View of On a Newton type method

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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.

(2)

,

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,)[email protected],@)

- 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)

(3)

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,,)

_],

(4)

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) [email protected]): 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

(5)

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.

àn,.,."*

arrrt rlrrir,si_

an aflit,tÌìai,ìr,c ¿ìns\\.el

to fltis

orras_

s alrcl 1,o

sive

i:iomo

clìa,],a.t."ir;iiì;i.

Inslilule oI Oulctilus llcpttbIicii,:)'l

PO l.lo¡: 63

l,l0t) Clu.j-Nullc:rt llontrinitt

2, lz, þ, nt) - [:ON\ ItX Iì{;\(:.t,tOÀiS

I)r';rrr''rl'r'ro'r

:.1.

,1,.'J

./:

10,

irl

1,j,,:..,,,,1t l.ì ,t);

1,,?/¿

c [0, 1]=.,{

T";, ;:'ì:î','oì,)r,lr'i ,'ï1"'1,,,(,t il' ,,í¡l..""'

''' 'rr

fr), ú

| 'ir

¿,,r1

c,ri'

ir :

(2.I) tft*

1- nt(.I -_

r)y) <

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