Reduced Recurrence Relations for the Chebyshev Method

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 98, No. 2, pp. 385-397, AUGUST 1998

Reduced Recurrence Relations for theChebyshev Method1

M. A. HERNANDEZ2

Communicated by I. Galligani

Abstract. In this paper, we give sufficient conditions ensuring the con-vergence of the Chebyshev method in Banach spaces. We use a newsystem of recurrence relations which simplifies those given by Kantorov-ich for the Newton method or those given by Candela and Marquinafor the Chebyshev and Halley methods.

Key Words. Chebyshev method, nonlinear equations in Banach spaces,third-order methods, recurrence relations.

1. Introduction

The study of third-order iterative processes has become more and moreimportant in recent years. Among these methods, the most famous are theChebyshev, Halley, and super-Halley methods. Many papers on third-ordermethods have been published: for the Chebyshev method, Refs. 1-2; forthe Halley method, Refs. 3-5; for the super-Halley method, Refs. 6-7. Mostauthors follow the technique developed by Kantorovich for the Newtonmethod (Refs. 8-9). In Ref. 10, a unified theory is presented for all theabove third-order methods. To sum up, in these papers the convergence ofthe sequence in Banach spaces follows from the convergence of a majorizingsequence, which is obtained by applying the real third-order method to acubic polynomial.

Another way to get convergence is by using recurrence relations. Thistechnique has been also used by Kantorovich and other authors for theNewton method (Refs. 11-13). In the same way, Candela and Marquina

1This work was supported in part by the University of La Rioja and Grants No. ULR-96PYB25MHV, No. ULR-API-97/B12, and No. ULR-API-98/B12..

2Professor, Department of Mathematics and Computation, University of La Rioja, Logrono,Spain.

3850022-3239/98/0800-0385$15.00/0 C 1998 Plenum Publishing Corporation

(Refs. 2,4) establish the convergence for the Halley and Chebyshev methods,respectively. Starting with two real parameters, they construct a system ofrecurrence relations consisting of four real sequences of positive real num-bers, yielding an increasing convergent sequence that majorizes the sequencein Banach spaces. The use of these relations affords some advantages,because we can reduce our initial problem in a Banach space to a simplerproblem with real sequences and real functions.

The aims of this paper are threefold. First, we simplify the well-knownrecurrence relations for convergence analysis of iterative processes in Banachspaces. Moreover, the error estimates obtained are not worse than others(Refs. 2, 14). For this, we construct a new type of recurrence relations forthe Chebyshev method consisting of two real sequences of positive realnumbers. We use them in order to prove its convergence under Kantorovich-type assumptions. In addition, we obtain a priori error bounds with thesame order as those obtained by Candela and Marquina (Refs. 2,4).

Then, we increase the cubic decreasing region obtained by other authors(Refs. 2,4), and then the region of accessibility (Ref. 14) is also increased.Finally, one of the most important advantages of this new system of recur-rence relations consists in relaxing the convergence conditions. So, conver-gence of the Chebyshev method is proved under a condition on the uniformboundedness of the second derivative of operator. In this manner, the con-vergence conditions considered by other authors (Refs. 1,10) are relaxed.Numerical examples are also given to illustrate the results.

2. Recurrence Relations

Let X, Y be Banach spaces, and let F: O<X -» Y be a nonlinear twiceFrechet differentiable operator in an open convex domain Q0 £ O. TheChebyshev method for solving the equation F(x) = 0 is given by

386 JOTA: VOL. 98, NO. 2, AUGUST 1998

where LF(x) is the linear operator defined as follows:

Let us assume that

exists at some x0 e O0 , where £(Y, X) is the set of bounded linear operatorsfrom Y into X. We assume in Sections 2 and 3 that:

JOTA: VOL. 98, NO. 2, AUGUST 1998 387

Let us denote

and define the sequences

where

Notice that

In this situation, we prove the following statements for n > 1:

Assuming that

we have

Then, F1 is defined and


On the other hand, we can write (1) in the form

and we obtain from the Taylor formula

Then, we have

for m = 0, and (II) is true for n= 1. To prove (III) and (IV) for n= 1, noticethat

Finally, we deduce easily that

Now, following an inductive procedure and assuming that

items (I)-(V) are proved for n arbitrary.So, we must analyze the real sequences {an} and {bn} to study the

sequence {xn} defined in a Banach space. To establish the convergence of{xn}, we have only to prove that this is a Cauchy sequence and the aboveassumption (3).


3. Convergence Study

In this section, we study the sequences {an} and {bn}, denned in Section2, to prove the convergence of the sequence {xn} given by (1). First of all,we give a technical lemma including results concerning functions of one andtwo variable that we need next.

Lemma 3.1. Under the previous notations, we have that f(x) is increas-ing and f(x) > 1 for x e(0,0.5). For a fixed xe(0,0.5), g(x,y) increases asa function of y, and for a fixed y>0, g(x, y) increases in (0,0.5). Moreover,if ye(0,1), then g(yx, r2y)<Y2g(x,y), for xe[0,0.5] and y > 0 .

Lemma 3.2. Let 0<a0<0.5, f(a0)2g(a0,b0)<1. Then, the sequences{an} and {b0} are decreasing.

Proof. From the hypothesis, we deduce that

since f(x) > 1 in (0,0.5). Now, we suppose that

Then,

Notice that

Consequently,

Now, the fact of proving bk+1 <bk is equivalent to proving that

Taking into account that bk+1 >0 and following the previous reasoning, theresult also holds. D

In the following lemma, whose proof is obvious, we give sufficient condi-tions so that the real sequences {an} and {bn} are decreasing.

Lemma 3.3. If 0 < a0 < 0.5 and b0 < (3/4)( 1 - 2a0)(8 -4a0 - a3), thenf(a0)2g(a0,b0)<1.


Let us prove that (3) holds. Under the hypotheses of the previouslemma, we have that

and this inequality is true since a0e(0,0.5). Next, we prove that(1 +an/2)\\TnF(xn)\\ is a Cauchy sequence. So, we note that

Next, we analyze the factor H n - 1 f (a k )g(a k , bk) by means of the followinglemma.

Lemma 3.4. Let us suppose that the hypotheses of Lemma 3.3 aresatisfied and define y=a1/a0 . Then:

Proof. Notice that (i) is trivial. Next, we prove (ii) following an induc-tive procedure. So, for n= 1.

if and only if f(a0)>1, and by Lemma 3.1 the result holds. If we supposethat (ii) for « arbitrary is true, then for n+1,

On the other hand, we have

if and only if

and this is true since f(an) > 1. Now, as


(ii) also holds. Moreover,

Finally, we observe that

and the proof is complete.

As a consequence of this, if we denote A= 1 / f (a 0 ) , it follows that

So, from A<1, we deduce that Hn-1f(ak)g(ak,bk) converges to zero byletting n->00.

Now, we are ready to state the following result on convergence for (1).

Theorem 3.1. Let X, Y be Banach spaces, and let F: Q c X-> Y be anonlinear twice Frechet differentiable operator in an open convex domainQ0sQ. Let us assume that

exists at some x0eQ0 and that conditions (i)-(iv) in Section 2 are satisfied.Let us denote a0=MBn and b0 = KBn2. Suppose that 0 < a0<0.5 and b0<(3/4)(1 -2a0)(8-4a0-a0). Then, if

where

the sequence {xn} denned in (1) and starting at x0 converges cubically to asolution x* of the equation F(x) = 0. In that case, the solution x* and theiterates xn belong to B(x0, Rn), and x* is the only solution of F(x) = 0 inB(x0 ,2/MB-Rn)^ O0.

Furthermore, we can give the following error estimate:

Proof. When 0 < a0 < 0.5, the convergence of the sequence {xn} followsimmediately from the previous lemmas. On the other hand, we consider


So, we obtain

for n = 0, and (4) by letting P->oo in (5).To prove that F(x*) = 0, notice that ||rnF(xn)|| ->0 by letting n-> oo.

As

and {||F'(xn)||} is a bounded sequence, we deduce that ||F(xn)|| -+0, andthen F(x*) = 0 by the continuity of F.

Now, to show unicity, suppose that

is another solution of F(x) - 0. Then,

Using the estimate


we have that the operator

has an inverse, and consequently, y* = x*.Finally, it is known (Ref. 2) that the Chebyshev method is of order

three, and the proof of the theorem is concluded. D

Next, we give two examples to illustrate the previous results. We usetwo functions quoted as a test in several papers (Refs..2,14).

Example 3.1. Let us consider F(x) =x3—10, x0 = 2, and denote by x*the positive root of F(x) = 0. We give an upper bound C to the number1011||x*-x2||, where x2 is the second iterate of (1). Starting in [1, 3], wehave that

Consequently,

Making the same decomposition as Candela and Marquina in Ref. 2, andcalculating the smallest value of n so that ||x*-x2|| is of order 10-11, weconsider

and obtain C= 130932. For the same function and iterative method, Candelaand Marquina obtained C= 142360.973 (Ref. 2).

Example 3.2. Let us consider the operator F: C[0, 1] ->• C[0,1] definedby

where C[0,1] is the space of all continuous functions defined on the interval[0,1], with the sup norm || • || = || • \\m ; see Ref. 14 for details.

If we choose x0 = x0(s) = s, then we obtain the upper bound C=17013500. Notice that, for the Halley method, Doring obtained C=82500000 (Ref. 14).

The error bounds that we have obtained for the Chebyshev method areof the same order as those obtained by Candela and Marquina and arebetter than the ones obtained by Doring.


Fig. 1. Cubic decreasing regions.

To complete this convergence study of the Chebyshev method, we seethat our second aim of the introduction is reached. The cubic decreasingregions (see Refs. 2,4 for definition) of the Chebyshev method are represen-ted in Fig. 1, where a0 and b0 are the coordinates. The dotted line representsthe curve

obtained by Candela and Marquina, and the solid line represents our curve

In consequence, our cubic decreasing region is bigger; therefore the regionof accessibility for the Chebyshev method has been increased.

4. Mild Convergence Conditions

Until now, necessary conditions for the convergence of (1) have beenestablished assuming that the second Frechet derivative of F satisfies aLipschitz condition,

for x and y in a suitable region of X; see Refs. 1, 10 for more information.The technique developed there is an extension of the technique followedby Kantorovich and other authors (Refs. 12,13) for studying the Newtonmethod.


Recently, Smale (Ref. 16) obtained the convergence of the Newtonmethod for analytic maps from data at one point, instead of the regionconditions (6) in the Newton-Kantorovich theorem.

Smale-like theorems for the convergence of iterative processes assumethat the following inequalities are satisfied at a point x0 :

The constant h is different for different processes (Refs. 17,18).Our goal in this section is to prove the convergence of (1) just assuming

that Fn is bounded and a punctual condition. We also show an examplewhere our conditions are fulfilled and the previous ones fail.

Theorem 4.1. Let X, Y be Banach spaces, and let F: QsX-* Y be anonlinear twice Frechet differentiable operator in an open convex domainQ0^Q. Let x0eQ0 be a point where the operator r0 = F'(x0)

-1 exists. Sup-pose that ||F0|| <B, \\r0F(x0) || < n, \\Fn(x)\\ <M, for all xeQ0. Define an+1 =anf(an)

2g(an), with a0=MBn, f(x) = 2/(2 - 2x - x2), g(x) = x(8 + 4x + x2)/8. Denote R = (1 +a0/2)/(l - A), with A=g(a0)

1/2 and

where r is the smallest positive root of the polynomial p(x) =2x4 + 7x3-4x2-24x + 8. If S(x0, Rn) is contained in O0, then the Cheby-shev sequence (1) is well defined, is contained in B(x0, Rn), and convergesto x*, a solution of F(x) = 0. Furthermore, x* is the unique solution inB(x 0 , 2 /Mp-Rn)^n 0 .

Proof. Notice that the hypothesis guarantees the existence of x1 in (1)and

It can be shown without difficulty that F1 F'(x0) exists and

Then, x2 is defined; taking into account (1), we have

by the Taylor formula. So,


On the other hand,

Finally, from (8) and f(a0)2g(a0) < 1, we have

Applying an inductive procedure, we can replace x1 by x2, then x2 by x3,and in general xn-1 by xn so as to obtain that there exists Yn and the followingrecurrence relations:

Consequently, taking into account that g(a0) < 1 and following a similar wayas in Lemma 3.3, we infer that {xn} is a Cauchy sequence and thereforeconverges to x*eB(x0, Rn). Finally, F(x*) = 0, and unicity follows as inTheorem 3.5. D

Next, we give an example where the conditions of Theorem 4.1 aresatisfied, while conditions (6) and (7) fail.

Example 4.1. Consider the function

defined in X=[-\, 1], and let x0 = 0. Notice that the derivatives h (k)(x) arenot defined in x0 for k>3. Then, the Smale-like condition (7) does notwork in this case. Moreover, hn does not satisfy the Lipschitz condition (6).Therefore, the Kantorovich-type conditions also fail. However,

and a0 = 0.3 < 0.326664 . . . . Therefore, (8) is satisfied and Theorem 4.1 canbe applied.

References

1. ARGYROS, I. K., and CHEN, D., Results on the Chebyshev Method in BanachSpaces, Proyecciones, Vol. 12, pp. 119-128, 1993.

2. CANDELA, V., and MARQUINA, A., Recurrence Relations for Rational CubicMethods. II: The Chebyshev Method, Computing, Vol. 45, pp. 355-367, 1990.

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4. CANDELA, V., and MARQUINA, A., Recurrence Relations for Rational CubicMethods, I: The Halley Method, Computing, Vol. 44, pp. 169-184, 1990.

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Reduced Recurrence Relations for the Chebyshev Method

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