Ë
ÍÂÛñåÇ Ë
ÍÂÛÜåÇ¯Ë
ÍÂÛñåÇ Ë
ÍÂÛÜåÇ¯Ë
áÜÉ³ÔÜË
áÜÉ³ÔÜË
\\
Þ¿ÖµáñËÂÛñÎÚáÜÛÜÙ³ÐçËÂÛÜÊØ
èµVÛ UôÛÜÒÛÜÊØ ^ÛÜØ¯ËÂÉ³åÉÍG`
ÞßÊ èÚÖµáÜÖµÒ
ÐèµèµÛñËÂÛñÊØ
ÒÖ _µËÍäÐçÙËÂÛÜÊØ
ÊÖ«ËÎÚÖ«Ë áÜÉ³ÔÜËÈÒÇµÛÜÔÜË
ÛÜØ«ÎÚÖ«Ë ÍÂÛÜåÇ¯Ë¹ÒÇµÛÜÔÜË
áÜÉÒÒ
åÌÍÂÉ³ÐçËÂÉÍ
áÜÉÒÒòÉ WôÖÚÐá
åÌÍÂÉ³ÐçËÂÉÍòXÉ WôÖÚÐá
ÉXW ÖÚÐáÜÛñË ì
ÛÜØµXÉ W ÖÚÐáÜÛñË ì
_ÚÛñË æ ÛñÒÉÏÐØµè
_ÚÛñË æ ÛñÒ
É P«ÊÌÍ
_ÚÛñË æ ÛñÒÉÓÊÌÍ
áÜÊåÛñÙ¬Ðá¤ÐØµè
áÜÊåÛñÙ¬Ðá ÊÌÍ
ÐÒîÒÛÜåØµÞßÉ³Ø Ë
Þ¿ÖµáñËÈÐÒÒÛñåØÚÞßÉØ Ë
èµVÛ U ÐÒÒîÛÜåØµÞßÉ³Ø Ë
ÞßÊ èÐÒîÒÛÜåØµÞßÉ³Ø Ë
Ðèµè ÐçÒÒÛÜåØµÞßÉ³Ø¯Ë
ÒÖ _ ÐçÒÒÛÜåØµÞßÉ³Ø¯Ë
ÍÂÒîÇÚÛñÔÜË ÐÒîÒÛÜåØµÞßÉ³Ø Ë
áÜÒîÇÚÛñÔÜË·ÐçÒÒÛÜåØµÞßÉ³Ø¯Ë
ÐØµè ÐçÒÒÛÜåØµÞßÉ³Ø¯Ë
P¥ÊÌÍ ÐÒîÒÛÜåØµÞßÉ³Ø Ë
ÊÌÍ·ÐÒÒÛñåØÚÞßÉØ Ë
ÒÉáÜÉ³ÙËÂÛñÊØ
IÉ P¥Ù³ÉÎµËÂÛÜÊØ
ÒÉ WôÖµÉ³ØµÙ³ÛÜØµå
é³é¬é
*
/
%
+
<<
>>
<
>
<=
>=
==
!=
&
^
|
&&
||
=
*=
/=
%=
+=
-=
>>=
<<=
&=
^=
|=
?
throw
,
\
\
\
K K \
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
áÜÉ³ÔÜË
áÜÉ³ÔÜË
áÜÉ³ÔÜË
áÜÉ³ÔÜË
áÜÉ³ÔÜË
áÜÉ³ÔÜË
áÜÉ³ÔÜË
áÜÉ³ÔÜË
áÜÉ³ÔÜË
áÜÉ³ÔÜË
áÜÉ³ÔÜË
áÜÉ³ÔÜË
áÜÉ³ÔÜË
áÜÉ³ÔÜË
áÜÉ³ÔÜË
áÜÉ³ÔÜË
áÜÉ³ÔÜË
áÜÉ³ÔÜË
ÍÂÛÜåÇ¯Ë
ÍÂÛÜåÇ¯Ë
ÍÂÛÜåÇ¯Ë
ÍÂÛÜåÇ¯Ë
ÍÂÛÜåÇ¯Ë
ÍÂÛÜåÇ¯Ë
ÍÂÛÜåÇ¯Ë
ÍÂÛÜåÇ¯Ë
ÍÂÛÜåÇ¯Ë
ÍÂÛÜåÇ¯Ë
ÍÂÛÜåÇ¯Ë
ÍÂÛÜåÇ¯Ë
ÍÂÛñåÇ Ë
áÜÉ³ÔÜË
Table 9: ¹£³±ÄIFµ³ ±Ä¬ ©F ©¯Ì±³ ©¯ DÌ»î¬ Â¶ ³£©¯£¦çÅ
³¢
\ Index
ËÂÇµÛÜÒ³í \ ^»áÜÛÜØµÉ_ÚÍÉ¬Ð a
] `´í \n
\ _ÚÛñËÈÒîìôÒîËÂÉ³Þ í \0
·ÍÛñËÂÇµÞßÉËÂÛÜÙ UÐáÜÖÚÐçËÂÛñÊØ
ÖÚáñÉ í ÐçÍÛñËÂÇµÞßÉËÂÛÜÙ·ÉIP¥ÎµÍÂÉ³ÒîÒÛÜÊØ¤í \
ÐçÍÛñËÂÇµÞßÉËÂÛÜÙ·ÊÌÎ ÉÍäÐçËÊÌÍÂÒ³í ÐçÍÛñËÂÇµÞßÉËÂÛÜÙòË ì Î ÉÌí
ÐçÍÛñË ìí ÐçÍîÍäÐ³ìí \
ÐÒÈÔ×ÖµØÚÙÄËÂÛÜÊØ ÐçÍåÖÚÞßÉØ Ë¬í !
èµÛÜÞßÉ³ØµÒÛÜÊØ¤í è«ÍäÐ³
æ _ ÐÙ ]¥Òí É³áÜÉÞßÉ³Ø Ë¬í \ Y P«É³èáÜÉ³ØµåÌËÂÇ¤í ÛÜØµÙ³ÊÞÀÎÚáÜÉËÉ·Ë ìôÎ ÉÌí \
ÛÜØµèµIÉ P í \
ÛÜØµÛñËÂÛÜÐáÜÛ ¬ÐÑËÂÛÜÊØ¤í \
ÛÜØµÛñËÂÛÜÐáÜÛ ¬ÐÑËÂÛÜÊØÀÔÜÍÂÊÞ ÒîËÍÛÜØµåáÜÛ²ËÂÉÍäÐá»í ÛÜØµÛñËÂÛÜÐáÜÛ ³ÉÄÍ áÜÛÜÒË¬í \
Þ¿ÖÚá²ËÂÛÜèµÛÜÞßÉ³ØµÒÛÜÊØÚÐá°í ÒîÛÜÞÖµá×ÐÑËÂÛÜÊØ _ôì ÊØÚÉèÚÛñÞßÉ³ØµÒÛÜÊØÚÐáÓÐçÍÍÂÐ¬ìí
^ ³ÉÍÂÊÀÙäÇÚÐçÍäÐçÙËÂÉÍG`´í ÐÙ³ÙÉ³ÒÒ
èÚÐçËäÐàÞßÉ³Þ_ ÉÍ¬í \ ÞßÉ³
Þ _ ÉÍ¹Ô×ÖµØµÙËÂÛÜÊØ¤í \ ÐÙ³ÙÉ³ÒÒòÍÉ³ÒîËÍÂÛñÙËÂÛÜÊØµÒ
ÎµÍÂVÛ UÐçËÂÉÌí \ ÎÚaÖ _ÚáÜÛñÙÌí \ ÐÙ³ÙÉ³ÒÒ·ÒÎ É³Ù³VÛ YÚÉÍ
ÎµÍÂVÛ UÐçËÂÉÌí \ ÎÚaÖ _ÚáÜÛñÙÌí \ Ù ]ÉÍÂÞ®ÐçØÚØ Ô×ÖµØÚÙÄËÂÛÜÊØ¤í \
ÐèµèµÛñËÂÛÜÊØ ÐÒÒîÛÜåØµÞßÉ³Ø Ë¹ÊÌÎ ÉÄÍäÐçËÂÊÌÍ¬í ÐèµèµÛñËÂÛÜÊØ ÊÌÎ ÉÍÂÐçËÂÊÌÍ¬í Ðèµè«ÍÂÉ³ÒÒ
ÊÔ ÐàÞßÉ³ÞßÊÌÍìßÙÉ³áÜáí ÊÔ Z
Ð UÌÐÑÍÂÛ×Ð _ÚáÜÉÌí \ Ðèµè«ÍÂÉ³ÒÒòÊÌÎ ÉÍäÐÑËÂÊÌÍ¬í K Ðè ÖµÒîËÂÞßÉ³Ø¯Ë
ÊÔ ÐçÍÍÂÐ¬ì ÐçÍåÖÚÞßÉØ Ë¬í !
ÐáÜåÊÌÍÛñËÂÇµÞ
Ù¬ÐÙäÇµÉ ºÊ _ÚáÜÛ U¥ÛÜÊÖµÒ³í ÐáñËÉÍÂØÚÐçËÂÉÓèµÉ³ØµÛ×Ðáí O " ê
ÍäÐØµèµÊÞ ØôÖµb
Þ _ÉÍÈåÉ³ØµÉÍäÐçËÂÊÌÍ³í \ ÐØ¯ËÂVÛ UÐáÜÉ³ØµÙ³ÉÌí ÐçÎµÎÚáÜÛÜÙ¬ÐÑËÂÛÜÊØßÎµÍÂÊåÌÍäÐÞ í ÐçÍÂåÖµÞßÉ³Ø¯Ë ºèµÉÎÉ³ØµèÚÉØ ËÈáÜÊôÊ ]ôÖ«Î¡í \a
·ÍÂÛ²ËÂÇµÞßÉËÂÛÜÙ UÐáÜÖÚÐçËÂÛÜÊØ ÖµáÜÉ í
·ÍÂÛ²ËÂÇµÞßÉËÂÛÜÙ UÐáÜÖÚÐçËÂÛÜÊØ ÖµáÜÉ \ í ØµÊÌË·ÒîÉ³áÜÔºèµÉ³ÒîÙÍÂÛV_ÚÛÜØµå«í K ÊÔ Î ÊÛñØ ËÂÉÍÒ³í ÊÖ«Ë ºÊÔ _ ÊÖµØµèóÛÜØµèµIÉ P í K
Î ÊÛÜØ¯ËÂÉÍ¹ËÂÊ«í ÍäÐØµèµÊÞ ÐÙ³Ù³ÉÒÒ³í \
ÒÉ³ÙÖµÍÛñË ì ÛñÒÒÖµÉ³Ò³í ÒÖ _ÚÒÙÍÂÛ²ÎÚË³í \
ÒÖ _ÚÒÙÍÂÛ²ÎÚËòÊÌÎ ÉÍÂÐçËÂÊÌÍ¬í \
ÖµØµèÚÉÄÍÂáñì¥ÛñØÚå Ë ìôÎ Éí \
³ÉÍÊ ËÂÉÍÞßÛÜØÚÐçËÂÉ³è¤í ÐçÍîÍäÐ³ì ËÊ Î ÊÛÜØ¯ËÂÉÍÈÙ³ÊØ UÉÍÂÒÛÜÊØ¤í K
" ¹êîêÃÙ³ÊôèµÉÌí ÐÒîÒÉÍËÂÛñÊØ¾í \
ÐÒîÒÛÜåØµÞßÉ³Ø Ë
\
ÞßÉ³Þ_ ÉÍ»æ ÛÜÒÉí \ ÊÔ ÐàÒîËÍÂÖµÙËUÌÐçáÜÖµÉÌí \ ÊÔ ÍÂÉ³Ô×ÉÄÍÂÉ³ØµÙ³ÉÌí \
Î ÊÛÜØ¯ËÂÉÍ¬í K ÐÒÒîÛÜåØµÞßÉ³Ø Ë¹ÊÌÎ ÉÄÍäÐçËÂÊÌÍ¬í ÐÒÒîÊôÙ³Û×ÐÑËÂVÛ UÉÓÊÌÎ ÉÍäÐÑËÂÛÜÊØ¤í
ÐÒÒîÊôÙ³Û×ÐÑËÂVÛ U¥Û²Ë ì
áÜÉ³ÔÜË¬í ÍÂÛÜåÇ¯Ë¬í ÐÒÒîÊôÙ³Û×ÐÑËÂVÛ U¥Û²Ë ìÀÊÔ ÊÌÎ ÉÍÂÐçËÂÊÌÍ¬í
ÐçËËÂÐÙ ]ÉÍ¬í ÐÖ«ËÂÊÞ®ÐçËÛÜÙ ÒîËÂÊÌÍÂÐåÉãèµÖ«ÍäÐçËÛÜÊØ¤í ÐÖ«ËÂÊÞ®ÐçËÊØ
èµÉËÂÉÍÂÞßÛñØÚÛñÒîËÂÛÜ
Ù YÚØµÛñËÂÉí Ð _ ì¥áÜÊØµÛ×ÐØ ÞßÉËÂÇµÊ è¾í \
_ ÐÒÉ
ÊÔ Z
Ð QÚÊÐçËÂÛÜØµåàÎ ÊÛÜØ¯ËòØ ÖµÞb_ ÉÄÍòÒì¥ÒîËÂÉÞ í " ê
ÎµÍÂÊåÌÍäÐÞßÞßÛÜØµåëáÜÐØµåÖ ÐçåÉÌí _ É³ÇÚXÐ U¥ÛñÊÌÍ
ÛÜÞÀÎÚáÜÉ³ÞßÉ³Ø¯ËäÐçËÂÛñÊØÀèµÉ YÚØÚÉè¾í \
ÖµØÚèµÉ YÚØµÉ³è¤í \
ÖµØÚÒÎ É³Ù³VÛ YÚÉ³è¤í \
_ÚÛÜØÚÐçÍì®IÉ PôÎ ÐØµÒÛÜÊØ
ÊÔ ØÚÐçËÂÖ«ÍäÐá Ø Öµb
Þ _ ÉÄÍ¬í ÊÔ ÍÂÉ¬Ðçá¾Ø Öµb
Þ _ ÉÄÍ¬í _ÚÛÜØÚÐçÍì®ÊÌÎ ÉÄÍäÐçËÂÊÌÍ¬í _ÚÛÜØÚÐçÍìßÍÂÉÄÎÚÍÉ³ÒÉ³Ø¯ËäÐçËÂÛÜÊØ
ÊÔ ÛÜØ¯
Ë UÐáÜÖµÉÌí ÊÔ ÖµØµÒÛÜåØµÉ³èÛñØ Ë UÌÐçáÜÖµÉí _ÚÛÜØÚÐçÍì®ÒÉ³ÐçÍÂÙäÇ¤í \ \ _ÚÛÜØÚÐçÍì ËÂÊ èÚÉÙ³ÛÜÞ®Ðá Ù³ÊØ UÉÍÒÛÜÊØ¤í _ÚÛÜØµÊÞßÛ×Ðá Ù³ÊôÉ Ù³ÛÜÉ³Ø¯Ë¬í \ \ _ÚÛñË¬í _ÚÛñË æ ÛÜÒîÉÏÊÌÎÉÍäÐçËÂÊÌÍÒ³í _ÚáÜÊôÙ ]í ) " í
_ Ê èµì
ÊÔ èµÊÒËäÐçËÂÉ³ÞßÉØ Ë¬í
ÊÔ Ô×ÊÌÍòÒîËäÐçËÂÉÞßÉ³Ø Ë¬í ÊÔ Ô×ÖµØµÙËÂÛÜÊØ¤í ÊÔ æ ÇµÛÜáñÉ ÒîËäÐçËÂÉÞßÉ³Ø Ë¬í \
_ Ê Êáí Ê
ÊáÜÉ¬ÐØ¤í Ê ÊáÜÉ¬Ð
Ø UÐáÜÖÚÐçËÂÛñÊØ ÚÖ áñÉÌí
Ê ÊáÜÉ¬ÐØ IÉ P¥ÎµÍÂÉ³ÒîÒÛÜÊØ¤í Ê ÊáÜÉ¬ÐØ Ô×ÖÚØµÙËÛÜÊØ¤í Ù³ÊÞÀÎÚáÜÉËÉ³ØµÉ³ÒÒ³í _µÍÂÉ¬Ð ] ÒËäÐçËÂÉ³ÞßÉØ Ë¬í
_µÍÂÖ«ËÂÉ Ô×ÊÌÍÂÙ³ÉâÐçÎµÎÚÍÊÐÙäÇ¾í _ÚÖ _a_ áñÉ ºÒÊÌÍîË¬í \\
_ÚÖµå«í \
_ÚÖµÛÜáñË ÛÜØ®Ë ì Î ÉÌí \ _ÚÖ«ÍÂÒîË¬í ÒîËÂÐØµè ÐÑÍÂè¤í \
Ù¬ÐçÙ´ÇµÉÌí Ù¬ÐçÙ´ÇµÉ Ê _ÚáÜÛVU¥ÛñÊÖÚÒÈÐáñåÊÌÍÂÛñËÂÇµÞ í Ù¬ÐçáÜá¾ÐçÍÂåÖµÞßÉ³Ø ËÒ³í Ù¬ÐçáÜá _ôì®ÍÂÉ³Ô×ÉÍÉ³ØµÙ³ÉÌí \ Ù¬ÐçáÜá _ô
ì UÌÐçáÜÖµÉÌí \ Ù¬ÐçáÜá¤ÒîËäÐÙ ]í \ Ù¬ÐçØÚÙÉ³áÜá×ÐçËÛÜÊØ
ÛÜØ Q ÊÐÑËÂÛÜØµåàÎ ÊÛÜØ ËÈÙ³ÊÞÀÎÚÖ«ËäÐçËÛÜÊØµÒ³í ¹ÐçÍÂèÚÐØµÊ Ò¹Ô×ÊÌÍÂÞ¿Öµá×Ðôí \ Ù¬ÐçÒîË ^»Ô×ÖÚØµÙËÛÜÊØÚÐá ØµÊÌËäÐçËÂÛÜÊØ `Äí Ù¬ÐçÒîË·GÉ P¥ÎµÍÂÉ³ÒÒîÛÜÊØ¤í Ù³ÉØ ËÍäÐçá¾ÎµÍÂÊ Ù³É³ÒÒÛñØÚå¿ÖµØµÛñË ^ `Äí ÙäÇÚÐçÍ
áÜÛñËÉÍäÐáí ÎµÍÂÊÞßÊÌËÂÛÜÊØßËÂÊ ^ÖµØµÒÛÜåØµÉ³è ` ÛÜØ Ë¬í Ë ìôÎ Éí ÙäÇÚÐçÍäÐÙÄËÂÉÍ¬í í
Ù³ÊØ¯ËÍÂÊáí áÜÛÜØµ
É _µÍÂÉ¬Ð ]í ÙäÇµÊôÊÒÛñØÚå¿ØôÖµ
Þ _ ÉÍÂÒ
åÐÞßÉÌí \\
ïÇôÖ«ÍÂÙäÇ¤í áÜÊ
Ø ³Ê ^ `Äí ïÇôÖ«ÍÂÙä
Ç Æ Ö«ÍÂÛñØÚå ËÂÇµÉ³ÒÛñÒ³í Ù³áÜÐÒÒ
ÐÙ³ÙÉ³ÒÒ·ÒÎ É³Ù³VÛ YÚÉÍ³í \ Ù³ÊØµÒîËÍÖÚÙÄËÂÊÌÍ¬í \ \
èµÉ YÚØµÛñËÂÛÜÊØ¤í \ ÛÜÞÀÎÚáÜÉ³ÞßÉØ ËäÐçËÛÜÊØ¤í \ ÞßÉ³
Þ _ ÉÍ¹èµÉ³Ù³áÜÐçÍäÐçËÂÛñÊØ¾í \ \ ÞßÉ³Þ_ ÉÍ¹Ô×ÖµØµÙËÂÛÜÊØ¤í \ ÞßÉ³
Þ _ ÉÍ¹Ô×ÖµØµÙËÂÛÜÊØ Ù¬ÐáÜáí \ ÞßÉ³
Þ _ ÉÍ¹ÊÌÎ ÉÍÂÐçËÂÊÌÍ¬í \ ÞßÉËÂÇµÊôè¤í \ ØµÉ³ÒîËÂÉ³èóË ì Î ÉÌí \ ÎµÍÂVÛ UÐçËÂÉÏÞßÉ³b
Þ _ÉÍ¬í \ ÎÚaÖ _ÚáÜÛñÙÏÞßÉb
Þ _ ÉÄÍ¬í \ Ù³á×ÐçÒÒòÒîÙ³ÊÌÎ ÉÌí \ Ù³á×ÐçÒÒÈË ì Î ÉÌí \ ïÊáÜá×ÐçË ÎµÍÂÊ _ÚáÜÉÞ í ïÊáÜá×ÐçË í ÊÌËÇ ÐÑÍ ^ `Äí
Ù³ÊÞßÞ®ÐØµè®áÜÛÜØµÉÏÐçÍÂåÖµÞßÉ³Ø¯Ë¬í Ù³ÊÞßÞßÉ³Ø¯Ë¬í \ Ù³ÊÞÀÎÚÛÜá×ÐÑËÂÛÜÊØ¤í ÒÉÎ ÐçÍäÐÑËÂÉÌí !
Ù³ÊÞÀÎÚÛÜáÜÉÄÍ¬í Ù³ÊÞÀÎÚáÜIÉ PßØôÖµb
Þ _ÉÍ¬í \ Ù³ÊÞÀÎÚáÜIÉ P¥ÛñË ì
ÊÔ ÐëÎµÍÂÊ _ÚáÜÉ³Þ í \
ÊÔ ÐØÐáÜåÊÌÍÂÛñËÇÚÞ í \
Ù³ÊÞÀÎ ÊÒîÛñËÂÉ IÉ PôÎÚÍÉ³ÒÒÛÜÊØ¤í \
Ù³ÊÞÀÎ ÊÖµØµè ÒîËäÐçËÂÉÞßÉ³Ø Ë¬í Ù³ÊÞÀÎÚÖ«ËäÐ _ÚáÜÉ Ô×ÖµØµÙËÂÛÜÊØ¤í Ù³ÊÞÀÎÚÖ«ËÂÉÍ
Þ®ÐÛÜØ ÞßÉ³ÞßÊÌÍìí ÞßÉ³ÞßÊÌÍìßÙÉ³áÜáí ÎµÍÂÊôÙ³ÉÒÒÊÌÍ¬í UÊØ O É³ÖµÞ®ÐØµØ¤í Ù³ÊØµèµÛñËÂÛÜÊØ
ÊÔ Ðëæ ÇµÛÜáÜÉÓÒîËÂÐçËÂÉ³ÞßÉ³Ø¯Ë¬í ÊÔ ÐØ ÛÜÔ ÒîËÂÐçËÂÉ³ÞßÉ³Ø¯Ë¬í Ù³ÊØµèµÛñËÂÛÜÊØÚÐáÊÌÎ ÉÍäÐçËÊÌÍ¬í Ù³ÊØµÒîË
ÞßÉ³
Þ _ ÉÍ¹Ô×ÖµØµÙËÂÛÜÊØ¤í \ ïÊØÚÒË
ÖÚÛñèÚÉáÜÛÜØµÉÌí \ Ù³ÊØµÒîË WôÖÚÐáÜÛ Y Éè Ë ìôÎÉÌí \ Ù³ÊØµÒîË »ÍÂÉ³Ô×ÉÍÉ³ØµÙ³ÉÌí \ Ù³ÊØµÒîË »Ë ìôÎ ÉÌí \ ÛÜØµÛñËÂÛ×ÐçáÜÛ ¬ÐçËÛÜÊØ¤í \ í \ Ù³ÊØµÒîËäÐçØ Ë¬í Ù³ÊØµÒîËäÐçØ ËòIÉ P¥ÎµÍÂÉ³ÒîÒÛÜÊØ¤í \
Ù³ÊØµÒîËÍÖÚÙÄËÂÊÌÍ¬í \ \
èµÉ³ÔªÐÖµáñË¬í \ ÉIPôÎ áñÛÜÙ³ÛñË¹Ù¬ÐçáÜáí \ ÛÜØµÛñËÂÛÜÐáÜÛ³ÉÄÍ¬í \ \
Ù³ÊØ¯ËäÐÛÜØµÉÍ³í K ÛñËÂÉÄÍäÐçËÂÛÜÊØ¤í K Ù³ÊØ¯ËÂÛÜØ ÖÚÉÏÒîËäÐÑËÂÉ³ÞßÉ³Ø¯Ë¬í Ù³ÊØ¯ËÍÂÊá¤ÙäÇÚÐçÍäÐÙËÉÍ¬í Ù³ÊØ¯ËÍÂÊá QÚÊÑæâí ÛñËÂÉÄÍäÐçËÂÛÜÊØ¤í ÖµÞÀÎ¡í
áÜÛÜØµÉ¬ÐÑÍ¬í ÒÉ³áñÉ³ÙËÂÛÜÊØ¤í Ù³ÊØ¯ËÍÂÊá¤ÒîËäÐÑËÂÉ³ÞßÉ³Ø¯Ë¬í Ù³ÊØ¯ËÍÂÊá UÐçÍÂÛ×Ð _ÚáÜÉí Ù³ÊØ UÉÍÂÒÛñÊØ
ÐçÍÍÂÐ¬ì®ËÂÊ¿Î ÊÛÜØ¯ËÂÉÍ¬í K
IÉ PôÎ áñÛÜÙ³ÛñË¬í QÚÊÐçËÂÛÜØµåàÎ ÊÛÜØ¯Ë¬í ÛÜÞÀÎÚáÜÛÜÙÛñË¬í ÛÜØ¯ËÂÉ³åÌÍäÐá»í ÎµÍÂÊÞßÊÌËÂÛÜÊØ¤í ÒîËäÐçØÚèÚÐçÍè¾í \ ÖµÒÉÍ èµÉ YÚØÚÉè¾í \ Ù³ÊÖ«Î ÊØÙ³ÊáÜáÜÉ³ÙÄËÂÊÌ!Í Ò ÎµÍÂÊ _ÚáÜÉÞ í ãí ÍäÐÞßÉÍ ÒïÍÂÖµáÜÉÌí ÙÍîìôÎµËÂÊÒîìôÒîËÂÉ³Þ í K
èÚÐçËäÐàÉ³ØµÙ¬ÐÑÎ ÒîÖÚáÜÐçËÂÛÜÊØ¤í \ èÚÐçËäÐàÞßÉ³
Þ _ ÉÍ
ÐÙ³ÙÉ³ÒÒ·Ô×ÊÌÍ³í \ ÊÔ ÒîËîÍÂÖµÙË¬í \ É )ÊÌÍÂåÐ
Ø ÒÈá×Ð³æ Ò³í èµÉ _ÚÖµååÛÜØµåÊÖ«ËÎÚÖ«Ë¬í \
èµÉ³Ù³ÛñÞ®Ðá ËÊ _ÚÛÜØÚÐçÍìÙ³ÊØ UÉÄÍÂÒÛÜÊØ¤í èµÉ³Ù³áÜÐçÍäÐçËÂÛñÊØ¾í áÜÊ Ù¬Ðáí \
ÊÔ ÐàÙ³á×ÐçÒÒòÞßÉb
Þ _ ÉÄÍ¬í \ ÊÔ ÐàÔ×ÖµØµÙËÂÛÜÊØ¤í ÊÔ Z
Ð UÐçÍÂÛ×Ð _ÚáÜÉí \ ÊÔ ÔÜÍÂÛñÉ³Øµè¾í \ ÒîËÍÖÚÙÄË¬í \ èµÉ³Ù³áÜÐçÍäÐçËÂÛñÊØ ÒîËÂÐçËÂÉ³ÞßÉ³Ø¯Ë¬í èµÉ³Ù³áÜÐçÍäÐçËÂÛ UÉÓÍÉ³åÛÜÊØ¤í \ èµÉ³ÔªÐÖµáñËòÐçÍÂåÖµÞßÉ³Ø¯Ë
ÊÔ ÐàÔ×ÖµØÚÙÄËÂÛÜÊØ¤í èµÉ³ÔªÐÖµáñËÈÙ³ÊØµÒîËÍÖÚÙÄËÂÊÌÍ¬í \ èµÉ³ÔªÐÖµáñËÈÛÜØµÛñËÂÛÜÐáÜÛ ¬ÐÑËÂÛÜÊØ
ÒîËÍÂÖµÙË³í \ èµÉ³ÔªÐÖµáñË ÛÜØµÛñËÂÛ×ÐçáÜÛ ¬ÐçËÛÜÊØ
_ôì èµÉ³ÔªÐÖµáñËÈÙ³ÊØµÒîËÍÂÖµÙËÊÌÍ¬í \ èµÉ YÚØµÛñËÂÛÜÊØ
ÊÔ ÐàÙ³á×ÐÒîÒ³í \ ÊÔ ÐàÔ×ÖµØÚÙÄËÂÛÜÊØ¤í ÊÔ Z
Ð UÌÐÑÍÂÛ×Ð _ÚáÜÉÌí \
ÒîËÍÂÖµÙË³í \ èµÉ³áÜÉËÉãIÉ PôÎµÍÂÉ³ÒÒÛñÊØ¾í \
èµÉ³ØµÊÌÍÂÞ®ÐáÜÛ ³É³è®ØôÖµ
Þ _ ÉÍ¬í \ \
èµÉÍÂÉ³Ô×ÉÄÍÂÉ³ØµÙ³ÉëÊÌÎ ÉÄÍäÐçËÂÊÌÍ¬í K èµÉÍÂÉ³Ô×ÉÄÍÂÉ³ØµÙ³ÛÜØµå«í K èµÉËÂÉÍÞßÛÜØµÛÜÒîËÂÛÜ
Ù Y ØµÛñËÉãÐçÖµËÊÞ®ÐçËÂÊØ¤í [ ëí èµÛÜÞßÉ³ØµÒÛÜÊØ
^Þ¿ÖÚá²ËÂÛÜèµÛÜÞßÉ³ØµÒÛÜÊØÚÐá ` ÐçÍÍäÐ³ìí èµÛñÍÂÉ³ÙÄËÂVÛ UÉ
ÛÜØµÙ³áÜÖµèµÉÌí \ ÖµÒÛÜØµå«í èµÛÜÒÙ¬ÐÑÍÂèµÛÜØµå¿ÙÊØÚÒË¬í \ èµÛÜÒÙÍÛÜÞßÛÜØÚÐØ Ë
ÊÔ Ð WôÖÚÐè«ÍäÐçËÛÜÙÓXÉ WôÖÚÐçËÂÛñÊØ¾í èµVÛ U¥ÛñèÚÉÏÐØµèóÙÊØ W ÖµÉÍ¬í \\
èµVÛ U¥ÛñÒÛÜÊØ ÐÒÒÛÜåØµÞßÉ³Ø¯Ë¹ÊÌÎ ÉÍäÐÑËÂÊÌÍ¬í èµÊ ÒîËÂÐçËÂÉ³ÞßÉ³Ø¯Ë¬í
_ Êôè«ìí
èµÊÞ®ÐÛÜØ
ÊÔ ÐàÔ×ÖµØÚÙÄËÂÛÜÊØ¤í \
èµÊÖ _ áñÉÌí è«ÍäÐØµè
ÍäÐØµèµÊÞ ØôÖµb
Þ _ÉÍÈåÉ³ØµÉÍäÐçËÂÊÌÍ³í \ è«ì¥ØÚÐÞßÛÜÙ ÞßÉÞßÊÌÍì®ÐáÜáñÊôÙ¬ÐçËÛÜÊØ¤í è«ì¥ØÚÐÞßÛÜÙ ÎµÍÂÊåÌÍÂÐÞßÞßÛÜØµå«í è«ì¥ØÚÐÞßÛÜÙ ÒËÂÊÌÍäÐåÉÏèµÖµÍÂÐçËÂÛÜÊØ¤í ì¥ØÚÐÞßÛÜÙ " ËÊÌÍäÐåÉ
ÖµÛÜèµÉ³áÜÛÜØµÉÌí É³èµÛñËÂÊÌÍ³í É É³ÙË
ÊÔ ÐàÔ×ÖµØÚÙÄËÂÛÜÊØ¤í \
ÊÔ ÐàÒîËäÐÑËÂÉ³ÞßÉ³Ø¯Ë¬í Ê Ô ÐØ ÉIP¥ÎµÍÂÉÒÒÛÜÊØ¤í \
É É³ÙË ^ÒÉ³Þ®ÐØ¯ËÂÛÜÙ¬ÐçáËÂÉÍÂÞ `´í \ É³áñÉ³ÞßÉ³Ø Ë
ÊÔ ÐçÍîÍäÐ³ìí \ É³ØµÙ¬ÐÑÎ ÒîÖÚáÜÐçËÂÛÜÊØ
ÊÔ èÚÐçËäÐ í \ ÉXW ÖÚÐáÜÛñË ì
Î ÊÛÜØ¯ËÂÉÍ¬í K ÍäÐçËÂÊÒËÂÇµÉ³ØµÉ³Ò " ÛÜÉ UÉÌí \ ÖµÙ³áÜÛÜèµÉ¬ÐØÐáÜåÊÌÍÂÛ²ËÂÇµÞ í \
ÖµáÜÉÍ¬í É³ÊØµÇ ÐÑÍÂè ^ `Äí É UÐáÜÖÚÐçËÛÜÊØ
ÊÔ ÐØ IÉ P¥ÎµÍÂÉÒÒÛÜÊØ¤í \
ÊÌÍÂèµÉÍòÊÔ ÊÌÎ ÉÍÂÐØµèÚÒí ÒÇµÊÌÍËòÙ³ÛñÍÂÙÖÚÛ²Ë¬í É UÐáÜÖÚÐçËÛÜÊØ ÒîXÉ WôÖµÉ³ØµÙ³ÉÌí P«Ù³Éá \ _ Öµå«í \
ÉIP¥É³Ù³Ö«ËäÐ _ÚáÜÉí ÉIP¥É³Ù³Ö«ËÂÛÜÊØ¤í ÉIPôÎÚáÜÛÜÙ³ÛñË¹ÙÊØ UÉÍÂÒîÛÜÊØ¤í ÉIPôÎ ÊØµÉ³Ø Ë
ÊÔ Z
Ð QÚÊÐçËÂÛÜØµåàÎ ÊÛÜØ¯ËÈØôÖµÞb_ÉÍ¬í ÉIPôÎµÍÂÉ³ÒÒÛñÊØ¾í \
ÐçÍÂÛ²ËÂÇµÞßÉËÂÛÜÙÌí \
ÊôÊáñÉ¬ÐØ¤í Ù¬ÐÒË¬í Ù³ÊÞÀÎ ÊÒîÛñËÂÉÌí \
Ù³ÊØµÒîËäÐçØ Ë¬í \
èµÉ³áÜÉËÉÌí \
É É³ÙË ÊÔºí \
É UÐáÜÖÚÐçËÂÛñÊØ ÊÔºí \
É UÐáÜÖÚÐçËÂÛñÊØ ÒÉ WôÖµÉ³ØµÙ³ÉÌí Ô×ÖµØµÙËÂÛÜÊØ Ù¬ÐáÜáí áÜÛñËÉÍäÐáí \ Vá UÐáÜÖµÉÌí ÞßÛ P¥É³è¤í ØµÉæëí a
ÊÔ Ë ìôÎÉ UÊÛÜè¤í ÊÌÍÂèµÉÍòÊÔ É É³ÙËÂÒ³í ÎµÍÂÛÜÞ®ÐçÍîìí \
Í UÐáÜÖµÉÌí Ë ìôÎ ÉÏÊÔºí \
\
UÌÐçáÜÖµÉãÊÔºí \
UÌÐÑÍÂÛ×Ð Ú
_ áÜÉÌí \ ÉIPôÎÚÍÉ³ÒÒÛÜÊØÒîËäÐçËÂÉÞßÉ³Ø Ë¬í ÉIPôÎÚÍÉ³ÒÒÛÜÊØ ËÍÂÉ³ÉÌí ÔªÐÙËÊÌÍÂÛÜØµåØ ÖµÞb_ ÉÄÍÂÒ³í \
ÔªÐÛñÍòèµÛÜÙ³ÉÌí \\
ÔªÐáÜÒîÉÌí [ VÛ _ÊØ ÐçÙ³Ù³Û¤ØôÖµ
Þ _ ÉÍÂÒ³í \ YÚáÜÉÌí !
YÚØµÛñËÂ
É QÚÊÐçËÂÛÜØµåàÎ ÊÛÜØ¯ËòØ ÖµÞb_ ÉÄÍòÒì¥ÒîËÂÉÞ í Y P«É³è Î ÊÛÜØ ËÈØ ÖÚ
Þ _ ÉÍ³í QÚÊÐçË¬í [ áÜÊÐÑËÂÛÜØµå ÊÛñØ Ë ·ÍÛñËÂÇµÞßÉËÂÛÜÙ ÖÚÛñèÚÉáÜÛÜØµÉ í [ áÜÊÐÑËÂÛÜØµå ÊÛñØ Ë ·ÍÛñËÂÇµÞßÉËÂÛÜÙ ÖÚÛñèÚÉáÜÛÜØµÉ \ í QÚÊÐçËÂÛÜØµåàÎ ÊÛÜØ¯ËÈÙ³ÊÞÀÎÚÖµËÂÐçËÂÛÜÊØµÒ
Ù¬ÐØµÙ³É³áñá×ÐçËÂÛÜÊØ¤í èµÛ ÉÍÉ³Ø ËòÒÛ ³ÉÒ³í XÉ WôÖÚÐáñÛñË ìßËÂÉ³ÒîË¬í ÍÂÉ³á×ÐÑËÂVÛ UÉÓÉÍÍÂÊÌÍ³í QÚÊÐçËÂÛÜØµåàÎ ÊÛÜØ¯ËÈÙ³ÊØ UÉÍÒÛÜÊØµÒ³í QÚÊÐçËÂÛÜØµåàÎ ÊÛÜØ¯ËÈØôÖµb
Þ _ÉÍ
èµÉ³ØµÊÌÍÂÞ®ÐáÜÛ É³è¤í \ \
IÉ P¥ÎÊØÚÉØ Ë¬í ÛÜØ YÚØÚÛ²Ë ìí \ \
Þ®ÐØ ËÛÜÒÒÂÐôí O Ð O í \ \
ØµÊÌÍÂÞ®ÐáÜÛ É³è¤í ÒÛÜåØ¤í ÒÛÜåØµVÛ YÚÙ¬ÐØµè¤í QÚÊÐçËÂÛÜØµåàÎ ÊÛÜØ¯ËÈØôÖµb
Þ _ÉÍÈÒîì¥ÒîËÉ³Þ í _ ÐÒÉÌí á×ÐçÍÂåÉÒîËÈIÉ P¥Î ÊØµÉ³Ø¯Ë¬í ÎµÍÂÉ³Ù³ÛÜÒîÛÜÊØ¤í ÒÞ®ÐáÜáñÉ³ÒîË¹IÉ PôÎ ÊØµÉ³Ø Ë¬í QÚÊÐçËÂÛÜØµåàÎ ÊÛÜØ¯Ë¹Ë ì Î ÉÌí QÚÊôÊÌÍÈÔ×ÖµØµÙËÂÛÜÊØ¤í Ô×ÊÌÍòÒîËäÐçËÂÉÞßÉ³Ø Ë¬í _ Êôè«ìí ÛÜØµÛñË ÒîËäÐçËÂÉÞßÉ³Ø Ë¬í ÛñËÂÉÍÂÐçËÂÛÜÊØ¤í ËÂÉÍÂÞßÛñØ ÐÑËÂÛÜÊØ¤í Ô×ÊÌÍÂÞ®Ðá¤ÐÑÍÂåÖµÞßÉ³Ø Ë¬í ÔÜÍäÐçÙËäÐáí \ í \
ÔÜÍÂÛñÉ³ØµèóèµÉ³Ù³á×ÐçÍÂÐçËÂÛÜÊØ¤í \ Ô×ÖµØµÙËÂÛÜÊØ
Ù ]ÉÍÂÞ®ÐçØÚØ¤í \
_ Ê èµìí Ù¬Ðáñáí Ù¬Ðáñá¡ÐÑÍÂåÖµÞßÉ³Ø ËÂÒí Ù¬Ðáñá _ ì®ÍÂÉ³Ô×ÉÍÂÉØÚÙÉÌí \ Ù¬Ðáñá _ ì UÌÐáñÖÚÉí \ èµÉ³Ù³á×ÐÑÍäÐçËÂÛÜÊØ¤í èµÉ³ÔªÐÖµáñËòÐçÍÂåÖµÞßÉ³Ø¯Ë¬í èµÉ YÚØµÛñËÂÛÜÊØ¤í èµÊÞ®ÐÛÜØ¤í \
É É³ÙË¬í Ô×ÊÌÍÂÞ®Ðá¤ÐÑÍÂåÖµÞßÉ³Ø Ë¬í Ô×ÊÌÍÂÞ®Ðá«Î ÐçÍäÐçÞßÉËÂÉÍÕÊÔÍÂÉ³Ô×ÉÍÉ³ØµÙ³É Ë ìôÎ ÉÌí \ Þ®ÐÛÜØ¤í \ Þ¿ÖµËÂÐçËÂÛÜØµå«í !\
Ê UÉÍÂáÜÊÐèµÛÜØµå«í \ Î ÊÒîËÙ³ÊØµèÚÛ²ËÂÛÜÊØ¤í ÎµÍÂÉ³Ù³ÊØµèµÛñËÂÛñÊØ¾í ÍÂÉ³ÙÖµÍÒVÛ UÉÌí \ \
ÍÂÉ³ÙÖµÍÒVÛ UÉâÙ³ÐáÜáí \ \
ÍÂÉËÖµÍØ _ôì ÍÂÉ³Ô×ÉÍÉ³ØµÙ³ÉÌí \ ÍÂÉËÖµÍØ _ôì UÌÐçáÜÖµÉÌí \ ÍÂÉËÖµÍØË ìôÎ ÉÌí ÍÂÉËÖµÍØ UÌÐçáÜÖµÉãÊÔ ÍÉ³Ô×ÉÍÂÉ³ØµÙ³ÉãË ì Î ÉÌí \ ÒÙ³ÊÌÎÉÌí ÒÙ³ÊÌÎÉâÊÔ Ô×ÊÌÍÞ®Ðá¤ÐçÍÂåÖµÞßÉ³Ø¯ËÂÒ³í ÒÛÜåØÚÐçËÖµÍÉÌí UÐáÜÖµÉÌí UÊÛÜè¤í Ô×ÖµØµÙËÂÛÜÊØ Ù¬ÐáÜá
WôÖÚÐáñVÛ YÚÉ³è¤í \a
ÖµaØ WôÖÚÐáÜVÛ YÚÉ³è¤í \a
Ô×ÖµØµÙËÂÛÜÊØ Ù¬ÐáÜá¤ÊÌÎ ÉÄÍäÐçËÂÊÌÍ¬í \
Ô×ÖµØµÙËÂÛÜÊØÚÐá ÊÌÎÉÍäÐçËÂÊÌÍòØµÊÌËäÐçËÂÛñÊØ¾í \
Ô×ÖµØµÙËÂÛÜÊØÚÐáÎµÍÂÊåÌÍÂÐÞßÞßÛÜØµåëá×ÐØµåÖÚÐåÉÌí \ Ô×ÖµØµÙËÂÛÜÊØÚÐáñÛñË ì
ÊÔ ÐàÒîËÍÖÚÙÄË¬í \ ÊÔ ÐëË ìôÎ Éí \ ÊÔ ÐØ ÊÌÎ ÉÍäÐÑËÂÊÌÍ¬í Ô×ÖµØµèÚÐÞßÉ³Ø ËÂÐáË ìôÎ ÉÌí \ \
åÐÞßÉ
ÙäÇÚÊ ÊÒÛÜØµå¿ØôÖµÞb_ÉÍÂÒ³í \\
åÐçÍ _ ÐåÉÓÙ³ÊáñáÜÉ³ÙËÂÛñÊØ¾í \
ÐçÖÚÒîÒ³í ÈÐçÍá [ ÍÂÛÜÉèµÍÛÜÙäÇ¾í åáÜÊ _ Ðá ÒÙÊÌÎ ÉÌí åáÜÊ _ Ðá UÐçÍÂÛ×Ð _ áñÉÌí åÊÌËÂÊ¿ÒîËäÐçËÉ³ÞßÉ³Ø Ë³í
åÌÍÂÉ¬ÐÑËÂÉ³ÒîËòÙ³ÊÞßÞßÊØ®èµVÛ U¥ÛÜÒîÊÌÍ¬í \
åÖµÛÜèµÉ³áÜÛÜØµÉ
ì¥ØÚÐÞßÛÜÙ " ËÂÊÌÍÂÐåÉÌí ÇÚÐáñËÂÛñØÚåàÎµÍÂÊ _ÚáÜÉ³Þ í í ÇÚÐçÍÂÞßÊØµÛÜÙ Ø ÖÚÞ_ ÉÍ³í ÇµÉ¬ÐèµÉÍ
YÚáÜÉÌí !
ÊÔ ËÂÇµÉÏÒîËäÐØµèÚÐçÍÂè áÜÛV_µÍäÐçÍìí \
ÛÜÊÒËÍÂÉ¬ÐÞ í \ ÇµÉ¬ÐçÎ¾í ÇµIÉ PµÐçèÚÉÙ³ÛÜÞ®ÐááñÛñËÂÉÍäÐçáí ÇµÛÜèµÛÜØµå
ÊÔ ØÚÐÞßÉÌí
ÛÜèµÉ³Ø¯ËÂÛVYÚÉÍ¬í \
ê Ù³ÊÞÀÎÚáÜÛ×ÐçØÚÙÉÌí \
ê ÒîËäÐçØÚèÚÐçÍè í ÐçÍÂÛñËÇÚÞßÉÄËÂÛÜÙ ÊÌÎ ÉÄÍäÐçËÂÛÜÊØµÒ³í èµÊaÖ _ÚáÜÉÓIÉ PôËÂÉ³ØµèµÉ³èóÎµÍÂÉ³Ù³ÛñÒÛÜÊØ¤í \
ÒÛÜØµåáÜÉÓIÉ PôËÂÉ³ØµèµÉ³èóÎµÍÂÉ³Ù³ÛÜÒîÛÜÊØ¤í \
UÌÐçáÜÖµÉÏÍäÐçØÚåÉí ê ÒîËäÐçØÚèÚÐçÍè í \
ÛÜÔ¡ÒîËäÐçËÂÉÞßÉ³Ø Ë¬í Ù³ÊØµèµÛñËÂÛÜÊØ¤í ÛÜÔ É³áÜÒÉÏÒîËäÐçËÉ³ÞßÉ³Ø Ë³í ÛÜÔ×Þ ÛÜØ¯ËÂÉ³åÉÍ¬í ÛÜÞÀÎÚáÜÉ³ÞßÉØ ËäÐçËÛÜÊØÀèµÉ YÚØµÉ³è _ É³ÇÚXÐ U¥ÛñÊÌÍ¬í \
ÛÜÞÀÎÚáÜÛÜÙÛñË Ù³ÊØ UÉÍÂÒÛñÊØ¾í ÒîËÍÂÉ³ÐÞ ËÊ _ Ê Êáí ÛÜØµÙ³áÜÖµèµÉÓèµÛñÍÂÉ³ÙËVÛ UÉÌí \ UÌÐÑÍÂÛ×ÐØ¯ËÈæ·ÛñËÂÇÐØµåáÜ
É _ÚÍÂÐÙ ]ÉËÂÒ³í ! UÌÐÑÍÂÛ×ÐØ¯ËÈæ·ÛñËÂÇ WôÖµÊÌËÂÉ³Ò³í !
ÛÜØµÙ³ÊÞÀÎÚáÜÉËÉ ÐçÍÍäÐ³ì®Ë ìôÎ Éí \
ÛÜØµÙ³ÊÞÀÎÚáÜÉËÉ·Ë ìôÎ ÉÌí \ ÛÜØµèµÉ³Ø ËÂÐçËÂÛÜÊØ¤í \ ÛÜØµèµÉIP
ÊÔ ÐçÍîÍäÐ³ì ÉáÜÉ³ÞßÉ³Ø¯Ë¬í \
ÛÜØµèµÛñÍÂÉÙËÂÛÜÊØ¤í K ÛÜØ YÚØµÛñËÂÉ áÜÊ ÊÌÎ¾í ÛÜØ YÚØµÛñËÂÉ ÍÂÉÙ³Ö«ÍÂÒÛÜÊØ¤í \ ÛÜØ Y P ÊÌÎ ÉÍÂÐçËÂÊÌÍÈØµÊÌËäÐçËÂÛÜÊØ¤í \
ÛÜØµÛñËÛ×ÐáÜÛ ³ÐçËÂÛÜÊØ
_ôì Ù³ÊØµÒîËÍÖÚÙÄËÂÊÌÍ¬í \ _ôì ³ÉÍÊ«í
Ù³ÊØµÒîË »Ë ìôÎ ÉÌí \ í \ ÞßÉ³
Þ _ ÉÍ »æ ÛÜÒîÉÌí \ ÊÔ ÐàÒîËÍÖÚÙÄ
Ë UÐáÜÖµÉÌí \ ÊÔ Z
Ð UÐçÍÂÛ×Ð _ÚáÜÉí ÊÔ ÐçÍîÍäÐ³ìí \
ÊÔ ÍÂÉÔ×ÉÍÂÉ³ØµÙ³ÉÌí \
Î ÊÛÜØ¯ËÂÉÍ¬í K ÛÜØµÛñËÛ×ÐáÜÛ ÉÍ
ÊÔ Ù³ÊØµÒîËîÍÂÖµÙËÂÊÌÍ¬í \ \
ÛÜØµÛñËÛ×ÐáÜÛ ÉÍ áÜÛñÒîË
ÐçÍÍÂÐ¬ìí \
ÛÜØ«ÎÚÖ«ËÈÊÌÎ ÉÍäÐçËÊÌÍ¬í \
ÛÜØ«ÎÚÖ«ËÈÒîËÍÂÉ¬ÐçÞ í \
ÛÜØ«ÎÚÖ«Ë ÊÖ«ËÎÚÖ«Ë¹É Ù³ÛñÉ³Ø ìí \
ÛÜØµÒÉÄÍË ºÒîÊÌÍË¬í \\
ÛÜØ¯Ë¬í
ÛÜØ¯ËÂÉ³åÉÍòèµVÛ U¥ÛñÒÛÜÊØ¤í
ÛÜØ¯ËÂÉ³åÉÍòèµVÛ U¥ÛñÒÛÜÊØ ÊÌÎ ÉÍäÐçËÊÌÍ¬í
ÛÜØ¯ËÂÉ³åÌÍäÐçá Ù³ÊØ UÉÍÒÛÜÊØµÒ³í ÛÜØ¯ËÂÉ³åÌÍäÐçáË ì Î ÉÌí ÛÜØ¯ËÂÉ³åÌÍÂÛ²Ë ì
ÊÔ ÍÂÉÄÎÚÍÉ³ÒÉ³Ø¯ËäÐçËÂÛÜÊØ¤í \ ÛÜØ UÌÐÑÍÂÛ×ÐØ¯Ë
ÊÔ ÐàÒîËÍÖÚÙÄË¬í \ ÛÜÊÒËÍÂÉ¬ÐÞ í \ ê " ê * ÒîËÂÐØµè ÐÑÍÂè \ í \
ÛñËÉÍäÐçËÂÛñÊØ¾í Ê UÉÍ Ð Ù³ÊØ ËÂÐÛÜØµÉÍ¬í K Ê UÉÍ ÐØóÐçÍÍäÐ³ìí K ÛñËÉÍäÐçËÂÛñÊØ ÒîËÂÐçËÂÉ³ÞßÉ³Ø¯ËÂÒ³í XÉ W ÖÚÛ UÌÐáñÉ³ØµÙ³ÉãÊÔºí ÛñËÉÍäÐçËÂÊÌÍ³í K í !\
çÊÒÉÎÚÇ
ÖÚÒ¹ÎµÍÂÊ _ÚáÜÉ³Þ í ÖµÞÀÎóÒîËäÐÑËÂÉ³ÞßÉ³Ø¯ËÂÒ³í
ØôÖ«ËÂÇ%)ÊÌÍîÍÂÛÜÒ ÍäÐÑËË ÐçáÜåÊÌÍÂÛñËÂÇµÞ í ]¥Ø Ö«ËÂÇ
ÍäÐØµèµÊÞ ØôÖµÞb_ÉÍÈåÉ³ØµÉÍäÐçËÂÊÌÍ³í \ ÊôÉ³ØµÛÜå¿áÜÊ Ê ]¥Ö«Î¾í \a
á×Ð³ìÊÖ«ËÈÊÔ ÎµÍÂÊåÌÍäÐÞ í \ áÜÉ¬ÐçÒîËÈÙ³ÊÞßÞßÊØ Þ¿ÖµáñËÂÛñÎÚáÜÉí
áÜÉ³ÔÜË·ÐÒÒîÊôÙ³Û×ÐÑËÂVÛ U¥Û²Ë ìí áÜÉ³ÔÜË ÐÒÒîÊôÙ³Û×ÐÑËÂVÛ UÉÌí áÜVÛ _µÍäÐÑÍìí ! ÒîËäÐØµèÚÐçÍÂè¤í \ ÛÜØµèÚÉØÚÞ®Ð³ìÉÍ¹Òì¥ÒîËÂÉÞ í \
ÐáñÎÚÇÚÐ _ ÉË³í \
ÔÜÍäÐÙËÂÐáí \
åÌÍäÐçÎÚÇµÛÜÙ¬Ðá ÛñØ ËÂÉÍîÎÚÍÉËäÐçËÂÛñÊØ¾í \!
ÛÜØµÛñËÂÛ×Ðçáæ¹ÊÌÍÂè¤í \
ÎµÍÂÊôèµÖµÙËÂÛÜÊØµÒ³í \
ÛÜØµèÚÉØÚÞ®Ð³ìÉÍ³í ·ÍÂÛÜÒËÂÛÜèµ
É ^ \ `Äí \ \ \
áÜÛÜØµ
É _µÍÂÉ¬Ð ] ÙäÇÚÐçÍäÐÙÄËÂÉÍ¬í áÜÛÜØµÉ¬ÐÑÍòÙÊØÚåÌÍÖÚÉØ ËÂÛ×Ðçá åÉ³ØµÉÍäÐçËÂÊÌÍ³í \ áÜÛÜØµÉ¬ÐÑÍòÙÊØÚåÌÍÖÚÉØ ËÂÛ×Ðçá ÞßÉËÂÇµÊôè¤í \ áÜÛÜØµÉ¬ÐÑÍòÙÊØ ËÍÂÊá QÚÊ¦æëí áÜÛÜØ ]ÉÍ³í !
ÛÜØô
Ö P í !
áÜÛñËÉÍäÐáí \ _ ÊôÊá»í ÙäÇ ÐÑÍ¬í èµÊaÖ _ÚáÜÉí QÚÊÐçË¬í ÇµIÉ PµÐèµÉ³ÙÛÜÞ®Ðáí í ÛÜØ Ë³í
áÜÊØµåèµÊÖ _ÚáÜÉÌí \
ÊôÙËÂÐáí ÒîËÍÂÛñØÚå«í ÖµØÚÒîÛÜåØµÉ³èÛÜØ¯Ë¬í áÜÊÐèµÉ³è èµÛÜÙ³ÉÌí \\
áÜÊ Ù¬Ðá èµÉ³Ù³á×ÐÑÍäÐçËÂÛÜÊØ¤í \
áÜÊ Ù¬Ðá ÒÙ³ÊÌÎÉÌí áÜÊåÛÜÙ³ÐáÎ ÐçÍÂÉ³Ø¯ËÂÇµÉ³ÒÉ³Ò³í áÜÊåÛÜÙ³Ðá ÊÌÎ ÉÍäÐÑËÂÊÌÍÂÒ³í áÜÊåÛÜÙ³ÐáÎ ÐçÍÂÉ³Ø¯ËÂÇµÉ³ÒÉ³Ò
áÜÉ¬ÐçèÚÛñØÚå ÊÌÎ ÉÄÍäÐØµè¤í ÒÉ³ÙÊØÚèÚÐçÍîìÊÌÎ ÉÄÍäÐØµè¤í áÜÊØµå¿èµÊaÖ _ÚáÜÉí \
áÜÊØµå¿ÛÜØ Ë³í áÜÊ Ê ]¥Ö«Î
ÐçÍÂåÖµÞßÉ³Ø¯Ë ºèµÉÎÉ³ØµèÚÉØ Ë¬í \a
áÜÊ ÊÌÎ¾í ÛÜØ YÚØµÛñËÂÉÌí ÎµÍÂÊåÌÍÂÉ³ÒîÒòËÂÊ¦æÈÐçÍÂèµÒïËÂÉÍÂÞßÛñØ ÐÑËÂÛÜÊØ¤í áVUÐáÜÖµÉÌí áVUÐáÜÖµÉ »ËÂÊ Í UÌÐáñÖÚÉÓÙ³ÊØ UÉÍÂÒÛñÊØ¾í )óÐÙ " í !
Þ®ÐÙäÇµÛÜØµÉ ÉÎÚÒÛÜáñÊØ¾í Þ®ÐÙäÇµÛÜØµÉ á×ÐØµåÖÚÐåÉÌí í !
Þ®ÐÙÄÍÂÊ«í \
Þ®ÐÛñØ Ô×ÖµØµÙËÂÛÜÊØ¤í \ Þ®ÐÛñØ ÞßÉ³ÞßÊÌÍîìí )óÐØµèµÉ³Vá _µÍÂÊÌËÈÒÉÄË¬í \ Þ®ÐØ¯ËÂÛÜÒÒÐ
ÊÔ Z
Ð QÚÊÐçËÂÛÜØµåàÎ ÊÛÜØ¯ËÈØôÖµÞb_ÉÍ¬í Þ®ÐçËÇÚÉÞ®ÐçËÂÛÜÙ¬Ðçá ÛÜØµèµÖµÙËÂÛÜÊØ¤í \ )Ù ¹ÐçÍËÇ ì Ò [ ÖÚØµÙËÛÜÊØ¤í \ \
ÞßÉ³
Þ _ ÉÍÈÐçÙ³Ù³É³ÒÒ
ÛÜØÐ¿ÒîËîÍÂÖµÙË¬í \ ÞßÉ³
Þ _ ÉÍÈÐçÙ³Ù³É³ÒÒòÊÌÎ ÉÍäÐÑËÂÊÌÍ¬í \ ÞßÉ³
Þ _ ÉÍ¹Ô×ÖµØµÙËÂÛñÊØ¾í \ ÐÙ³ÙÉ³ÒÒ·Ô×ÊÌÍ³í \ ÐØµèóÙ³ÊØµÒîË³í \ Ù¬Ðáñáí \ ÛÜÞÀÎÚáÜÛÜÙÛñË Ù¬ÐçáÜá¤ÐçÍÂåÖµÞßÉ³Ø¯Ë¬í \ ÞßÉ³
Þ _ ÉÍ¹ÊÌÎÉÍäÐçËÂÊÌÍ
ÊÔ Ù³áÜÐÒÒ³í \ ÞßÉ³
Þ _ ÉÍ¹ÒÎ É³Ù³VÛ YÚÙ¬ÐÑËÂÛÜÊØ
ÊÔ ÐàÒîËÍÖÚÙÄË¬í \ ÞßÉ³
Þ _ ÉÍ æ ÛñÒÉ ÐÒÒîÛÜåØµÞßÉ³Ø Ë¬í \ ÞßÉ³
Þ _ ÉÍ æ ÛñÒÉ ÛÜØµÛñËÂÛÜÐáÜÛ ¬ÐÑËÂÛÜÊØ¤í \ ÞßÉ³ÞßÊÌÍîìßÙ³É³áÜá»í Ðèµè«ÍÂÉ³ÒÒí ÞßÉ³ÞßÊÌÍîìßáÜÉ¬Ð ]í ÞßÉÍåÉ ºÒÊÌÍîË¬í \\
Ù³ÊÞÀÎÚáÜIÉ P¥ÛñË ìí \
)ÉÍÒÉ³ØµØµÉãÎµÍÂÛÜÞßÉ³Òí ÞßÉËÂÇµÊ è
ÊÔ Ù³á×ÐçÒÒ³í \ ÞßÛÜØµÛÜÞ¿ÖµÞºÒÊÌÍË³í \
Ù³ÊÞÀÎÚáÜIÉ P¥ÛñË ìí \
ÞßÛ P¥É³è GÉ P¥ÎµÍÂÉ³ÒÒîÛÜÊØ¤í ÞßÊôèµÖµá×ÐçÍÛ ¬ÐçËÂÛñÊØ¾í !\
ÞßÊôèµÖµáÜÖµÒ¹ÐÒÒîÛÜåØµÞßÉ³Ø Ë¹ÊÌÎ ÉÄÍäÐçËÂÊÌÍ¬í ÞßÊôèµÖµáÜÖµÒïÊÌÎ ÉÍÂÐçËÂÊÌÍ¬í
Þ¿ÖÚá²ËÂÛÜèµÛÜÞßÉ³ØµÒÛÜÊØÚÐáµÐÑÍÍäÐ³ìí èµÛÜÞßÉ³ØµÒÛÜÊØ¤í ÒÛÜÞ¿Öµá×ÐçËÂÛÜÊ
Ø _ôìÓÊØµÉ³èµÛÜÞßÉ³ØµÒÛÜÊØÚÐáÐçÍÍäÐ³ìí Þ¿ÖÚá²ËÂÛñÎÚáÜÛÜÙ¬ÐÑËÂÛÜÊØÀÐçÒÒÛÜåØµÞßÉ³Ø¯Ë¹ÊÌÎ ÉÍäÐçËÊÌÍ¬í Þ¿ÖÚá²ËÂÛñÎÚáÜÛÜÙ¬ÐÑËÂÛÜÊØ ÊÌÎÉÍäÐçËÂÊÌÍ³í í Þ¿ÖµËÂÐçËÂÛÜØµåàÔ×ÖµØµÙËÂÛÜÊØ¤í !\
ØÚÐÞßÉ
Ù³á×ÐÒîÇ¾í \ ÇµÛÜèµÛÜØµå«í
ÊÔ ÐàÙ³á×ÐÒîÒ³í \ ÊÔ ÐàÔ×ÖµØÚÙÄËÂÛÜÊØ¤í ÊÔ ÐëË ìôÎ ÉÌí \ ÊÔ Z
Ð UÌÐÑÍÂÛ×Ð _ÚáÜÉÌí \ ÊÔ Ô×ÊÌÍÂÞ®Ðá¤ÐÑÍÂåÖµÞßÉ³Ø Ë¬í WôÖÚÐáÜÛ Y Éè¾í \ ÖµØ W ÖÚÐáÜVÛ YÚÉ³è¤í \ ØÚÐÞßÉ³ÒîÎ ÐÙÉÌí \ ØÚÐÞßÉ³ÒîÎ ÐÙÉÏÒîÙ³ÊÌÎ ÉÌí ØµÉ³ÒîËÂÉèóË ì Î ÉÌí \ ØµÉæ IÉ P¥ÎµÍÂÉÒÒÛÜÊØ¤í a
ØµÊÌÍÂÞ®ÐáÜÛ ³É³è QÚÊÐçËÂÛÜØµåàÎ ÊÛÜØ¯ËòØ ÖµÞb_ ÉÄÍ¬í ØôÖµáÜá Î ÊÛñØ ËÂÉÍ³í K
ØôÖµáÜá Î ÊÛñØ ËÂÉÍ UÐáÜÖµÉÌí K
ØôÖµáÜá¤ÒîËÂÐçËÂÉ³ÞßÉ³Ø¯Ë¬í ØôÖµÞßÉÍÛÜÙÓáÜÛÜÞßÛñËÂÒ
ÊT
Ô QÚÊÐçËÂÛÜØµåàÎ ÊÛÜØ¯ËïË ìôÎ É³Òí \
ÊÔ ÛÜØ¯ËÂÉ³åÌÍäÐáË ì Î É³Ò³í Ê _ ÉÙË¬í \
ÖµØÚØÚÐÞßÉè¾í \
Ê _ ÉÙË·ÙÊôèµÉÌí !
ÊÌÎ É³ØÒÊÖ«ÍÂÙ³ÉâÒîÊÔÜË æÈÐÑÍÂÉÌí ! ÊÌÎ ÉÍÂÐØµè¾í É UÐáÜÖÚÐçËÂÛÜÊØ ÊÌÍÂèµÉÍ¬í ÊÌÎ ÉÄÍäÐçËÂÛÜØµå ÒîìôÒîËÂÉ³Þ ^ " `´í !
ÛÜØô
Ö P í !
)óÐÙ " í !
·ØµÛ Pí !
S¼ÛÜØµèµÊÑæ·Ò³í !
ÊÌÎ ÉÄÍäÐçËÂÊÌÍ
ÐèµèµÛñËÂÛÜÊØ¤í ÐèµèµÛñËÂÛÜÊØ ÐÒÒîÛÜåØµÞßÉ³Ø Ë¬í Ðèµè«ÍÂÉ³ÒÒí K ÐçÍÂÛ²ËÂÇµÞßÉËÂÛÜÙÌí ÐçÍÂÛ²ËÂÇµÞßÉËÂÛÜÙ ÐçÒÒÛÜåØµÞßÉ³Ø¯Ë¬í ÐçÍÂÛ²Ë ìí ÐÒÒîÛÜåØµÞßÉ³Ø Ë¬í ÐÒÒîÊôÙ³Û×ÐÑËÂVÛ U¥Û²Ë ìí
_ÚÛÜØÚÐçÍìí _ÚÛÜØµèµÛÜØµå«í
_ÚÛñË æ ÛÜÒîÉÌí Ù³ÊØµèµÛñËÂÛÜÊØÚÐá»í Ù³ÊØ UÉÍÂÒÛÜÊØ¤í \ èµÉÍÂÉ³Ô×ÉÄÍÂÉ³ØµÙ³ÉÌí K èµVÛ U¥ÛñÒÛÜÊØ ÐÒÒÛÜåØµÞßÉ³Ø¯Ë¬í Ô×ÖµØµÙËÂÛÜÊØ Ù¬ÐáÜáí \
Ô×ÖµØµÙËÂÛÜÊØÚÐá ØµÊÌËäÐçËÛÜÊØ¤í \
Ô×ÖµØµÙËÂÛÜÊØÚÐáÜÛ²Ë ìí ÛÜØ Y P ØµÊÌËäÐçËÂÛÜÊØ¤í \
ÛÜØ«ÎÚÖ«Ë¬í \
ÛÜØ¯ËÂÉ³åÉÍòèµVÛ U¥ÛÜÒîÛÜÊØ¤í
áÜÉ³ÔÜË ÐÒÒîÊôÙ³Û×ÐÑËÂVÛ UÉÌí áÜÊåÛÜÙ³Ðáí ÞßÊôèµÖµáÜÖµÒ³í
ÞßÊôèµÖµáÜÖµÒ¹ÐÒÒîÛÜåØµÞßÉ³Ø Ë¬í Þ¿ÖÚá²ËÂÛñÎÚáÜÛÜÙ¬ÐÑËÂÛÜÊØ¤í í Þ¿ÖÚá²ËÂÛñÎÚáÜÛÜÙ¬ÐÑËÂÛÜÊØÀÐçÒÒÛÜåØµÞßÉ³Ø¯Ë¬í ÊÌÎ ÉÍÂÐØµè¾í ÊÖ«ËÎÚÖ«Ë¬í \
Ê UÉÍÂáÜÊÐèµÛÜØµå«í \ Î ÊÒîË ºèµÉ³ÙÍÂÉÞßÉ³Ø Ë¬í Î ÊÒîË ºÛÜØµÙÍÂÉÞßÉ³Ø Ë¬í ÎµÍÂÉ èÚÉÙÍÂÉ³ÞßÉ³Ø¯Ë¬í ÎµÍÂÉ ÛÜØµÙÍÂÉ³ÞßÉ³Ø¯Ë¬í ÎµÍÂÉ³Ù³ÉèÚÉØÚÙÉÌí
ÍÂÉ³áÜÐçËÂÛÜÊØÚÐáí ÍÂÉËÖµÍØ UÌÐçáÜÖµÉÌí \
ÒÖ _ÚÒÙÍÂÛñÎµË¬í \ í K
Ò Ö _µËÍäÐÙËÛÜÊØ¤í Ò Ö _µËÍäÐÙËÛÜÊØóÐÒîÒÛÜåØµÞßÉ³Ø Ë³í ÂË ÉÍÂØÚÐçÍîìí ÖµØ ÐÑÍìí ÖµØ ÐÑÍì ÞßÛÜØôÖµÒ³í ÖµØ ÐÑÍì®Î áñÖÚÒí ÊÌÎ ÉÍÂÐçËÂÊÌÍ¹ËÂÊ ]É³Ø¤í Ê UÉÄÍÂáÜÊÐèµÉ³è¤í ÊÌÍÂèµÉÍòÊÔ É É³ÙËÂÒ³í " í !
ÊÖ«Ë ºÊÔ _ ÊÖµØµè ÐçÍÍäÐ³ì ÛÜØµèµIÉ P í K
ÊÖ«ËÎÚÖ«ËòÊÌÎÉÍäÐçËÂÊÌÍ³í \
ÊÖ«ËÎÚÖ«ËòÒËÍÂÉ¬ÐÞ í \
ÊUÉÍ QÚÊÑæ
ÊT
Ô UÐáÜÖµÉ ÍäÐØµåÉÌí \
ÊUÉÍÂáÜÊÐèµÛÜØµå
ÐçÍÂåÖµÞßÉ³Ø¯Ë ºèµÉÎ ÉØÚèµÉ³Ø¯ËÈáÜÊôÊ ]ôÖµÎ¾í \a
_ É³ÒîËòÞ®ÐçËÂÙäÇ¤í \a
ÊÔ Ô×ÖµØµÙËÂÛÜÊØµÒ³í \ ÊÔ ÊÌÎ ÉÍÂÐçËÂÊÌÍÂÒ³í \ ÊUÉÍÂáÜÊÐèµÛÜØµå ÍÂÉ³ÒÊáñÖµËÛÜÊØ¤í \a
Ð ØµÛÜØµÛí \ \
Î ÐçÍäÐáñáÜÉ³á Ù³ÊÞÀÎÚÖ«ËÂÉÍ¬í \
Î ÐÒîË »ËÂÇµÉ ºÉØÚè Î ÊÛÜØ ËÉÍ¬í K *) " í
Î ÉÍÞÖ«ËäÐçËÛÜÊØ¤í Î ÉÍîÎ ÉËÂÖÚÐá¤Ù¬ÐáñÉ³Øµè ÐÑÍ¬í ÎÚÛñÎ ÉÌí ÎÚá×ÐçËÂÔ×ÊÌÍÞ í !
Î ÊÛÜØ¯ËÈÊÔ èµÉ³Ù³á×ÐÑÍäÐçËÂÛÜÊØ¤í í
Î ÊÛÜØ¯ËÂÉÍ¬í K\
ÐèµèµÛÜØµå ÐØÛÜØ ËÂÉåÉÍ¬í K ÐçÍÂÛñËÇÚÞßÉÄËÂÛÜÙÌí K ÐÒÒÛñåØÚÞßÉØ Ë¬í K Ù³ÊÞÀÎ ÐçÍÂÛÜÒîÊØ¾í K XÉ WôÖÚÐáñÛñË ìí K ÛÜØµÛñËÂÛ×ÐçáÜÛ ¬ÐçËÛÜÊØ¤í K ØôÖµáÜáí K
ØôÖµáÜá UÐáÜÖµÉÌí K
Î ÐÒîË »ËÂÇµÉ ºÉ³Øµè¤í K ÒÖ _ÚÒÙÍÂÛñÎµËòÊÌÎ ÉÍäÐÑËÂÊÌÍ¬í K
Ò Ö _µËÍäÐÙÄËÂÛÜÊØ¤í K
ËÂÊ ÐçÍÍäÐ³ìí Ë ìôÎ Éí K\
Î ÊÛñØ ËÂÉÍ¹Ë ìôÎÉ
ÖµØµèÚÉÄÍÂáñì¥ÛñØÚå Ë ìôÎ Éí K\
Î ÊÌÍîËÂÛÜØµå«í Î ÊÒË ºèµÉ³ÙÍÉ³ÞßÉ³Ø ËÈÊÌÎ ÉÄÍäÐçËÂÊÌÍ¬í Î ÊÒË ºÛÜØµÙÍÉ³ÞßÉ³Ø Ë¹ÊÌÎÉÍäÐçËÂÊÌÍ³í Î ÊÒËÂÙ³ÊØµèµÛñËÂÛÜÊØ
ÊÔ ÐàÔ×ÖµØµÙËÂÛÜÊØ¤í Î ÊÌËÉ³Ø ËÂÛÜÐá ÒÙ³ÊÌÎ Éí ÎµÍÂÉ èµÉ³ÙÍÂÉ³ÞßÉØ ËòÊÌÎ ÉÍäÐçËÊÌÍ¬í ÎµÍÂÉ ÛÜØµÙÍÂÉ³ÞßÉØ ËÈÊÌÎ ÉÍäÐÑËÂÊÌÍ¬í ÎµÍÂÉ³ÙÉ³èµÉ³ØµÙ³ÉëÊÔ ÊÌÎÉÍäÐçËÂÊÌÍ³í
ÎµÍÂÉ³ÙÛÜÒÛÜÊØ
ÊÔ Z
Ð QÚÊÐçËÂÛÜØµåàÎ ÊÛÜØ¯ËÈØôÖµb
Þ _ÉÍÈÒîì¥ÒîËÉ³Þ í ÎµÍÂÉ³ÙÊØÚèµÛñËÛÜÊØ
ÊÔ ÐàÔ×ÖµØµÙËÂÛÜÊØ¤í ÎµÍÂÉ³èµÛÜÙ³ÐçËÂÉÌí \
ÎµÍÂÛÜÞ®ÐÑÍìßIÉ PôÎÚÍÉ³ÒÒÛÜÊØ¤í \
ÎµÍÂÛÜÞßÛ²ËÂVÛ UÉ·ÍÂÉÙ³Ö«ÍÂÒÛÜÊØ¤í \ ÎµÍÂVÛ UÐçËÂÉ
Ù³á×ÐçÒÒòÞßÉb
Þ _ ÉÄÍ¬í \ ÎµÍÂVÛ UÐçËÂÉ ñí \ ÎµÍÂÊ Ù³É³èµÖ«ÍäÐá ÎµÍÂÊåÌÍäÐÞßÞßÛÜØµå«í ÎµÍÂÊ Ù³É³ÒÒÊÌÍ³í ÎµÍÂÊ èÚÖµÙËÛÜÊØ
ÊÔ Ð ÛÜØµèÚÉØÚÞ®Ð³ìÉÍÈÒîìôÒîËÂÉ³Þ í \
ÎµÍÂÊåÌÍäÐçÞ
á×Ð³ìÊÖ«Ë¬í \ UÐáÜÛÜè¤í \ ÎµÍÂÊåÌÍäÐçÞ ÒËäÐçËÂÉÌí ÎµÍÂÊåÌÍäÐçÞßÞßÛÜØµåëá×ÐØµåÖÚÐåÉÌí Ô×ÖµØµÙËÂÛÜÊØÚÐáí \ ÎµÍÂÊÞßÊÌËÂÛñÊØ¾í _ Ê Êá¤ËÊÛÜØ¯Ë¬í ÙäÇÚÐçÍòËÊ ^ÖµØµÒÛÜåØµÉ³è ` ÛÜØ Ë³í QÚÊÐçË¹ËÂÊ èµÊÖ _ÚáÜÉÌí ÎÚÒÉ³Öµè«ÍÂÊÐØµèµÊÞ ØôÖµ
Þ _ ÉÍÂÒ³í \ ÎÚÖ _ÚáÜÛÜÙ
Ù³á×ÐçÒÒòÞßÉb
Þ _ ÉÄÍ¬í \ ÎÚÖ _ÚáÜÛÜÙ ²í \ ìôËÂÇÚÐåÊÌÍÉ¬ÐØ ËîÍÂÛñÎÚáÜÉÌí WôÖÚÐáñÛVYÚÉ³è ×Ô ÖµØµÙËÂÛñÊØóÙ¬ÐáÜá»í \a
WôÖÚÐáñÛVÚ
Y É³è Ú
Ø ÐÞßÉí \ WôÖÚÐØ¯ËÂÖµÞ Ù³ÊÞÀÎÚÖµËÉÍ¬í \
í ÍäÐØµèµÊÞ ÐÙ³Ù³ÉÒÒ
ÛÜØóÐÑÍÍäÐ³ìí \
ÍäÐØµèµÊÞ ÐÙ³Ù³ÉÒÒ·ÞßÉ³ÞßÊÌÍîì ^ ) `Äí ÍäÐØµèµÊÞ ØôÖµ
Þ _ ÉÍ¬í \ ÍäÐØµèµÊÞ ØôÖµ
Þ _ ÉÍÈåÉ³ØµÉÍäÐçËÊÌÍ¬í \ O " ê í \ è«ÍäÐØµè í \ ]¥Ø ÖµËÇ í \ ÍäÐØµåÉ
ÊÔ Î ÊÛÜØ¯ËÂÉÍÂÒí ! ÍäÐçËÛÜÊØÚÐá ØôÖµ
Þ _ ÉÍÂÒ³í \ \
ÍÂÉ³ÙÖµÍÒÛÜÊØ
ÛÜØ YÚØÚÛ²ËÂÉÌí \ ÎµÍÂÛÜÞßÛñËÂÛ UÉÌí \ ËäÐÛÜá ºÉ³Øµè¤í \ ÍÂÉ³ÙÖµÍÒVÛ UÉâÙ³ÐáÜáí \ \
ÍÂÉ³ÙÖµÍÒVÛ UÉâÔ×ÖµØµÙËÂÛñÊØ¾í \ \
Ù³ÊÌÍÍÂÉÙËÂØµÉ³ÒÒ³í \ ËÂÉÍÂÞßÛñØ ÐÑËÂÛÜÊØ¤í \ ÍÂÉ³ÔªÐçÙËÂÊÌÍÂÛÜØµå«í ÍÂÉ³Ô×ÉÄÍÂÉ³ØµÙ³ÉÌí \
ÐÒÒÛñåØÚÞßÉØ Ë¬í \
ÛÜØµÛñËÂÛ×ÐçáÜÛ ¬ÐçËÛÜÊØ¤í \
ÉÔ×ÉÍÂÉ³ØµÙ³É ÖµÛÜèµÉ³áÜÛÜØµÉÌí \ ÍÂÉ³Ô×ÉÄÍÂÉ³ØµÙ³ÉâË ìôÎÉÌí \
ÍÂÉ³áÜÐçËÂÛÜÊØÚÐáÊÌÎÉÍäÐçËÂÊÌÍÒ³í ÍÂÉ³áÜÐçËÂVÛ UÉÓÉÍÍÊÌÍ
ÛÜØ QÚÊÐçËÛÜØµå ÎÊÛÜØ ËÈÙ³ÊÞÀÎÚÖ«ËäÐçËÂÛñÊØÚÒí ÍÂÉ³ÒîÉÍ UÉ³è ØÚÐÞßÉÌí \
ÍÂÉËÖµÍØ _ôì ÍÂÉ³Ô×ÉÍÉ³ØµÙ³ÉÌí \ ÍÂÉËÖµÍØ _ôì UÌÐçáÜÖµÉÌí \ ÍÂÉËÖµÍØóÒîËäÐçËÉ³ÞßÉ³Ø Ë³í í
ÍÂÉËÖµÍØË ìôÎ ÉÌí ÍÂÉËÖµÍØ UÌÐçáÜÖµÉÌí IÉ UÉÍÂÒÉ ¡ÊáÜÛÜÒÇ O ÊÌËäÐçËÂÛñÊØ¾í ÍÂÛÜåÇ¯ËòÐÒÒÊ Ù³Û×ÐçËÂÛ U¥ÛñË ìí ÍÂÖµØ ËÛÜÞßÉ ÉÍÍÂÊÌÍ³í Í UÐáÜÖµÉÌí )
" ÐçÍîÍÂÖµÒ «ÍÂÖµáÜÉÌí ÒÙÊÌÎ É
åáÜÊ _ Ðáí áÜÊ Ù¬Ðáí ØÚÐÞßÉ³ÒîÎ ÐÙÉÌí ÊÔ ÐàèµÉ³Ù³á×ÐÑÍäÐçËÂÛÜÊØ¤í ÊÔ ÐàÔ×ÖµØµÙËÂÛÜÊØóèµÉ³Ùá×ÐçÍäÐçËÛÜÊØ¤í ÒÉåÞßÉ³Ø ËäÐÑËÂÛÜÊØ ÔªÐÖµáñË¬í K
ÒÉáÜÉ³ÙËÂÛñÊØ¾í ÒÉáÜÉ³ÙËÂÛñÊØÒîËäÐÑËÂÉ³ÞßÉ³Ø¯ËÂÒ³í ÒÉáÜÔ ºèµÉ³ÒÙÄÍÂVÛ _ÚÛÜØµå«í K ÒÉÞ®ÐØ ËÂÛñÙ¬Ðá UÐáÜÖµÉÓÍäÐØµåÉ
ÊÔ ÐàÒîËÍÖÚÙÄË¬í \ ÒÉÞ®ÐØ ËÂÛñÙ³Ò³í \ ÒÉØ ËÂÛÜØµÉ³á»í ÒÉÄÎÐÑÍäÐçËÂÉÏÙ³ÊÞÀÎÚÛÜá×ÐçËÛÜÊØ¤í !
ÒÉ WôÖµÉ³ØµÙ³ÉâÎ ÊÛñØ Ë¬í " ÇµÉ ÉÍ ÒîËîÍÂÊ ]ÉÌí " ÇµÉ ÉÍ¬í ·É³Ø«Í
ì ) é ^ ! `Äí ÒÇµÊÌÍËòÙ³ÛñÍÙ³ÖµÛñËÈÉ UÌÐçáÜÖÚÐçËÂÛÜÊØ¤í ÒÇµÊÌÍËòÛÜØ¯Ë¬í ÒÇµÊÌÍËÉ³ÒîËÈÎÐÑËÂÇÎµÍÂÊ _ÚáÜÉ³Þ í ÒÛñèÚÉÏÉ É³ÙË¬í \
" ÛÜIÉ UÉâÊÔ ÍäÐçËÊÒîËÂÇµÉ³ØµÉ³Ò³í \ ÒÛñåØ
ÊÔ Z
Ð QÚÊÐçËÂÛÜØµåàÎ ÊÛÜØ¯ËÈØôÖµb
Þ _ÉÍ¬í ÒÛñåØ ÐÑËÂÖ«ÍÂÉÓÊÔ ÐàÔ×ÖÚØµÙËÛÜÊØ¤í ÒÛñåØÚÉèóÙäÇ ÐÑÍ¬í ÒÛñåØÚÛ Y Ù³ÐØµè
ÊÔ Z
Ð QÚÊÐçËÂÛÜØµåàÎ ÊÛÜØ¯ËÈØôÖµb
Þ _ÉÍ¬í " ÛÜØµåáÜ
É )Ê èÚÛ Y Ù³ÐçËÂÛÜÊØ ÖµáÜÉí ÒÊÖ«ÍÂÙÉ³Ù³ÊôèµÉÌí XÐ UÌÐçÛÜá×Ð _ÚÛÜáÜÛ²Ë ìí !
ÒîÎ ÐåÇµÉËîËÂÛ¤Ù³ÊôèµÉÌí ÒîÎÉ³Ù³Û×Ðá¤ÙäÇÚÐçÍäÐÙÄËÂÉÍ¬í ÒîËÂÐØµè ÐÑÍÂèÙ³ÊØ UÉÍÂÒÛñÊØ¾í \ ÒîËÂÐØµè ÐÑÍÂèÉÍîÍÂÊÌÍ¬í \
ÒîËÂÐØµè ÐÑÍÂèÛÜØ«ÎÚÖ«Ë¬í ÒîËÂÐØµè ÐÑÍÂèáÜÛ _ÚÍÂÐçÍìí \ Þ®ÐçËÂÇµÉ³Þ®ÐÑËÂÛÜÙ¬ÐáÚÔ×ÖµØµÙËÂÛÜÊØµÒ³í \
ÒîËÂè Ù³ÉÍîÍ¬í \
ÒîËÂè Ù³ÛÜØ¤í \
ÒîËÂè Ù³ÊÖ«Ë¬í \
ÒîËÂè YÚáÜáí ÒîËÂè Î Ê¦æëí !
ÒîËÂè Ò W ÍîË¬í !
ÒîËäÐçØÚèÚÐçÍèÊÖ«ËÎÚÖ«Ë¬í \
ÒîËäÐÑËÂÉ³ÞßÉ³Ø¯Ë¬í _µÍÂÉ¬Ð ]í
Ù³ÊÞÀÎ ÊÖµØµè¤í Ù³ÊØ ËÛÜØôÖµÉÌí Ù³ÊØ ËîÍÂÊáí èµÉ³Ù³á×ÐçÍÂÐçËÂÛÜÊØ¤í èµÊ«í
IÉ P«ÉÙ³Ö«ËÂÛÜÊØ¤í IÉ P¥ÎµÍÂÉÒÒÛÜÊØ¤í Ô×ÊÌÍ¬í åÊÌËÂÊ«í
ÛÜÔºí ÛÜÔ ºÉáÜÒÉÌí ÛñËÂÉÍÂÐçËÂÛÜÊØ¤í ÖÚÞÀÎ¾í
ØôÖµáÜáí ÍÂÉËÂÖ«ÍÂØ¤í í
ÒÉ³áÜÉÙËÂÛÜÊØ¤í Òîæ Û²ËÂÙäÇ¾í æ ÇµÛÜáÜÉí ÒîËäÐÑËÂÛÜÙÓÒîËÂÊÌÍäÐçåÉãèµÖ«ÍäÐçËÂÛñÊØ¾í ÒîËäÐÑËÂÛÜ
Ù UÌÐÑÍÂÛ×Ð _ÚáÜÉÌí í
ÒîËÂè Ù³ÉÍîÍ¬í \
ÒîËÂè Ù³ÛÜØ¤í \
ÒîËÂè Ù³ÊÞÀÎÚáÜIÉ Pí \ ÒîËÂè Ù³ÊÖ«Ë¬í \
ÒîËÂè Ò W¯ÍË¬í !
ÒîËÂÊÌÍÂÐåÉãèµÖ«ÍäÐçËÛÜÊØ¤í ÐÖ«ËÂÊÞ®ÐçËÂÛñÙÌí è«ì¥ØÚÐÞßÛÜÙÌí ÒîËäÐçËÛÜÙÌí ÒîËÍÛÜØµåáÜÛ²ËÂÉÍäÐá»í ÒîËÍÛÜØµåÞ®ÐçËÙ´ÇµÛÜØµå«í ØôÖ«ËÂÇ %)ÊÌÍÍÛÜÒ ÍäÐçËîË ÐáñåÊÌÍÂÛñËÂÇµÞ í Ê _ UôÛÜÊÖµÒ·ÐáñåÊÌÍÂÛñËÂÇµÞ í ÒîËÍÖÚÙÄË¬í \ ÐÒÒÛñåØÚÞßÉØ Ë¬í \ èÚÐçËäÐàÞßÉ³
Þ _ ÉÍ¬í \ èµÉ³Ù³á×ÐçÍÂÐçËÂÛÜÊØ¤í \ èµÉ³ÔªÐÖµáñËÈÛÜØµÛñËÂÛÜÐáÜÛ¬ÐÑËÂÛÜÊØ¤í \ èµÉ YÚØµÛñËÂÛÜÊØ¤í \ Ô×ÖµØµÙËÂÛÜÊØÚÐáÜÛ²Ë ìí \ ÛÜØµÛñËÂÛÜÐáÜÛ ¬ÐÑËÂÛÜÊØ¤í \ ÞßÉ³
Þ _ ÉÍÈÐÙÙ³É³ÒÒ³í \ ÞßÉ³
Þ _ ÉÍ¹ÒîÎÉ³Ù³VÛ YÚÙ¬ÐçËÛÜÊØ¤í \ ÖµØµèÚÉÄÍÂáñì¥ÛñØÚå Ë ìôÎ Éí \ UÐáÜÖµÉÏÍÂÐØµåÉÌí \ ÒîÉ³Þ®ÐØ ËÛÜÙ¬Ðáí \ Òì¥Ø ËÂÐÙËÂÛÜÙ³Ðáí \ ÒÖ _ÚÒÙÍÛñÎµË
ÊÔ ÐçÍîÍäÐ³ì ÉáÜÉ³ÞßÉ³Ø¯Ë¬í \
ÒÖ _ÚÒÙÍÛñÎµË·ÊÌÎ ÉÄÍäÐçËÂÊÌÍ
ÐçÍÍÂÐ¬ìí \
Î ÊÛÜØ¯ËÂÉÍ¬í K
ÒÖ _µËÍäÐçÙËÂÛÜÊØÐÒÒÛÜåØµÞßÉ³Ø¯Ë¹ÊÌÎ ÉÍäÐÑËÂÊÌÍ¬í ÒÖ _µËÍäÐçÙËÂÛÜÊØ ÊÌÎ ÉÍäÐçËÊÌÍ¬í " ÖµèµÊ ]¥Ö¤í \ ÒîæÈÐÑÎÚÎÚÛÜØµå
ÊÔ ÞßÉ³ÞßÊÌÍîìí Òîæ·ÛñËÂÙäÇÒËäÐçËÂÉ³ÞßÉØ Ë¬í ÒîìôØ ËäÐçÙËÂÛÜÙ¬Ðçá UÐáÜÖµÉÓÍäÐØµåÉ
ÊÔ ÐàÒîËÍÖÚÙÄË¬í \ ÒîìôØ ËäÐ P í \ ÒîìôØ ËäÐ PÉÄÍÍÂÊÌÍ¬í \ ËäÐçÛÜá ºÉØÚè®ÍÂÉ³ÙÖµÍÒÛÜÊØ¤í \ ËÂÉÞÀÎ ÊÌÍäÐçÍìÀÊ _ É³ÙË³í \ ÍÂÉ³Ô×ÉÄÍÂÉ³ØµÙ³ÉâËÂÊ«í \ ËÂÉÄÍÂØÚÐçÍì ÊÌÎ ÉÍäÐÑËÂÊÌÍ¬í ËÂÊÌÎÊáÜÊåÛÜÙ¬ÐáÒÊÌÍËÂÛÜØµå«í Æ Ê¦æ¹ÉÍÂÒÈÊÔ ÐçØÚÊÛ»í \ \
ËÍÖÚÉí Æ Ö«ÍÂÛñØÚå¿Þ®ÐÙäÇµÛÜØµÉÌí Æ Ö«ÍÂÛñØÚå«í á×ÐØ ^ \ `Äí
ËÂÖ«ÍËáÜÉÓåÌÍäÐçÎÚÇµÛÜÙ³Ò³í \
Ë æ¹Ê ÒÈÙ³ÊÞÀÎÚáÜÉ³ÞßÉØ Ë¬í Ë ìôÎÉÌí \ ÐçÍÂÛ²ËÂÇµÞßÉËÂÛÜÙÌí
_ Ê Êáí _ÚÖµÛÜáñË ÛÜØ¤í \ ÙäÇÚÐçÍ¬í Ù³á×ÐçÒÒ³í \ Ù³ÊØµÒîË¬í \ Ù³ÊØµÒîË WôÖÚÐáÜÛVYÚÉ³è¤í \ èµÊaÖ _ÚáÜÉí QÚÊÐçË¬í QÚÊÐçËÂÛÜØµåàÎ ÊÛÜØ¯Ë¬í Ô×ÖµØÚÙÄËÂÛÜÊØÚÐáÜÛñË ìßÊÔºí \ Ô×ÖµØÚèÚÐÞßÉØ ËäÐá»í \ ÛÜØµÙ³ÊÞÀÎÚáÜÉËÂÉí \ ÛÜØµÙ³ÊÞÀÎÚáÜÉËÂÉ ÐçÍÍäÐ³ìí \
ÛÜØ Ë³í
ÛÜØ ËÉ³åÌÍäÐáí áÜÊØµåèµÊÖ _ÚáÜÉÌí \
áÜÊØµåÛñØ Ë¬í ØÚÐÞßÉÓÊÔºí \ ÊÔ Z
Ð UÌÐÑÍÂÛ×Ð _ÚáÜÉÌí \ ÊÔ ÐØ IÉ P¥ÎµÍÂÉ³ÒîÒÛÜÊØ¤í \
ÊÔ Ô×ÊÌÍÂÞ®Ðá¤ÐÑÍÂåÖµÞßÉ³Ø Ë¬í Î ÊÛÜØ¯ËÂÉÍ¬í K\
ÍÂÉ³Ô×ÉÍÉ³ØµÙ³ÉÌí \
ÒÇµÊÌÍËòÛÜØ Ë¬í ÒÛÜåØµÉ³èóÙäÇÚÐçÍ¬í ÖµØÚèµÉÍáñì¥ÛÜØµåÐàÒîËÍÂÖµÙË¬í \ ÖµØÚèµÉÍáñì¥ÛÜØµåÐØóÐÑÍÍäÐ³ìí \
ÖµØÚÒîÛÜåØµÉ³èÙäÇÚÐçÍ¬í ÖµØÚÒîÛÜåØµÉ³èÛÜØ¯Ë¬í \
ÖµØÚÒîÛÜåØµÉ³èáÜÊØµå¿ÛÜØ Ë³í ÖµØÚÒîÛÜåØµÉ³èÒÇµÊÌÍËòÛÜØ¯Ë¬í UÌÐçáÜÖµÉÏÍäÐçØÚåÉÏÊÔºí \ UÊÛÜè¤í Ë ìôÎ ÉèÚÉÔºí \ ÖµØÚÐçÍì ÞßÛÜØôÖµÒïÊÌÎ ÉÄÍäÐçËÂÊÌÍ¬í ÖµØÚÐçÍì ÊÌÎ ÉÍäÐÑËÂÊÌÍ¬í ÖµØÚÐçÍì®ÎÚáÜÖµÒÈÊÌÎÉÍäÐçËÂÊÌÍ³í ÖµØµèÚÉÙ³ÛÜèÚÐ _ÚáÜÉ ÎµÍÂÊ _ÚáÜÉ³Þ í ÖµØµèÚIÉ Y ØµÉ³è _ É³ÇÚXÐ U¥ÛÜÊÌÍ¬í \
ÖµØµèÚÉÄÍ QÚÊÑæ
ÊT
Ô UÐáÜÖµÉ ÍäÐØµåÉÌí \
ÖµØµèÚÉÄÍÂáñì¥ÛñØÚå Ë ìôÎ É
ÊÔ ÐëÎ ÊÛÜØ¯ËÂÉÍÈË ì Î ÉÌí K\
·ØµÛ Pí !
ÖµØµØ ÐçÞßÉ³è Ê _ É³ÙË¬í \
ÖµaØ WôÖÚÐáÜVÛ YÚÉ³è Ô×ÖµØµÙËÂÛÜÊØ Ù¬ÐáÜáí \a
ÖµØaWôÖÚÐáÜÛ Y Éè ØÚÐÞßÉÌí \ ÖµØµÒÛÜåØµÉ³è Ù´ÇÚÐçÍ³í ÖµØµÒÛÜåØµÉ³è ÛÜØ Ë¬í \
ÖµØµÒÛÜåØµÉ³è áÜÊØµåÛÜØ¯Ë¬í ÖµØµÒÛÜåØµÉ³è ÒÇµÊÌÍË·ÛñØ Ë¬í ÖµØµÒîÎ É³ÙVÛ YÚÉ³è _ É³ÇÚÐXU¥ÛñÊÌÍ¬í \
ÖµÒÉÍ ºèµÉ YÚØµÉ³è Ù³ÊØ UÉÄÍÂÒÛÜÊØ¤í \ ÖµÒÛÜØµå¿èµÛñÍÂÉ³ÙËVÛ UÉÌí UÐáÜÛÜè ÎµÍÂÊåÌÍäÐÞ í \ UÐáÜÖµÉ
ÊÔ ÐZUÐçÍÂÛ×Ð _ÚáÜÉí \ Ê Ô ÐØ ÉIP¥ÎµÍÂÉÒÒÛÜÊØ¤í \
UÐáÜÖµb
É ^ÒÉ³Þ®ÐØ¯ËÂÛÜÙ¬ÐçáËÂÉÍÂÞ `´í \ UÐáÜÖµÉÓÍäÐØµåÉ
ÊÔ ÐàÒîËÍÖÚÙÄË¬í \ ÊÔ ÐëË ìôÎ Éí \ ÊÔ Ë ìôÎÉ _ Ê Êáí ÊÔ Ë ìôÎÉãèµÊÖ _ÚáÜÉÌí ÊÔ Ë ìôÎÉ QÚÊÐçË¬í ÊÔ Ë ìôÎÉãÛÜØ¯Ë¬í ÊÔ Ë ìôÎÉãÖµØµÒÛÜåØµÉ³è ÛÜØ Ë¬í Ê UÉÍ QÚÊÑæâí \
ÒÉ³Þ®ÐçØ ËÂÛÜÙ³Ðáí \ ÒîìôØ ËäÐÙÄËÂÛÜÙ¬Ðá»í \ ÖµØµèÚÉÄÍ QÚÊÑæâí \
UÐçÍÂÛ×Ð _ áñÉÌí \ Ðèµè«ÍÂÉ³ÒÒòÊÔºí \ Ù³ÊØ¯ËÍÂÊáí åáÜÊ _ Ðáí ØÚÐÞßÉÓÊÔºí \ ÒîËäÐÑËÂÛÜÙÌí í
Ë ìôÎ ÉÏÊÔºí \ UÐáÜÖµÉãÊÔºí \ UÐçÍÂÛ×Ð _ áñÉÏèµÉ³Ùá×ÐçÍäÐçËÛÜÊØ¤í \ UÐçÍÂÛ×Ð _ áñÉÏèµÉ YÚØµÛñËÂÛñÊØ¾í \
UôÛÜÒVÛ _ÚÛÜáÜÛ²Ë ì
ÊÔ ØÚÐÞßÉÌí UÊÛÜè¤í UÊÛÜèÔ×ÖÚØµÙËÛÜÊØ¤í UÊØ O É³ÖµÞ®ÐØµØ Ù³ÊÞÀÎÚÖ«ËÂÉÍ³í æ·ÇÚÛñáÜÉ ÒîËäÐçËÉ³ÞßÉ³Ø Ë³í
_ Êôè«ìí æ µÇ ÛñËÉ³ÒîÎ ÐÙ³ÉÌí S¼ÛÜØµèµÊÑæ·Ò³í !
æ Û²ÍÂÉ³ÔÜÍäÐÞßÉ ÞßÊ èµÉ³áí \ )
åÌÍäÐçÎÚÇµÛÜÙ³Ò¹Ô×ÊÌÍÂÞ®ÐçË¬í ³ÉÍÊ ºÛÜØµÛñËÂÛÜÐáÜÛ¬ÐÑËÂÛÜÊØ¤í
³ÉÍÊ ËÂÉÍÞßÛÜØÚÐçËÂÉ³è ÐçÍÍäÐ³ìí

Download Report

[PDF]

Paperzz.com

Your Paperzz