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:202

ELFNA

PoPovIcIu

10

-t,

%

2 2,

l'ensemble d'es fonctions dans les définitions que nous avons aid.e des différences divisées en

utili-

MAI.IIEMATICÀ

-

REVUD D'ANAI,YSD NUI,IDRIQUE

El'

DE TTTEONTE DD L'APPROXIMÀTION

L'nNALYSE

NUMERIQUE

ET I,A THÉORIE DE

L'APPROXIMATION

'fome

B, No

2,

1979, PP.

203-214

BIBLIOGRAFID]

Aswþra unei

gene

noliuníi de conueri'

(cluj)

vI'

65 -7

Sn,-u'"

gén'órøtr

s fonctions aonueres, (1e5e)

¿ii-

"ihi'o

nøtematicd çi legdturø lor cu teoria s cotuuefres, Paris' 1945'

ON A MODIFIED SECANT METHOD

by

II, A. POTRA

(IÌncureçti)

Abstract' Irr this p]l)er we apply tlrc

rnebod

of v.

e,r.{.r

-([4],

t5])

to ttiã"riuày of th"-."iruirg.n"" oi'a-rnodified

secant method" We prove

irr.i- lrt ríte of .otir'.,g.í"e of this

method

is of the

form

Reçú le 8. x. 1978.

SLr. Dorobanlilor 40 3400 Cl.uj-Nøþoco

Roumøni¿ cu(z)

:

iCt, + d -2

Hza.z

-

dr)

rvlrere

a, d, H

and'

r

a;::

pc;ilive

nuirrbers d'epending on,,the

initial

condi-

tions. We

also

girr. îituip-ã'ti*ut"' for thË

distaãce

llx' - xxll'

n'

: l'

;, . .,

where

(*-)i:, is [itt

sec{Llence

obtained by the

rnodified' secant rrrcthod and

x* is its lirnit'

1"

Thc Intluotion

I'heorem

,lhe

methorl

of

Nonc]'isc:-cte Matlrematical

Induction, introduced

by

in t

nce

in

bY

t6l. pos

the

'l)'

.o(z)

-

r', ttt''rL(r¡

:

o'1o"'1'¡"¡

,U*'ï:i't;:jt:::"'

DEFTNTTToN

l.l.

Tttz fnnction- o>'. d'efined'

oy T' is

called'

ø røte "f

,on îrli"ti,--¡i ¡¡

søtisfies" the fotlo'øing þroþerties :

(1) a

maþs

T into

itself

;

æ

t2) for

each

r e T

tke series

Ð, t"(t) is

conuergent'

(2)

204 F. A. POTRA

The

sum

of the

above series, o(r)

:

following functional

ecluation :

À.'(t),

obviously satisfies the

(4)

z(0) :0 ! z(,)-.

No$', r,r,e can state

the Induction 'fheo¡ern

[4]

IHEOREM

l'1. If

(5) z(r) CU (z(t':(r))' r)'

lor

each r

e T,

tlten

(6) z(r) CU (Z(0)'

o(r)),

for

each

r-7.

^-- -W"

shall

sketch below

how

the method of nondiscrete mathematical

<¡f

the

convergence of

iterative

pro- complete

metric

space

X into

itself, se

that

we can aftach

to

the Pair (F,

åï"?!itã,loo"'il,,itd

.a ramiiY or

(7) x¡e

Z(ro),

(B)

x

e Z(r)+ F(x) e

t) (x,

r) ) Z(a(r)) for

each r

e

T"

llren the Induction

Theorem assures

the fact tÍrat Z(0),* Ø'

On^tl-te

otrr"ir1á"a

(8) implies

that

each elenent \,o1 Z(0) is a fixed elenent o{ the

"-"ñt"g

F \.ä. ¡'tgl :

E.

rt

also follows

that via the iterative

procedttre :

(9) x¡*t :

þ

(*,), n :0, l, 2,

' '

',

rË''obtain

a! sequence

(*,)i-o

whlgh convelges -!o-

"t

element x*

e

z(0),

.rr"h-tttut the

?ollowing 'inequalities

are

satisfied:

(10)

d'(xn¡r,

x^)3 a"(r), rt:0,I,2, "

'

(11)

d(x,,, x*)

5

o(o"(zo))

n:0,1,2' "

'

.3 oN 4' MoDIFIED SECANT METHOD 205

Frorn

(10) one obtains the following estimates of

the

distance between the

n' th iteiate x,

an'd'

the ,,starting poinl"

xo:

(12)

d'(x,,,

x) 5

"(ro)

-

o(o"(ro))'

apriori

estimat proced'ure (9) a

fied

by

the fact

the iterative

Procedure'

Suppose,thatloracertainne{1,2,""}'^onehasalreadycotnputed

X1, X2, , , ,

., xr. If

(13)

xn-r

ê Z(d(x,,,

xn-')),

thcn it can

easily

be

proved.

that the

follorving inecluality

is

satisficd:

(14) t\(x,,,

xå')

3

o(a(d.(x^,

x,-r)) :

o(d'(x,,,

x,-t)) -

d'(xn' xu-1)'

lhe

above estimate

will be

call

'can aPos

tive

procedure

(.c)) a

sequence

(x*)i-o which

converges

to

a

lixedpoint

øx

of'tir,

anclfôr.

9a9hìø;

{0, 1,

2''''-\theinequali-- lião 1ìo¡-(12) are '"o,,"', if for a certain ne..{l'?t?' ."'t}

thc condititn'(13) then for g¡is n, the inequality (ta) is

also lrr1liie d.

.tr'he above corol,l,øry

will

be

the

basis

of the prof of tire'Ihcorcm

3.1,

.cotrcerning

the

couver-gence

of the rnodified

secãnt

rnethod, which will be

gi'rzen

in

Section 3.

2. Divided

dilferenccs o-[

an

opcraÍor

,lhe

notion

of

divided. clifference

of a

(nonlinear) operator

is

an exten- .sion of the usual notion of d,ivid.ed. difference of a function, in

the

same sense

in

rvhich the F'r¿àUet d.erivative of an operator

is

an extension of the classi- ca1

notion of tf*-ã.tivative of a function. 'lhis uotion

was introduced-

by by

,t. sÞRcEEv

[9] and ¡.

sc:trurDr [7]

thod for the

iterativc

solution of the non-

h

sprlc-s.

"s. iV"

shall denote

by L(8,

-iì) the ecl oPerators,

from E into It t"et ¡ F,

anal

lel x

alnð' Y be

two

different

oÞFrNrlroN 2'l' A

bouud'ed'

linear

operator

Ae L(E'^F) is

called

I a di;ã;lAiffãi",r""

of

the

operator

/

on the-pornts

x

a'.d

y, l1:

I

I '{rs) tL(x-ù:r@)-f(Y)'

2

æ

(3) o(r):rlo(a(r))'

we

shall

justify the

name

of

,,rate

of

convergence", given

to the

function

co,

after stating the Induction 'I'heorem' '

-' i;l (X,

a¡'A" a complete rnetric-space.

If

,4 is a subset of

X,

and

x

an elernent

àt'X',

rn,e

shal1'c by

ct(-x,,4)

the g'l.b. oI

th-c set

.{d\r, y);

i ¿ Àl b"'

z

'*'" shall

dõnote

bv

U

(1,

Ò,

'l:

:11

\* = X;

clcnrent

of X,

u'c shal1 rvrite

for

simpltctty

ùþ, r) j,

,).

I,et ih"'iot"trr"1 ]0, ro] of the

real

line,

aud

for

cach

r

= î,- lel Z(r)

represent

a certain

strbset-of

X. We

sha11 use

the

follo- u,ing

notation'for ttt" limit

of

the Íamily

Z(.).

(3)

206

F. A.

PorRA

4

Iu the

scalar case

the divided

clilference

of a luuction is

uuiq.ue, but. irr

;h;;;;"t;

case

this

assertion is not true.

r.et

us exanrine as an illtrstraLiou

itr" Ë.r"

rvhere

E : F : R2.In this

case, a nonlinear

operator/is

charac-

;;ild

b),

t*o

¡eal lunctions of tu'o real

variables/, and/'

i'e'

(v)

ø

:

(**',)

=*', r@) :(:;,:å'' .;"]\'

'lhe[

eacir

of the linear operators,4,

and

'4,

givcn

b]'the

follou'irrg tn'o rnatriccs

satisfy

(15) :

b oN A MoDIFIED SECANT METHoD 207

In [9] ore

assunes

that the-napping !',.1'): l*, ?i /l is

symm-etric

i'e'

¡';,ii jl:

¡5,,,

x; Íl.In17l.t¡ii

couaitión

is

no longer.required.

I,et

us'

remark

that in oo, åi"rnplå 2,

and

A,

are

not

sym¡retric,

while ;1, a'd

L¿.+!A"or".

2'2

In both ol the

above

cited

papers, ol1e stlpposes,

in

order

to

assure

sufficient londitions for the

conriergence

of thé

secant method,

that

the

;;;;;ü

(x,

j)* li, ll;ll

salisfies a"Lipschitz conctitiou

at

least. Wc shall

writc tlris

condition urldcr

thc lorln

:

(17) lllr, y; fl - lu, ,; Íils nllx -

ull

+

lb'

-

ull)'

Itíseas¡.toprovetlratiftheaboveinequalityisfulfiledloraTlx,y,

,t/,

1)eÚ-Û1r','rri¡, *'ith tt*1'9yd u+a, then lor

each

ø=Uthere

exists the lirnit lini"¡r', ll; il, írd it

eqttals

the

Fréchet

derivativ" Í'(x)'

We

have

th"rt' '-'

(18) lf'@) - f'b')lls 2Hllx - )'ll' x' )' e u

The

above

remark

allows

us to take by definitíon lx, x; ll:f'(x)

for

each x

e X. Thus

(18)

implics

(17)'

--- nã.,"irety, if U* àpÉ.tài /

is'Fr1échet differentiable

for

each

xe

U,

urra

li- iiÚ í.'

satisfiedi

then

ihere..exists

a

mapping

-9 *

U

= \*'y)- --'ii,

y',

il= 4n, i1 v'hich

satisfies(16)

and

(17)'

\\re can take' for

exarnple,

This

rernark

will

be usecl

to obtain the theoren

concerning

the

con-

vcrgùce of

's process

[3]

as a cotlsequence of the theo- renr

concer oi th" trloaiti"tl

secant method

lvich u'ill be

proved

3.

The

motlifierl

secant melhod

Thesalneasintheprece<lirrgsectiolr,let/bearronlinearope-ratorlrom the

Banach

,pu""

.E

inio the lianach

lPace 4', ^314

let the

sphere

U-:,

:--uø;:,-;;rj ri" in"l"¿ed iuto its

ciomaiir

of definition. We

supposc

that the¡e

exists

a

maPPing.

U x U = (x, 3t)-

lx,

Y; fl ¿ I-(E' F)'

r"hi.lt

satisfics (16) aud (17).

Let.r'o bc a-poiut.of

U',Io.r u'hich

thc'lirrcar

;ñ;;;-i;,,- oì,ilìr-¡àrrí¿"ãìy ii,.,,"'tibie.

Thc modificd sccant rncthod,

*iL ur:"

gã;äg

io'Júd; "o"sistt of the

following

interative

procedure :

(19) x¡t*L:

x^

-

lxo,

*u', Íl-t

-f

(^'"), n:0'1'2'

'

"

f,(xr, yr)

-

f'(Y', Y,) fr(xr, xr) - fr(xr, Y")

frz-tz A

Az:

Ít-lt xr, !z) - .fz(

f"( lt.,1,"\ f,(x,

-

ItU',

fit-lt

fr(xt, xr) - fr(y', x")

xz-lz fr(yt, xr) - ft(Yt' Yr)

/"

- \t"

f,(yr, xr)

-

f,(Y', Yr)

, rz) - Í"(x', Y")

,z-tz it-lt

tlt - tt

I

Il / is

differentiable and

its

Fréchet d, rivatives

/'

is coltinuous or1

,h"

,t]:g-

Àént [r, 5t]: {tx +

(1

- t)y;

t

e f0, 1l}, then the linear

operator grvcn by

1, :

i -f'(x *

t(5t

-

x))dt,

also satisfies (15).

That

means

that

A2,

As, are

devided differences

of t

Mõreover, anY convex combination o

x

and y.

Let

us nolv Lettrrn

to the

genera

divided

differences see

[1].

Concern

spaces

sec

f

l0]. I,ct us

sulll,losc

th

includcd

into the

dotnain of

tbc

ope

D : {(x,

1,)

e U x U', x¡ 1,}.

\4/e

D =

(x,

y)-

lx, 1,

, fl = L(8, þ)

where,

for any pair (x, !) = D, the linear operatorlx,l'| Í) is a

diviciecl

diffcrencc of

/ on thc points r

and

J'

r.e' :

(16) lx,1; f)(x - )'):Í(*) -Í(:')

l-x, )' ;

Í) --

.f

'(, I t(y -

t))dt,

(4)

.20g

F. À'

PorI{A

6

For the study of the

convergence

of the

seçluence

(7-)i-'

yielded

by

l1g). rve need soine results concðrning

the

behaviour

of

such

a

selluence

ì;"iir"";";iã"roi

"u." *h"t" / is a cerîai.

real quadratic polinomial'

LII}IMA3'|,Ud,H,qoønd'roareþositiaenumbcrssati'sfyingtkecond'i-

.tion,s

(20) 6ln +

^tq,

¡

^Y

< !

tlten tlte function'

(2t) ct(r):+(u, +

d

-211øa'Ç-8il)

is a

rate

of

conaergence on the interaøl

7r:10, rol,

ønd the corresþond'ing fu,ncti'otc

c

'is giaen bY

7 oN A MoDIFIED SECANT METHoD 209

Taking into

account

the fact

til:at

f is a

convex

function,

we

infer that

d' >

f'(xo!o}

>-

Í'@o(')) for anv r e

10, ro)'

Thus, for

each

r e f0, ro], we

shall

obtain, via the iterative

procedure

25\. a

seouence (x.)î--0, decrcasingly converging

to x*'

Tn

this

case

it

is

iíäi ìn"Ëi|;-1.t;ËiffiÁ o, and ol

defined

ãs

ábove, reprcsent

a rate

of

;;;;"ú;;"e- and the function

related

to it.

The

folloq'ing

equalities are obviously satisfied :

(28)

xo

- x, :

o(r)

-

o(co"(r))'

(2g)

xn

- %,,+t: ^"(r),

(30) x, - x* :

o('n(z))'

Now,

we are able

to

state

our result

concerning

the

modified secant method:

THEOREM

3.1. If tlte

cond,itions (16). -a,ud. (17.) are,søtisfied'

for

aII

*, y,'-"iî'J U :

Ll(xo, nt'),

ønd if

tlt'c

following

inequalities:

(31) lllro, 'o' ¡.t-t¡1-rz

d"

(32) ll*, - xolll

4o,

(3s) lll*0, xo; fl-'f(xo)ll <

,o'

o(r) : 4r+ -r -

d &,

(22) H wlt,u'c, .(23)

(26)

":*"Jffi

Proof.

First,

rve observe

that

the

incquality

(20) implies

that

the quan-

titv ¡ndËr the squalããrt

sign

frorn

(23)

is

nonnegative.

Let

us consicler

the real

poliuominal

(24) Í(*) :

H(xz

-

ø2)'

It is

casy

to

1lr-ove,

that

for .any startiug.

Point

chosen

in the

interval

1a, t

.o' | , arid

tor

oäy

poritivc

n*rnb".

7,^b"lott

g to tlie

interval lf

'@o) '

i-

.o L,

thô itcrative

Proccdrtrc

(25) nn*r: x, - f

"(x)¡d

yielcls

a

seqLlence

(**)i:0,

decreasingly converging

to the root 5* :

Q

ót tt1"

equation

f(x) :0.

Sctting for an¡' r e f0,

ro)

are

fw then the

seq'bence

(7,)i:0,

obtained'.

ll tnl

üeratiae þroced'ure

(19),

c s to o ,oot x* 't:it

equøt'ion

f (*)-: 0,

and' the Jol'l'owing

inequø re satisfied:

-

(36) ll*, -

øoll

S 6(rò - o"(o"(zo)), n:0, 1,2, "', (97)

llx,

- r*ll <

o(<on(zo)), m

:0,

1,

2,

' '

',

(3S) llr^ - x*ll S

o(llx^

-

x,_,rll)

- llx, -xn_,l[ n:1, 2, 3, " ',

wløere a¡ and' o q're giuen resþectiuely

by

(22) ønd' (23)'

e

Corollary stated

in

Section

I

and

sent section. The

iterative

procedure

u!,

t*

ot,*

o;,i'¡r! I1lrl;-J";n"i ;"ít

c¡uation

Í(x) :0. We attach to

the / - L'analyse numérique et la théolie de I'approximation - Tome 8, No 2' 1979

(34) (35)

(^ln +

^ln + n)'

=

m 2

o(ro),

v

d

xo:

xo(r)

: a"*-r,

d

H

we have

xo) x*, aú'f(xo)ld': ¡..-'r'aking.P^(Í):f(xr)ltl

anð"

o(r):

vo

- - r* *" äbtoi"'the

t<irmirlas (2'2)

and

(23)'

Denoting

*r:-irlrrl + q;,

àoa

"o*pitihg t¡e .divided

differcnce of

trr" ro""iio""¡ å"

thö'päints^xo(ro)

and ro

rve obtain

,t27) lx¡,

ño:

fl :

d'

(5)

210

pair (F, r0) the rate of

convergence

r,i giver by

(22) and

the family

of sets :

Z(r\ : {x e E; lllxo, xo; -fl-'f@)ll

S

r, ll* -

xoll

S

"(ro)

-

(39)

(

-o(r)\, r=)0,rol

It is

clear

that

z(ro)

:

{xo}, so

that

condition (7)

of the

above mentioned Corollary

is

satisfied.

We

shal1 prove

that condition

(B)

is

also satisfiecl.

I.et x be an

element

of z(r),

and le

(40) x' : F(x) : x - lxo, xo; fl'f

@).

Using

(3)

we

can r,vrite

(41)

llx'

- *oll

<-

ll*' -

xll

t

llx

- xolls

z

f

6(ro)

- "(r) :

:

o(ro)

-

o(co(z)).

From

(16)

and

(40) we

infer that

,,2¡'i:,,.'

);,1,1, ; ",o ;,,r,,'uï) _;',) :

Accordingto the conditions (17), (31) and (32),

the

above equality yields:

lll*0, xo; fl-'"f(x')ll =! Qll* -

roll

* ll*' -

xll

f

llro

- t,ll)llr'-*ll.

Using

(22),

(23),

(39)

and

(40),

we

obtain

lllxo,

xo;

f)-'f (x')ll (.(r).

This

relation together

with

(41)

imply

tlnat x,' =Z(at(r)) so

that

condition

(6) is

also

fulfiþd. It

follor,vs

that by the iterative

procedure (19), one

obtains

a

seçl11ence

(x,)i:o

rvhich convefges to.

a root

ø'r'

of the

equatio_n

l@):0.

Moieover

foi

each

n e {0, I, 2, ,..}^the

ineclualities

(10)-(12) aie

satisfied.

But the

inequalities (11)

and (i2),

correspond respectively

to the

inequalities (37) and (36), while

from

(10),

(I2),

and frorn

the

fact

tha| o is

án increasing

function on

10,

,ol

we

infer that

llx,,-, -

xoll

I

o(ro)

-

o(llx,,

- x,-'.ll), n: l, 2, 3, ...

The above relation shorvs

that x,,-,

=

Z(llx,,- x,_,ll) lot 'n: l, 2, ,3, ..

'

so

that the condition

(13)

of the corollary is

fu1filed. consequent.ly t4.9

aposteriori estimate

(38), r,vhich correspond

to the inequality

(14),

will be

satisfied

lor

%

: l, 2, 3,

. .

.

i.

Concerning

the

hypotheses

of the

above theorem, we have

to

note

thal, in practìcal

applications,

the number

qoÍro,m

the left side of

the

inequalitf (32)

can 6ô

taken

as sma11 as wanted, because

having

an

ini- tial- appioximation xs, oîe can take lor xr.a small perturbation of it

(for

exampTe

xo:

(1

f

o)øo).

The key

condition

of our theorem is

re-

g oN A MoDIFIED sEcANr METHoD 211

oresented.

bv the inequality

(34).

This inequality

can

be

satisfied

only if i"ìïJåär;ä";"h;ihä[ ir,it lr'é ioiti"t

app.roximation

x, is

gootl enough-

rrowever, we can ptonã

tnát

tne condition (3-'4) is

in

some sense

the

weakest

""r.ì¡ì".'f"deecl,'lei h, H,

qo anð.

ro be'some

positive numbers,

and let

i.r""""ri¿". the'real

functioñ

f

given

by the

formula

f(x) :

Hxz

-

dro

- fi ø -

Hqo)'.

.l.lre d.ivid.ed.

dillerence of the

function

f , wi)l

-obviously

satisfy

(16) ancl

ila. îfr" i""qualities (31)-(33)

are also verified',

if

we take

rr: +, fro:o *"!ro'

.

Ilowever,

if the

condition (34)

is not verified, then dïo) ) ø -

Hq')''

and thus the

equatio"

f@):0

has

no

solution'

In

the following we-shall show

that

the estimates

(36)_(38),

obtained

in theorem

3.1,

arä in

some sense,

the

best possible'

pRopo,srrroN

3.1. (96)_(93)

a.re

s !"!!-

ense I d, H'

4q Q'nil

r¡'. e

xnø-

isal

nd' ø .Pàir . of.

O9;

whick

tke

and'

Jor wlt'ich''th'e (36)-

&Ye

qualitY '

Proof,Theproofoftheabovepropositiorrisaconsequenceofthe proof of

Lemma 3.1.

From

(36)

it

rollorvs tinal

llx*.- xrll5 õ'(f.)'

We

shall

prove

!h"! .f*

is

trre

äid;.';.";";f tt* "qü"tiot f'('il: b"itt o

neighbourhood'

of

the

point ro. I'etT

denote

the

open sphere

with

cenLre

xo

and' rad'ius o'(/0)

+

*

2a.

PRoPoSIrroN

3'2' If

ttt'a ineqwølity Qa) fro;ry'. Th'eorem

3'1 is

strict'

tnrn]îìiìilt i*, ,kor, ,iirtrnæ

i,s-guørântbed.' by

"this

theorem,

is

tkeunique solulion, of th'e eqnøtion f (x)

:

0

'in

the set U

^

V

eroo¡. Firi,"rve ,toie'that if the t""n"-"tll"^(U,t) tt strict' the ø;0'

so

fhat x* = UO

1r. luet

Y* b: an

eletneut

of

U

(-)

lZ,

such that /(yr) :

0'

Using

(16) we

obtain the relation:

(41) x* -!* -

lxo,

*oi Íl-'(lxo, fro; fl- l**, y*; fl)(x* -y*)'

Now taking into

account (17)

we

obtain :

,(42) llx* -

y*

ll < 1 (tt*, - x*ll

-F llto

-

1*ll)ll*^'

- v*ll'

F. A. POTRA Õ

(6)

272 F. A, POTRA

On

the

other hand,

from

(22), (31)

and

(32), we

infer that

((43) T Urr,- x*ll*

llro

- r*ll) <ft {z"tr,) *

i

?o)

:

1

Finally the inequalities

(42)

and

(43)

imply that x* : y*,

so

that

the'

proof of the

propositíon

is

completed.

d.

Thc" modified Newton's method

As we have anticipated

in

Section

2, the

results concerning

the

modi-

fied

Newton's method can

be

obtained, as

a limit

case,

from the

rcsults concerning

the

modified secant method.

In the

following, we shall trans-

cribe the results obtained in the

preceding section

for the

case where.

fro: *o and

çro

:

Q.

r,ElrMÂ 4.1.

IÍ d, H

and.

to

&/e three þosiliae n,untbers søtisþing th,e:

inequøl,ity ;

'(44'¡

4Uro

3

d.,

then, -'

11

l*lrDGoRÞM 4.1.

cond,itioto

(lB)

hold's

for

each'

x, ! e U : U(xo,'wl

ønd,

i,f

th,e

follouing

'in,equalities :

(48) lllf'@o)1-r¡¡-t¿

d

(4e) lllÍ'@o)l-'Í(xòll s

ro,

(50)

4Hro

3

d,

(sl)

ttt,

/

or(r,)

: L""A

- ^IF-Z ufi),

øre

Julfiled,

then'

tlte

sequ,ertce (47), converges

to a root

x'N

ineq'u,al.il,ies arc satisfied :

(*,)i:o

obtain,ed' b),

tht

iterafiae þroced'u,re

of

th,c cqtr,al;iott

Í(*) :0,

and

the

following

(52) llr, - roll S

or(zo)

- c'(aü(ro)), n :0,

1,

2,

. . .,

(53)

llx,,

-

xnll

S o'(coi(ro)), n:0,

7,

2,

.

'., (54) llx, - r'tll < orlì(r, -

xn-t ll)

-

lìø,,

- ,r,-'li, 7L: l, 2, 3, ...,

uherc

o,

and.

o,

a.re giaert, resþectiuely

by (aQ

and (46).

From

Propositions

3.1

and 3.2

we obtain the lollowing two

proposi-

tions,

concerning

the

sharpness

of the

estimates

(58)-

(60) and

thc

uni- qrleness

of the root x*:

rRoposrrroN

4.1

. The

estimates

(52)-(54) nre

sloørþ

in'

the

folloa' uiqg

sense: tsor any three þosit'iue nutttbers

d, H

and ro satisfyingthe inequa-

li,ty-(s})

tlrere exists a function

f, ukich

søtisfies the kyþotheses

of

T'hcorem

4.7,

nnd

for

uhich tke ineqr.taliti,es

(52)-(54)

are aerified

uith

equ,al'ity.

pR.oposrrro¡t 4.2.

If

tløe inequal,ity

(50) of

T'hertrt"rn.

4.1 is

strict,

tltcn úhe

root

x'i', wltose existence

is

quørønteed b), Tlteorenr, 4.1,.is the'wn'ique

solulion of tlte

eqn,øtion

Í(*) : 0 in

th'e set U

)

l'

, ulterc l,' is

tlrc oþen

sþhere

- In uilh the

end, center 1et xo

us

end

note

rad"itts

that the

ot(ro) results stated

|

2ø.

in

this section repre- sent a slight improvement oT

the

results obtained

by tls in [3].

Nanemely

the condìtion (tB) of the

present

paper in

weaker

than the

condition

lf"@)ll

= ZU, x = (J,

irnposed

there.

Moreover

the

aposteriori estimate (54), I'ronr Theorem 4.1,

is

nerv.

ITIlIlERENCIìS

l0r ON A MODIFIED SECANT METLIOD zl'3

(45) .,(r) : ilnr+d-"lF@)

ìs a

rate

of

conuergence

on

thc

intetaal I: ]0, rol

and. the corresþonã'ing

Junction o, is

giuen by :

c,(r):lÇlF--zmp;4 -

1

I

(46) dz

-

4

lfdr

Ncu,, as

in

the pre cee ding

tlvo

sections,

let/be a

nonliuear operator wliich, maps

the

-sphere

U :

A

(*0,

m,)

of the

Banach space

E into the

Banach space

1ì. We

suppose

that / is

Fréchet differentiable

on U

and

that

the

condition (lB)

holds. Then, according

to the remark

made

in

Section 2,.

there

cxistò

a

rnapping

U x U =

(x,

),),-

Lx,

),, Íl e I'(8, n)

such

that

(16) and (17) hold. Moreover

lor

e ach

x e U

u'e have

lx, x; f):'

: f'

(x).

I,et us

suppose

nou' that thc

Fréchet

derivative.f'(xo) is

boundedly

invertiblc.

We

may thcn

consider

the following iterative

procedure:

(47) r¡tt:

x^

- lf'(xo))-'.f(*"), n:0, \, 2' ...

which is

called

the

rnodified Newton's method.

'lhis

procedure

may

be.

regarded. as

a limit

case

oÍ the modified secant

urethod so

that

frorn-

Theorem

3.1 we

can derive

the

follorving theorem:

[] Il a 1 a z. s, G., O inear sþøccs'

lìe e et d 5-9 (1973)'

[2] Colclne su'þra aeiþentræ

rcz onale ..Ccrc. ìfat.,

tom P0, 7 (1968).

(7)

214 F. A. POTRA t2 t\{a1.HÞn{,{Trc,{

-

RB\¡UE D' aNÀrrY lsIl NUNIÍIRT QUrl

DE 1'III]ORIE DE ]T'APPIIOXII{Á'TION

I,'ANALYSE NUIIÉRIQUE ET I,A TIIBORIE ÐE

I;,APPROXIMATIOiT l'ome B,

l\o 2,

1S79, PP.

215-227

[3] P o t¡ a, Il. -1.., 'flrc ral,e of conuergçnce of a øtod,ified Newton's þlocess. Preptint series

' in rirathematics lo. 36/1978 INCREST.

[4] P t á k, \¡., Nond,i.screte matlrcnatical induction and ilevøtiue e,vístence þroofs. I,inettt algebra ancl its applications 13 (1976), 223-238

[5] P t á k, V., 'I-hc rate of conuergence of Neuton's þrocess, Nunr. l\{athetn, (1976), 279- t6l

L7l i8l lel

IJO]

285.

P t á

k, Y.,

lVhat should be o rate of conuevgcnce ? R.A.I.R.O. Analyse Numérique 1l' 3(le?7)

p

27e 286.

S c 1r trr i d

t,

J., Eine iibertragung d,e/ Ilegltla. Falsi auf Gleicltungen in Banachrøu.n+. I,

II,

Z. Anges'. ìfath. Mec., 43 (1963), p.

l-8,

97-110.

Sclrröc1

er, J,

Ni,clttl,i,neaye Majorønten beirn. Verfalwen rler scJt.vittucissen Ncihcrung, Arch. ùIath. (Basel) 7, 471-484.

CepreeB, A. C. O uentoôe xopô. Cu6up ilameu.

X,2

(1961),282-289,

V ¡ ¡ rrr, C. 06 o6o6qeHru,ß pas)ercHHbl; pasHocmnx)

I,

II, I4AH 3CCP, (luaøxa, ltare- uarur<a,

l6

(1967), 13-26, 146-156.

Received 12. III. 1979.

Il/CRES'I' - Bttcuresli

oN INTERPOLATION OPERATORS (II)

(A PROOF OF TIMAN',S THEOREM FOR DIFFERENTIABLE FUNCTIONS)

by

R. B. SÂXENA antt K, B. SRIVASTÀVÀ (Lucknow, India)

1. In continuation of our

stu

I,et

-1 < r < 1,

cos

/: x,

and cos /¿a

:

xÞn

with

(1.1) lhr:-'":-- , h:0,fl*,

2n*l

n :7,

2,

Frrrther lot

Jt,

: -n,

11,,

let

ri"- 2n*l

2 (t-t¡")

(r.2) I

*,(t) :

(2n

!

1)

sin-

(l

-

f¡")

t ¡ : O,ø stanils fo¡ h - 0, 1, 2, t.

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