View of On a modified secant method

: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'

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:[email protected])-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(.).

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

:

_(**',)

=*', ^{[email protected]) :(:;,: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,

.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* ^{oÍ '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'

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-'[email protected])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

[email protected]):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"

[email protected]):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

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,) *

^2ø

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.

IÍ

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-"[email protected])

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).

214 ^{F. A. POTRA} t2 t\{a1.HÞn{,{Trc,{

-

^{RB\¡UE D' aNÀrrY lsIl NUNIÍIRT QUrl}

Eî

^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, 2í ^(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.}