Ostrowski-Traub nonlinear iterative methods are not optimal in the sense of s-order of convergence

Appl. Math. Lett. Vol. 5, No. 3, pp. 51-52, 1992 Printed in Great Britain. All rights reserved

CR393-9659192 $5.00 + 0.00

Copyright@ 1992 Pergamon Press Ltd

OSTROWSKI-TRAUB

NONLINEAR ITERATIVE METHODS ARE NOT OPTIMAL SENSE OF s-ORDER OF CONVERGENCE

IN THE

DONG

CHEN

Department of Mathematical Sciences, University of Arkansas Fayetteville, Arkansas 72701, U.S.A.

(Received

November

1991)

Abstract-We provide an iterative method which is of S-order 5, but N-order 4. We also give a numerical example to illustrate the theoretical conclusions.

1. INTRODUCTION Ostrowski-Traub iterative method with order 4 is a well-known multipoint iteration for finding the solution of nonlinear equations [1,2]. King introduced a parameter Q to generalize it to a family of fourth-order methods [3]. We denote King’s family of iterative methods as Q(X, o). In this short report we try to determine the parameter o so that the sequence {X,} generated by \E(X, o) is the best iteration in the following sense:

U(X, a*) = sup {p(o) : for all a over R & p = S-order 2. S-ORDER

OF CONVERGENCE

FOR

of *(X,(Y)}

OSTROWSKI-TRAUB

METHOD

We apply the original definition [4,5] of S-order of convergence which was introduced by the author to the Ostrowski-Traub method and the King family. Then we can not only find the order of convergence without assuming the existence of the solution X* but also get more order in the sense of S-order than that of R-order. (i) Ostrowski-Traub

Iteration.

yn= x, - *; The Ostrowski-Kantorovich q&+1)

J

=

xn+l

n

representation

)

Y, - 2P($_xa(x

=

n

n

P(Yn).

[1,6,7] is given in [8] as follows:

1 .“(K

+ q&+1

Y,))(l -

-

t) q&a+1

-

KY

0

-

1

Wta) .“‘(& 2P(Y,)- P(Xn>J 0

+ i(Yn - Xn)) t(1 - t)

&(Y,

- X,)?

Thus, s(h+1)

Hence, ps = PQ =p~

=

;

=

CO-T(tn, %)(Sn - Q4.

=4

(h+1

d2

=

$

[K(sn

_

t ; + g’(tn)]2 n

(Sn -

Q4

and

CO-j+*) (ii) Fifth-order

-

Ostrowski-Traub

= nlimwCO-T(h, (King family

sn) =

K3p2 8(1-

with a = -0.5)

2h) ’ iteration

[3]. Typeset by d.@-T)$

51

52

D. CHEN

The Ostrowski-Kantorovich

representation

J P”(% +

[8,9] is

1

P(Xn+l) =

0

J

2P(yn) +

P(L)

5P(Y,)

P(Y,)2

1 +

-

- xl))

~(&a+1

J;

- Y,)2

1

P”‘(%

0

P”(X,

5 P/(X,)

(1 - t) dt(X,+1 +

+ t(Yn

-

2P(X,)

t(Y,

Xn))

-

Xn))

di(Y,

(1 - t) &(Y, -

- X,)3

X,)

.

- 5P(Yn)

so CF_O-T(t*)

=

s(tn+1>

lim n--roo (Sn -

t,)4

;K3 =

ii&

Jg’(t*)2

-

=

O*

Hence, ps 2 5. But pg = pR = 4. CONJECTURE.

s(tn+d

,,!t.t.(Sn where,

y is a nonzero

=

=

y(l

“$2

’

real number. 3. THE

We take P(X) X* = 2.094551481.

_ t,)5

X3

Then

-

NUMERICAL

EXAMPLE

2X - 5, X0 = 2.0, Eo = 0.94 x 10-l, we have the following numerical results:

El(a)

= (1X1(0) - X*11 and

Table 1.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.

A.M. O&row&i, Solution of Equations and Systems of Equations, Third Edition, Academic Press, New York, (1973). J.F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, Englewood Cliffs, N.J., (1964). R.K. King, A family of fourth-order methods for nonlinear equations, SIAM J. Namer. Anal. 10, 876-879 (1973). D. Chen, On a new definition of order of convergence in general iterative methods I: One point iterations, (Preprint), (1991). D. Chen, On a new definition of order of convergence in general iterative methods II: Multipoint iterations, (Preprint), (1991). L.V. Kantorovich and G.P. Akilov, Functional Analysis in Normed Spaces, Oxford: Pergamon Press, (1964). J.M. Ortega and W.C. Rheinboldt, Iterative SolzLtion of Nonlinear Equations in Several Vakbles, Academic Press, (1970). D. Chen, On the convergence and optimal error estimates of King’s iteration procedures for solving nonlinear equations, Intern. J. Computer Math. 26, 229-237 (1989). D. Chen, Kantorovich-type theorem and optimal estimates for a family of iterative procedures, Computer Engineering d Designs 4, 55-59 (1983).