View of A new convergence theorem for the Steffenssen method in Banach space and applications

REVUE D'ANALYSE NUMÉRIQUE ET DE THÉORIE DE L'APPR,OXIMATION Tome 29, No 2, 2000, pp. 119-122

A NEW CONVERGENCE THEOREM FOR THE STEFFENSEN

METHOD IN BANACH

SPACE

AND APPLICATIONS

IOANNIS K. ARGYROS

Abstract. We approximate a locally _unique solution of a nonlinear operator eqlation in a Banach space using the Stefiensen method. A new semilocal convergence theorem is provided usi[g Lipschitz conditions on the second F]échet-derivative of the operator involved.

Earlier results have used Lipschitz conditions only on the first divided difference. This way our conditions are different from earlie¡ ones. Hence, they have theoretical and practical value.

A

numerical example is also provided to show that our results apply to solve a nonlinear equation, where earlier ones fail.

Key word,s _and, phrases: Banach space, Steffensen method, divided difference, F!échet- derivative, Newton-Kantorovich hypothesis.

1.

INTRODUCTION

_ .J" ^{this stud¡}

^{lve are concerned}

^{with the}

^probrem

^of

approximating a

locally _unique solution

z*

of the equation

(1) F(ø) :s,

where

tr'is

a twice FYéchet-differentiable operator defined on a convex subset

D

of. a Banach space .E

with

values on itself.

'We use the Steffensen method

(2) rntr : rn - lr,,B (r,) ; Fl-t F @,) ^{(ro e D) ("} ì

⁰⁾

to

generate a sequence

{*"} (, )

⁰⁾ converging

to r*.

Here,

[r,a;F]

^d.enotes

a divided difference of order one

at

the points

r,u _{€ D,}

which is an element

of L(8,.8), the

space of bounded linear operators

from

-Ð

into itself. B

is a continuous operator defined on

D with

vâlues

in .8,

usually related

to f'

^by

B(")--r-F(") ("e D).

Sufficient convergence conditions

for the

Steffensen method were given

in

_{[1], [2], L4l,}

_[5]'

[6], [9]-[11] ^using

mainly

Lipschirz-rype condirions on the AMS subject classification: 65805, 47HLT, 4gDt5

L20 Ioannis K. Arsvros

first

divided difference

of F. In our

study, we use

Lipschitz

conditions on the second tr'réchet-derivative

to

obtain a u.ew semilocal convergence theorem

for the

Steffensen

method. l'his

way, our convergence conditions differ from earlier ones. Therefore our results have theoretical as well as practical value.

Finally, we complete this study by providing a numerical example to show

that

the Steffensen method

starting

from an

initial

^guess :ú0 converges

to r*,

whereas the same is

not

guaranteed by existing conditions [1], [2], [4], [5], ^[6], lsl-[11]

2.

CONVERGENCE ANALYSIS

From now on we

will

set

A(r) :

_[r,

B (r)

_;

F] r € D,

and

further

denote

A("")

by

A,

^(rr.

>

0)

for

simplicity.

Let

a, b, c, d,.R be given nonnegative constants

with

c

e

_[0, ¹⁾

, ^{rs e D}

such

that Al' :

A

(ro)-t € L (8, E), f

be an increasing real function, wh.ich is continuous and nonvanishing on [0,.R]. Define the polynomial

p

by

That

completes the proof of Lemma

l. ¡

Lpvtua 2.

Assu,me condi,ti,on

(7)

holds, and

(e)

_.f

(t) # ^0, /(ú) ^<

^p,(t)

for

^{arr ú}

e

_{[0, 11]} ^.

Then

iteration {t,r} (" 2 ⁰⁾

^giuen

^{by (10) is}

monoton,ically i,ncreasing anrl conuerges

to

1.

Proof. Define function g by

(10) su):t ^e(t)

r ^(t) Then by differentiating function g we get

(11)

_s'(t)

: ^{f (t)U(t) - p'(t))+ l'(tþg-.}

r

þ)2

It

^follows

from the proof of

Lemma

l,

(B) and

(g) that p,(t) <

0,

p(f) >

0,

f

^{(t)_< 0_, and}

l'(t) ^>

⁰

^{for all}

^ú

^€ [0,r1].

^Hence,

by

(11) function

g

increases on

[0,r1].

^So,

if t¡

e. [0,11] for some Æ, hen

:ttc+7,andf¿-,.1 (i ,r- p(tk) - p(rt)

lftù:rr- r1¡:r,

A Theorem for the Steffensen Method _T2T

p(tn)

I ^(t*)

That

completes the proof of Lemma

2. !

Remark, 7.

It

can easi,ly be seen by (11) that cond,'ition

(g)

can be replaced,

by the ueaker

f ^(t) + ^0, Í (t)(f ^(t) - ^{p,(t)) + Í'(t)p(t)} t

0

for

^au

t e

_[0,

"r]

^.

We can nov/ prove the semilocal convergence theorem

for

Steffensen me-

thod

(2).

TnpoRprvr

7. Let F : D c E -+ E

be

a

twice Fréchet-rtifferentíaile oyterator. Assume:

(a)

there erists

r0 e D

such

that Aot : A(*o)-r e L(E,E);

(b)

for allr e U(*o,A) : {reElllr-*oll <Ã}

^there

^e,ist

^constants

u,b,c

such that

(12) _Il¿1,

_@,,

(*) _ p,,("0))ll (

o

llr _

"oll ^,

(13) lltlrp,,("0)ll (

^ó,

(14) ll11', @' (") _ A("))ll _<

";

(c)

condi,tions

(7) and (9) are

sat,isfi,ed,

for

some cont,inuous monoton,ically 'increas'ing and nonuanishi,ng function

f

^on 10,11] such that

(15) llt;'çq@)-

^Ao)ll

^< f fll"-roll +1) < 1, _f oratr reu(ro,rt)

2 ò

(3)

the constants

a,

_{B by}

(4)

p(t) :

lot' +Iur' - ^(r -

^c)t

⁺

^d,,

2(1-

c)

b+Jæ+2a(1-c)'

t¡1t¡-

(r.:

(5) þ: ^(L- c)a-loo'-ï0c,,

and the

iteration {t"} ^(" l0),

^by

(6) ! ! p(t")

tn-tr : ," - Vö,

^to

:0

(n

>

0) ^. We need the lemmas:

LnttvlR 7.

The real polgnomi,al p has two positiue zeros

rr,

¹² with 11

{

¹²

and

a

negatiue zero

-r3 ("¡ > 0) if

and only i,f

(7) d<p.

Proof. Polynomial

p

has

a

negative zero -'t'3) ^since

p(0) - ^{d > 0,}

^and

p(t) < 0asú -+--oo.

Moreover,

p'(0): _-(1 -q) < 0,andp'(ú) ) 0as

ú -+

+oo.

^Hence, there exists

azero

of pt

in (0,-),

which by the form of

p

is given

by

(4). Thus,

p

has two positive zeros

if

and only

if

(B) p(a) <0,

which is equivalent

to

condition (7).

t22 and (16)

(d)

the followi,ns hold:

(17) (18)

(

ie)

Ioannis K. Arsyros

a

< lla;1r ("0)ll

_;

c

e

_[0,1),

rz(Ã,

U

(ro,R) ç n,

llr"+t-r"ll(ún+1

-úr¿

llr"- ^{r-ll <} rt-tn

A Theorem for the Steffensen Method

4 5 r23

where 11 o,nd, T'2 are the positi,ue zeros

of

equati,on

p(t) :0,

and polynomi,al p is giuen bA

@.

Then, the Steffensen method,

{r"} ⁽ⁿ )

⁰⁾ generated by (2) i,s wetl defined, rema'ins i,nU (r.6,11)

for alln)0,

and conaerges to a

solut'ionr* e

U

(rs,r1) of

equati,on

-F(z) : g. If rt < rz the

solution

r*

^{'is un,ique}

in

U

(ro,rz),

uhereas

if 11:

12,

r*

^{is un'ique}

in

U

(16,r).

Moreoaer, the following error ^{bounds hold}

for

^aII

ⁿ )

⁰

(20) and (21)

Proof

.

We first show linear operator ,4

(r)

is invertible for all

r e

U (ro,

a),

where

a

is given bV

(a). It

follows

from

(15) the Banach lemma on invertible operators [4], [B], the estimate

llz';, ^çq(") - ^{Áo)ll < ¡} fl" - _"oll) ⁺

¹

^<

^1,

that

A(")-t e L

(Ez,.E1) and

(22) _ll, f"l-'r,ll . -r 11"-,olt)-' < -/ ^(o)-'.

We must show

that

estimate (20) holds

for all n > 0. First,

note

that

11 is defined,

andbyusing

(2), (6), and (16)

weget llrl-"oll : ll-{t¡("0)ll ^<

d,

: tt -

^ú¡,

^which

^{shows (20)}

^{for n} : 0. It

follows

from

(22)

that

linear operator

A(*t)-t e L(E,-E),

and hence

12 cal

then be defined

by (2).

Let

r e lrs,r1]: {r:r:\rL ⁺

⁽¹

-À)"0,

⁰

^<

^À

( 1}. By

Taylor's formula _[3], l4], [7]

for

a twice Fréchet-differentiable operator _{G on}

D,

we can write

(23)

G

(*) :

G (*o)

+

G'

(rs) (* -

^*o)

*ï"" ^("0) (, - ^,o)' *

+

j ,""

^(a)

- G"(,.)l (, -

^{a) da.}

æo

Using approximation (23) for G

(r) : _{A,rF @) (" € D),}

_{we can get}

(24) AlrF @) : Ao'F ("0) + (* -

^*o)

^{+,4tt (F,}

^(*o)

-

^Ao)

⁺

+;Alt

1 ^Ftt @ù @

- ^*r)' *

* j or' (p"

(y)

- ^{F', (ro)) (*} -

^{a) dy.}

1O

Let Àd:

s, then

by

using (B), (12)-(14), _(16), approximarion (24) gives

(25) ll4'¡(")ll ^<

⁽¹

-À) d+c^d+)a*a2+

fo,r3as:p(s)

(since

ï - ro:

^À

^(rr - ^ro) : _-À,4;1f

(ø¡)).

Moreover,

by

(2), (22) _and (25), we get

il*r-zril ( llar",)-',4'll ^{.ll.4t'F (*t)ll} . -ffi ^:tz-tr.

similarl¡

\rye can show (20) for

all n ) ^0.

^Estimate (20) and Lemma 2 imply

that

steffensen method

{""}

^(?_2 0) is cauchy

in

a'Banach space,E, urrd u, such

it

converges

to

some

r* et (r0,")

(since

u (*o,s)

is a closed set). Fyom (25) and the

continuity of

.F' we get

F (r.):

O.

'f,"rtirermoïe,

estimíte

(Zt) f,o_lbyl immediately

from

(20) by using standard majorization techniques'[B], [4],

[8].

^{To show} uniqueness,let

z e

U

(rs,r2) with f _lr¡ :0.

_Using

_iZ+¡'t*

r :0,

we obtain,

.xr-z : Al'(F'

("0)

- ^Ao)(r-ro) f ï,OOto,,(*o)("-*o)2 ⁺

(26) * I ^o.'

lp" (y)

- F" (ro)l (, -

^{a) da.}

aDo

We-Setll,-"oll (r-r-úoif.zeU ("0, r), and,llz-roll :þ(rz_úo),0 <u<

7,if z

€.U

(rs,rz).

Hence,

by

(21) and (26), wà get

foi

alt

n)0:

_ll'i

^_

""'ll <

,!. tn,if. z e t(r9,r1),

and

llr-",11 { t"?i-tn),

^if.

^{z e} U('*0,r2)-."In

either case, ^rñ/e gut

J]lg rn:

zt which yields

t* :

z.

That

completes the proof of Theorem

1. !

We now state a theorem by pavaloiu _[g] for comparison:

TnnoRpru

2. LetB,F: D c E

-+

E

becontinuous operatorswi,thdiuid,ed, di,fferences of order one

fr,y; Bl

and,

[*,A; Fl

respecti,aely. Assume:

(u)

(27) B(r):r-F(")("eD);

A Theorern forthe Stellensen \4ethod ^L25

6 7

124 Ioannis K. Arevros

(28)

(2e ₎

(b)

There eï'ists

ro € D

suclt'

thatls - [r¡, ^B (*ù; ¡]-1 e L(E'E)

^and

llroll ( h';

(")

rnax{llr1 -

^roll

^{, llr,} - ^{A (z¡)ll}} (

^ho;

Set also

B(r): r-F(') ^{(*€D) ''r'hen'}

^using

^{(4)' (5)'}

⁽¹²⁾

^-(16)'

(27)

-

^{(30) and (33), we obtain}

l¿r

:

1.1566 265,

h2:?, no:

'5391566

, h3:

'4157352

)

'25'

a:!..1566265,Ó:'"3855422,c:'0119947'cl:'2891566 o :

1.0155688, B

:'5392822'

condition

(31) of Theorem 2 ^{cloes nob} hold.

That

is, Theoretn 2 cannot guar- antee

that

Steffensen methocl

starting

from z6

:

0 converges

to

a solution of

"qrrution

Fþ;) :0,

wher-e frtnction

F

is gìr'en

by

(32)'

-However: all conditions of Theorem

l

are satisfiecl. Indeecl

f.orrt"tlt"

^{above we have}

that

condition ⁽⁷⁾ is satisfiecl.

I)efl'e

funcrion

,/

^so

that 1+.i

^(¿)

: _l¿;t (A(t) - ^¡o)l

!¿

e 'D)' It

^can

easily be ^secn thal, tlrc left-hand ^siclc inequalil,.y

in

(15) is satlsilccl ^as equallry' whereas the right-hand ^{sicle is smaller} than 1. Fur-thelrnore

it

is sirnple algebra

to

shorv

that

^condition

⁽⁹⁾

^holds

^also. (

oncliiions (18)

and

⁽¹⁹⁾ are ^needecl

to

show uniqueness

or

trrá sorution

r* \n (J(ro,r2). 'rhey

^can

^ccrtainly

^be

Uniqueness is then guarantecd only

(1) : -.3f33065, it

follows

that

for potheses

of

Theorem 1 are satisfied'

arting

fi'om

ø¡ : 0,

^converges

to

^a

I'(r):0,

_',r'Ìrere

^F

^{is given}

^b)'

^(3?)

ae'iith

the results

in

_{[1], [2]' [5]'} [6]' [10], ^[11].

Remarlt2.Cond,i,tion(7)canbereplaced'bgastronger'hrt'teasiertocheck Newton-I{antorouich type iy7tothesi's

l4),

lynomr'als

p:'

b, ^ ":,'- ".'. , .,','

¹

ft bapt(t):ït' -

⁽¹

- c)ttct, f¡(t;) ;ad+t:

^and

bz

:

1

- ^c. lt

can eas'ily ^lte seen that

-the

^{"l nom}

ⁱⁿ

^the

u1"r^'Vt"r,iof,üdr,í5),

prouid'erl'

that

hllpothesis

2t4ct<(L-")'

holds, ^{ancl 14,}

rs

(r+

{r5)

^{are tlte} nonnegatiue ^zerr¡s of the equation ^p1(¿)

:

0'

S;;r;p

^(¿)

< pt"(i) for

^{atl ¿}

€

_{[0, r2f}

,

^{ue haue}

r'1(

r¿

(

rs

{ ^rz'

Remark

3.

^{The results} obtai,ned''in Theorern 2 can ^{be ertend'ed}

for

steJJensen- Ai,tken metltod

(33)

!)n,*r

: an-lBt(a,),8,(aù; ^{Fl-t F} (aò ^{(n2 0)'}

whe,e

Bt,Bz: D ^C E -+ E

^are continuous operators related

fo F _[4]'

[10]'

[11].

^Si'rnply

,"¡tt*" O(")-: _lr,B ^(t:); Fl ^{(r e D) to}

^obtairt'

^a

^Theorern

^I'

h,old,i'ng

for

method (33)' (d)

(30) lll*,rt;t3l-la,u; ^Blll (hzllr-ull 'forallr'v'ue

^D;

(") (31) o,nd

hs:

^holt'Jt',

^<

_4;1

(f)

U(ro,2ho)çD'

Then,

stefJer.tsen ^,metl¿od çteneratecÌ,

by (2)

i,s

well

defined, remains ^'in

U'(ro,2lt'L)for"oltrr>0,anrlcon'uergestoasolutionu*ofequat'i'onF'(r):g' Màràor"r,

ihe ¡ottowing error bounds hold

for

^all

n)'

^0:

ll'-: ""ll " #'

and

ll"- - ,,ll {

hTlrzllr"

- ^{*n,-tll ll"-} - ^B ('"-r)ll

' u:here

hT

>lll,n,R(,"); Fl-'ll _{t', >}

^ol

o"*n(;î:;ill¡r,y;Blll

(

^r¿¿

( r fu

^aII

^{r,v ee} u(ro,2hs),

^thert'

r*

^{'ís tÍ¿e}

u,ni,qui solution of ^equati,on

(1) i'nU (r¡,2hs)'

we

provide an example

to

show

that

rrnder the conditions of Theorem l, steflensen method ^converges to ^a soluiion

r*

ofequation (1)' whereas the same is

not

guaranteed under the conditions of Theorem 2'

Exampte. Let

E:

^{IR, IJ}

: [-1,1], ro : 0'

Define function

f'

on

D

by

(32) ^{F (r) :} _å,'* i* ^-

ancl the clivided difference

lr,y; ^Fllry

lr,a;

^þ'l

: ^F

^{(')^}

^- !-("), _{r,u e} D,, I

^u

u-'e

-r-l- 51 64

126 Ioannis K. Arsvros ₈

I

^{A Theorem for the} Steffensen Method L27

Remarlc

4.

Cond,i,ti,on (11r) can be replaced by

ll¿tt (r'

@)

- A("))ll _<

co

t

^ct

ll" - _"oll ^for

^{some co}

^{2 0, ct 2}

⁰

and

allU(rs,R).

We can also set

c: cl+qR.

Cond,ition (18) can be replaced, by

a { ^R,

but uniqueness

is

then guaranteed only

in U(rs,a).

Remarlc

5.

The results obtained ,in Theorem

1

can be ertend,ed, so as to hold a rnore general setti,ng as follows:

(a)

Let cs, c1 be nonnegatiue constantsi

ut,

u2 be posi,ti,ue monoton,ically increas'ing.functions

of

one uariable

on[0,R]

wi,th

]g3rr

^(ú)

:

]ïu2(t) :

^g

such that

lltl,

^@,,

^{(*) _} F,,(".))ll ( ,r

^(l1"

^_ roll)

,

and

ll¿tt (¡' ^(") - A("))ll _<

cs

*

c1u2|1"

- _"oll)

for

^all

r €U (ro,R).

(b) Functi,onp giuen by t

p(t): I

_,l

A-lu1(r)ar+]ut2-

⁽¹

- c0-cru2(ú))¿+

d,

on ll,Rl,

0

has a un'ique zero e¡

€

_[0,

R],

and

p(rt) <

^0.

Moreover, set

e¡:

¹¹ , and

R:12.Furthermore,

replace conditions (12), (Ia)

by (u)'

(7) by (b), and polynomial p by function p above. Then, under the rest of the hypotheses, as

it

can easily be seen from the proof, the conclusions of Theorem

t

hold in this more general setting. Call such a result Theorem 1/,.

Finally,

note

that for u1(t) : at,

co

: _ct

and u1(¿)

:

O

(or

c1

:

0),

¿

€

_{[0, rR]} function

p

reduces

to

polynomial

p

and

rheorem

1//

to

Theorem 1.

[6] ^CÀTINA$, E., On Some Steffensen-tgpe ,iterat,iue methods for a class of nonl'i,near equa- úions, Revue d'analyse numerique et de theorie de l'approximation, 24, nos. L-2, pp.37-43, 1995.

[7] ^GRAVES, L, M., Riemann integration and Taylor's theorem in general analysis, Trars.

Amer. Math. Soc., 29, pp. t63-L77,1927.

[8] KANTOROVICH, L. V. and AKILOV, G. P., Functional Analysis, Pergamon Press, Oxford, 1982.

[9] PÄVÄLOIrJ,L, Sur Ia méthode de Steffensen pour Ia résolut'ion d,es équations- opéra- tionnelles non I'inéaires, Rev. Roum. Math. Pure et Appl.,

XIII,

no. 6, pp. 857-861, 1968.

[10] PAVÄLOIU, I., Sur une généralisation d,e Ia m,ethod,e d,e Steffensen, Revue d'analyse numerique et de theorie de l'approximation, 21, no. 1, pp. 59-65, 1992.

[11] PÀ.VÀLOIU,

I.,

B,i,Iateral approrirnat,ions for ^{the solutions} of scalar equations, Ftevre d'analyse numerique et de theorie de l'approximation, 23, no. 1, pp. 95 100, 1994.

Received January 27, 2000 Cameron Un'iuersity

D epartrnent of M athematics

Lauton, OK 73505, U.S.A.

E -mail : ioannisa@cameron. edu

REFERENCES

[1] ^ARGYROS, I. K., On the solution of nonl'inear equo,tions wi,th a nonilifferentiable term, Revue d'analyse numerique et de theorie _de 1'approximation,22, no. 2, pp. l2b-18b, 1993.

[2] ARGYROS, I. K., An error analysis for the Steffensen method under generalized, Zabrejlco- Nguen-type assumptions, Revue d'analyse numerique et de theorie de L'approximatìon, 25, nos. 7-2, pp.t7-22, 1996.

[3] ^ARGYROS, I. K., Polynomial Operator Equat,ions

in

Abstract Spaces and, Applications, CRC Press LLC, Boca Raton, Florida, 1gg8.

[4] ^ARGYROS,

L

K. and sZIDARovszKY, F., The theory and Applications of lteration Method,s, C.R.C. Press, Boca Raton, Ftorida, 1gg3.

[5] BALAzs, M. ^{and GOLDNER,} G., on the approrimate solut'ion of equations in ^Hilbert

space by a Steffensen-type method,, Revue d'analyse numerique et de theorie de 1,ap- proximation, L7, no. 1, pp. 19-23, 1gB8