View of On a Newton type method

ll1ì\'f iì r)';rN.\t.tstì ¡-uNÌìnIQUIì Iì,I rltì

].iltioIìlIì

I)rt L',\t'ì,ltoxlìt.\1.ro\

'l-cnre 23, No P, lf)94, _ltp, l$7- 1221

ON A NE\ /TON TYPII. I{äTHOI)

IO]IN r..\zÃtì

(Cìu j-N:r¡toca)

I

.

IN'l.ltoì)u(;T _l0N

rype :*:'åiä.il: 3d;,ä:îi*îi"ï,^"1î3il"iîå"å ",Ì"ìffiì

also

_eiarnlrle.

the

r,oots

of a

single

nonlincar

_{equation .f(n)}

:

0,

Moser [2]

has proposed

the

follorving- it-eratió;"s

fin+r: r" - !/nÍ(rò, lJn+t:'!J" - l/n(f'(ròy* -7),

^t¿

:0,I,...

the first iteration is

sirnilar:

to

Ne eclna,l

Lo 7lf'(n"). _'Ihe

seconcl

iter

that _:I$2... rhe

orcrcr

or

con\¡er.sc,ìc"

r""";lÏJiil:,:"1ä:i"t1-'?î Ì'¡;;T;":

.An

irnproved scrrc're \\'as proposecr

by u|rn _[4]

ancl

Hafcl f1l.

tr'or' 1,he cquation

(1.1)

_F(;n)

:

o,

*.ltete

n' : D c -\'_- I, _-L a'tl y

^a,re tn,o B¿r,n:r,ch spaces ancl

/) c -l a'

open set, the a'thors ha'e

consiclerecl

ilre

iteratioî,s

(1

.2)' _r;n+t:

_e:,,

-

^A,,F(ur),,

(1.3)

An+t

: _an

- A^(n,(c,n*t)An- ₁₎ : _a,,(2I _

x,'(e:,,nr)A,),

n:0, 1,..., n,ilìr

LIt.c'

jnifial

_qrr(rsses

"roe

f)

^{arrr-l _zl}

_oe

/,(

I,,

_\.), ^qlrer,c

!-(y,.\') Y into

,ìí.^rlcrrolcs

^llre

^.Barraelr

^{*iir.o ui} 1l;"31*"ìì;ìi iiiLit;

o¡cr,aror,s ¡r,orr

. ,Jterations (1'2).antl (1.8)

rreep

the

propertics

of

n{oser's iterations ancl

thc

seqnencc (ø)_conr-e'gj"*

q.ä,l"atiäall'i'to a

snr..tion

of (1.1)

(scc Zehncler.

[5]

^ancl

rráia p

_11.

,

^Oll a Ncrv[o¡r '-l')'pc _l\.I0tììod

168 I. ^Lazirr 169

Since,

by

(2.1)

I]ncler l(a'tor.or.iCh t¡'pc

zìsisurnlltions)

si.rilarl¡',.rvit'h

^{l,hose 01}

givc a

cónvelgence theorem

for

the bouncls are sharper t'han t'hose

from

th:rrr in ì{en'tonts

llllocess.

2. (lONYlìIìGlÌ\i;lì 1'lIIìolllìll

[)oncorningccluatiorr(1.1)ancltlreiter.ations(1.2)_arrcì11.3)l'ehar'e llrtnolturt

^2.1.

II ^{It is þ'róclta} 'tttfeì'ltti"iìL''"" ^51'o' ì'; - _!Y-t 'Y'

l,

r--a,orl < r'l c n',i,l,t'lni'"

r'o,1,,íi'"ì,'r"

tí, ír ^o

^ttttd r¡ >"tJ

thc _'fdlouing

c¡tn, tl'í,t i,c¡ns h,oltl

(2.

la) ^{Ao is}

inuet't'ible utttl

-4t'

^e

t

^(-\

'

^{Y) ;}

(2.

1b)

llr{oLi(øo))ll

< I

^;

(2,

1c)

ll

Á0(/¡'(r)-lr"1v¡¡ll < I

ll

n-all,

^V

^{n'v eS(ro'}

^{r') ;}

2.

1rt)

^li

^r -

Àozr"(rco)

ll <

^{q ;}

I

^¿¿,

(2.

1e) d::lt'rt* ^q < t +Íl

^ct'tttl

^{t" >}

^'1t

+ hvfFl

rtcr (r',,)

is

^wrIL tLe,finett by

(l

_'2¡ nrrrl (-l '3)'.rettmin's irr'

S(rn' r)

ttt tt'

stlttliott' t:* ^oJ

,1ql"ìtlo'i

(l

_' ^l

)'

^'l'l¿'is solwl'iot¡

is

^tr'n'iqu'e

r), iJ r <(1-q)/tu'

tire ioltotoitìg

u þriori ^enr¡r

esl'ì'¡t'¡tt'tes

¡ ^{a crÐr*!, , IA,} - It''(r'r¡-tll <

cz(2d)'¿"'

n:r'2'"'

tntd the

^tr, Ttosteri'ot'i errot' estitt¿u'tes

(2.3)

li e,,

- ^{T'i'li <}

(2cl¡'"-t _ll"to

- r"-tll'

^n'

:2t 3¡"'

r ^¡3l I ^tr)

_{ll /.11.}

wherc

Ct = .'- ^'-,'

^-

^'ntttl

^C"

-

7,,1

t -ar12) o

I -sdz

ProoJ'

Fot'rr

^n'nrl _"1,

^givcn b¡'('l'2) lnrl

^{(1'3) rvc shatl}

tinrl sitnilat

lelal iotts

to

a*stttt'ptions (2'1)' ,

Irrorn (1.3)

ib

^{lollorvs i}

ltat

Arrt)

: (2I - Ao\''(rr))Aow, lor all

w

eÍ' I -

tlrÞ''(rut)

: (I - ^Aul'(n'))z

I-A'A;1 -AoE'(n')-I'

(2.7) _llI -

Aop,(ør)lJ

< lll _ Aolt,(ur)ll + _ll

Ao(I.,(ao)

_ _IÌ,(ar))ll

< tul *

e

: _it,

_u,e

_obtain,

using (2.6), _ll

I ^_

A Á_å i/

< d < ^1, vhich irnplies that il

îhis ^{.,rere exists}

^{(,4r,4;1) e}

^{L(X) ancl} ll(:lr:l;,¡_r _{il <}

_{f /(1} __ a¡.

(2.8)

_{-4, is} invertible, _{_4¡r e}

L (X, y)

_{ancl /1,4f,}

_ll < li-A;, IyÍ _

_d,).

}-rom e.4)

^ancl

^(2.7)

tve obtain

(2.s)

llArî(Ør)ll <

⁽¹

+

^r¿)il

Aor- (ur)ll.

and, by

(1.2)

Ao,(nr) : Aoln(rr) _F(ro) _F,(u)(ø1_no)l

+ Aol,(ro) | F,(øo)(rr_nòl

:

A _{olF (} ur) _ _{tr, ( u}

o) _ ]., ^(. r _o)(

ut*

n ò

I ^_f ll

^{_ A} oF, ⁽ r _{o) JA op( n} r).

But, fi'orn the

assnmplions

(2.1) it

follorvs

that _lhe

is

dirrercntiabte _on

Sü;;^;);;ir',i. j',,

Lipschirz,

so operator' ÁoF(.)

ll

AolÛ(nr) _

F(æo)

_ _F,(a)(n, _ _nòlll

_{_.}

₁

ll

nr_rollr, thus, frorn the

above

inequality

_{and (2.g)}

(2.10) lla,r'(æ,)it <

⁽¹

* ,[* n, + tj:t \u ^(d,_q,).

üre

also

obtain from

(2.4),

(2.7)

_{and (2.1c)}

(2'11)

_{ll Area'(æ)}

- x'(v))li <(1 +

d,)r!ao@,@)

- ^{p,(E)), <Ãi(l} *

^d)

lrn*y

ri,

;iîlii?íue

^{8(ø0, r'),}

(2'72) lI - Ar?,(#l)il _< lll _,qop,(nr)ll,

<

^ar.

n"o'r"

uJ$"inIT:?,åtqJ' r0

_{-_12)}

are simitar to

assumptio

i*,,

r,r,i,årï;îîïi;T)113J ,'l"HT;î1i ^{z¡ ana (r} ãi";;;"#;u'- á|ittl"ä,",,i1î

** ,"rilil_r, let _us

consider ttre"sequences (/c,),

(n), (q^) and(d,)

_definetl

ho:'k, ïo: I;

^Qo

- ^gt do:

d,

(z rB) r::_':,i;Iî,:^_,,,

!" :

dÊ-r, n^

:-rjþ _kti-'

-

^etr-t),

^{n ='r,}

^{2,. . .}

(2.4) (2.5) ancl (2.6)

77 (\

\yhcncc it, folloti's th¿r,t

(2.16) anrl

[, [,az-irl 4

Jr'or' (rln) \vc obatirì 1,he follorving

recrLlrcrt

t'elal,ion

(9.11)

^d17

:1(.1 f

1 r/,,-r)2(r[u,,

--

^c1i

,) !

^tLi_r,

ancl,

nsirg

(2.1e) wc caÌr pror.c

by intlucliou

tha,t

(2,15) I,4 ^il,u(

^(1,-r

' <

_-

_7+V2 -l -= ^rtntl

^cl,

^{4 2tti} ,, fol all n ^>

^L,

r\ow, _b¡.

_(2.18c)

_and

(2.19)

llr,,,*, - ^rnll " "'f'

_.n,

<Qd,)'_'

.r ^--

^(2¿¡r,,,,,

¡!,, 4.lt _{I _(2¿ty}

!t,) i. ^a

^{Carrchv sr:}

J,cL x:,r,

--ìì;ì,"ä,.

ñt'qllcncc' _so

it is

corrvelgort.

rorn tJrc abovc

iuctlrratiiy"ü

Ona

1/t,

Ncrvt.ou TJ'pc ì{eilrocl

(2¿¡'"

7

-

(2cl)2" ¹

777

,

for all

tt,

tn ÞIt

flrer.efote

follorvs _ilrai, for,

all lr, I,

(2.20)

llr,-r* _//< I

.1,

<

.1 (!r1)r" _, l

(:'l i)

(1,, tli,;t

< I _,1., < l

_(2tr\.¿,

t,' -t; ' _I/

foL' a1t n

>7..

So,

it

calì bo shou,n lrv ilclrLctiotr tlli¡l,

fot all

_{n, Þ}

I

(2.

tLla) r,

^{e rS( rrru, t') :}

(2,,t3b) thclc

crisl.s

,l;r

e

,(-\) I) arxl ij,1;'i <

l,l;.1,

lii(l ^--rl,-t)) (2.18c)

liÄ,,11(r',,) I

{

^{.4, ,}

(2.1,!tl)

^), ,4,,(11'(a;)

- lt"(ll)) i(

^{/:,,' ,.,}

-'!l

^1,

^{fol :ll[ nt ll eB(ro,}

^{r') ;}

(2.

tEt,)

ii

/

':1,,11"(:r,,) '

a

^rtu.

Lr'<.r' r¿

.:

1, (2.18)

holl.

\\¡c suplri)se ttrril,

(2.t8)

holcl

lol

i,

: I,

...,,u,

SitL<rc,

fot

r¡11

j, n >

¹

iiDt t .,,,4-7.''i'l ',:l¡ll:'( t 'i,'(Zt¿)-ittt¡-

,8, ' ^{' 4t;} 1¡ ^.ri

^,r1,

(l'

^I

l))

ed)' 't

^{12¿ ¡''}

''

rl l;: (2rl)t'

zr,rcl _f/ cr;

,-

_noli<

//

,i ^_

noll _{1_ ll} n,

_n*

_/l

_<

_{.n _L}

_d2

[Jr¿Í, is _a,'n.e S(ro,

r,). ^¡ \ '/ '

_{fu(.1 -_ J¿-i)}

*

^r''

, l.ru,thct,

_¡is

,,e r'riltcn ,

^{,, w}

^prol¡e

^1,Itat

^{s,r is}

^¿t,

solution of (1.1).

_Silc

c (L2)

_catt

il,.is

sufficie't to sh

A;',(n,*t

-

^nu)

:

-F(u,,),

Jlrrsi;, b1,

(2.1g)

^ow

^{tJi t the}

^{sequence (//}

A;til)

is bou¡clecl.

(2.27)

_!!

t ^_ tl,,F,(#*)

ii

<

ll

I ^_

A,,L.,(c),,)

ll

+

li

A,(,.,(n,)

__ Lt,(rx))

< ^q,

+_ l;,Jl a,,

_

_ø* ¡¡

i/, for

al_ rt,

¿

J, ancl,

br. (Z.llj) _anrl (2.1õ)

^")

.tn+tl-4,t

-- ,:(, -l

el,,_,)(r:,,

l_

rt)],_r¡<

1.

tl"

^-- 1'' '4"

- -

_,,.,r-

^< ¿/,, !t:L lt _ ^r!',4, _ hl;

l

l\-lìelì _{cr_-,}

tl'h

ich

iu Lplics

¡/Ð o.), ,, æ

\a'4¿)

L

r,,

_{,,* i]}

_<

X

2 'r¡, 1

1-rt {

[r,'au, I'r,t' ,¡¡1 _tt , ^ttt ) ^{.l .}

?u r-ra

(

_4,, 1

,[

^{-r) '4,,,} 1ìrl a,ll n

> l"

¡¡t:O2'r.

'-So, 1,hc estin¿l,tes (2.2.L) becorne

it

^{[o]lo u's t'[ttr,t}

li ,r',.,-,

-rii,li< r,-t _2, \i ^{.'r, i _,,, ^rl:1 ,., <r,

Ircnce

íril1.1

c:s(oo,r'), ^-.,r,rt ,- ã.'s*l ^r,,,'

^{o''.' ,,,}

,ootlr'ono ,rrì

rr'o\.o (2..LBlr--e) fot'

,i,'=

_{tL -l-}

1, as

rve

ltavc plovetl

(2.8) and (2.'1.0 -2.12).

(2.23) l!r -.1,,11,(or,)i/<

q,,i_2/i:,,.r,,,_-2d,,,_

{lu{Bct)l_,=lo{ruir,,

13v

(2.18') a'ci (2,'5)

_rvc

'¡ìvc ltA;t]l<i/;t¡r llle__(ty,,a,nr[

_rising

(2,28)

llA;t -

^I1t,(nt,¡

i/<//;l;ril,ll ¿

^_ tL,,n,,(nr,)J/< tlì _ir

,.r,,

^(2t1¡2,,

tvlreltcc lt

,l

^.r

-

^rttt/ ^,;i.\

u ,\ , 4

^'

"o l'

₍₇

-¡.¡''

^t

*,:::,:i".'ii,";-í',Ííl,l[

rì.9',,"i'i ¡,un,,,,,

¡.il,;, i

Í1_f,,',ur,,- .r,c,.,rrrtrrJ,

lil.rr -

^{*'o == rrt. -_ tl} ,rl!r(n,r) ^--_a,N ₌₌

=. l,I -_.A,,'ltr,(n,) l(an

_

_{n4) _1. tI,l P(e*) ._}

I(n,,)

^__ [tj,(n,,)(6,t,

_

_n,,)

_],

en+t{ (+ ⁺

^e,,)r,,,

ancl, b¡'

(2.22)

(2-24) dn+r

(

(lt^n^

*

^q.)en

:

rl,en,

for all n > 1, thus, using

(2.16)

,L-1

., * _(tt, ,,) ,, _* + e,t)':"' ^,, :rff,,,

-r, flol. Q.20) there follows

^1,he

filst part of

(2.2). tror b5,Q2B) for

n :1

thele lollorvs _I

I _A,tì,,(u*) ll ì

^{SAh .<}

t

isl,s

.lAr7'("iJl_-l

^e

L(X) antr

_ll ArV,1s*¡-r

ll i ^{l¡1r _B(t\',}

(2.4) and

(2.7)

ll.Ér'(¿x¡-ril

<1/(1 -ldr)ll1,ll ^< I+d

ll/^¡.

r_3d2

^"

Finally,

because A,,

- ^l'1¡*)-t : - lI - A,fÌ,(a*)lP,(n*)-r, _using

_(2.28)

rve ge!

ll

A,,-fi'1ø'*)-t ll 4

Or(2d,)2''.

ifhe a

posteriori

örror

bound (2.S) follorvs

florn

au

4

^{dn_t €n_L¡ €,,_1}

4 2.qn¡ ancl

(2.16).

.

- ithe-uniq,*eness

of ilre solution u*, if r _< (r -rr)lt¡,

^follo.rvs

^from tlrc fact

thal, 1,he operator

p(r) : r _-A9!(ø¡ i!_coìtrac"ii""-óu" Sir;;

;,i:

rnclecd,-

₊ ll/o(¡"( p ^no)- _is

c'rifferenriabre, ^F'(n)) ll

<

^/Lr

I

_ìl

¡,'(r)

^q

^{< r,}

_ll

: ^if

_¡it ^{ø e S1ä0,}

-!rn'irjl

^'r¡.

^ãìl-¡-

,+rn;i*"ry,

í

772 1. I.az.irr

by (2.18) rve

get

3. NUì,IIìRIC^1, lìXr\tr[I,LE

Jret

us

considel

the

l_lammerstein

integral

ectrual,ion

r'et _ñ : (ø');rro'

_D=

:

_á(s,)ilo.

_{so thc}

s¡'ste'r (3.3) can be _*,rittclr (3.4) I@): e - ^{lf )e} _ _{- _ 0,}

_rvhere

?: _{Àr+, +}

_lJ.v1r.

tr-or ø- e ll.\,

lr _ll,e

_{set ,7}

:

orr l?À' ^r

j t,,(ã) :

./

_

f7,1åjr"à,,

'ìâx :2 _{,ra* È",}

eniç.r.

,so ^'sircll Y¡

-

^â.,/

^<

^(2,v

(3.4) _rve shall use

lro :1

(rhe

,.,""*,,Tli,ii"1i îî,i;,?

(3'5)

,r-(,{rr)

_-

_ùth)

- ^Arpl¡ut¡,

(3'6)

_t7¿,¡¡

-

^A,_t(21

_

I1t,(ù(i))Ai_1)

u,

^.

_

₁₁

_2, ...,

lt.

5,?j1_:t_

eonrputing ilre rnatr,jx,4,

ivvely. _{So rve musù}

ps.

^sat'c

^lhe

^Fréchtt

of application of (8.5)

_ana

tes for

qo, c/.

i

csrirnatäs (å.'Bi1:,. "il

rl,T"l

6 7

On a

^\c\\,ton

t of arìilunetic

_operations

\{/i'iting

_s

--

_,s,

jn

_(8.2)

for _the

rnethod G,2,8) applieri

to

solrre

I'e

obl,ain

the

nonlineâ,r _s¡rsN6¡n

ø¿

- ErI(si,,e¡, ñ¡)ru¡:

_ó(s,),

l:

o¡1r..

'l'l'pc lIcilìod atìl0un

(3.2)

(3.3)

,/Y

(3.1) ø(s)

-

^{s?(u(t))z U,}

::os,

s e

[0,

^1],

in C

_[0,

1], anrllet H

(s,

f,

ø)

:

stttz, ö(s)

:

9120 s.

^ ^Using

^{t_he repea-ted} trapezoidal

rule rvlth

fr,¡

:i-lM,

s,

: _iffl, [:

I 0',1',.-.., l, w .

^llyt

i - !,...,T11

^and,_wo

:

r,oN

:

n,*¡2,

*iaþpr*

ximate the

exact solution _ø(s)

:

sl2

of

(S.1)

byihe sölution'oî iheËci;_

tion

(3.2)

ø(s)

- r&t(r,

^s¡,

^ã(s¡))w.:

^{ó(s), s e [0, 1].}

v[e are not

concernecl

l'ith the

existence

of the

solutions

of (3.1)

_and

(3.2) or

wil,h

the approximatiou error but we

stratt,

rai¡ãr

estirnate the

114 I, I.ttzät' _tJ NIì\'UIì ì},.,\N'\I,YSII NUI.IIìIQUIì ]'.f ])I] TIII¡ORII, ])]I I,!\I'PROXIIìI,{TION

^V a plior i

cstinute.s

a p _os tclior i

cs tiìnâ Lcs

'lorle _{2J, No} 9, lgg¿, pp. f Zã.-f7fì

(lo do

4 16 0.1

1.32 .10 ¹

1 .25 ^.10-1 7.24.70-1

2 .96 .10-1 2 .81 .10-1 2.78 ^.1C-1

2 . 1r) '10-s

1 .32 ^.10-3 1.29.10-3

1.03 ^.10-ã 3.31 ^.10-a 3.12 ^.10-0

6 .62 ^.10-3

4 .03 ^.10-'r 2.51 ^.70-É

ON (u,

_{p, m.)}

-- _{CON\/trX trUNCTIONS}

lìItIrItP. Ilr\CllS \/,\SILI' J\,III I-ESAN

(Cìu j-Na¡roca)

'1' I'l'trr, S, Ott tttc Ilcratiue llcll¡octs utillt SinttútantotLs ,\¡t¡tro:uitnaliot't oI tlte Inuersc rsf ltc

O1tcrrLlor. Iz.r'...\cad. Na[ì< ]jsLoDsltoi S.S.lì., 1,{:,4,,IOi_ ^,111 (1g67).

õ. Zclrtrtìcr', .lj. .L ¡! llcn¡.¿r'li ttbotLL \4olt¡tt's lleiluttt'. (ìonrl¡. l.)r¡.c ¡\ppl. ttaf tt., 32, 3G1_366

(1 e74).

1, t\:l.Itotllr(;1.t{)ì\

lìccr.ivc(l lã XII 1903

Ilr

₁₃

l ^*'c trcfillcrl a

cr:ìs:i ol,gerrr*itrizc(-[

c.,r\.c\

l,lru(.ti..,rr)t \vrlich

å,;ll1:ì rl,i,,Tr1":,:]l,o

,ro,,urnl,iä'i,,n,,"r"*i,,s., _i^,ì_l,i;;n,'r.

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