We present some new conditions which assure that the Aitken-Steffensen method yields bilateral approximation for the solution of a nonlinear scalar equation. The auxiliary functions appearing in the method are constructed under the hypothesis that the nonlinear application is not differentiable on an interval containing the solution

Ion Păvăloiu

Directory of Open Access Journals

REVUE D’ANALYSE NUMÉRIQUE ET DE THÉORIE DE L’APPROXIMATIONRev. Anal. Numér. Théor. Approx., vol. 31 (2002) no. 2, pp. 195–198ictp.acad.ro/jnaatAITKEN-STEFFENSEN-TYPE METHODSFOR NONSMOOTH FUNCTIONS (II)∗ION PĂVĂLOIU†Abstract. We present some new conditions which assure that the Aitken-Stef-fensen method yields bilateral approximation for the solution of a nonlinear scalarequation. The auxiliary functions appearing in the method are constructed underthe hypothesis that the nonlinear application is not differentiable on an intervalcontaining the solution.MSC 2000. 65H05.Keywords. Aitken-Steffensen iterations.1. INTRODUCTIONIn this note we continue the study of the convergence of the Aitken-Steffen-sen-type iterations(1) xn+1 = g1 (xn)−f (g1 (xn))[g1 (xn) , g2 (xn) ; f ], n = 0, 1, . . . , x0 ∈ Ifor solving(2) f (x) = 0,where f : [a, b]→ R, a, b ∈ R, a < b.The functions g1 and g2 in (2) are chosen such that equationsx− gi (x) = 0, i = 1, 2,(3)to be equivalent to equation (1).Under supplementary assumptions we shall show, as in [7], that (2) ge-nerates three monotone sequences, converging to the solution x̄ of (1).Regarding the monotonicity and convexity of f we shall consider the notionsintroduced in [3]. We shall also use Theorem 1 and Lemma 2 from [8].For defining the functions g1 and g2, we shall consider α, β ∈ R such thata < α < β < b and f (α) < 0, f (β) > 0, defining then:g1 (x) = x− f(x)[β,b;f ] , x ∈ [α, β] ,(4)g2 (x) = x− f(x)[a,α;f ] , x ∈ [α, β] .(5)∗This work has been supported by the Romanian Academy under grant GAR 45/2002.†“T. Popoviciu” Institute of Numerical Analysis, P.O. Box 68–1, 3400 Cluj-Napoca, Ro-mania (pavaloiu@ictp.acad.ro).196 Ion Păvăloiu 2Regarding f we shall make the following assumptions.i. f (α) · f (β) < 0;ii. f is increasing on [a, b] ;iii. f is convex on [a, b] and continuous at a and b;iv. if x̄ ∈ (a, b) is the solution of (1), then f is differentiable at x̄.Remarks. 1◦ Hypothesiis iii. ensures the continuity of f on [a, b], andtherefore the existence of x̄. Hypothesis ii. ensures the uniqueness of x̄.2◦ From hypotheses ii, iii and [8, Lm. 1.1], it follows that for any u, v ∈(α, β) one obtains(6) [u, v; g1] > 0 and [u, v; g2] < 0,i.e., g1 is increasing and g2 is decreasing on (α, β). Let x0 ∈ (α, β) be such thata) f (x0) < 0b) g2 (x0) < β.2. THE CONVERGENCE OF THE AITKEN-STEFFENSEN-TYPE ITERATIONSWe shall study in the following the convergence of the sequences (xn)n≥0(g1 (xn))n≥0 to the solution x̄. We obtain the following result.Theorem 1. If the function f verifies assumptions i–iv., the functions g1and g2 are given by (4) and (5), and x0 ∈ (α, β) verifies a) and b), then thesequences (xn)n≥0 , (g1 (xn))n≥0 and (g2 (xn))n≥0, generated by (2) satisfy:j. (xn)n≥0 is increasing;jj. (g1 (xn))n≥0 is increasing;jjj. (g2 (xn))n≥0 is decreasing;jv. for all n ∈ N, one has(7) xn < g1 (xn) < xn+1 < x̄ < g2 (xn) .Proof. By [β, b; f ] > 0 and f (x0) < 0 it follows g1 (x0) > x0, while byx0 < x̄ and the fact that g1 is increasing it follows g1 (x0) < g1 (x̄) = x̄, i.e.g1 (x0) < x̄.Now, since x0 < x̄, g2 is decreasing one gets g2 (x0) > g2 (x̄) = x̄, i.e. thefollowing relations hold:(8) x0 < g1 (x0) < x̄ < g2 (x0) .Let x1 = g1 (x0)− f(g1(x0))[g1(x0),g2(x0);f ] , since f (g1 (x0)) < 0 implies g1 (x0) < x1.From the identity3 Aitken-Steffensen-type methods for nonsmooth functions 197f (x1) =f (g1 (x0)) + [g1 (x0) , g2 (x0) ; f ] (x1 − g1 (x0)) ++ [x1, g1 (x0) , g2 (x0) ; f ] (x1 − g1 (x0)) (x1 − g2 (x0))taking into account the convexity of f and relation (1), we get f (x1) < 0, i.e.x1 < x̄.By the above relations and by (8) it followsx0 < g1 (x0) < x1 < x̄ < g2 (x0) ,which shows that (7) is verified for n = 0.Repeating this reason, the induction shows that (7) holds for n ∈ N. Thisattracts in turn that the sequences (xn)n≥0 and (g1 (xn))n≥0 are increasing,i.e., statements j and jj.We show next that (g2 (xn))n≥0 is decreasing. Indeed, by xn < xn+1 forn ∈ N we get g2 (xn) > g2 (xn+1) since g2 is decreasing. Inequalities xn < x̄,n ∈ N, show that g2 (xn) > g2 (x̄) = x̄.Let us notice that the sequences (xn)n≥0, (g1 (xn))n≥0, (g2 (xn))n≥0 aremonotone and bounded, so they converge. Let l1 = lim xn, l2 = lim g1 (xn)and l3 = lim g2 (xn). We show that l1 = l2 = l3 = x̄.We prove first that l1 = l2. Assume the contrary, l1 6= l2, e.g. l1 < l2.Obviously, l1 = supn∈N {xn} and l2 = supn∈N {g1 (xn)} . Let 0 < ε < l2 − l1be a positive number. Then there exists nε ∈ N such that g1 (xn) > l2 − ε forn > nε.This implies thatxn+1 > g1 (xn) > l2 − ε > l1so l1 is not the exact upper bound of the elements of the sequence (xn)n≥0.Hence, clearly, l1 = l2 = l,l = lim xn = lim g1 (xn) = g1 (l) ,i.e., l = x̄. Since lim xn = x̄, it follows thatlim g2 (xn) = g2 (x̄) = x̄,since x̄ is the unique solution of equation x− g2 (x) = 0. The above relations show that we have a control of the error at each iterationstep, justified byx̄− xn+1 < g2 (xn)− xn+1, or x̄− xn+1 < g2 (xn)− g1 (xn) , n = 0, 1, . . .The identity0 = f (x̄) =f (g1 (xn)) + [g1 (xn) , g2 (xn) ; f ] (x̄− g1 (xn)) ++ [x̄, g1 (xn) , g2 (xn) ; f ] (x̄− g1 (xn)) (x̄− g2 (xn))relation (7), and the hypotheses of the above theorem lead to(9) x̄− xn+1 = −[x̄, g1 (xn) , g2 (xn) ; f ][g1 (xn) , g2 (xn) ; f ](x̄− g1 (xn)) (x̄− g2 (xn)) .198 Ion Păvăloiu 4Further, by Lemma 2 from [8] we getx̄− g1 (xn) = [xn, x̄; g1] (x̄− xn)(10)x̄− g2 (xn) = [xn, x̄, g2] (x̄− xn)(11)and, taking into account (4) and (5),[xn, x̄; g1] = 1− [xn,x̄;f ][β,b;f ] < 1−[a,α;f ][β,b;f ] = 1− q < 1.Analogously,[x̄, xn; g2] = 1− [x̄,xn;f ][a,α;f ] > 1−[β,b;f ][a,α,;] =[β,b;f ][a,β;f ]([a,α;f ][β,b;f ] − 1)= 1q (q − 1)where we have denoted [a, α; f ] / [β, b; f ] = q > 0 and by Lemma 2 from [8]q < 1. This relation, together with the decreasing of g2 lead to − [x̄, xn; g2] <1q (1− q) , i.e.,∣∣[x̄, xn; g2]∣∣ < 1q (1− q) .Denoting M = maxu,v∈[α.β]∣∣ [x̄, u, v; f ] ∣∣ and m = [a, α; f ], by (9) we get|x̄− xn+1| < M(1−q)2mq |x̄− xn|2 , n = 1, 2, . . .which characterizes the convergence order of the studied methods.REFERENCES[1] Balázs, M., A bilateral approximating method for finding the real roots of real equa-tions, Rev. Anal. Numér. Théor. Approx., 21, no. 2, pp. 111–117, 1992.[2] Casulli, V. and Trigiante, D., The convergence order for iterative multipoint proce-dures, Calcolo, 13, no. 1, pp. 25–44, 1977.[3] Cobzaş, S., Mathematical Analysis, Presa Universitară Clujeană, Cluj-Napoca, 1997(in Romanian).[4] Ostrowski, A. M., Solution of Equations and Systems of Equations, Academic Press,New York, 1960.[5] Păvăloiu, I., On the monotonicity of the sequences of approximations obtained bySteffensens’s method, Mathematica (Cluj), 35 (58), no. 1, pp. 71–76, 1993.[6] Păvăloiu, I., Bilateral approximations for the solutions of scalar equations, Rev. Anal.Numér. Théor. Approx., 23, no. 1, pp. 95–100, 1994.[7] Păvăloiu, I., Approximation of the roots of equations by Aitken-Steffensen-type mono-tonic sequences, Calcolo, 32, no. 1–2, pp. 69–82, 1995.[8] Păvăloiu, I., Aitken-Steffensen-type methods for nonsmooth functions (I), Rev. Anal.Numér. Théor. Approx., 31, no. 1, pp. 111–116, 2002.[9] Traub, F. J., Iterative Methods for the Solution of Equations, Prentice-Hall, Inc.,Englewood Cliffs, N.J., 1964.Received by the editors: January 16, 2002.

Aitken-Steffensen-type methods for nonsmooth functions (II)

https://www.ictp.acad.ro/jnaat/journal/article/download/2002-vol31-no2-art10/Pavaloiu-pdf

Aitken-Steffensen-type methods for nonsmooth functions (II)

Abstract

Similar works

Full text

Available Versions

Directory of Open Access Journals