4
Srolf| Ixqfwlrq Lwhudwlrq
Wklv lv txlwh vlpsoh lq wkh ghwhuplqlvwlf prgho/ lq idfw vr vlpsoh wkdw lw pd| eh
kdug wr xqghuvwdqg1
Wdnh rxu prgho
X
@
[4 fw
@
D+nw , . +4 ,nw nw.4
w@3
w orj+f ,
w
Wkh vroxwlrq wr wklv sureohp lv wkh vwdwlrqdu| srolf| ixqfwlrq nw.4 @ j +nw ,1 Wkh
vwhdg| vwdwh lv=
@ @ D+n ,4 . +4 ,
Vwhdg| vwdwh frqvxpswlrq lv/ f @ i +n , n / dqg xwlolw| rz/ x @ x+f ,/
4
dqg wkhuhiruh ydoxh=
Y @
[4 x
w
w@3
@
x
4
dqg zh fdq xvh wklv ydoxh +d vfdodu , wr lqlwldol}h dq dojrulwkp wkdw zloo surgxfh
d ixqfwlrq ri fdslwdo/ y +n,1
Wkh sureohp lq g|qdplf surjudpplqj irup lv zulwwhq
y+n, @ pd{ ix+n> n3 , . y+n3 ,j
n3
Wkh srolf| lwhudwlrq vwduwv zlwk dq lqlwldo srolf| jxhvv/ n3 @ j3 +n,1 Jlyhq rxu
nqrzohgjh wkdw wkhuh lv d vwhdg| vwdwh dqg wkdw wkh vorsh ri wkh srolf| ixqfwlrq
dw wkh vwhdg| vwdwh lv ohvv wkdq rqh/ d xvhixo uvw jxhvv iru wkh srolf| yhfwru
lv jlyhq e|
n3 +m , @ n+m , .
4
5
+n
n+m ,,
iru hdfk hohphqw m ri wkh fdslwdo vwdwh yhfwru1
414
Srolf| Lwhudwlrq Orrs iru d ghwhuplqlvwlf sureohp
Khuh zh vwduw zlwk wkh lqlwldo jxhvv j3 dw wkh srolf| ixqfwlrq=
nw.4 @ j3 +nw ,
Wkhq zh frqvwuxfw xwlolw| jlyhq wklv lqlwldo jxhvv
x3 +n> n33 , @ orj ^D+nw , . +4 ,nw j3 +nw ,`
4
Qrz zh xvh wklv ixqfwlrq wr frpsxwh d uvw jxhvv dw wkh ydoxh ixqfwlrq e|
grlqj
y3
@
y3 +n,
@
x3 +n> j3 +n,, . y3
x3 +n> j3 +n,,
4
Dqg qrz wklv lv xvhg dv qh{w shulrg*v ydoxh ixqfwlrq dv zh qlvk wkh
orrs e| jxhvvlqj wkh qhz srolf| ixqfwlrq j4 =
j4 +n, @ duj pd{ ix+n> n3 , . y3 +n3 ,j
n3
exw zkhuh fuxfldoo|/ wkh xwlolw| ixqfwlrq lv qrz xquhvwulfwhg dqg frq0
wdlqv qr jxhvv1 kdylqj grqh wklv zh jr edfn xs wr wkh ehjjlqlqj ri wkh orrs1
415
Vwrfkdvwlf sureohp
Qrz wkhuh lv d elw ri d frpsolfdwlrq ehfdxvh zh kdyh wr lqyhuw wkh h{shfwdwlrq
rshudwru1 Vxssrvh qrz wkdw whfkqrorj| iroorzv d Pdunry surfhvv rq wzr srlqwv/
D 5 ^D4 > D5 `/ zlwk wudqvlwlrq pdwul{
S
@
s44 s45
s54 s55
zkhuh slm @ Su+Dw.4 @ Dm mDw @ Dl , lv wkh suredelolw| ri prylqj iurp vwdwh l
wrgd| wr vwdwh m wrpruurz1
Qrz wkh Ehoopdq htxdwlrq +dqg wkh srolf| ixqfwlrqv, kdyh wzr vwdwh ydul0
deohv/
y+D> n,
n
D/
3
ix+D> n> n , . Hy+D > n ,j
3
@
pd{
@
j +D> n,
n3
3
3
Zh duh uhdg| wr vwduw1
Iluvw zh qhhg lqlwldo jxhvvhv iru wkh srolf| ixqfwlrq/ rqh iru hdfk hohphqw ri
nw.4
nw.4
@
@
j34 +D4 > nw ,
j35 +D5 > nw ,
dqg d xvhixo lqlwldo jxhvv vlpso| wdnhv wkh ghwhuplqlvwlf vlqjoh srolf| ixqfwlrq
dqg vkliwv lw xs ru grzq uhodwlyh wr wkh vwhdg| vwdwh e| d shufhqwdjh idfwru wkdw
lv htxdo wr wkh glhuhqfhv lq D1
Vhfrqg/ frpsxwh wkh xwlolw| ixqfwlrq jlyhq wklv srolf|
x43 +D4 > n> j34 ,
x53 +D5 > n> j35 ,
@
@
D4 +nw , . +4 ,nw j34 +D4 > nw ,
5
orj D5 +nw , . +4 ,nw j3 +D5 > nw ,
orj
5
Wklug/ xvh wklv wr frpsxwh wkh uvw jxhvv ri wkh ydoxh ixqfwlrqv1 Wklv lv
zkhuh zh pxvw lqyhuw wkh h{shfwdwlrq1 Exw lw lv qrw wkdw kdug1
y34
y35
@
@
x43 . H4 +y3 , @ x43 . s44 y34 . s45 y35
x53 . H5 +y3 , @ x53 . s54 y34 . s55 y35
dqg zh fdq zulwh wklv dv d pdwul{ v|vwhp
y34
y35
@
y3
+L S ,y3
y3
@
@
@
x43
x53
.
x3 . S y3
x3
4
+L S , x3
s44 s45
s54 s55
y34
y35
Irxuwk/ zh lwhudwh rq wkh srolf|/ zklfk iru vwdwh rqh lv jlyhq e|
j44 +D4 > n,
@
duj pd{
H4
@
S +4> =,
n3
ix+D4> n> n3 , . H4y3+n3 ,j
zkhuh rqfh djdlq/ wkh ydoxh zh mxvw frpsxwhg lv qrz wdnhq wr eh wrpruurz*v
ydoxh ri fdslwdo/ dqg wkh xwlolw| ixqfwlrq grhv qrw lqfrsusrudwh dq| jxhvvhv1
Rqh yhu| lpsruwdqw wklqj wr qrwh khuh/ lv wkdw/ iru wkh qxphulfdo lpsohphq0
wdwlrq/ wkh lqyhuvlrq ri wkh pdwul{ +L S , lv grqh rxwvlgh wkh lwhudwlrq orrs1
Lq wklv sduwlfxodu sureohp dw ohdvw/ wklv lv qrw d yhu| frvwo| surfhgxuh1 Lqyhuwlqj
pdwulfhv lqvlgh d orrs fdq eh yhu| frvwo|1
6
© Copyright 2026 Paperzz