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# 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

IO]IN r..\zÃtì

(Cìu j-N:r¡toca)

I

### .

IN'l.ltoì)u(;T l0N

eiarnlrle.

r,oots

single

equation .f(n)

0,

has proposed

### the

follorving- it-eratió;"s

t¿

sirnilar:

Ne eclna,l

seconcl

orcrcr

con\¡er.sc,ìc"

### .An

irnproved scrrc're \\'as proposecr

ancl

### Hafcl f1l.

tr'or' 1,he cquation

F(;n)

o,

*.ltete

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

a,re tn,o B¿r,n:r,ch spaces ancl

consiclerecl

iteratioî,s

(1

e:,,

A,,F(ur),,

An+t

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

LIt.c'

"roe

arrr-l _zl

/,(

_\.), qlrer,c

,ìí.rlcrrolcs

.Barraelr

### *iir.o ui 1l;"31*"ìì;ìi iiiLit;

o¡cr,aror,s ¡r,orr

rreep

propertics

### of

n{oser's iterations ancl

### thc

seqnencc (ø)_conr-e'gj"*

snr..tion

(scc Zehncler.

ancl

11.

(2)

### ,

Oll a Ncrv[o¡r '-l')'pc l\.I0tììod

168 I. Lazirr 169

Since,

(2.1)

zìsisurnlltions)

l,hose 01

### givc a

cónvelgence theorem

### for

the bouncls are sharper t'han t'hose

### th:rrr in ì{en'tonts

llllocess.

2. (lONYlìIìGlÌ\i;lì 1'lIIìolllìll

2.1.

l,

r'o,1,,íi'"ì,'r"

ttttd r¡ >"tJ

### thc 'fdlouing

c¡tn, tl'í,t i,c¡ns h,oltl

(2.

### la) Ao is

inuet't'ible utttl

e

(-\

Y) ;

(2.

llr{oLi(øo))ll

;

(2,

ll

ll

V

r') ;

2.

li

Àozr"(rco)

q ;

¿¿,

(2.

ct'tttl

'1t

rtcr (r',,)

### is

wrIL tLe,finett by

### (l

'2¡ nrrrl (-l '3)'.rettmin's irr'

ttt tt'

,1ql"ìtlo'i

' l

### )'

'l'l¿'is solwl'iot¡

tr'n'iqu'e

tire ioltotoitìg

esl'ì'¡t'¡tt'tes

cz(2d)'¿"'

### tntd the

tr, Ttosteri'ot'i errot' estitt¿u'tes

li e,,

(2cl¡'"-t ll"to

n'

ll /.11.

wherc

-

C"

7,,1

ProoJ'

n'nrl "1,

(1'3) rvc shatl

lelal iotts

### to

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

Irrorn (1.3)

lollorvs i

Arrt)

w

tlrÞ''(rut)

Aop,(ør)lJ

Ao(I.,(ao)

e

u,e

using (2.6), ll

A Á_å i/

(,4r,4;1) e

f /(1 __ a¡.

### (2.8)

-4, is invertible, _4¡r e

ancl /1,4f,

d,).

ancl

tve obtain

(1

r¿)il

(1.2)

### :

A olF ( ur) _ tr, ( u

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

n ò

### I _f ll

_ A oF, ( r o) JA op( n r).

assnmplions

follorvs

### is

dirrercntiabte on

Lipschirz,

ll

F(æo)

_.

ll

above

and (2.g)

(1

also

(2.4),

and (2.1c)

ll Area'(æ)

d,)r!ao@,@)

d)

ri,

8(ø0, r'),

ar.

n"o'r"

-_12)

assumptio

i*,,

### ** ,"rilil_r, let us

consider ttre"sequences (/c,),

definetl

Qo

d,

dÊ-r, n^

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

t'elal,ion

d17

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

c1i

tLi_r,

ancl,

### nsirg

(2.1e) wc caÌr pror.c

tha,t

(1,-r

-

cl,

L,

(2.18c)

(2.19)

.n,

<Qd,)'_'

(2¿¡r,,,,,

Carrchv sr:

J,cL x:,r,

ñt'qllcncc' so

corrvelgort.

rorn tJrc abovc

### iuctlrratiiy"ü

Ona

1/t,

Ncrvt.ou TJ'pc ì{eilrocl

(2¿¡'"

7

(2cl)2" 1

777

,

tt,

### tn ÞIt

flrer.efote

follorvs ilrai, for,

(2.20)

.1,

.1 (!r1)r" , l

(2tr\.¿,

>7..

So,

### it

calì bo shou,n lrv ilclrLctiotr tlli¡l,

n, Þ

(2.

### tLla) r,

e rS( rrru, t') :

crisl.s

e

l,l;.1,

liÄ,,11(r',,) I

.4, ,

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

/:,,' ,.,

1,

r') ;

(2.

ii

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

rtu.

Lr'<.r' r¿

1, (2.18)

### holl.

\\¡c suplri)se ttrril,

holcl

i,

...,,u,

SitL<rc,

r¡11

1

,r1,

I

12¿ ¡''

rl l;: (2rl)t'

zr,rcl f/ cr;

noli<

//

noll 1_ ll n,

/l

.n _L

### _d2

[Jr¿Í, is a,'n.e S(ro,

fu(.1 -_ J¿-i)

r''

¡is

,, w

1,Itat

¿t,

Silc

catt

il,.is

nu)

Jlrrsi;, b1,

ow

sequence (//

is bou¡clecl.

!!

ii

ll

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

ll

li

__ Lt,(rx))

+_ l;,Jl a,,

ø* ¡¡

al_ rt,

J, ancl,

")

.tn+tl-4,t

el,,_,)(r:,,

rt)],_r¡<

-- 1'' '4"

,,.,r-

l

l\-lìelì cr_-,

tl'h

iu Lplics

¡/Ð o.), ,, æ

L

,,* 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',.,-,

íril1.1

o''.' ,,,

### ,ootlr'ono ,rrì

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

tL -l-

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,,,_

13v

rvc

rising

I1t,(nt,¡

### i/<//;l;ril,ll ¿

_ tL,,n,,(nr,)J/< tlì ir

(2t1¡2,,

tvlreltcc lt

.r

rttt/ ^,;i.\

'

(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)

e,,)r,,,

(2.22)

(2-24) dn+r

(lt^n^

q.)en

rl,en,

(2.16)

,L-1

1,he

### filst part of

(2.2). tror b5,Q2B) for

thele lollorvs I

SAh .<

isl,s

e

ll ArV,1s*¡-r

(2.7)

ll.Ér'(¿x¡-ril

"

because A,,

(2.28)

ll

Or(2d,)2''.

posteriori

### örror

bound (2.S) follorvs

au

dn_t €n_L¡ €,,_1

(2.16).

### .

- ithe-uniq,*eness

follo.rvs

### from tlrc fact

thal, 1,he operator

;,i:

rnclecd,-

### + ll/o(¡"( p no)- is

c'rifferenriabre, F'(n)) ll

/Lr

ìl

q

ll

¡it ø e S1ä0,

'r¡.

,+rn;i*"ry,

772 1. I.az.irr

### by (2.18) rve

get

3. NUì,IIìRIC^1, lìXr\tr[I,LE

Jret

considel

l_lammerstein

ectrual,ion

D=

á(s,)ilo.

rvhere

lJ.v1r.

tr-or ø- e ll.\,

set ,7

orr l?À' r

./

f7,1åjr"à,,

eniç.r.

,so 'sircll

â.,/

### <

(2,v

(3.4) rve shall use

(rhe

,r-(,{rr)

ùth)

t7¿,¡¡

A,_t(21

I1t,(ù(i))Ai_1)

.

11

lt.

5,?j1_:t_

### eonrputing ilre rnatr,jx,4,

ivvely. So rve musù

sat'c

Fréchtt

ana

qo, c/.

### i

csrirnatäs (å.'Bi1:,. "il

6 7

On a

^\c\\,ton

operations

s

,s,

(8.2)

### for the

rnethod G,2,8) applieri

solrre

obl,ain

### the

nonlineâ,r s¡rsN6¡n

ø¿

ó(s,),

### l:

o¡1r..

'l'l'pc lIcilìod atìl0un

(3.2)

(3.3)

(3.1) ø(s)

s?(u(t))z U,

s e

1],

[0,

(s,

ø)

stttz, ö(s)

9120 s.

### ^ Using

t_he repea-ted trapezoidal

fr,¡

s,

llyt

and,_wo

r,oN

n,*¡2,

### ximate the

exact solution ø(s)

sl2

(S.1)

ø(s)

s¡,

### ã(s¡))w.:

ó(s), s e [0, 1].

concernecl

existence

solutions

and

wil,h

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

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

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

orras_

s alrcl 1,o

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,.'J

10,

1,,?/¿

fr), ú

¿,,r1

ir :

1- nt(.I -_

1¡:r,1.,,(t)

te)J,1t¡¡Str,

'11

,,)-conr.c,r

### .ru'cbio's

on ,,,,, ,¡rÏ*Il,;IosÌ'f'to'N

f,et

Lt]. lt'tretr.

,i,f

Í{,,,!1,)(,,1

### ..:Ii_ll) _

"_!:l'!!:)_

(t.__ 1¡¡1¡?,

,is ittcreusittq ou, (,nLu, l).1.

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