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// Raise a number to the eighth power .
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int main ()
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// output b * b , i .e . , a ^8
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// Raise a number to the eighth power ,
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Raise a number to the power 8.
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main
ifm::integer
// Program : power8 .C
// Raise a number to the eighth power .
int main ()
{
// input
cout < < " Compute x ^8 for x =? ";
int a;
cin > > a;
// computation
int b = a * a ; // b = a ^2
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// b = a ^4
// output b * b , i .e . , a ^8
cout < < a < < " ^8 = " < int main []{ int a ; int b; int c; std :: cin > > a ;
cin > > b;c = a * b ; std :: cout < < c *c; return 0;}
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return 0;
-
int main ()
{
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5
9 * celsius / 5 + 32
32
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(9 * celsius) / 5
((9 * celsius) / 5) + 32
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H
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1 * 15 + 32
15 + 32
47
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Unary additive operators.
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9 / 5
1
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b
b
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#
a = (a
a / b
a
/
fahrenheit.C
15
+
+
--
--
--
1
--
++
--
--
--
++
# 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;
}
+ 1
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1
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1
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1
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%
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limits
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+
b

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7
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* H
, H
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=
#
Assignment operators.
b−1
, −2
int
, %
1 ( * 1
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& !
) ,
1 1
, ( * ! 1 *
H H 1 * /
12 * * * , ! 1 1
,
!
* !
*
& IH
H ) &
!
+=
1
1 b−1
{−2
!
+=
2b
Prefer pre-increment over post-increment.
// Program : limits .C
// Output the smallest and the largest value of type int .
# include < iostream >
# include < limits >
" Minimum int value is "
std :: numeric_limits < int >:: min () < < " .\ n"
" Maximum int value is "
std :: numeric_limits < int >:: max () < < " .\ n";
Minimum int value is -2147483648.
Maximum int value is 2147483647.
/
%
2147483647+1
int
2.2.6 The type unsigned int
int

5
69
5
7
6
8
4
9
7
8
8
67
-
,
? 1 H !
* ! H
* 1
, H , , !
!
! &
;
**
!
'
'
a
3
b
4294967295
int
int
Conversion binary → decimal.
k
X
i=0
bi 2 i = b 0 + 2
k−1
X
i=0
+
bi+1 2i = . . . = b0 + 2(b1 + 2(b2 + 2(
H
1
1
=:n
{z
1
) *
C
|
F
*
i=0
∞
X
b k . . . b0
+ 2bk ) . . . ))
H H
bi+1 2i+1 = b0 + 2
!
n
-
bi
,
H
! ) *
1
!
13 = 1 20 + 0 21 + 1 22 + 1 23 .
C F ) H *
*
!
H
!
&
{0, 1}
i=0
∞
X
bi 2 i = b 0 +
,
*
i=1
∞
X
* - *
* & * *
1
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, ! * i=0
!
*
bi 2 i = b 0 +
H
A
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1
1
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n
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1
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b
1
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?
%
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1
int
13
* ,
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, 1 1
!
1
unsigned int
*
Conversion decimal → binary.
∞
X
b−1
*
bi
b0
n
bi i 1
n = (n − b0 )/2
n = 14
b0 = 14
2 = 0
n = (14 − 0)/2 = 7
n=7
b1 = 7
2=1
n = (7 − 1)/2 = 3
n=3
2=1
n = (3 − 1)/2 = 1
n = b3 = 1
14 b3 b2 b1 b0 = 1110
i=0
∞
X
*
n
n =
,...,2
C
!
−2
H
-
2
-
n=
b2 = 3
1
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1
,
H
)
F
unsigned int
int
b
b−1
int a = 3 u ;
int b = 4294967295 u;
unsigned int
n
H
0
!
1
+
H
H *
*
! **
*
!
limits.C
unsigned int
17+17 u −→ 17 u +17 u −→ 34 u
int
, +
F
unsigned int
F * ) C ! unsigned int
1
int
1
*
Minimum value of an unsigned int object is 0.
Maximum value of an unsigned int object is 4294967295.
H
int
unsigned int
? 1
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,
int
,
,
, ?1
)
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!
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int
1
unsigned int
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1
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*
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1
+
b
1
H
?
HI
*
*
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, ! b
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2.2.8 Binary representation
− 1,
0, . . . , 2b − 1.
bi 2 i ,
1101
bi+1 2i
}
5
69
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b
1
b=3
{2b−1 , . . . , 2b − 1}.
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n
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0
1
2
3
1
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(((b4 2 + b3 ) 2 + b2 ) 2 + b1 ) 2 + b0 = (((1 2 + 0) 2 + 1) 2 + 0) 2 + 0 = 20.
-
b
000
001
010
011
100
101
110
111
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2b unsigned int
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1 A !
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1
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100
101
110
111
000
001
010
011
b
n
b=3
110
111
1101
101
−3
n
n
n
b
unsigned int
2b
unsigned int
n + 2b
unsigned int
2.2.9 Integral types
unsigned int
"
5
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int
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int
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)
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H
)
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1
C
*
+
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&
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1
!
)
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** H *
2 * C 1
1 *
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7 011
2.2.10 Details
x4
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x4 )
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x4 ))).
2
13
l
x3
x3 )
x3 )
x3 )),
unsigned int
b
unsigned int
5
69
5
7
6
8
7
4
9
8
8
67
i
+!
int
11
i
12
++ i + i
i
6
1
! !
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C
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17
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C
long int
H
H
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&
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,
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int
unsigned int
)
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*
" --(-a)
a+(+b)
?
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5
,
signed char short int int
1
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,
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!
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(a++)+b
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H ,
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H 1
0
1 H A
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1
* H
a+++b
a++b
*
'
'
a+++b ---a
a+(++b)
a+(+(+b)
-(--a) --(-a)
-(-(-a)
Order of effects and sequence points
-
, *
!
!
/
, H
1
)
1
+
Other integral types.
* &
-
* ! H
1
H
1
-
, ?
--(-a)
+
-
- * C 1
* !
! 1
-
*
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int
,
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*
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unsigned char unsigned short int
7
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? !
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1
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+
,
!
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)
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+
)
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C
H ,
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,
)
+
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int
i = ++ i + 1
i
++i
long int
i
++i
5
7
6
i
1

#
+- * / %
6 3 7 1
# 6 $ 3
7 ! #
2
3
2
7
6
c
!
1
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#
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# * ) * - C * )
int
unsigned int
5 b
0" #
a/b/c
c
0
b, c > 0
a b
(
a
#
!
/
!
)
c=-a=b
b*=++a+b
a+3*--b+a++
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31/4/2

$
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H
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:
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H
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, H * *
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b)
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* c=a+7+--b
a-a/b*b
7+a=b*2
$
Exercise 15
Exercise 14
$
(a
+
unsigned int
%
, *
Exercise 13
-
, B
2.2.11 Goals
unsigned int
*
*
, B
H std::cout
-2-4*3
5u+5*3u
!
!
%
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Exercise 12
, *
!
%
A *
nextvalue
<
, int
H
1
,
, *
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6 * /
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*
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, 7
!
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( std::cout
H
1
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:
( nextvalue *= 5) += 3
$! # 0
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,
1
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)
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,
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7
(nextvalue *= 5)
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H
H
H
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Exercise 11

7
A
nextvalue = 5 * nextvalue + 3
-
1
7
int
int
- 2
*
*
HI 1
,
? G
nextvalue
B
7
6
i = ++i + 1
)
6
int
* 1
unsigned int
-
! * B7 )
#
*
3
Operational.
7
6
Dispositional.
, 1
* 3

Single Modification Rule:
unsigned int
6-3+4*5
-1-1u+1-(-1)
c=a=-b
a-a*+-b
b+++--a
-
(bc).
"
"
5
69
5
7
6
8
4
9
7
8
8
67
k = 3
3 " $
1
# !
1
B
1
2 8 2
k
a
3
2
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/
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0
4
8
7
6
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6 #
3 7 !
2
a
# (
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!
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#
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$ 0 ;
0 # 1
0 0
( , #
" ,
1
x + y/9
13 4/9
−1 − 1/9 = −10/9
/ !
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0
2.2.13 Challenges
4
1
" #
!
2
projecteuler.net/
" #
$ % " ! 2 4 ! " k
0
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$ 6 3 7 ! {1, . . . , 41}
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(3, 4, 5)
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Exercise 22
a2 + b 2 = c 2
pythagoras.C
1000
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243,112,609 − 1
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||
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2.3.4 Details
5
69
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7
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x =
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g : {0, 1}2 → {0, 1}
bitwise operator ϕg
P
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2
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H
ϕ
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2.3.5 Goals
bool
bool
2.3.6 Exercises
)
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5
5
5
69
5
7
6
8
7
4

69
8
{ int main ()
{
# include < iostream >
1
1
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2
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7
}
/
# 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;
}
<
z==2
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1
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5
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x y
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4
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b
c
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{
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unsigned int a;
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5
5
5
69
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2.4 Control statements
2.4.1 Selection: if– and if-else statements
5
5
5
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unsigned int n;
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// computation of sum_ {i =1}^ n i
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return 0;
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do
do
bool
for
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0
int a ;
// next input value
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n
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continue
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}
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int main ()
{
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}
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return 0;
}
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int main ()
{
int x ;
std :: cin > > x ;
int s = 0;
int i = -10;
do
for ( int j = 1;;)
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while (++ i <= x );
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* *
1 " H 7
!
!
e
H
1
!
?
G
82.4 = 824 10−1 ,
0.0824 = 824 10−4 .
.
@
!
*
* *
,
C
*
A
H & HI
)
?
,
!
" !
/
7
H
F
)
82.4
0.0824
)*
G
1
3
* H
H - ?
unsigned int
double
%=
double
int
float
, double
,
*
,
!
int
82.4
{ | } { ,
. * ! !
G
6 H !
1
7 * " ) !
H *
1 double
)* H *
, , 1 ! *
*
float
* A
H *
, ,
float
10
, ,
double
A
0
) &
,
H
, 1
, !
H -
! ! IH
G
* 1
' '
$
!
H *
H , ,
,
, * (
*

6 int
float
double
1.23e-7
.1
E
double
e
∞
X
1
= 2.71828 . . .
i!
i=0
"
5
69
5
7
6
8
4
9
5
7
8
6
67
6
5
7
t /= i
H
H
+
*
A
H * , )* , A , * ! ,
,
<
*
,
)
)
H
float
(unsigned int)(1.6f)
!
float
T x =
H
−1
6
H
H
H
A
* ( *
- ,
! ) !
* A
+
2
!
!
, * !
! +
* H
,
<
float
*
!
* H *
* (

H
*
, *
,

)
t
) ,
,
,
&
! 1
! ! !
-
H *
* * ! float
double
double
−1.6
*
1
i
x
A
"
!
double
G
, 1
, *
,
! H !
*
, * +!
float
G
)
H
,
*
,
unsigned int
float
x
−2
float
*
,
&
+
1 , ,
7 )
H double
float
,
6
1
A
,
+
?
* H H 7
* , & ,
float
,
#
F ,
) 1
* , ! G
, , 1
*
*
* double
int
int(-1.6f)
,
!
H
? A
*
double
*
)
,
!
1
H
-
H * A
!
*
,
,
int
)
7
) 1
* * H
,
!
* * ,
, ,
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- )
) ! )
* *
, H
*
/ !
H
int i = -1.6f;
H 7
6
1/i!
float
float
unsigned int
2.5.2 Mixed expressions, conversions, and promotions
e
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
7
H
}
return 0;
* e += t /= i
float
H 7
) !
*
+
{|
}
* * 1
)
! )
? *
)
, 6 - 1
}
F *
) * D* *
,
* E !
7 H
7
*
6
7
7 H
&
A *
int celsius
*
,
H 7 H
1/(i − 1)!
)
i
,
!
)
?
1/i!
!
&
, ,
1
*6
) !
unsigned int
!
G , +
1 * H H !
, , - -
HI G
)
* 1
) HI &
1
* , * , e += (t /= i)
<
,
1
1
! G
)
) 1
, 6
* (

7 - &
,
*
int
,
! ,
,
Program 11:
,
i
* euler.C
unsigned int
fahrenheit.C
9 * celsius / 5 + 32
*
- +!
* )
, *
IH
!
?
&
* *
t
e += t /= i ;
// compact form of t = t / i ; e = e + t
std :: cout < < " Value after term " < < i < < " : " < < e < < "\n ";
double
int i = -1.6 f;
i
float
2.5.3 Explicit conversions
unsigned int
float celsius
5
69
5
7
6
8
7
4
9
5
8
6
67
6
5
7
HI
d0 = 0
0
0.1
-
H
!
H
c
,...,e
" * *
C
i
e {e
d0 d1 . . . dp−1
β = 10
!
1
H
" ! !
!
1
1
*
!
1
6
!
H
/
" 7
%
&
H 7 1
!
H
6 &
1
!
/ ! 1
7 6 " !
7 $
6 ! e
, A A ! %
!
* 1
*
A * *
A
G * * 1 1 *
*H - *
* H A
*
;
*
*
*
1
* <
!
1
)
1
H
{0, . . . , β − 1}
)
p
!
,e
1
1
* 1
/
!
G {−1, 1} di
s
e
0.1 100 0.01 101
1
!
1 H
1* 1
C *
1
* , H
)
,
!
H
First number
=?
Second number
=?
Their difference =?
Computed difference
βe
H
-
,
!
,
,
+
H * H , , 1
? * G ! 1
!
!
< + ! 1
, 1
1
*
,
7
F " G 1
!
H
1 HI D
E
1
< &
6
H
F
G
- *
* 1 , !
*
! * * *
,
?
First number
=?
Second number
=?
Their difference =?
Computed difference
H
G
(β, p, e
,e
(β, p, e
i=0
p−1
X
s
=? ";
- s
β
e
2
// Computation and output
std :: cout < < " Computed difference - input difference = "
< < n1 - n2 - d < < " .\ n" ;
return 0;
-
=? ";
float d ;
std :: cout < < " Their difference =? ";
std :: cin > > d;
F
)
# include < iostream >
// Program : diff . C
// Check subtraction of two floating point numbers
*
!
)
1 0 H std :: numeric_limits < double >:: max ()
}
1.79769
, 1 !
7 : 6 * H 1
) * 6
7 * )* 1
H
1
1 A
! 1 float n2 ;
std :: cout < < " Second number
std :: cin > > n2 ;
G
*
*
/ 1
* /
*
! A *
* A
1.79769e+308
<
} { | } { int main ()
{
// Input
float n1 ;
std :: cout < < " First number
std :: cin > > n1 ;
2
Program 12:
{|
10308
"
{ {{ {
{
double
!
}{
!
1
*
1
* *
)
2.5.4 Value range
1.5
1
1.5
1.0
0.5
- input difference = 0.
1.1
1.0
0.1
- input difference = 2.23517e -08.
1.1
2.5.5 Floating point number systems
di β−i βe ,
}
d0 .d1 . . . dp−1 βe .
1.0 10−1
$
5
69
5
7
6
8
4
9
5
7
8
6
67
6
5
7
i=0
xi = (xn+1 − 1)/(x − 1)
65, 535
1.1
)
77.1
850
100, 000
(2, p, e
,e
H
)
1
*
=:x
!
0
1 X
bi−1 2i .
2 i=−∞
| {z }
bi i
x = 2(x − b0 )
H
0
,
-
bi−1 2i−1 = b0 +
,e
1
- 1
! (2, p, e
x = 1.1
p, e
e
b0
b−1
b−2
b−3
b−4
b−5
b−6
→
→
→
→
→
→
→
..
.
1.1
0.2
0.4
0.8
1.6
1.2
0.4
1
i=−∞
0
X
*
bi 2 i = b 0 +
x
−1
+!
)
F H !
1
*
, 1
)
!
<
C
*
1
, 1
* * 1
H ,
1 H
) &
)
F
*
N
2k
!
# H
1 C
! 1 ) * &
* 1 ! H
C *
* ! C x = 1.1
b0
k
-
! * ? , ? 1 1
! * ,
*
! C
A *
* !
! =
=
=
=
=
=
1
c
i=−∞
−1
X
1
double
0.1
0.2
0.4
0.8
0.6
0.2
2
2
2
2
2
2
1.1
1 G 1 1
)
1 H !
& * 1
* & < * ( * *
?
1
* =
=
=
=
=
=
,
*
* ,
1 6 A ,
7 1 G ? 1
p
, 1 * &
H H G
H
2(1.1 − 1)
2(0.2 − 0)
2(0.4 − 0)
2(0.8 − 0)
2(1.6 − 1)
2(1.2 − 1)
A
HI
1
* c
y
float
bi 2 i = b 0 +
x = y + 2k
* :
#
1
H 1
, , i=−∞
0
X
x
*
x
i.
H
H * ! F
)
; H
* * ,
*
H
,
+
x =
,
6 * * * * !
7
1 x<2
1
!
)
βe+1 < βe+1 .
e
0.1
i
- H
1
, *
,
1 B 1
1
G
? D !
E x
G
% 1 *
?
*
1
)
*
*
, D
!
* E
*
, A
)
/5
d0 = 0
e
x
"
1.1
bi
i−i+1
x=1
s
2
)
i−i0
* H
1 !
* H ,e
1
-
/ * 1
)
, !
, 1
*
! (
* *
*

1 & * 1
$
/ ! 1
G
1
Computing the floating point representation.
x
i=0
i−i
X
!
-
i=0
p−1
X
#
(2, p, e
1.25 = 1 2−2 + 1 20 .
*
(β − 1).(β − 1) . . . (β − 1) βe =
bi−i 2i−i =
-
{0, 1}
)
−∞
di := bi−i
i=0
i−i
X
bi 2 i =
1
1
bi
i
e
6
1 ! H " 1
H
1
" &
! *
0
)
*
x
, ! H ! 1 )
!
!
!
1 * & IH C
bi 2 i ,
Pn
+
*
i=i
i
X
)
i=−∞
∞
X
x=
x=
1.0 . . . 0 β = β ,
e
y < 2
x<2
=1
=0
=0
=0
=1
=1
=0
1.00011
)
{∞}
x = 0.1
The Excel 2007 bug.
0.1

5
69
5
7
6
8
4
9
5
7
8
6
67
6
5
7
!
!
-
Arithmetic operations.
x
^
H
1 * 1
!
* G
! 1
) *!
G
2−p
*
*
HI
H A
1
*
*
?
A
0
, 1
1
* ) 1 1 , G 1
1 1
; F
! * , H
* * H , , !
! * ! +
! !
* ,
H * &
2
* , 1
,
,
1
! H
3
4
5
x
^
H
6

1
)*
!6
*
!
* -
7
1
G
H
!
)
H
x
- , 1 !
H
C
!
, )
*
1
G
?
)*
1 D &
E 1
1 * *
, G ? 1
1 ,
1
*
1 ,
)* < A * 1
G
H
1
!
2−p
H
*
)
0
% H
!
H
* * ! 1
- x
1 , ! 7 1
- * C * !
* 1
G
1 1
* ! 6 ,
D ,
IH *
E * * 1 :
! * ! G H 7 ?
1
**
! 1 * H 1
1
* / )
* 1 1 !
1 ,
H H H
)
1
1
)
- H
*
* *
, H
H A
<
*
*
1
!
!
x x
2i−p+1
H H !
H
- ,
1 1 H ! * :
$ !
! *
; * 1
* * H 1
H
1
* , , x
* 1
0
F (2, 3, −2, 2)
1.11 22 = 7
[1/4, 7]
* 6
G !
H
1 G
1
x
7
}
1
,...,e
x = x + 2i−p+1
{e
1
1
G
0
!
< H 1
*
H
1 * H H
, ! 2i = 2i−p+1 .
1.00 2−2 = 1/4
A
x
2−p ,
2i−p
H
i=−∞
|x − x
^|
|x|
#
bi 2 i .
x
^
x
G *
1 ,
!
1
B
!
6 * * 1
1
C * !
* : |x − x
^|/x
G
1
bi−p+1 = 0
x
2i−p+1
)
#
x
x
^
p−1
p
+
!
1
1
)*
C
, * * A
* /
&
! )
H 7
! , !
1
*
1
*
65, 535
H
x
p
,e
i−p
i=i−p+1
i
X
bi , i
G ( 1
1 , 1 ) H
A )
) 1
77.1
bi 2i = bi .bi−1 . . . 2i ,
-
H !
)
1 77.1
$
*
x = bi .bi−1 . . . bi−p+1 2i =
, H *
1
bi 2 i
i−p
X
(2, p, e
x
850
65, 535
1
2i−p+1
2i−p+1
i=−∞
%
0.1
#
x−x=
i−p
X
i=−∞
i
X
x=
bi−p+1 = 1
Relative error.
1.1
65, 535
65, 535
65, 535
x
x
2i
2−p ,
x
^
x
x = 0.
x
bi = 1
8
x
x
x
5
69
5
7
6
8
4
9
5
7
8
6
67
6
5
7
, 1
*
,
, G
1
*
*
!
,
H !
G
* H , + 1
) H * H ) 1
* *
:
7 ,
H
*
, !
, 1
)*
6
(
?*
!
* ,
A
*
,
G 1
!
* H
'
1 + *
1
?*
'
-
* &
1
HI 2
A
- ,
*
*
H
* C
H , ,
* & 6 * * 1 < 1
, , c - ! !
1 , H
1
*
H 7 +! ?
1
*
0
* * 1 1
!
1
, 1 H
* ) ,
!
*
!
1
F
, ! 1
1
*
1
* IH
1
1 IH
,
)
H
I
* &
, C
1 *

1
1
H **
!
)
+
H -
&
D " E
! 1
1 *
HI * , <
)
! H
*
)
* 1 , 1
HI )
1 C
!
1
1
!
1
H
euler.C
1
G
1
double
* *
*
IH *
, H
6 7 , ? 1
1
H G , 3 IH !
1 !
1 , * float
,
*
H *
F
)
+
dp−1 = 0
Floating Point Arithmetic Guideline 1:
!
& D
!
*E
G
1
H
F Requirements for the arithmetic operations.
H !
!
!
)*
G
*
, +
< * )
*
H 1
1 1 1
* &
1 IH 1.1
double
F (2, 53, −1022, 1023)
double
C
'
'
*
1
! , * , !
,
1
,
*
,
float
float
-
2.5.6 The IEEE standard 754
+
1.001 20 .
0
*
!
p=4
float
A - )
! !
* *
* ,
IH
! HI *
, 1
1.00101 20 .
F (2, 24, −126, 127)
+
100.101 2 .
! *
* * H !
G , 6
1 !7
5
5
5
H !
, ?
−2
! * G
1 7 - , 1
IH
! *
1
*6
?
!
"
!
1.111 2−2
+ 10.110 2−2 .
0
float
HI
*
1
,
H 1.111 2−2
+ 1.011 2−1 .
* &
&
!
H
1
G
1
H
*
- *
* & +
C H 1
1 !
B * * * 1
!
1 H
1
G
* *
+
&
H 1
, 1
1
-
( * *
, ?
1 * 0
B ! )
32
#
1
!
H *
!
G
Value range.
double
double
p=4
32
254 = 28 − 2
8
2046 = 211 − 2
64
2.5.7 Computing with floating point numbers
1.1
"
5
69
5
7
6
8
4
9
5
7
8
6
67
6
5
7
Hn =
i=1
i
.
Compute H_n for n =? 100000000
"
1 / i
*
*
A
)* ! , 2
<
"
!
* &
1
G
*
H
1
&
- ) )*
, 1
HI
G , 1
1
1
*
H *
C
H *
)
*
HI 1 * HI !
1
* ,
)
*
6
7
,
H !
:
)
) -
IH
)*
H ,
?
G
H
*
1
)*
*
F
*
H
* &
H 1 * 7 ! * 1 6 , 1
{ }{ { { {{ { {| } { | } !
{
*
G
HI *
! H )
* * )
*
, H H
, 1
, 3
!
!
*
*
! )
A G ( * *
C ! C
, !
H A 1 , *
*
H - A
/
*
1 H
* 0
H
HI
, !
* H C
, H 1
/
, HI
*
C
, * @
* B * A ! 1
H
n
,
* 1
Program 13:
*
/
* &
-
n
n
2p + 2
/
2p
p+1
)
bi 2 i ,
A
!
p
(
*
1
(
: * H 1
Adding numbers of different sizes.
2p
1
Hn
p
-
βp

{
p
(bp , bp−1 , . . . , b0) = (1, 0, . . . , 0, 1)
2p + 1
2p
* / * 6 1
! *
* +
H
H 7 HI !
==
-
-
E * & ! G
1 6
) 1
H !
/
7 DA
i=0
p
X
2
* - , * /
*
H * & *
*/
H *
1
!=
(x + y) * (x - y)
n
X
1
n
& * !
& * * * * 1
! 6
! *
* * * 1
1
!
! 7 F !
&
H ? 1 8 !
IH 2p + 1 =
IH x * x - y * y
n
1
1
1
!
!
69
7
6
5
5
*
8
7
4
9
5
8
6
67
6
5
7
)
*
+
* , ,
)
,
, G
1
Hn
1
n
harmonic.C
// Program : harmonic .C
// Compute the n - th harmonic number in two ways .
# include < iostream >
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;
}
Compute H_n for n =? 10000000
Forward sum = 15.4037
Backward sum = 16.686
2p−1 − 1, c = 2p−1 + 1
1
b = 2p , a =
4
0
,
* +
6 H 7 &
1 / G
! 1
!
!
,%
! 1
G *
* 1
!
C
1
, +
* *
*
0 ! H
!
1
*
*
-
H
%
*
* H , ) * C ,
, * C
* &
) H 1
H
1
* 1 * *
C * *
*
:
H *
?
!
*
*
* &
,
*
* & <
!
* * *
*
*
! H +
!
, *
H IH 1 * * &
& C
1
?@!
H !
!
1 * -
*
! )
H
6 ? ! 7 1 G
* H 1
- 1
, 1
! * 6
H 7
1
*
* * H
*
!
*
)
1
7
6
!
C
1
H
1
1
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p = 64
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double
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// 2^23 + 1
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std::numeric_limits<float>::digits
std::numeric_limits<float>::min_exponent
std::numeric_limits<float>::max_exponent
+1
+1
,
0
−∞
NaN
2.5.9 Goals

5
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0
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5
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for ( float i = 0.0 f ; i < 100000000.0 f ; ++ i)
std :: cout < < i < < "\n ";
x 0<
1
float
s
1
s
xn = (xn−1 +
).
2
xn−1
s.

5
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Exercise 62
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.
"

5
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2.6 Arrays and pointers
bool
n
// Program : eratosthenes.C
// Calculate prime numbers in {2 ,... ,999} using
// Eratosthenes ’ sieve .
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;

5
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2.6.4 Arrays are not self-describing
int a [5] = {4 ,3 ,5 ,2 ,1}; // array of type int [5]
int b [5];
b = a;
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a
p, p + s, p + 2s, . . . , p +
p
4−5
s
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
2.6.6 Pointer types and functionality
The dereference operator.
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!=
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* p = false ; // * p is the element pointed to by p
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new
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n
// 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 ]) {
$
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7
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H
int * i = new int ;
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new
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7 *
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.
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new
new
:
new
n
-
7
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// cross out all proper multiples of i
for ( unsigned int m = 2* i ; m < n ; m += i )
crossed_out[ m ] = true ;
// * i is undefined
// * j is 6
delete j;
delete i;
new
delete

5
69
5
7
6
8
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5
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97
’\0’
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char
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1
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Dynamic Storage Guideline:
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char
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The type char.
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9
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Memory leaks.
int
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$
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delete[]
for ( char c = ’a ’; c <= ’z ’ ; ++ c)
std :: cout < < c;
for
abcdefghijklmnopqrstuvwxyz
std :: cout < < ’a ’ + 1;
char
’a’+1
’a’+1
{0, . . . , 255}
char text [] = { ’b ’ , ’o ’ , ’o ’ , ’l ’}
char text [] = " bool "
0
’\0’
’l’
5
5
69
5
7
6
8
4
9
5
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7
8
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6
7
int main ()
{
// definition and initialization : provides us with
// Booleans crossed_out [0] ,... , crossed_out [999]
bool
* &
! ?
int main ()
{
// search string
char s [] = " bool ";
# include < iostream >
std::cin
E
IH
1 !
*
,
*
H
* # include < iostream >
// Program : eratosthenes. C
// Calculate prime numbers in {2 ,... ,999} using
// Eratosthenes ’ sieve .
)*
std::cin
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eratosthenes.C
"bool"
16
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0
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:
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i
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
// find pattern in the text being read from std :: cin
std :: cin > > std :: noskipws ; // don ’t skip whitespaces!
unsigned int w = 0; // number of characters read so far
unsigned int i = 0; // index where t logically starts
// cyclic text window of size m
char * t = new char [m ];
& - )
+
* H // Program : string_matching. C
// find the first occurrence of a fixed string within the
// input text , and output the text so far
// determine search string length m
unsigned int m = 0;
for ( char * p = s ; * p != ’\0 ’ ; ++ p ) ++ m;
1 7 6 !
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2, 3, . . . , m + 1
1, 2, ..., m
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;
}
i+m
std::cin >> t[i]
std :: cin > > std :: noskipws ; // don ’t skip whitespaces!
"
5
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5
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6
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4
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n
n2
0
nk
new
n
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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 ;
}
@
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5
69
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7
6
8
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9
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67
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//
//
//
//
//
// 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 )
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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;
)
#
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Figure 15:
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i = 23
// read floor dimensions
int n ; std :: cin > > n ; // number of rows
int m ; std :: cin > > m ; // number of columns
21 20 19 20
)
*
// 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];
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17 18
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23 22 21 22
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Figure 14:
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8 12
------X - - - - -XXX - -X - - - - --SX - - - - - - - ---X - - -XXX - ---X - - -X - - - ---X - - -X - - - ---X - - -X -T - -------X - - - -
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;
S
5
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<<
<<
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}
!
// 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 ;
# include < iostream >
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int main ()
{
// read floor dimensions
int n ; std :: cin > > n ; // number of rows
int m ; std :: cin > > m ; // number of columns
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else if ( floor [r +1][ c ] == d ) ++ r;
else if ( floor [r ][ c -1] == d ) - - c;
else
++ c ; // ( floor [r ][ c +1] == d )
// 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 ||
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for ( int r =0; r <n +2; ++ r)
delete [] floor [r ];
delete [] floor ;
’S ’;
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’o ’;
’X ’;
’-’;
*
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<<
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<<
,
// 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 ";
}
<
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Program 17:
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} | } } } }
| } { | {
{ }
// 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)
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floor [r ][ c ] = i ; // label cell with i
progress = true ;
}
5
69
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// computation and output
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 ;
# include < iostream >
int main ( int argc , char * argv [])
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// Calculate prime numbers in {2 ,... ,n -1} using
// Eratosthenes ’ sieve .
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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;
n
const unsigned int n = 1000;
n = 10000; // error : can ’t assign to a constant
const unsigned int n ; // error : uninitialized constant
1000
// 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
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Exercise 65
argv[1]
# 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;
n=7
int
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Exercise 73
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
n
# include < iostream >
of a [0][0]
of a [0][1]
of a [1][0]
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19
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{
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// b^ e = (1/ b )^( - e)
b = 1.0/ b ;
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// the remainder is as before
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3.1.3 Function calls
pow(2.0,-2)
3.1.2 Function definitions
2
pow(2.0,-2)
3.1.4 The type void
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// outputs 2
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// in the scope of declaration in line 3
int main ()
{
f ();
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int f ( int i ); // scope of f begins here
1
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i
// 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)
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int i = 5; // invalid ; i hides formal argument
return i ;
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{
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{
int i = 5; // ok ; i is local to nested block
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n
n
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1
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// 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 < > n;
A
2
Program 20:
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void fill_n ( int a [] , int n , int value );
// 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 );
10
fill_n
fill_n
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// iteration by index
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a[ i ] = value ;
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return 0;
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std :: cin > > i;
h (i );
17
return 0;
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double f ( double x )
{
return g (2.0 * x );
}
double inverse ( double x)
{
double result ;
if ( x != 0.0)
result = 1.0 / x;
return result ;
}
bool g ( double x)
{
return x % 2.0 == 0;
}
17
bool is_even ( int i)
{
if ( i % 2 == 0) return true ;
}
void h ()
{
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3 7
int g ( int i)
{
return i * f (i ) * f( f(i ));
}
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{
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for ( int k = i ; k <= j ; ++ k )
r += 1.0 / k;
return r;
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{
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else
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// input
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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;
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3.2.1 A warm-up
n!
// POST : return value is n!
unsigned int fac ( unsigned int n)
{
if ( n <= 1) return 1;
return n * fac (n -1); // n > 1
}

69
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H
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1
f(3)
f(n-1)
f(n-1)
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#
void f ()
{
f ();
}
3
2
1
n
3.2.3 Basic practice
(0, n) = n

69
5
74
6
8
7
4
6
59
4
8
*
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b)
b 0
b =
- ! * b
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fib(n-2)
F45
// 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;
(a, b) → (b, a
Fn
n
fib(50)
F48
fib(n-1) F47
,
!
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C
!
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1
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H H
n = 30
+!
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F1 := 1,
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*
1
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, !
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G
HI
*
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F46
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1 ,
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- @
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)
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unsigned int gcd ( unsigned int a , unsigned int b)
{
if ( b == 0) return a ;
return gcd (b , a % b ); // b != 0
}
6
*
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H
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b)
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k
a
*
a
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Proof.
n > 1.
// 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
}
n = 50
Fn
1
Fi , i < n−1
3.2.4 Recursion versus iteration
b)
"

69
5
74
6
8
7
4
6
59
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! *
fib(50)
Fi , i
f
fib2
f
f
unsigned int f ( unsigned int n)
fib
fib
!
-
fib2
fib2
n
fib2(50)
F50
fib
fib2
3.2.5 Primitive recursion
m
n

 n + 1,
A(m, n) =
A(m − 1, 1),

A(m − 1, A(m, n − 1)),
A
20, 000
A(n, n)
A
!
A(4, 1)
A(4, 2)
A(4,3)
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?
3 A(m, n)
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1
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&
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+
, // 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 ;
}
Fi
a
*
, *
1 !
,
*
!
*
!
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1
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1
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!
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1
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i
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f
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+
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*
H
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!
a
!
!
*
f
}
return a ;
,
' '
! * A
H
*
* F !
) *
H ! * * !
* *
1
! *
a = b;
b = a_prev % b ;
{
if ( n == 0) return 1;
return f (f(n -1) - 1);
}
}
n
f
Fn
A(m, n)
A
// 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));
}
m=4

69
5
74
6
8
7
4
6
59
4
8
B
// 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 )
{
*
0
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n
!
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1
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1
2
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.
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!
n
1
*
*
H
1 + 2 + ...n − 1 =
0
!
$ minimum_sort
n
p_min = q
++q != last
F 1
!
!
1
H
*
C
!
* *
?
!
, Observation 1
*
c1 , c2
c1 n(n − 1)/2
n
! * *
* &
C ! *
! 6 *
) H ;
* *
? !
,
1
*
*
*
* * ,
!/ 3
#
n+3
1
)
*
B
, * H 7
, )
C ! C !
IH *
* ) *
1
! ! A
H *
) , !
1
1 +
* &
*
F 1 ** ) 7
#
2
.
[first, last)
!
D
, 1
E
* (

* (

H
, !
3.2.6 Sorting
*
*
C
B
!
!
1
6
..
2| 2{z } −3
/ −3
1
0
HI
* ,
65536
$
4
8

69
5
74
6
8
7
4
6
59
n
n+1
n+2
2n + 3
2n+3 − 3
*
, 3 C ) * * , ! -
6 A *
H 7 A
!
1 & 1
4 13 65533 265536 − 3 22
&
Table 6:
3
4
5
9
61
-
&
! C C
* *
1 ; H H < 1
2
3
4
7
29
H
*
C
! 1
2
3
5
13
m
0
1
2
3
! *
H
1
* %
1 C
! *
*
1
1
n
0
1
2
3
5
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 );
}
}
std::min_element
minimum_sort
n
n−1
while
n(n − 1)
2
int* q = p
n(n − 1)/2
n(n − 1)/2 + n
c1 n(n − 1)/2 + c2 n
c2 n
#
Gcomp=
merge
// PRE : [ first , last ) is a valid range
* ,
!
*
!
* * 7
1
A !
/
*
IH A
H
!
1
1
!
7
@
HI
*
H
*
1
1
"
H 1
A
IH 1
1
A
H
*
?
1
1
H
1
, !
*
, 1
H
* %
*
* - !
1 ! * 1
* !
) 1
C
*
C
H
1
- *
H
!
,
1
1 ! 6 ! * * 3 7 !
* , 3 * 1 *
!
IH !
*
Time
C
!
1
, 3
A
H H 1 H * 1
F H HI *
,
, ! n
1
- ,
C
H
& * 1
!
! 3
*
6
H ? 7
A
HI *
*
10−9 n(n − 1)/2 Time
sec/Gcomp
H
* 1
1 , "
1
1 1 < !
1
6
, -
A
+
#
?
!
#
*
! * 1
C
H ! *
1 (
*
* !
* * ! 1
1
!
!
1 H !
<
-
<
<
"

<
<
<
H
"
H H
H

H H n
Gcomp
Time (min)
sec/Gcomp
1
, * A
H
H
*
+ ! H 7 * A
1
1 *
* , , 6
* 1 A ? ? *
Table 7:
, 1
,
@
IH
!
, 1
H
A
D *
)
*
E H
2 1 C !
* % *
Tcomp
!
,
*
F
$
A
A !
0 1
1
A
* - * A *
*
, A !
*
) *
Tcomp
A
Merge-sort.
!
* // sort into ascending order
minimum_sort ( a , a+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;
// is it really sorted ?
for ( int i =0; i <n -1;++ i )
if ( a[ i ] != i ) std :: cout < < " Sorting error !\ n";
*
* * A * A
!
!
< *
* IH <
* H A , ! * ! 1
A ! * ! B 1 -
6 1 *
A &
* , A * ! A ;
std :: cout < < " Sorting " < < n < < " integers ...\ n";
delete [] a;
*
* B 7
6 int main ()
{
int n = 100000; // number of values to be sorted
int * a = new int [n ];
return 0;
H
n
" 1 * 1
'
'
H
!
,
*
* C * !
* ! ,
,
!
main
0, n−1, 1, n−2, ...
}
,
!
Ttotal
*
C
+
n Gcomp
IH 1
!
? 3
109
, minimum_sort
0, 1, . . . , n − 1
1
6 * H 7 ** F
, ,
1 , !
H
!
1 *
H
1
B **
D*
, 1 E
! Tcomp
%
%
-
n
*
Ttotal
Gcomp
sec/Gcomp

69
5
74
6
8
7
4
6
59
4
8
*
T (0) = T (1) = 0
T (2) = 1
1
*
!
0
A
* , !
*
* H
&
*
!
* 1
!
*
* A 1
! * C
! *
merge
0
1
n
#
n
+ * * A
!
, 1 7 !
*
1
6
* !
!
! , *
* !
merge
)
*
* * *
A ! * ) A * ;
1 !
H *
1
&
! !
! 6
* 7
1 *
*
,
H
%
! %
*
H *
F " 1 H * ! 1
!
H H 1
A 1
%
!
** 1
H C * *
! 1
1
!
1 *
*
C
//
//
//
//
1
1
3
* right ++;
* left ++;
* left ++;
* right ++;
*
!
2
merge_sort
=
=
=
=
A
#
*
A * A
+
H & ,
* * *
A / ) ; *
! !
! !
#
3
*d
*d
*d
*d
* 1
/
* A
* 1
A
H 7 A 6
! * &
**
- 1
!
*
* if
( left == middle )
else if ( right == last )
else if (* left < * right )
else
n
T (n)
2
(n − 1)
0
.
B
Analyzing merge-sort.
6
&
Theorem 2
Proof.
n/2
0
5
* 1
A * C
*
!
merge
HI *
,
*
!
*
!
9
" "
4
3
2
1
!
x
!
*
[first, last)
n/2
x x
// 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
x
// 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
}
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
H n/2
4
6
merge_sort
! C
*
F ) * H *
*
*
H H
* * !
!
! ! H " 1 H +
x
[middle, last)
x x
*
3
5
+!
4
6
n/2
merge
n/2
[first, middle)
2
3
5
!
*
n
H
H E7 H !
1
H
C D 6 C
6
E7 D
Figure 19:
*
+
+! % ,
1 H ) 1
, *
, C
H
*
H 1
&
1
, 1
4
6
*
2
3
5
H
!
*
1
4
6
H
A * A
*
* IH A
* , ,
& 1
! ,
+
!
1 ,
! * 1
2
1
3
2
1
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 ;
n
n
n−1
n
2
1

69
5
74
6
8
7
4
6
59
4
8
&
1
,
* A 1
! - H 1 * < H H
H " H H

!
* * 1
& +
* IH ! ;
3 * -1 H 1
!
H
sec/Gcomp
2
n
6
minimum_sort
sec/Gcomp
n
n = 51, 200, 000
Σ = {F, +, −}
Σ
F + F+
Σ
P:Σ→Σ
P
*
2
* *
*
(n − 1)
?
n = 1, 600, 000
minimum-sort
merge_sort
n
*
merge_sort
(n − 1)
2n
minimum_sort
n
* minimum_sort
n
!
F
1 1 1
H *
,
H
* 1
H
*
1
*
* 1
H
H
1
1
*
1
* !
* 1
*
C
* *
;
* ! *
*
2
n(n − 1)/2
, A !
1 -
!
,
! - +!
* 1 H H A
1
, H * * ! * !
1 ! n
(n − 1)
!
, ?
*
!
1
#
#
!
7
6
* -
Table 8:
! * 1
IH
1
* 6 A 1
*
* !
3*
H
:* 1 7 !
*
7
H
F
H
<
!
*
*
, *
/ C
! A
*
!
* A ! * 7
* +
! * ! H * * *
*
7
H 1 ,
* *
) 6 , &
* 1 1 & A -
* "
H " " "

H " H
$ H " "
H $
"
H n
Mcomp
Time (sec)
sec/Gcomp
merge_sort
) *
* ! H
* H A *
* H
*
, * *
? ? merge_sort
!
* B H ! 1
1 H
! 1
* ,
! *
3*
! * 1 * ) + 1 77
!
6
n = 1
C
6
HI ! *
* C
*
- !
1 ! * *
C
! ? )
1
2
6
6
*
, 7
*
, 1
- ! !
1 6 1 H !
*
* ! 1
!
*
n
&
*
* 7
)
*
&
!
7
T (3) = 2
H 7
6
1
*
1
* H *
*
H * * !
! 1 !
;
n/2 , n/2
+
n=1
F
)
T (n)
merge_sort
n−1
! 1
*
,
HI *
1 1 !
+
!
1 T (n)
H
H
! !
2

H
* *
, ,
*
!
!
* , (
*
F "
1
, ?
2
n
n
T(
) + T(
) + n − 1.
2
2
H 7
**
H
n
)
* * * A
! 1
n
n
T(
) + T(
)+n−1
2
2
n
n
n
n
(
− 1)
+(
− 1)
+n−1
2
2
2
2
2
2
n
n
(
− 1)(
− 1)(
2 n − 1) + (
2 n − 1) + n − 1
2
2
n
= (n − 2)(
n
−
1)
+
n
−
1
n
=
+ n2
2
2
(n − 1)(
n
−
1)
+
n
−
1
2
= (n − 1)
2n .
6
*
, 7
F
HI
, F
T ( n/2 )
merge
106
Time
*
*
T ( n/2 )
*
! !
!
1
,
2
C
H *
*
1
!
T (1) = 0 = (1 − 1)
n
2
{1, . . . , n − 1}
!
1
%
T
:
*
!
T (n)
1
merge_sort
Mcomp
Mcomp= 10−6 (n−1)
*
( * * * 1
* , A
* /
!
1
T (n)
H
2
n
% ! 0
)
1 H H
H & &)
*
0
) &
!
*
*
,
! 6
!
)
,
! * ,
1 , 0 ! ,
& * * * 1
!
C
!
! 1
H IH )
!
* 1 )
*
A7
*
T (3)
3
n
n
n
n
n
n
n
1, 600, 000
n(n − 1)/2
50
minimum-sort
3.2.7 Lindenmayer systems
Σ
Σ
"

69
5
74
6
8
7
4
6
59
4
8
* - , F
* *
6
1 ! C H 7
* * !
1
0
!
*
Turtle graphics.
F +
−
F
6 !
7 - * * !
H F
90
{F, +, −}
, * A
{
−
p
+
90
F + F+
// 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);
//
}
ifm
1
forward left
# include < iostream >
# include < IFM / turtle >
is drawn
{
F
w_ {i -1}^ F
+
w_ {i -1}^ F
+
turtle
7
6
H
!
P i (+) = +, P i (−) = −
&
1
? !
1
1
6
? !
1
1 H 7
- - * * *
)
(
!
+ → +, − → −
// Prog : lindenmayer.C
// Draw turtle graphics for the Lindenmayer system with
// production F - > F +F + and initial word F .
!
A
* 1
HI -
σk
i
!
1
) H ? !
, 1
! ) !
1
**
* +
! w2 = wF2 = F + F + +F + F + +
!
wi−1
P(σ) = σ1
right
*
* 6 * ?A F
F
H
wσi
&
wσi
1
σ1
Σ
1
k
wσi−1
, . . . , wσi−1
wσi
σ
- *
!
1
H 1
* *
* *
!
σ
(
*
!
-
σ
D
, E
*
, H
+.
−
wFi−1
+
)
*
!
" !
#
#
.
σk
!
=
wFi−1
1
!
$ H !
A !
1
) !
*
σ1
wFi
F + F+, + → +, − → −
+
1
wσi = wσi−1
wi =
H *
*
!
!
1 0
H *
, <
*
wi
-
s=F
wσi
+
!
*
H
*6
7 *
* A
* 7
F + F+
*
p
F,
F+F+
F + F + +F + F + +
F + F + +F + F + + + F + F + +F + F + + +
wi−1
1
Σ
s = w 0 , w1 , . . .
: H
!
)
$
! $
- (
*
) ! s
- F
& H
C ) &
*
!
* 0
= (Σ, P, s)
,
wi−1
H 1
1
H )
+
→ P(σ)
→ F + F+
→
+
→
−
H E
=
=
=
=
..
.
+
w0
w1
w2
w3
)
D
-
σ
F
+
−
* H
1
Figure 20:
p
} { | } *
* A ! E - 1 H
H
- * ! *
H E D
1 C ,
* *
! * D !
C * * H E
* *
A A
1 IH H
1 1 ! ) D
* ) *
H ! ?
*
!
Recursively drawing Lindenmayer systems.
σ
i
w+
i = +
i
σk
(i − 1)
k
wσi−1
.
wσi
wi
s
F →

69
5
74
6
8
7
4
6
59
4
8
{ { {
}{
// draw w_n = w_n ( F)
f(n );
return 0;
<
$
6

7
, !
!
!
*
-
0
*
90
n=5
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;
&
!
) *
G A ! 1
/
!
Additional features.
α
!
w∞
# 1 #
<$ ,
-
w∞ = w ∞ + w ∞ + .
// 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);
//
}
}
{
C
-
!
H
* n=∞
// 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 >
{ { { }{ { { {{ { {| } { | } , H
1 * !
*
*
B
% H * n
%
{
!
2
"
H
*
n = 14
F
!
Program 29:
{
}
// draw w_n = w_n ^ F ++ w_n ^F ++ w_n ^F
f(n );
// w_n ^ F
69
5
74
6
8
7
4

$
6
59
4
8
int main () {
std :: cout < < " Number of iterations =? ";
unsigned int n;
std :: cin > > n;
{
{ {{ {
{|
}
// POST : w_i ^ X is drawn
void x ( unsigned int i ) {
Lindenmayer systems.
#
1
! %
!
1
* H 7 6 1 * 1
7 1
*
2
"
<
"
!
Program 31:
* !
H A
,
H
26
!
int main () {
std :: cout < < " Number of iterations =? ";
unsigned int n;
std :: cin > > n;
}
*
1
0
H
H
H F
)
*
1 G
)
1
H
*
* )
B 1
!
1
! F
)
H -
2
<
3
0
"
!
} { |
| } { | {
{ { { }{ { { {{ { {| } {
7
6
H
//
//
//
//
//
// necessary: x and y call each other
f
]
if ( i > 0) {
x(i -1);
ifm :: left (90);
y(i -1);
ifm :: forward ();
ifm :: left (90);
}
// 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 >
F H
90
0
X
6
2
1
*
A
H 1
7
+
* *
Σ
void y ( unsigned int i );
w14
"
Program 30:
*
return 0;
0
* * 6
* ?
*
7
, !
*
}
{ Σ = {F, +, −, X, Y}
++
w_n ^ F
++
w_n ^ F
}
X → X + YF +
Y → −FX − Y
//
//
//
//
0
$6

=
$
7
$
"
| ifm :: left (120);
f(n );
ifm :: left (120);
f(n );
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
}
}
// draw w_n = w_n ^ X
x(n );
return 0;
[
F
3.2.8 Details

69
5
74
6
8
7
4
6
59
4
8
-
(
-
(
-
(
(-
4
mccarthy(99)
mccarthy(91)
mccarthy(101)
mccarthy(100)
#
2
#
*
)
1
"
<
$
:*
#
0" $A F * 2 #
#
n > 100
n 100.
9
! #
mccarthy
#
"
n − 10,
M(M(n + 11)),
0 #
#
M(n) :==
=
3 <
0 3
#
"
0 #
#
7
6
*
*
7
, A
, !
*
!
6
3 7
B *
, !
*
*
H *
, !
!
1
*
6
"3 7
, !
*
,
!
*
B
*
*
*
3
7
6
, !
*
, !
B
*
, 6
3 7
17
&
? !
*
&
'
'
!
*
, !
*
*
B *
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unsigned int f ( unsigned int n)
{
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unsigned int A ( unsigned int m , unsigned int n ) {
if ( m == 0) return n +1;
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{
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}

69
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//
exists
int * binary_search ( int * first , int * last , int x)
{
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100
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{
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add
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int n ;
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3
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rational v [3];
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struct
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* 1 ! ! H * * & ! )
B * * * * /
H
H
* , H H * ! *
, ! !
, * *
* -
rational t = subtract ( multiply (p , q ) , multiply (r , s ));
,
, D
E
A
<
A ! !
6
- 1
$
%
* )!
!
* 7
1
A add
*
, rational t = r + s ;
F
,
rational t = add (r , s );
)
,
add
!
, 1
H ,
H
1
6 & 7 D * * E ! !
* +
!
H
1
,
*
* !
#
" +
1
! * , , H * IH ! )
H
<
* * A
* ) & 1
! * *
* +
, ,
H 1 *
* , H *
! *
1 ! A
H
*
*
+
// POST : return value is the product of a and b
rational operator * ( rational a , rational b );
int
H
extended_int
,
,
*
t := r + s
H * * * & ! * <
* !
*
!
* *
!
square
!
*
H
4.1.5 User-defined operators
rational
// 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 ;
}
rational t = r + s ; // equivalent to rational t = operator + ( r , s );
// POST : return value is the difference of a and b
rational operator - ( rational a , rational b );

$
7
8
7
6
8
9
5
45
4
7
65
5
74
79
9
*
*
B
unsigned int
1
double
B
/
H
1
*
?
1
double
!
* , ,
?*
,
*
!
+
*
,
1
,
*
) ,
!
*
* 1
6
H & * * H
1
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* *
1 *
" H 1 H * *
7
HI
H
H 6
!
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1
*
!
*
%
g
-
7
& ! * 7
H
* ! ! !
*
*
6
A
C
*
H
,
* < * !
, *
! *
* 6 A *
*
1 * $ ! *
*
1
+!
*
f
* float
*
1
&
,
* H
+
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!
7
!
*
*
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%
*
H
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*
g
int
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6
&
f
C
*
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C
,
1
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67
&
* ,
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*
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1
*
*
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67
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6
7
HI A ! !
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! float
H HI
*
1
*
6
! * H
1 * &
! ! *
! * !
* * * * * ! 1 ! * * !
*
! ? +
int
1
*
1
*
H 7
*
+
* HI
*
! *
1
!
f
/
Dispositional.
%
+!
H
H <<
bool
*
* A
7
*
H
f
*
!
Overloading resolution.
H
!6 ) *
A
4.1.6 Details
* *
*
*
!
*
*
&
std
7
!
*
* *
F
Resolution: Finding the best match.
*
1 * *
C !
*
& )
& - !
! !
)& Argument-dependent name lookup (Koenig lookup).
std::sqrt(2.0)
H ! *
1 * H , !
! , C
!
A
* 1
!
f
-
* * add (r, s)
7
A
'
'
!
*
*
1
,
// 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;
}
double
g
f
g
void foo ( double d );
void foo ( unsigned int u );
float f = 1.0 f ;
foo (f );
unsigned int
int i = 1;
foo (i );
4.1.7 Goals
*
7
6
*
-
!
!

7
8
7
6
8
9
5
45
4
7
65
5
9
74
79
+
) 1
H C
z
{0, . . . , 6}
x=y+z
x−y
// POST : return value is the difference of a and b
Z_7 operator - ( Z_7 a , Z_7 b );
3 3 3
0 0 0
3 7
%
Exercise 112
extended_int
" #
$
#
0 "
! $
!
$ 2 !
1
0
" #
3 1
0
" #
$
2
!
$
%
extended_int
extended_int
int
6
3 7 6
3 7 !
$
"
0 #
*
H **
, !
*
,
*
**
!
*
? * , *
!
, &
*
7
6
+
1
1
H
* 5
5
6
0
1
2
3
4
#
&
B &
!
* B
3
4
4
5
6
0
1
2
3
,
, !
*
6
3
3
4
5
6
0
1
2
Exercise 110
rational
Exercise 111
rational
2
6 $
6
76 7 2
3
7 !
3

3
6 0
"
0 #
!
3
7
2
2
3
4
5
6
0
1
0
" " 2
! 6
3 7
1
1
2
3
4
5
6
0
3 # #
0
3 #
-
(
!
0
0
1
2
3
4
5
6
// POST : return value is the sum of a and b
Z_7 operator + ( Z_7 a , Z_7 b );
+
0
1
2
3
4
5
6
7
$
" ! !
#
!
%
2
( "%
!
# - #
$ # #
$ # $ 0 "
! !
$
2
!
" # ( ! !
# 6 / 3 7 6 3 7 1
!
0
3 # '
// POST : returns x OR y
Tribool operator || ( Tribool x , Tribool y );
1 7
9
-
(
// POST : returns x AND y
Tribool operator && ( Tribool x , Tribool y );
$
/7
2
" $
4.1.8 Exercises
=
76 3 7 7
0
" !
3 3 # 3
0 0
0
∨
3
2
Z_7
#
∨
6
# # # #
0
0
Tribool
(
2 1
3 #
Tribool
"
!
$
-
∧
0 #
Exercise 109
!
*
* ∧
$ $ ! !
! " #
! Exercise 108
" Operational.
6
6
0
1
2
3
4
5
// 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 );
// 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 );
Z_7
// POST : return value is true if and only if a >= b
bool operator >= ( rational a , rational b );

7
8
7
6
8
9
5
45
4
7
65
5
74
79
9
z = 0
1/
3
(x, y, z)
z
v = (vx , vy , vz )
!
-
! 0
" #
#
2
(
"
$
"
α
3
$
$! x
y
α
α
(x, y)
2
α −
α
*
(x, y)
!
- 2
$ #
3
0
(
!
2 #
! $
! $ $
!
$ #
" 1 " -
#
$ $
!
3
0
$ $
0 # "! " foo(1, 1u)
*
(x , y )
! " !
0
-
foo(1.0, 1)
=
x
y
"
!
4
# 2
4.1.9 Challenges
" # ! 0
1 - ! ! / ( !
2
" 2 2
$&
" ! $
$
$
$
$ #
! 0
"
!
0! $ !
0
3
0
!
"
!
"
foo(1.0f, 1.0)
.
foo(1, 1.0f)
0
1
3 # ! ! # 1
" $ ) " 2 3
#-
2
#
$
#
0" #
! " ! " 2 0
/ " 5
! #
$ #
(
.
-
/
void foo ( double , double )
{ ... }
void foo ( unsigned int , int )
{ ... }
void foo ( float , unsigned int ) { ... }
" 0 # ! / $ ! # 3 0
3 " !
!
0
/ 2 . $
0
Exercise 114
0
. ! ! " 2 #
2
#
"
0 #
<
Exercise 113
0 "
0 !
// 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 );
// function A
// function B
// function C
A, B, C
foo(1, 1)
foo(1u, 1.0f)
libwindow
.
z
74
79
"

5
7
8
7
6
8
9
5
45
4
7
65
9
r += s
H
1
, 1 * H ! * *
!
H
-
A
1/2
5/6
H
!
r
r
r += s
5/6
operator+= (r, s)
*
// 1/3
rational s ;
s.n = 1; s .d = 3;
// 1/2
)
rational r ;
r.n = 1; r .d = 2;
1
*
0
*
!
< !
+=
!
,
!
!
HI
2
2
! #
"
" " $ #
1 2 #
!
# !
! $ ! 2 " 0$ # ? 1$
! $ 1 0
# 3
0
! " #
$0
0
!
/
# #
"
$ $
!
!
! 2
1 .
# "
*
! 0
$
2
2 4
$
!
!
!
$
2 -
$
$#
(
!
! /
#
$ "
#
" ! #
p = (x, y, z)
operator+=
-
operator+=
b
H
s
a
, C 6
, , H
7
r
* (
@
!
vz = z
, 1
,
< $
z
.
vz − z
*
H
* !
1 ! * * !
!
*
!
.
t=
vz > 0
4.2 Type Variants
(x − t(vx − x), y − t(vy − y)),
4.2.1 Reference types
rational
rational operator += ( rational a , rational b ) {
a .n = a. n * b.d + a. d * b.n ;
a .d *= b .d;
return a ;
}
r += s;
std :: cout < < r .n < < "/ " < < r. d < < "\ n";
s
74
79

8
7
6
8
7
9
5
45
4
7
65

5
9
!
*
1 , !
B ! * !
* !
*
,
H
H *
*
! ?
4.2.2 Call by value and call by reference
int & increment ( int & i )
{
, r
// j becomes an alias of i
// k becomes another alias of i
r
%
*
!
1
!
*
*
H * * , !
-
// error : j must be an alias of something
// error : the literal 5 has no address
G
)
1
H 7 *
*
!
*
, !
++
, 6
H
,
!
12
*
!
IH
1
+ ! !
*
! 1 , !
* !
1 *
! H * , 1 $ ! 1
6 ! * <* 7 * *
, ! * !
H
!
B
*
? 1
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+
!
- 1
* * * , 1
!
+! *
!
* !
*
* 1 C
, *
6
, )
*
i
-
rr
H
j = 6
i
r
i
* *
1
*
*
*
!
%
!
* !
!
*
C , !
*
H H
,
! 1 $ 1 #
?
+
* !
// changes the value of
// outputs 6
!
* 12
j = 6;
std :: cout < < i < < "\n" ;
+
!
, 1 * !
! *
* H
H *!
! *
, ! !
H 1
* // j becomes an alias of i
! )* * 1 $ *
H 1 * #
*
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* !
H ! * 1
,
* * , * ) 6
! ! , * !
A
H
*
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* $
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H
1 ! *
7

+!
< * !
!
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*
*
1 ( * 7
,
- +
A ) H
,
!
7
, 1
!
* !
, 1 12
H * !
int i = 5;
int & j = i ;
, int i = 5;
int & j = i ;
int & k = j ;
*
!
! A
* A
%
)
!
! !
* *
1 , A !
*
!
* !
1 int & j ;
int & k = 5;
1 H
<
j
A
! * !
# , < 1 * 2 - !
! 12
*
) 1
, H
*
6
!
1
* 7
#
-
7
,
*
1
,
*
$
#
*
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6
* 7
7 7D
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D
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7
E
7
+
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7 !
*
* * H 7
7
7
H 7
+!
Definition.
A
+
*
*
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H
)
1
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*
*
H +
!
*
*
,
!
* ,
! * !
* B (
* H ! *
*
!
operator+=
void increment ( int & i )
{
++ i;
}
int main ()
{
int j = 5;
increment ( j );
std :: cout < < j < < "\ n" ; // outputs 6
}
return 0;
4.2.3 Return by value and return by reference
increment

$
8
7
6
8
9
5
45
4
9
7
67
8
7
D E H
! *
1 H H
% IH !
1 1 A *
*
A/B
3.4
H H
!
1
H
!
&
-
s
6 6
!
!
7 !
!
* ! 7
*
%
*
+
H H
! , 1
?
-
std::cin >> r.d;
1 A
!
* <
!
H std::cin >> r.n;
*
*
1
*
!
* 12
H * 12
,
!
&
* 12
H 1
,
!
)
!
*
!
<*
!
7
std::ostream
*
1
!
)
&
-
HI
*
1
A
* ,
!
!
*
3
)
H
, H
* *
1 <
&
, !
*
6 6
* (
1
1
7
+
!
!
)

)
!
* ! !
*
A
-
!
1
- H *
!
*
*
<<
F 1
, , +
!
* , 1 H
* ,
, !
* ( ,
, %
!
H A
!
!
!
2
A
H A
A
H +
!
*
( H *
*
1
, < !
* ! *
)
!
!
*
1 2 * HI -
!
! 1 !
! $
*
1
* 2
*
0
add (r, s)
H
*
1
*
A
Rational numbers: input and output.
add
operator+
r + s
*
// 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 ;
}
*
a
+
+=
!
* *
*
-
return ++ i;
* HI 1 1
A * A
1
0
4.2.4 More user-defined operators
C
increment
H
int i = 3;
int & j = foo (i );
// j refers to expired object
std :: cout < < j < < "\n" ; // undefined behavior
**
*
!
% *
H , *
! * * +
* ) &
!
!
!
1
1
!
Rational numbers: addition assignment.
rational
i
1
*
!
int & foo ( int i)
{
return i ;
}
H
!
H
<*
Reference Guideline:
1
*
-
- * * H * A
* ( H ?)
j
!
!
,
}
std :: cout < < " Sum is " < < t.n < < "/" < < t.d < < "\n" ;
std :: cout < < " Sum is " < < t < < "\ n";
// POST : a has been written to o
std :: ostream & operator < < ( std :: ostream & o , rational r)
{
return o < < r.n < < " /" < < r .d;
}
std :: cin > > r;
std::istream
// 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 ;
}
1/2
operator<<

8
7
6
8
9
5
45
4
9
7
67
8
7
&
< &
!
,
*
B & &
!
* *
H
& ? H
*
*
*
{|
{ {{ {
<
2
"
!
{
{
}
%
$
#
%
{
#
#
r + s
$
9
8
7
6
5
45
8
// 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 ;
}
return 0;
// Program : rational .h
// Define a type for rational numbers , and declare
// operations on it .
# include < iostream >
2 // the new type rational
struct rational {
int n;
int d ; // INV : d != 0
};
}
!
// Program : rational .C
// Define a type rational and operations on it
// computation and output
std :: cout < < " Sum is " < < r + s < < " .\ n" ;
operator+
{
Program 34:
std :: cout < < " Rational number s :\ n";
rational s;
std :: cin > > s;
Program 33:
}
int main ()
{
// input
std :: cout < < " Rational number r :\ n";
rational r;
std :: cin > > r;
// 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 : userational2.C
// Add two rational numbers .
# include < iostream >
# include " rational .C "
// POST : return value is the sum of a and b
rational operator + ( rational a , rational b );
*
) *
A
*
, H
1
1 1
*
1 * 1
1 , 1
) *
$
H
ifm
&
% !
B
&
H * rational
rational.C
userational2.C
rational
??
// the new type rational
struct rational {
int n ;
int d ; // INV : d != 0
};
"
D
*
E
! !
* ( *
+
rational.h
namespace ifm {
} { | &
! , * H
!
)* A
1
*
*
*

7
4
9
67
8
7
0
,
,
* !
*
A
main
int
// 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;
*
1
rational
*
H B
1
,
H
*
!
*
* H ,
+
int
rational operator - ( const rational & a)
{
const
/
-a
a
, , )
<*
C
* 1
* !
1 H
!
*
+! * ? 1 ? ! *
1
*
* !
B H * )
* *
* * ,
<
2 "
!
+
H
1
* , A
*
, 1 D * * A
* ) E& ! 1
1 !
! , * , H * !
* 1 H
operator+
>
*
7
* (
A
1
H
, 1 &
*
H
,
* !
+
! !
*
! * ,
!
* * 0
HI
*
!
, B
* H 6
! ) 7
(
H H
* !
,
H
H H
* A
A
!
) H
*
! ) 1
, , ! *
1
, !
1 *
H
! * *
HI
,
1
!
*
!
!
*
*
, 1
6
*
7
1
HI
<
operator+
*
!
!
a + b + c
)
} { | {
{ { { }{ {
rational
? ,
HI '
'
*
6
0
* *
! * * 1
A
* !
, 1
! 1 !
* ! * H %
?
*
4.2.5 Const-types
*
* * , !
%
// POST : a has been written to o
std :: ostream & operator < < ( std :: ostream & o , rational a )
{
return o < < a.n < < "/" < < a.d ;
}
>?
D ,
** 1
!
*E H
1
* ,
/ /
*
* !
,
!
// 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 ;
}
!
Rational number r:
1/2
Rational number s:
1/3
Sum is 5/6.
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)
{
char c ; // separating character , e.g . ’/ ’
return i > > a.n > > c > > a. d;
}
! ,
* )
1 * * * A
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+
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Program 35:
, *
)
}
a, b, c
operator+
rational
// POST : return value is -a
rational operator - ( rational a)
{
a .n = - a .n;
return a ;
}
rational operator - ( rational & a)
{
a .n = - a .n;
return a ;
}
"

8
7
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7
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#
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const
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const int & i = n ;
int & j = n ;
i = 6;
j = 6;
&
*
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!
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*
*
a
n
*
const int n ; // error : uninitialized constant
const int
!
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*
n
4.2.6 What exactly is constant?
-
n = 6
H
const int n = 5;
n = 6;
*
const
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1
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0
operator+
#
Definition.
a .n = - a .n ; // error : a was promised to be constant
return a ;
n
const int n = 5;
int i & = n ; // error : const - qualification is discarded
i = 6;
const
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
const
4.2.7 Const-references
operator+
// 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 ;
}
b

8
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! *
result
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;
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const
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Const Guideline:
!
H 1
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? H +
A
H * !
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< ! ?
+! * ?
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! * ! * ! * !!
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operator-
* * * ?
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& !
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* * !
*
1 &
* * H
H 7 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 ;
}
operator+
!
const
&
1
A +
*
const
const rational&
? !
const
rational
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1
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return
const
i
result
const
4.2.9 When to use const?
4.2.8 Const-types as return types.
const
const
const
const
"

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8
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9
#
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#
2 - 2
const int&
!
$
int int&
const int&
int const int int&
%
#
#
0" #
// 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 );
3 7 1
76
6
2
#
# ! 2
gcd
1
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2
Exercise 118
0
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6
Exercise 117
!
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Exercise 116
Hint:
$ # $
4.2.11 Exercises
! &
* 3 , 7
6
4.2.10 Goals
T foo (S i)
{
return ++ i;
}
#
#
$ #
"
0 #
<
Exercise 115
" 0
7
3
Operational.
76
1
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6
3 7
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3
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2 #
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t
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1
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7
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1 *
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!
const
foo
int a = 5;
int b = 6;
// here comes your function call
std :: cout < < a < < "\n" ; // outputs 6
std :: cout < < b < < "\n" ; // outputs 5
21
−14
−3
.
2
// POST : r is normalized
void normalize ( rational & r );
// POST : return value is the normalization of r
rational normalize ( const rational & r );
"

8
7
6
8
9
5
45
4
9
7
67
8
7
int main ()
{
int i = 5;
const int & j = foo ( i );
}
int main ()
{
const int i = 5;
int & j = foo ( i );
}
!
/
2
)
std::complex<double>
std::complex<double>
#include <complex>
r
$
$
$
!
#
" !
"
0
"
! 0 * $
?
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0
.
=<
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Exercise 121
double
$ 1
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3
7
6
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-
(
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$ 1 (
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3
6
7
{(2, 1), (0, 2), (0, 0), (3, −4)}.
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0
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3 # /
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$
2
22
- 1 $ # const int & bar ( int & i ) {
return i += 2;
}
8
main
int foo ( int & i ) {
return i += 2;
}
Exercise 120
$ 3 2 ! # " 3
(
#
2 $
#
#
0 " 0 "
0 !
"
#
/
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const
-
Exercise 119
!
#
"
!
#
.
$ #
(a, b)
int main ()
{
int i = 5;
const int & j = bar ( foo ( i ));
}
int main ()
{
int i = 5;
const int & j = foo ( bar ( i ));
}
int main ()
{
int i = 5;
const int j = bar (++ i );
}
4.2.12 Challenges
std :: complex < double >(r ,i ); // r and i are of type double
std::sqrt std::abs std::pow
double
// 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 ,
-
"

8
7
6
8
9
5
45
4
9
7
67
8
7
,
rational r
*
0
!
? %
- HI H
1
*
, ! !
, 1
?
!
1
1/2
!
,
H %
* *
(
r
,
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IH 7
!
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!
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6
?
1
H
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6
* !
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+
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*
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, 6 &
! * 1 7
* ,
rational
;
2
A H ,
)
! *
* 1
+
rational
??
B
! *
*
B
*
A
*
1 %
rational
Issue 1: Initialization is cumbersome.
r
,
1
$ # 0 # #
/
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2 0 0 ! #
! !
$
3
#
.
2
# 2
#
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2
1
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4
9
2 A
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$
0
!
/
$ 3
$
$
-
/
#
$
!
$ $
2
"
!
4
& 3
0
0
"
# # $ 0 0 !
! 0 " $ &
3
ax3 + bx2 + cx + d = 0
2
Hint:
/ ! $ #
! ( 0
"
:
$ std :: complex < double >& s1 ,
std :: complex < double >& s2 ,
std :: complex < double >& s3 );
4.3 Classes
4.3.1 Encapsulation
rational r ; // default - initialization of r
r.n = 1;
// assignment to data member
r.d = 2;
// assignment to data member
r
r
""

8
7
6
8
9
5
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7
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8
7
H
*
A
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,
6 7 A
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rational.h
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bool
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1
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unsigned int
bool
%
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%
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,
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*
-%
H !
1
1
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,
, + 1
!
,
H
1
Issue 3: The internal representation cannot be changed.
%
&
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B
H !
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H
1
!
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1 A B
, 1
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&
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1
&
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1
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r.d
,
*
create_rational
H A
, *
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1 !
* !
*
*
1 0
!
!
*
create_rational
!
H ! * 7 A
6
* rational r = create_rational (1 , 2);
!
?
// 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 ;
}
?
H
*
* 6
r.d = 0
, * * 1 *
H *
* < 7 *& * 1
-% ** ,
*
*
6
, %
A
,
%
-
!
*
H
@
1 A
1
1
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1
! <
)
% !
* ! 1
rational r ;
r.n = 1;
r.d = 0;
H -
H
)
F
!
*
! , !
rational
*
rational r;
H 1/2
r
@
,
1 rational
int
rational
, 1 (
*
* A !
*
*
- A
!
! *
* * H 1 1
* * , r.d
, *
1
+
!
*
H ,
* Issue 2: Invariants cannot be guaranteed.
, ! !
H 1 1
!
6
1 1
1
7
1 !
+
H 0
rational_vector_3
r
struct rational {
unsigned int n ;
// absolute value of numerator
unsigned int d ;
// absolute value of denominator
bool is_negative; // sign of the rational number
};
create_rational
rational r ;
r.n = 1;
r.d = 2;
rational r = create_rational (1 , 2);
.d
"

8
7
6
8
7
4
9
99
4
9
5
45
"

8
#
!
1
.
H
?
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1
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*
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)
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-
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*
1
,
1
!
+!
rational
**
1
// get numerator of r
// get denominator of r
H
,
/
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* H ?
* A
!
*
,
!
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*
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!
*
1
!
7 * <
$
* (
d
The implicit call argument and *this.
public
int n = r. numerator ();
int d = r. denominator ();
A
H
,
H
class rational
1
*
* 1
* & !
*
1 + 0
!
*
!
* H
1
!
*
H 1
!
! 1!
* <
*
*
/
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!
*
1 ! ,
*
* !
H &
B ! D
! E
1
1
*
!
6 *
A *
' !
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A
*
, 1 1
!
*
HI
1
?)
+
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+
!
*
HI
*
* r
1
G
)
1
B H *
*
H ! * !
* *
1
1
* !
- A !
! ** 1
<
*
!
public:
!
r.d = 0
rational r ;
r.n = 1;
// error : n is private
r.d = 2;
// error : d is private
int i = r. n ; // error : n is private
*
< private
numerator
,
* C 6
1
* *
H % * , + !
* ! , 1 7 H 1 1
** 1 !
! *
1 * IH 1
H
!
+
class rational {
private :
int n;
int d ; // INV : d != 0
};
!
* H 1
n
* " !
, *
H
) 1 6
& 1 #
H
+! 1
, H
1 *
1 1
H * *
*
1
*
* * 1
H
private:
H
H
% *
& H 1
A
! !
* !
& *
1 +
! !
struct rational
&
!
+!
+
* 7 * H * !
* * &
*
6
& H (
* 1 $
1
1 1
* !
1 ! , *
1 ! +
!
*
class
class rational
, 1
**
public
*
+
* * < H
* *
1
*
1
6
* 1 H 7
1
, 6
7
struct
operator+
public:
class
, * A
1 * * !
**
! * *
1 HI # !
*
4.3.2 Public and private
! * * 7
! class
rational
-
%
* * 1
# ) ,
!
4.3.3 Member functions
const
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
};
denominator

"
$
8
7
6
8
7
4
9
99
4
9
5
45
8
class rational {
public :
// POST : return value is the numerator of * this
1
)*
H 1
-
H
7
6
H 7
6
H
+
1
*
*
!
!
6
H H
7
6
H
H
6 H H 7
<
67
6
7
7
76
7
7
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int numerator () const ;
// POST : return value is the denominator of * this
int denominator () const ;
private :
int n;
int d ; // INV : d != 0
};
rational.h
int rational :: numerator () const
{
return n ;
}
int rational :: denominator () const
{
return d ;
}
4.3.4 Constructors
8
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rational
H
double
rational double
operator double
// POST : return value is double - approximation of * this
operator double ()
A
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rational
6 rational
-
rational r = rational (1 , 2);
rational r ;
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* * // PRE : d != 0
// POST : * this is initialized with numerator / denominator
rational ( int numerator , int denominator)
: n ( numerator) , d ( denominator)
{
// somehow check that d != 0
}
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// POST : * this is initialized with 0
rational ()
: n (0) , d (1)
{}
rational r
rational r;
4.3.6 User-defined conversions
// POST : * this is initialized with value i
rational ( int i)
: n ( i ) , d (1)
{}
double
8
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1
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1
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class rational
H
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1
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int
+
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*
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denominator
,
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operator+
+
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A
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0
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&
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rational & operator += ( rational & a , rational b)
{
return a = a + b ;
}
+
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operator+
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* // 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 );
}
!
*
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1
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H 4.3.7 Member operators
A &
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,
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+
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*
1 // POST : b has been added to * this ; return value is
//
the new value of * this
a
*this
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2. operator + ( r )
a + b
numerator
operator
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6
return double (n )/ d;
!
A
{
rational & operator += ( rational b)
{
n = n * b.d + d * b. n;
d *= b.d ;
return * this ;
}
operator+
// POST : return value is the sum of a and b
rational operator + ( rational a , rational b)
{
return a += b;
}
operator+
// POST : * this is initialized with value i
rational ( int i );
operator+=
rational
r + 2
r. operator + (2)
int
8
7
6
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4
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99
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1
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!
*
, rational::rat_int
int
−
typedef int rat_int ;
*
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// 1
// 2
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1
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& * +
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* A
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
};
typedef ifm :: integer rat_int ;
+!
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typedef rational :: rat_int rat_int ;
rational r (1 ,2);
rat_int numerator = r. numerator ();
rat_int denominator = r. denominator ();
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4.3.8 Nested types
rat_int
denominator
4.3.9 Class definitions
class

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random
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Program 36:
operator()
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1
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!
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knuth8
H
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x0
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H
m = 28 = 256,
H
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H
i > 0.
H
H
H
H
H
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H
m,
H
H
H
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H
H
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H
m
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H
!
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H
H
H
H
H
H
H
H
x1 , x2 , . . .
H
H
H
H
H
c = 187,
H
H
H
H
H
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)
xi = (axi−1 + c)
H
H
H
H
H
a = 137,
H
H
H
%
c
H
H
187 206 249 252 151 138 149 120 243 198 177 116 207 130 77
240 43 190 105 236 7 122 5 104 99 182 33 100 63 114 189 224 155
174 217 220 119 106 117 88 211 166 145 84 175 98 45 208 11 158
73 204 231 90 229 72 67 150 1 68 31 82 157 192 123 142 185 188
87 74 85 56 179 134 113 52 143 66 13 176 235 126 41 172 199 58
197 40 35 118 225 36 255 50 125 160 91 110 153 156 55 42 53 24
147 102 81 20 111 34 237 144 203 94 9 140 167 26 165 8 3 86 193
4 223 18 93 128 59 78 121 124 23 10 21 248 115 70 49 244 79 2
205 112 171 62 233 108 135 250 133 232 227 54 161 228 191 242 61
96 27 46 89 92 247 234 245 216 83 38 17 212 47 226 173 80 139
30 201 76 103 218 101 200 195 22 129 196 159 210 29 64 251 14
57 60 215 202 213 184 51 6 241 180 15 194 141 48 107 254 169 44
71 186 69 168 163 246 97 164 127 178 253 32 219 238 25 28 183
170 181 152 19 230 209 148 239 162 109 16 75 222 137 12 39 154
37 136 131 214 65 132 95 146 221 0 187
H
x1 , x2 , . . .
H
C
-
*
C
!
6
H * ,
1
H
F
4.3.10 Random numbers
m
[0, 1)
xi
// 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

$
8
7
6
8
7
4
9
99
4
9
5
45
8
#
unsigned int
/
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1
1
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1
1
)*
1 :
2
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{|
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,
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1 7
7
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[0, 1)
m = 248 ,
pi
pi = 1/6
i
)
+
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1
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1
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xi
1
operator()
# include < IFM / random .h >
@
,1
IH
! C <
* A ) 1 1
1 ,
1 H *
// Prog : random .C
// implement a class for pseudorandom numbers .
1
@
;
)
) H *
*
, * , ! * ) 1
! ,
!
*
C
The game of choosing numbers.
/
* 1
* H
*
1 &
1 ! 1 * * C
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 )
{}
random
*
1 *
7 1
1 *
/
&
,
* 6
1
!
*
* ! &
6 *
6
1 !
H 7
* *
* , ?
*
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
} { | } { random
{0, . . . , 232 −1}
double
F
drand48
1
*
+
* 1
*
)
H ? A
**
x_i
!
1
!
H *
Program 37:
c = 11,
operator()
// Prog : loaded_dice.h
// define a class for rolling a loaded dice .
# include < IFM / random .h >
{ | } { }
#
m
a = 25214903917,
random
1
*
)
A
!
*
<
67
6
7
1
!
*
H H
67
76
operator
!
, 1
knuth8
i
{1, . . . , 6}
loaded_dice
[0, 1)
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 ,

8
7
6
8
9
5
45
4
9
99
4
7
8
!
-
)
6 77
- * 1
*
H
H
=0
7
6
p_upto_6 = 1
p_upto_1
p_upto_5
operator()
i
x < p1 + . . . + pi .
p_upto_0
i
F
0.12
7
) *
H

*
& H
1
# include < iostream >
# include < IFM / loaded_dice.h >
/
*
!
2
<
"
Program 39:
* H &
* 1
1
*
! * 1
,
i
p6
{ { {
{
}{
6
pi
* + !
1
H
!
* 1 1
* 1
1
,
, 6
2
"
!
{{
{| {
// 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)
{}
{ {{ { {| } { | } { {
operator()
6
1
+!
[0, 1)
p5
p 1 , . . . , p 5 p6 =
p4
H
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
* *
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return value is the outcome of rolling a loaded
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induced by p1 ,... , p6
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//
//
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double p5 , ifm :: random & 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
// Prog : choosing_numbers.C
// let your loaded dice play against your friend ’s dice
// in the game of choosing numbers .

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7
// output the result :
std :: cout < < " Your total payoff is "
< < your_total_payoff < < "\n ";
,
*
// 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 ());
}
7
// 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 );
<
// 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 );
2
"
"
}{
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 );
7
Program 40:
!
} { | } { | {
{ { {
// 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
}
4.3.11 Details
operator>>
operator>>
private
rational
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
};
.d
4.3.12 Goals
return 0;
"

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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 )
{}
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{
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;
t
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