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Метод Дэвидона-Флетчера-Пауэлла

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Ieienoa?noai iaoee, aunoae oeieu e oaoie/aneie

iieeoeee ?inneeneie Oaaea?aoeee.

Iiaineae?neee Ainoaea?noaaiiue

Oaoie/aneee Oieaa?neoao.

?aoa?ao ii enneaaeiaaieth iia?aoeee ia oaio

«Iaoiae Aeyaeaeiia – Oeao/a?a – Iaoyeea».

Aa?eaio ?2.

Oaeoeueoao: AAO.

Eaoaae?a: ANO.

A?oiia: AN-513.

Nooaeaio: Aieei Eiinoaioei Aiaoieueaae/.

I?aiiaeaaaoaeue: ?aiei Na?aae Aaneeueaae/.

Aeaoa: 19 ieoya?y 1997 aiaea.

Iiaineae?ne

Aaaaeaiea.

f(y). Iai?aaeaiea a?aaeeaioa yaeyaony, oaeei ia?acii, ioeeiiaiiui a
?acoeueoaoa oiiiaeaiey ia -Dj , aaea Dj – iieiaeeoaeueii ii?aaeaeaiiay
neiiao?e/aneay iao?eoea ii?yaeea n ( n, aii?ieneie?othuay ia?aoioth
iao?eoeo Aanna. Ia neaaeothuai oaaa iao?eoea Dj+1 i?aaenoaaeyaony a
aeaea noiiu Dj e aeaoo neiiao?e/aneeo iao?eoe ?aiaa iaeei eaaeaeay. A
nayce n yoei noaia eiiaaea iacuaaaony noaiie ei??aeoeee ?aiaa aeaa.

Aeai?eoi Aeyaeaeiia – Oeao/a?a – Iaoyeea.

?anniio?ei aeai?eoi Aeyaeaeiia – Oeao/a?a – Iaoyeea ieieiecaoeee
aeeooa?aioee?oaiie ooieoeee ianeieueeeo ia?aiaiiuo. A /anoiinoe, anee
ooieoeey eaaae?aoe/iay, oi, eae aoaeao iieacaii iicaeiaa, iaoiae
au?aaaouaaao nii?yaeaiiua iai?aaeaiey e inoaiaaeeaaaony iinea auiieiaiey
iaeiie eoa?aoeee, o.a. iinea iienea aaeieue eaaeaeiai ec nii?yaeaiiuo
iai?aaeaiee.

Ia/aeueiue yoai.

(0 – eiinoaioa aeey inoaiiaee. Aua?aoue oi/eo o1 e ia/aeueioth
neiiao?e/aneoth iieiaeeoaeueii ii?aaeaeaiioth iao?eoeo D1. Iieiaeeoue y1
= x1, k = j = 1 e ia?aeoe e iniiaiiio yoaio.

Iniiaiie yoai.

f(yj) e acyoue a ea/anoaa (j iioeiaeueiia ?aoaiea caaea/e ieieiecaoeee
f(yj + (dj) i?e ( ( 0. Iieiaeeoue yj+1 = yj + (jdj. Anee j ( n, oi
ia?aeoe e oaao 2. Anee j = n, oi iieiaeeoue y1 = xk+1 = yn+1, caiaieoue
k ia k+1, iieiaeeoue j=1 e iiaoi?eoue oaa 1.

Oaa 2. Iino?ieoue Dj+1 neaaeothuei ia?acii :

, (1)

aaea

pj = (jdj, (2)

f(yj). (3)

Caiaieoue j ia j + 1 e ia?aeoe e oaao 1.

I?eia?.

?anniio?ei neaaeothuoth caaea/o :

ieieiece?iaaoue (x1 – 2)4 + (x1 – 2×2)2.

?acoeueoaou au/eneaiee iaoiaeii Aeyaeaeiia – Oeao/a?a – Iaoyeea
i?eaaaeaiu a oaaeeoea 1.

Oaaeeoea 1. ?acoeueoaou au/eneaiee ii iaoiaeo Aeyaeaeiia – Oeao/a?a –
Iaoyeea.

k xk

f(xk) j yj

f(yj) (( D dj (j yj+1

1 (0.00, 3.00)

(52.00) 1

2 (0.00, 3.00)

(52.00)

(2.70, 1.51)

(0.34) (-44.00, 24.00)

(0.73, 1.28)

50.12

(44.00, -24.00)

(-0.67, -1.31) 0.062

0.22 (2.70, 1.51)

(2.55, 1.22)

2 (2.55, 1.22)

(0.1036) 1

2 (2.55, 1.22)

(0.1036)

(2.45, 1.27)

(0.0490) (0.89, -0.44)

(0.18, 0.36) 0.99

0.40

(-0.89, 0.44)

(-0.28, -0.25) 0.11

0.64 (2.45, 1.27)

(2.27, 1.11)

3 (2.27, 1.11)

(0.008) 1

2 (2.27, 1.11)

(0.008)

(2.25, 1.13)

(0.004) (0.18, -0.20)

(0.04, 0.04) 0.27

(-0.18, 0.20)

(-0.05, -0.03) 0.10

2.64 (2.25, 1.13)

(2.12, 1.05)

4 (2.12, 1.05)

(0.0005) 1

2 (2.12, 1.05)

(0.0005)

(2.115, 1.058)

(0.0002) (0.05, -0.08)

(0.004, 0.004) 0.09

(-0.05, 0.08)

0.10 (2.115, 1.058)

Ia eaaeaeie eoa?aoeee aaeoi? dj aeey j = 1, 2 ii?aaeaeyaony a aeaea

f(yj), aaea D1 – aaeeie/iay iao?eoea, a D2 au/eneyaony ii oi?ioeai (1)
– (3). I?e

k = 1 eiaai p1 = (2.7, -1.49)T, q1 = (44.73, -22,72)T. Ia aoi?ie
eoa?aoeee

p1 = (-0.1, 0.05)T, q1 = (-0.7, 0.8)T e, iaeiiaoe, ia o?aoueae eoa?aoeee

p1 = (-0.02, 0.02)T, q1 = (-0.14, 0.24)T. Oi/ea yj+1 au/eneyaony
iioeiecaoeeae aaeieue iai?aaeaiey dj i?e ia/aeueiie oi/ea yj aeey j = 1,
2. I?ioeaaeo?a inoaiiaeaia a oi/ea

y2 = (2.115, 1.058)T ia /aoaa?oie eoa?aoeee, oae eae ii?ia ((f(y2) ((=
0.006 aeinoaoi/ii iaea. O?aaeoi?ey aeaeaeaiey, iieo/aiiay iaoiaeii,
iieacaia ia ?enoiea 1.

?enoiie 1. Iaoiae Aeyaeaeiia – Oeao/a?a – Iaoyeea.

Eaiia 1 iieacuaaao, /oi eaaeaeay iao?eoea Dj iieiaeeoaeueii ii?aaeaeaia
e dj yaeyaony iai?aaeaieai nionea.

Aeey aeieacaoaeuenoaa eaiiu iai iiiaaeiaeony :

Oai?aia 1. Ionoue S – iaionoia iiiaeanoai a An, oi/ea x ( cl S. Eiionii
aiciiaeiuo iai?aaeaiee a oi/ea x iacuaaaony iiiaeanoai D = {d : d ( 0, x
+ (d ( S i?e anao ( ( (0, () aeey iaeioi?iai ( > 0}.

Ii?aaeaeaiea. Ionoue x e y – aaeoi?u ec An e (xTy( – aaniethoiia
cia/aiea neaey?iiai i?iecaaaeaiey xTy. Oiaaea auiieiyaony neaaeothuaa
ia?aaainoai, iacuaaaiia ia?aaainoaii Oaa?oea : (xTy( ( ((x(( ((y((.

Eaiia 1.

f(yj), a (j yaeyaony iioeiaeueiui ?aoaieai caaea/e ieieiecaoeee f(yj +
(dj) i?e ( ( 0. Ionoue, e?iia oiai, aeey

f(yj) ( 0 aeey

j = 1, …, n, oi iao?eoeu D1, …, Dn neiiao?e/aneea e iieiaeeoaeueii
ii?aaeaeaiiua, oae /oi d1, …, dn – iai?aaeaiey nionea.

Aeieacaoaeuenoai.

I?iaaaeai aeieacaoaeuenoai ii eiaeoeoeee. I?e j = 1 iao?eoea D1
neiiao?e/aneay e iieiaeeoaeueii ii?aaeaeaiiay ii oneiaeth eaiiu. E?iia
oiai,

f(y1) ( 0, oae eae D1 iieiaeeoaeueii ii?aaeaeaia. Oiaaea ii oai?aia 1
aaeoi? d1 ii?aaeaeyao iai?aaeaiea nionea. I?aaeiieiaeei, /oi
ooaa?aeaeaiea eaiiu ni?aaaaeeeai aeey iaeioi?iai j ( n – 1, e iieaaeai,
/oi iii ni?aaaaeeeai aeey j+1. Ionoue x – iaioeaaie aaeoi? ec En, oiaaea
ec (1) eiaai

(4)

Oae eae Dj – neiiao?e/aneay iieiaeeoaeueii ii?aaeaeaiiay iao?eoea, oi
nouanoaoao iieiaeeoaeueii ii?aaeaeaiiay iao?eoea Dj1/2, oaeay, /oi Dj =
Dj1/2Dj1/2. Ionoue

a = Dj1/2x e b = Dj1/2qj. Oiaaea xTDjx = aTa, qjTDjqj = bTb e xTDjqj =
aTb. Iiaenoaaeyy yoe au?aaeaiey a (4), iieo/aai :

(5)

Ii ia?aaainoao Oaa?oea eiaai (aTa)(bTb) ( (aTb)2. Oaeei ia?acii, /oiau
aeieacaoue, /oi xTDj+1x ( 0, aeinoaoi/ii iieacaoue, /oi pjTqj ( 0 e bTb
( 0. Ec (2) e (3) neaaeoao, /oi

f(yj)]. (6)

f(yj) ( 0, e Dj iieiaeeoaeueii ii?aaeaeaia, oae /oi

f(yj) ( 0. E?iia oiai, dj – iai?aaeaiea nionea, e, neaaeiaaoaeueii, (j
( 0. Oiaaea ec (6) neaaeoao, /oi pjTqj ( 0. E?iia oiai, qj ( 0, e ,
neaaeiaaoaeueii, bTb= qjTDjqj ( 0.

Iieaaeai oaia?ue, /oi xTDj+1x ( 0. I?aaeiieiaeei, /oi xTDj+1x = 0. Yoi
aiciiaeii oieueei a oii neo/aa, anee (aTa)(bTb) = (aTb)2 e pjTx = 0.
I?aaeaea anaai caiaoei, /oi

(aTa)(bTb) = (aTb)2 oieueei i?e a = (b, o.a. Dj1/2x = (Dj1/2qj. Oaeei
ia?acii, x = (qj. Oae eae x ( 0, oi ( ( 0. Aeaeaa, 0 = pjTx = ( pjTqj
i?ioeai?a/eo oiio, /oi pjTqj ( 0 e ( ( 0. Neaaeiaaoaeueii, xTDj+1x > 0,
o.a. iao?eoea Dj+1 iieiaeeoaeueii ii?aaeaeaia.

f(yj+1) ( 0 e Dj+1 iieiaeeoaeueii ii?aaeaeaia, eiaai

f(yj+1) Nienie eeoa?aoo?u. Iaoiae Aeyaeaeiia - Oeao/a?a - Iaoyeea. Aieei Eiinoaioei. PAGE PAGE 8

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