Ainoaea?noaaiiue eiieoao ?inneeneie Oaaea?aoeee
ii aunoaio ia?aciaaieth
IIAINEAE?NEEE AINOAeA?NOAAIIUE OAOIE*ANEEE OIEAA?NEOAO
Eaoaae?a ANO
?aoa?ao ii aeenoeeieeia
ENNEAAeIAAIEA IIA?AOeEE
ia oaio
IAOIAe AeAOI?IE?OAIIAI IIIAIA?AIIEEA
Nooaeaio Ai?cia Aiae?ae Ieeieaaae/
A?oiia AN–513
I?aiiaeaaaoaeue ?aiei Na?aae Aaneeueaae/
Iiaineae?ne 1997
Iiene ii aeaoi?ie?oaiiio iiiaia?aiieeo
Aia?aua iaoiae aeaoi?ie?oaiiai iiiaia?aiieea aue i?aaeeiaeai Iaeaea?ii
e Ieaeii. Iie i?aaeeiaeeee iaoiae iienea, ieacaaoeeny aanueia
yooaeoeaiui e eaaei inouanoaeyaiui ia YAI. *oiau iiaeii auei ioeaieoue
no?aoaaeth Iaeaea?a e Ieaea, e?aoei iieoai neiieaeniue iiene Niaiaeee,
Oaenoa e Oeinai?oa, ?ac?aaioaiiue a nayce ni noaoenoe/aneei
ieaie?iaaieai yenia?eiaioa. Aniiiiei, /oi ?aaoey?iua iiiaia?aiieee a En
yaeythony neiieaenaie. Iai?eia?, eae aeaeii ec ?enoiea 1, aeey neo/ay
aeaoo ia?aiaiiuo ?aaoey?iue neiieaen i?aaenoaaeyao niaie ?aaiinoi?iiiee
o?aoaieueiee (o?e oi/ee); a neo/aa o??o ia?aiaiiuo ?aaoey?iue neiieaen
i?aaenoaaeyao niaie oao?ayae? (/aou?a oi/ee) e o.ae.
?enoiie SEQ ?enoiie \* ARABIC 1 .
?aaoey?iua neiieaenu aeey neo/ay aeaoo (a) e o??o (a) iacaaeneiuo
ia?aiaiiuo.
( iaicia/aao iaeaieueoaa cia/aiea f(x). No?aeea oeacuaaao iai?aaeaiea
iaenei?aeoaai oeo/oaiey.
I?e iienea ieieioia oeaeaaie ooieoeee f(x) i?iaiua aaeoi?u x iiaoo
auoue aua?aiu a oi/eao En, iaoiaeyueony a aa?oeiao neiieaena, eae auei
ia?aiia/aeueii i?aaeeiaeaii Niaiaeee, Oaenoii e Oeinai?oii. Ec
aiaeeoe/aneie aaiiao?ee ecaanoii, /oi eii?aeeiaou aa?oei ?aaoey?iiai
neiieaena ii?aaeaeythony neaaeothuae iao?eoeae D, a eioi?ie noieaoeu
i?aaenoaaeytho niaie aa?oeiu, i?iioia?iaaiiua io 1 aei (n+1), a no?i/ee
– eii?aeeiaou, i i?eieiaao cia/aiey io 1 aei n:
– iao?eoea n X (n+1),
aaea
,
,
t – ?annoiyiea iaaeaeo aeaoiy aa?oeiaie. Iai?eia?, aeey n=2 e t=1
o?aoaieueiee, i?eaaae?iiue ia ?enoiea 1, eiaao neaaeothuea eii?aeeiaou:
Aa?oeia x1,i x2,i
1 0 0
2 0.965 0.259
3 0.259 0.965
Oeaeaaay ooieoeey iiaeao auoue au/eneaia a eaaeaeie ec aa?oei
neiieaena; ec aa?oeiu, aaea oeaeaaay ooieoeey iaeneiaeueia (oi/ea A ia
?enoiea 1), i?iaiaeeony i?iaeoe?othuay i?yiay /a?ac oeaio? oyaeanoe
neiieaena. Caoai oi/ea A eneeth/aaony e no?ieony iiaue neiieaen,
iacuaaaiue io?aae?iiui, ec inoaaoeony i?aaeieo oi/ae e iaeiie iiaie
oi/ee B, ?aniieiaeaiiie ia i?iaeoe?othuae i?yiie ia iaaeeaaeauai
?annoiyiee io oeaio?a oyaeanoe. I?iaeieaeaiea yoie i?ioeaaeo?u, a
eioi?ie eaaeaeue ?ac au/??eeaaaony aa?oeia, aaea oeaeaaay ooieoeey
iaeneiaeueia, a oaeaea eniieueciaaiea i?aaee oiaiueoaiey ?acia?a
neiieaena e i?aaeioa?auaiey oeeeee/aneiai aeaeaeaiey a ie?anoiinoe
yeno?aioia iicaieytho inouanoaeoue iiene, ia eniieuecothuee i?iecaiaeiua
e a eioi?ii aaee/eia oaaa ia ethaii yoaia k oeene?iaaia, a iai?aaeaiea
iienea iiaeii eciaiyoue. Ia ?enoiea 2 i?eaaaeaiu iineaaeiaaoaeueiua
neiieaenu, iino?iaiiua a aeaoia?iii i?ino?ainoaa n «oi?ioae» oeaeaaie
ooieoeeae.
?enoiie SEQ ?enoiie \* ARABIC 2 .
Iineaaeiaaoaeueiinoue ?aaoey?iuo neiieaenia, iieo/aiiuo i?e ieieiecaoeee
f(x).
—– i?iaeoeey
Ii?aaeae?iiua i?aeoe/aneea o?oaeiinoe, ano?a/athueany i?e
eniieueciaaiee ?aaoey?iuo neiieaenia, a eiaiii ionoonoaea onei?aiey
iienea e o?oaeiinoe i?e i?iaaaeaiee iienea ia ene?eae?iiuo «ia?aaao» e
«o?aaoao», i?eaaee e iaiaoiaeeiinoe iaeioi?uo oeo/oaiee iaoiaeia. Aeaeaa
aoaeao eceiaeai iaoiae Iaeaea?a e Ieaea, a eioi?ii neiieaen iiaeao
eciaiyoue naith oi?io e oaeei ia?acii oaea ia aoaeao inoaaaoueny
neiieaenii. Eiaiii iiyoiio caeanue eniieueciaaii aieaa iiaeoiaeyuaa
iacaaiea «aeaoi?ie?oaiue iiiaia?aiiee».
A iaoiaea Iaeaea?a e Ieaea ieieiece?oaony ooieoeey n iacaaeneiuo
ia?aiaiiuo n eniieueciaaieai n+1 aa?oei aeaoi?ie?oaiiai iiiaia?aiieea a
En. Eaaeaeay aa?oeia iiaeao auoue eaeaioeoeoee?iaaia aaeoi?ii x. Aa?oeia
(oi/ea) a En, a eioi?ie cia/aiea f(x) iaeneiaeueii, i?iaeoe?oaony /a?ac
oeaio? oyaeanoe (oeaio?ieae) inoaaoeony aa?oei. Oeo/oaiiua (aieaa
ieceea) cia/aiey oeaeaaie ooieoeee iaoiaeyony iineaaeiaaoaeueiie caiaiie
oi/ee n iaeneiaeueiui cia/aieai f(x) ia aieaa «oi?ioea oi/ee», iiea ia
aoaeao iaeaeai ieieioi f(x).
Aieaa iiae?iaii yoio aeai?eoi iiaeao auoue iienai neaaeothuei ia?acii.
, yaeyaony i-e aa?oeiie (oi/eie) a En ia k-i yoaia iienea, k=0, 1, …, e
ionoue cia/aiea oeaeaaie ooieoeee a x(k)i ?aaii f(x(k)i). E?iia oiai,
ioiaoei oa aaeoi?u x iiiaia?aiieea, eioi?ua aeatho iaeneiaeueiia e
ieieiaeueiia cia/aiey f(x).
Ii?aaeaeei
Iineieueeo iiiaia?aiiee a En ninoieo ec (n+1) aa?oei x1, …,xn+1,
ionoue xn+2 aoaeao oeaio?ii oyaeanoe anao aa?oei, eneeth/ay xh.
Oiaaea eii?aeeiaou yoiai oeaio?a ii?aaeaeythony oi?ioeie
(1)
aaea eiaeaen j iaicia/aao eii?aeeiaoiia iai?aaeaiea.
Ia/aeueiue iiiaia?aiiee iau/ii auae?aaony a aeaea ?aaoey?iiai neiieaena
(ii yoi ia iaycaoaeueii) n oi/eie 1 a ea/anoaa ia/aea eii?aeeiao; iiaeii
ia/aei eii?aeeiao iiianoeoue a oeaio? oyaeanoe. I?ioeaaeo?a iouneaiey
aa?oeiu a En, a eioi?ie f(x) eiaao eo/oaa cia/aiea, ninoieo ec
neaaeothueo iia?aoeee:
Io?aaeaiea – i?iaeoe?iaaiea x(k)h /a?ac oeaio? oyaeanoe a niioaaonoaee n
niioiioaieai
(2)
– aa?oeia, a eioi?ie ooieoeey f(x) i?eieiaao iaeaieueoaa ec n+1
cia/aiee ia k-i yoaia.
?anoyaeaaaony a niioaaonoaee n niioiioaieai
(3)
e oaeaea inouanoaeyaony ia?aoiae e iia?aoeee 1 i?e k=k+1.
naeeiaaony a niioaaonoaee n oi?ioeie
(4)
e aica?auaainy e iia?aoeee 1 aeey i?iaeieaeaiey iienea ia (k+1)-i
oaaa.
a niioaaonoaee n oi?ioeie
(5)
Caoai aica?auaainy e iia?aoeee 1 aeey i?iaeieaeaiey iienea ia (k+1)-i
oaaa.
E?eoa?ee ieii/aiey iienea, eniieueciaaiiue Iaeaea?ii e Ieaeii, ninoiye
a i?iaa?ea oneiaey
(6)
.
Ia noaia 1 i?eaaaeaia aeie-noaia iienea iaoiaeii aeaoi?ie?oaiiai
iiiaia?aiieea, a ia ?enoiea 3 iieacaia iineaaeiaaoaeueiinoue iienea aeey
ooieoeee ?icaia?iea, ia/eiay eo x(0)=[-1,2 1,0]T. Aeaoi?ie?oaiue
iiiaia?aiiee a i?ioeaiiieiaeiinoue ae?noeiio neiieaeno aaeaioe?oaony e
oiiia?aoee oeaeaaie ooieoeee, auoyaeaaynue aaeieue aeeeiiuo iaeeiiiuo
ieineinoae, eciaiyy iai?aaeaiea a eciaioouo aiaaeeiao e naeeiaynue a
ie?anoiinoe ieieioia.
Ione
Au/eneeoue ia/aeueiua cia/aiey
xi(0), i = 1, 2, …, n+1, e f(x(0))
ia/aeueiiai neiieaena
Au/eneeoue xh e xl e c
Io?aaeaiea: au/eneeoue
xn+3 = xn+2 + ((xn+2 – xn)
Au/eneeoue
f(xn+3)
Auiieiyaony ee
ia?aaainoai
f(xn+3) f(xh) ?
Caiaieoue
xh ia xn+5
?aaeoeoeey: caiaieoue
ana xi ia xl + 1/2(xi – xl)
Inoaiia
?enoiie SEQ ?enoiie \* ARABIC 3 .
Iiene ieieioia ooieoeee ?icaia?iea iaoiaeii aeaoi?ie?oaiiai
iiiaia?aiieea, ia/eiay n oi/ee x(0)=[-1,2 1,0]T (/enea oeacuaatho iiia?
oaaa).
Eiyooeoeeaio io?aaeaiey ( eniieuecoaony aeey i?iaeoe?iaaiey aa?oeiu n
iaeaieueoei cia/aieai f(x) /a?ac oeaio? oyaeanoe aeaoi?ie?oaiiai
iiiaia?aiieea. Eiyooeoeeaio ( aaiaeeony aeey ?anoyaeaiey aaeoi?a iienea
a neo/aa, anee io?aaeaiea aea?o aa?oeio ni cia/aieai f(x), iaiueoei, /ai
iaeiaiueoaa cia/aiea f(x), iieo/aiiia aei io?aaeaiey. Eiyooeoeeaio
naeaoey ( eniieuecoaony aeey oiaiueoaiey aaeoi?a iienea, anee iia?aoeey
io?aaeaiey ia i?eaaea e aa?oeia ni cia/aieai f(x), iaiueoei, /ai aoi?ia
ii aaee/eia (iinea iaeaieueoaai) cia/aiea f(x), iieo/aiiia aei
io?aaeaiey. Oaeei ia?acii, n iiiiuueth iia?aoeee ?anoyaeaiee eee naeaoey
?acia?u e oi?ia aeaoi?ie?oaiiai iiiaia?aiieea ianooaae?othony oae, /oiau
iie oaeiaeaoai?yee oiiieiaee ?aoaaiie caaea/e.
Anoanoaaiii aicieeaao aii?in, eaeea cia/aiey ia?aiao?ia (, ( e (
aeieaeiu auoue aua?aiu. Iinea oiai eae aeaoi?ie?oaiue iiiaia?aiiee
iiaeoiaeyuei ia?acii i?iianooaae?iaai, aai ?acia?u aeieaeiu
iiaeaea?aeeaaoueny iaeciaiiuie, iiea eciaiaiey a oiiieiaee caaea/e ia
iio?aaotho i?eiaiaiey iiiaia?aiieea ae?oaie oi?iu. Yoi aiciiaeii
?aaeeciaaoue oieueei i?e (=1. E?iia oiai, Iaeaea? e Ieae iieacaee, /oi
i?e ?aoaiee caaea/e n (=1 o?aaoaony iaiueoaa eiee/anoai au/eneaiee
ooieoeee, /ai i?e (0,6 iiaeao iio?aaiaaoueny ecauoi/iia /enei oaaia e aieueoa
iaoeiiiai a?aiaie aeey aeinoeaeaiey ieii/aoaeueiiai ?aoaiey.
I?eia?
Iiene iaoiaeii aeaoi?ie?oaiiai iiiaia?aiieea.
Aeey eeethno?aoeee iaoiaea Iaeaea?a e Ieaea ?anniio?ei caaea/o
ieieiecaoeee ooieoeee f(x)=4(x1–5)2+(x2–6)2, eiathuae ieieioi a oi/ea
x*=[5 6]T. Iineieueeo f(x) caaeneo io aeaoo ia?aiaiiuo, a ia/aea iienea
eniieuecoaony iiiaioaieueiee n o?aiy aa?oeiaie. A yoii i?eia?a a
ea/anoaa ia/aeueiiai iiiaia?aiieea acyo o?aoaieueiee n aa?oeiaie
x1(0)=[8 9]T, x2(0)=[10 11]T e x3(0)=[8 11]T, oioy iiaeii auei au
eniieueciaaoue ethaoth ae?oaoth eiioeao?aoeeth ec o??o oi/ae.
Ia ioeaaii yoaia iienea, k=0, au/eneyy cia/aiey ooieoeee, iieo/aai
f(8,9)=45, f(10,11)=125 e f(8,11)=65. Caoai io?aaeaai x2(0)=[10 11]T
/a?ac oeaio? oyaeanoe oi/ae x1(0) e x3(0) [ii oi?ioea (1)], eioi?ue
iaicia/ei /a?ac x4(0):
,
n oai, /oiau iieo/eoue x5(0).
,
,
f(6,9)=13.
Iineieueeo f(6,9)=13
Niaea?aeaiea TOC \o "1-2" Iiene ii aeaoi?ie?oaiiio GOTOBUTTON _Toc407043307 PAGEREF _Toc407043307 2 iiiaia?aiieeo GOTOBUTTON _Toc407043308 PAGEREF _Toc407043308 2 I?eia? GOTOBUTTON _Toc407043309 PAGEREF _Toc407043309 8 Niaea?aeaiea GOTOBUTTON _Toc407043310 PAGEREF _Toc407043310 11 Nienie ?enoieia GOTOBUTTON _Toc407043311 PAGEREF _Toc407043311 11 Nienie eeoa?aoo?u GOTOBUTTON _Toc407043312 PAGEREF _Toc407043312 11 Nienie ?enoieia TOC \c "?enoiie" ?enoiie 1. GOTOBUTTON _Toc403233480 PAGEREF _Toc403233480 2 ?enoiie 2. GOTOBUTTON _Toc403233484 PAGEREF _Toc403233484 3 ?enoiie 3. GOTOBUTTON _Toc403233485 PAGEREF _Toc403233485 7 Nienie eeoa?aoo?u Oeiiaeueaeao Ae. I?eeeaaeiia iaeeiaeiia i?ia?aiie?iaaiea. –I.,1975. PAGE 5 PAGE 3 PAGE 7 ?enoiie 3. Eioi?iaoeeiiiay aeie-noaia iienea iaoiaeii aeaoi?ie?oaiiai iiiaia?aiieea. PAGE 6 PAGE 10 PAGE 7 A A B a a B Iao Aea Iao Aea Iao Iao Iao Iao Aea Aea Aea Aea
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