TEOREMA O SHODIMOSTI METODA N^TONA

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М. Н. Яковлев, Теорема о сходимости метода Ньютона, Зап. научн. сем. ПОМИ, 1998, том 248, 242–246

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6 ноября 2022 г., 11:02:09

seminarov POMI Tom 248, 1998 g.

M. N. kovlev

TEOREMA O SHODIMOSTI METODA N^TONA

Pust~ ^P

(

^x

)

operator, destvuwi iz prostranstva Banaha

X v prostranstvo Banaha ^Y. V izvestnyh teoremah, sm., napri- mer, 1], o shodimosti metoda N~tona rexeni operatornogo uravneni

P

(

^x

) = 0

predpolagaets suwestvovanie proizvodno Frexe ^P0

(

^x

)

opera- tora ^P

(

^x

)

, kotora, po opredeleni, vlets pri kadom^xli- nenym ograniqennym operatorom

(

^P0

(

^x

)

^{2 L}

(

^{X Y}

))

. to zasta- vlet pri rassmotrenii, naprimer, differencial~nyh uravne- ni vvodit~ metriku, v kotoro operator ^P

(

^x

)

vlets nepre- ryvnym operatorom. Privodima nie teorema pozvolet izbe- at~ togo usloneni. Krome togo, klass operatorov, rassma- trivaemyh v nieprivedenno teoreme, soderit vse operato- ry, dl kotoryh operator ^P0

(

^x

)

, udovletvoret lix~ lokal~no- mu uslovi Gel~dera s nekotorym pokazatelem^s

(0

^<s6

1)

.

Rassmotrim uravnenie

Tx

+

^F

(

^x

) = 0

⁽¹⁾

gde^F

(

^x

)

{ operator iz banahova prostranstva^X v banahovo pro- stranstvo^Y. Predpoloim, qto operator^F

(

^x

)

nepreryvno dif- ferenciruem po Frexe v nekotorom xare ^S

(

^x0r

)

^X radiusa

r. Pust~ ^T { additivny i odnorodny operator s oblast~

opredeleni ^D

(

^T

)

^X i oblast~ znaqeni ^R

(

^T

)

^{Y. Pust~}

x

0

2D

(

^T

)

^.

Teorema. Pust~ vypolneny uslovi.

1.

Suwestvuet lineny (opredelenny na vsem

^Y

) operator

^T

+

^Fx0

(

^x

)]

^;1

dl vseh

^x2S

(

^x0r

)

^\D

(

^T

) .

2.

Dl

^x

,

^z2S

(

^x0r

)

^\D

(

^T

) verno neravenstvo

^T

+

^Fx0

(

^z

)]

^;1

^F

(

^z

)

^;F

(

^x

)

^;Fx0

(

^x

)(

^z;x

)]

^6Mskx;zk1+s

0

^<s6

1

^: (2)

3.

Dl

^x2S

(

^x0r

)

^\D

(

^T

) verno neravenstvo

^T

+

^Fx0

(

^x

)]

^;1

^Tx0

+

^F

(

^x0

)]

^{6k :} (3)

4. ^q

=

^{k M<}

1 .

(4)

5. ^r>r0

=

^{kP+1k =0q(1+s)k;1:} (5)

Togda suwestvuet rexenie

^x

uravneni

(1)

, prinadle awee

S

(

^x0r0

)

^\D

(

^T

) k kotoromu shodits iteracionny process

x

n+1

=

^xn;

^T

+

^Fx0

(

^xn

)]

^;1

^Txn

+

^F

(

^xn

)]

ⁿ

= 0

1

^:::

priqem

kx

n

;x

k6 k q

(1+s) n

;1

1

^;qs(1+s)n

(

ⁿ

= 0

1

^:::

)

^{: (6)}

Dokazatel~stvo.

Poloim

;(

^x

)

^T

+

^Fx0

(

^x

)]

^;1

;

ⁿ

;(

^xn

)

^P

(

^x

)

^Tx

+

^F

(

^x

)

^PnP

(

^xn

)

^:

Pokaem snaqala, qto

x

n

2S

(

^x0r0

)

^\D

(

^T

)

^:

Imeem

kx

1

;x

0

k

=

^k

;

^0P0k6k<r0:

Dalee

;

^1P1

= ;

^1P1;

;

¹

^T

(

^x1;x0

) +

^Fx0

(

^x0

)(

^x1;x0

) +

^Tx0

+

^F

(

^x0

)] =

= ;

¹

^Tx1

+

^F

(

^x1

)

^;T

(

^x1;x0

)

^;Fx0

(

^x0

)(

^x1;x0

)

^;Tx0;F

(

^x0

)] =

= ;

¹

^F

(

^x1

)

^;F

(

^x0

)

^;Fx0

(

^x0

)(

^x1;x0

)]

i sledovatel~no

k

;

^{1P1k6Mskx1;x0k1+s:}

Dal~nexie rassudeni provodim metodom polno matemati- qesko indukcii.

Pust~ ue dokazano, qto

x

n

2S

(

^x0r0

)

^\D

(

^T

)

i qto spravedlivy ocenki

kx

n

;x

n;1 k6k q

(1+s) n;1

;1 (7)

k

;

^{nPnk6Mskxn;xn;1k1+s:} (8) Pokaem, qto togda

kx

n+1

;x

n k6k q

(1+s) n

;1

(9)

otkuda

x

n+1

2S

(

^x0r0

)

^\D

(

^T

)

(10) i qto

k

;

^{n+1Pn+1k6Mskxn+1;xnk1+s:}

Destvitel~no iz (7) i (8)

kx

n+1

;x

n

k

=

^k

;

^{nPnk6Mskxn;xn;1k1+s6k q(1+s)n;1:}

Formula (9) dokazana. Dalee imeem

Tx

n

+

^F

(

^xn

) +

^T

+

^Fx0

(

^xn

)](

^xn+1;xn

) = 0

^:

to pozvolet ocenit~ ^k

;

^n+1Pn+1k:

;

^n+1Pn+1

= ;

^n+1Pn+1;

;

^n+1Pn

+

^T

+

^Fx0

(

^xn

)](

^xn+1;x1

)

=

= ;

^n+1F

(

^xn+1

)

^;F

(

^xn

)

^;Fx0

(

^xn

)(

^xn+1;xn

)

sledovatel~no

k

;

^{n+1Pn+1k6Mskxn+1;xnk1+s}

i neravenstvo (10) take dokazano.

Teper~ ustanovim fundamental~nost~ posledovatel~nosti

fx

n

g. Iz neravenstva treugol~nika i ocenok (9) imeem

kx

n+p

;x

n k6k

n+p;1

X

k =n q

(1+s) k

;1

: (11)

Otsda

x

n

!x

2S

(

^x0r0

)

^:

Imeem

x

n+1

=;

^nFx0

(

^xn

)

^{xn; F}

(

^xn

)

=;

^nFx0

(

^xn

)

^{xn; F}

(

^xn

)

^{; Fx0}

(

^x

)

^x

+

^F

(

^x

)

+

;

^n;

;(

^x

)

^Fx0

(

^x

)

^x;F

(

^x

)

+ ;(

^x

)

^Fx0

(

^x

)

^x;F

(

^x

)

^{: (12)}

Dalee

k

;

^nk6k

;

^n;

;(

^x

)

^k

+

^k

;(

^x

)

^k6

6k

;

^nkkF0

(

^xn

)

^;F0

(

^x

)

^kk

;(

^x

)

^k

+

^k

;(

^x

)

^k:

Otsda, v silu nepreryvno differenciruemosti operatora

F

(

^x

)

v xare ^S

(

^x0r

)

, pri dostatoqno bol~xihⁿ

k

;

^nk6k

;(

^x

)

^k=

1

^;k

;(

^x

)

^kkF0

(

^xn

)

^;F0

(

^x

)

^k<

2

^k

;(

^x

)

^k

k

;(

^x

)

^;

;

^nk6k

;

^nkkF0

(

^x

)

^;F0

(

^xn

)

^kk

;(

^x

)

^k6

6

2

^k

;(

^x

)

^k2kF0

(

^xn

)

^;F0

(

^x

)

^k:

Otsda

k

;(

^x

)

^;

;

^nk!

0

pri ^n!1:

Takim obrazom prava qast~ sootnoxeni (12) stremits pri

n!1 k vyraeni

;(

^x

)

^F0

(

^x

)

^x;F

(

^x

)

^:

Otsda, v qastnosti, sleduet, qto ^x2D

(

^T

)

i

T

+

^Fx0

(

^x

)

^x

=

^Tx

+

^Fx0

(

^x

)

^x

=

^Fx0

(

^x

)

^x;F

(

^x

)

t.e.

Tx

+

^F

(

^x

) = 0

^:

Perehod k predelu pri ^p!1 v ocenke (11), poluqim

kx

n

;x

k6k 1

X

k =n q

(1+s) k

;1

6 k q

(1+s) n

;1

1

^;qs(1+s)n

poskol~ku

(1 +

^s

)

^k;

1

^{>k s} pri ^s>;

1

.

Sledstvie. Pust~ vypolneny uslovi.

1.

Suwestvuet lineny (opredelenny na vsem

^Y

) ope- rator

^T

+

^Fx0

(

^x

)]

^;1

dl vseh

^x2S

(

^x0r

)

^\D

(

^T

) .

2.

Dl

^z

,

^v

,

^w2S

(

^x0r

)

^\D

(

^T

) verno neravenstvo

k

^T

+

^Fx0

(

^z

)]

^;1

^Fx0

(

^v

)

^;Fx0

(

^w

)]

^{k6Lkv;w ks}

(0

^<s6

1)

^{: (13)}

3.

Dl

^x2S

(

^x0r

)

^\D

(

^T

) verno neravenstvo

k

^T

+

^Fx0

(

^x

)]

^;1

^Tx0

+

^F

(

^x0

)]

^{k6k :}

4. ^q

=

^{k;1+sL 1=s<}

1 .

5.

Verno neravenstvo

(5)

Togda verny vse zaklqeni teoremy.

Dokazatel~stvo.

Imeem v silu neravenstva (13)

k

^T

+

^Fx0

(

^z

)]

^;1

^F

(

^z

)

^;F

(

^x

)

^;Fx0

(

^x

)(

^z;x

)]

^k

=

1

Z

0

T

+

^Fx0

(

^z

)

^;1Fx0

(

^x

+

^t

(

^z;x

))

^;Fx0

(

^x

)

(

^z;x

)

^{dt 6}

1 +

^Lskz;xk1+s:

Potomu mono poloit~

M

s

= 1 +

^Ls:

Ostaets primenit~ teoremu.

Zameqanie.

Esli pri lbyh ^{x 2 X} imeet mesto ^Tx

= 0

, t.e.

T

0

i suwestvuet vtora proizvodna ^F00

(

^x

)

, ograniqenna v xare^S

(

^x0r

)

, qto vleqet vypolnenie neravenstva (13) s^s

= 1

, to my poluqaem teoremu, dokazannu I. P. Mysovskih 2].

Literatura

1.V. A. Trenogin,Funkcional~ny analiz. M. (1993).

2.I. P. Mysovskih,O shodimosti metoda L. V. Kantoroviqa dl rexeni nelinenyh funkcional~nyh uravneni i ego priloeni. | Vestn. LGU

11(1953), 25{48.

Yakovlev M. N. A convergence theorem for the Newton method.

The convergence of the Newton method is established dor equations of the form

^Tx

+

^F

(

^x

) = 0, where

^T

is an unbounded operator, and the Frechet derivative

^F0

(

^u

) of the operator

^F

(

^u

) satises Holder's condition.

Postupilo 3 nobr 1997 g.

S.-Peterburgskoe otdelenie Matematiqeskogo instituta im. V. A. Steklova RAN