On the convergency of a Steffensen-type method

9 years ago

Abstract

Let $X_{1},X_{2}$ be two Banach spaces, $f:X_{1}\rightarrow X_{2}$ a nonlinear mapping. We study the convergence of the Steffensen method for solving $f\left( x\right) =0$: \[x_{n+1}=x_n-[x_{n},g(x_{n});f]^{-1}f(x_n), \quad n=1,…\] Under some simple Holder type conditions on the divided differences of order one of $f$, of the form \[\|[y, u; f] − [x, y; f]\| ≤ l_1 \|x − u\| ^p + l_2 \|x − y\|^p + l_3 \|y − u\|^p\] we give some error estimations and we determine the convergence order.

Authors

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)

Keywords

Steffensen type method; Holder conditions on divided differences; nonlinear equations in Banach spaces; iterative methods

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Cite this paper as:

I. Păvăloiu, On the convergency of a Steffensen-type method, Research Seminars, Seminar of Mathematical Analysis, Preprint no. 7 (1991), pp. 121-126.

About this paper

Journal

Seminar on mathematical analysis,
Preprint

Publisher Name

“Babes-Bolyai” University,
Faculty of Mathematics,
Research Seminars

DOI

Not available yet.

References

[1] Argyros, I.K., The secant method and fixed points of nonlinear operators, Mh. Math. 106, 85–94 (1988).

[2] Pavaloiu, I., Sur la methode de Steffensen pour la resolution des equations operationnelles non lineaires, Revue Roumaine des Mathematiques pures et appliquees, 1, XIII, 149–158 (1968).

[3] Pavaloiu, I., Introduction in the Theory of Approximation of Equations Solutions, Dacia Ed., Cluj-Napoca 1976 (in Romanian).

[4] Pavaloiu, I., Remarks on the secant method for the solution of nonlinear operatorial equations, Research Seminars, Seminar on Mathematical Analysis, Preprint no. 7, (1991), pp. 127 132.

[5] Ul’m, S., Ob obobscenie metod Steffensen dlea resenia nelineinih operatornih urnavnenii, Journal Vicisl., mat. i mat.-fiz. 4, 6 (1964).

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"BABEŞ-BOLYAI" UNIVERSITY

Faculty of Mathematics

Research Seminars

Seminar on Mathematical Analysis

Preprint nr.7, 1991, pp.121-126

On the Convergency of a Steffensen-type Method

Ion Păvăloiu

1. In the paper [2] I.K. Argyros adopts for the divided difference of the mapping $f : X_{1} \to X_{2},$ where $X_{1}$ and $X_{2}$ are Banach spaces, the following definition:

Definition 1.

One calls divided difference of the application $f$ at the points $x, y, \in X_{1}$ a linear application $[x, y; f] \in ℒ (X_{1}, X_{2})$ which fulfils the following conditions:

(a)

$[x, y; f] (y - x) = f (y) - f (x)$ for every $x, y \in D \subseteq X_{1};$
(b)

there exist the real constants $l_{1} > 0, l_{2} > 0,$ $l_{3} > 0, p \in (0, 1]$ such that for every $x, y, u \in D$ the following inequality holds:

$‖ [y, u; f] - [x, y; f] ‖ \leq l_{1} {‖ x - u ‖}^{p} + l_{2} {‖ x - y ‖}^{p} + l_{3} {‖ u - y ‖}^{p} .$

In [5] there are obtained refinements of Argyros results concerning the secant method applied to the solution of the equation:

(1)

f (x) = 0

where $f : X_{1} \to X_{2}$

2. We shall study further down the convergence of Steffensen’s method for the solution of equation (1), namely the convergence of the sequence ${(x_{n})}_{n \geq 0}$ generated by means of the following procedure:

(2)

x_{n + 1} = x_{n} - {[x_{n}, g (x_{n}); f]}^{- 1} f (x_{n}), x_{0} \in x_{1}, n = 0, 1, \dots,

where $g : X_{1} \to X_{1}$ is an operator having at least one fixed point which coincides with the solution of equation (1).

Obviously, the sequence ${(x_{n})}_{n \geq 0}$ can be generated by means of the procedure (2) if at each iteration step there exists the mapping ${[x_{n}, g (x_{n}); f]}^{- 1} .$

For our purpose observe firstly that the following identities:

(3)		$x_{n} - {[x_{n}, g (x_{n}); f]}^{- 1} f (x_{n}) =$
(4)		$= g (x_{n}) - {[x_{n}, g (x_{n}); f]}^{- 1} f (g (x_{n})) f (x_{n + 1})$
	$= f (g (x_{n})) + [x_{n}, g (x_{n}); f] (x_{n + 1} - g (x_{n}))$
	$+ ([g (x_{n}), x_{n + 1}; f] - [x_{n}, g (x_{n}); f]) (x_{n + 1} - g (x_{n}))$

hold for every $n = 0, 1, \dots$

Let $x_{0} \in X_{1}$ be an element, and consider the nonnegative real numbers: $B, ε_{0}, ρ_{0}, p \in (0, 1], α, β, q \geq 1, l_{1}, l_{2}$ and $l_{3}$ , where

ρ_{0} = β α (l_{1} B^{p} + l_{2} B^{p} + l_{3} B^{p} α^{p} {‖ f (x_{0}) ‖}^{p (q - 1)})

and

ε_{0} = ρ^{1 / (p + q - 1)} ‖ f (x_{0}) ‖ .

Denote $r = \max {B, β}$ and suppose that $S \subseteq D,$ where:

S = {x \in X_{1} : ‖ x - x_{0} ‖ \leq \frac{r ε_{0}}{ρ_{0}^{1 / (p + q - 1)} (1 - ε_{0}^{p + q - 1})}} .

The following theorem holds:

Theorem 1.

If the constants $B,$ $ε_{0},$ $ρ_{0}, p, α, β, q, l_{1}, l_{2}, l_{3},$ the mapping $f$ and $g$ , and the initial element $x_{0} \in X_{1}$ , as well, fulfil the conditions:

(I)

for every $x, y \in S$ there exists ${[x, y; f]}^{- 1}$ , and $‖ {[x, y; f]}^{- 1} ‖ \leq B$
(II)

for every $x \in S$ , $‖ f (g (x)) ‖ \leq α {‖ f (x) ‖}^{q};$
(III)

for every $x \in S, ‖ x - g (x) ‖ \leq β ‖ f (x) ‖;$
(IV)

the divided difference of the mapping $f$ fulfils the conditions (a) and (b) specified in the definition given in Section 1;
(V)

$ε_{0} < 1,$

then the sequence ${(x_{n})}_{n \geq 0}$ generated by the procedure (2) is convergent, and, if we denote $\bar{x} = lim x_{n}$ , then $f (\bar{x}) = 0$ and the following delimitation holds:

‖ x - x_{n} ‖ \leq \frac{r ρ_{0}^{{(p + q)}^{n}}}{ρ_{0}^{1 / (p + q - 1)} (1 - ε_{0}^{p + q - 1})} .

Proof.

Consider $x_{0} \in X_{1}$ for which the condition (V) is fulfilled. Taking into account the condition (b) and the procedure (2), from the identities (3) and (4) it results:

‖ x_{1} - x_{0} ‖ \leq B ‖ f (x_{0}) ‖ \leq \frac{B ε_{0}^{1 / (p + q - 1)}}{ρ_{0}^{1 / (p + q - 1)}} ‖ f (x_{0}) ‖ \leq \frac{r ε_{0}}{ρ_{0}^{1 / (p + q - 1)} (1 - ε_{0}^{p + q - 1})},

from which follows $x_{1} \in S$ .

Here was used the inequality:

‖ g (x_{0}) - x_{0} ‖ \leq β ‖ f (x_{0}) ‖ \leq \frac{r ε_{0}}{ρ_{0}^{1 / (p + q - 1)} (1 - ε_{0}^{p + q - 1})},

from which follows that $g (x_{0}) \in S$ .

Now, considering the above results, we have:

	$‖ f (x_{1}) ‖$	$\leq ‖ [g (x_{0}), x_{1}; f] - [x_{0}, g (x_{0}); f] ‖ \cdot ‖ x_{1} - g (x_{0}) ‖$
		$\leq β α [l_{1} B^{p} + l_{2} B^{p} + l_{3} B^{p} α^{p} {‖ f (x_{0}) ‖}^{p (q - 1)}] {‖ f (x_{0}) ‖}^{p + q}$
		$= ρ_{0} {‖ f (x_{0}) ‖}^{p + q}$

This inequality leads to:

ρ_{0}^{1 / (p + q - 1)} ‖ f (x_{1}) ‖ \leq ρ_{0}^{1 / (p + q - 1)} {‖ f (x_{0}) ‖}^{p + q}

or, using the notation $ε_{1} = ρ_{0}^{1 / (p + q - 1)} ‖ f (x_{1}) ‖$ :

ε_{1} \leq ε_{0}^{p + q}

From this inequality follows that $‖ f (x_{1}) ‖ \leq ‖ f (x_{0}) ‖,$ and if

ρ_{1} = β α (l_{1} B^{p} + l_{2} B^{p} + l_{3} B^{p} α^{p} {‖ f (x_{1}) ‖}^{p (q - 1)})

then $ρ_{1} \leq ρ_{0} .$

Suppose now that the following properties hold:

( $α$ )

$x_{p} \in S;$
( $β$ )

$‖ f (x_{p}) ‖ \leq ‖ f (x_{p - 1}) ‖;$
( $γ$ )

$ε_{p} \leq ε_{0}^{{(p + q)}^{p}}, ε_{p} = ρ_{0}^{1 / (p + q - 1)} ‖ f (x_{p}) ‖, p = 1, 2, \dots, k$

From (2) for $n = k$ we obtain:

‖ x_{k + 1} - x_{k} ‖ \leq B ‖ f (x_{k}) ‖ \leq \frac{r ε_{k}}{ρ_{0}^{1 / (p + q - 1)}} \leq \frac{r ε_{0}^{{(p + q)}^{k}}}{ρ_{0}^{1 / (p + q - 1)}},

which leads to:

	$‖ x_{k + 1} - x_{0} ‖$	$\leq \frac{r}{ρ_{0}^{1 / (p + q - 1)}} (ε_{0} + ε_{0}^{p + q} + ε_{0}^{{(p + q)}^{2}} + \dots + ε_{0}^{{(p + q)}^{k}})$
		$\leq \frac{r ε_{0}}{ρ_{0}^{1 / (p + q - 1)} (1 - ε_{0}^{p + q - 1})},$

namely $x_{k + 1} \in S$ .

Here was used the inequality:

‖ g (x_{k}) - x_{k} ‖ \leq B ‖ f (x_{k}) ‖ \leq \frac{r ρ_{0}^{1 / (p + q - 1)}}{ρ_{0}^{1 / (p + q - 1)}} ‖ f (x_{k}) ‖ \leq \frac{r ε_{0}^{{(p + q)}^{k}}}{ρ_{0}^{1 / (p + q - 1)}},

from which follows immediately:

‖ g (x_{k}) - x_{0} ‖ \leq \frac{r ε_{0}}{ρ_{0}^{1 / (p + q - 1)} (1 - ε_{0}^{p + q - 1})},

that is, $g (x_{k}) \in S$ .

As to $‖ f (x_{k + 1}) ‖$ we have:

‖ f (x_{k + 1}) ‖ \leq β α (l_{1} B^{p} + l_{2} B^{p} + l_{3} α^{p} B^{p} {‖ f (x_{k}) ‖}^{p (q - 1)}) {‖ f (x_{k}) ‖}^{p + q}

namely

‖ f (x_{k + 1}) ‖ \leq ρ_{0} {‖ f (x_{k}) ‖}^{p + q},

which yields:

ε_{k + 1} \leq ε_{k}^{p + q} \leq ε_{0}^{{(p + q)}^{k + 1}} .

By virtue of the above proved results follows that the properties ( $α$ )–( $γ$ ) hold for every $p \in ℕ$ .

We prove further down that the sequence ${(x_{n})}_{n \geq 0}$ is a fundamental sequence. Indeed, we have:

	$‖ x_{n + s} - x_{n} ‖$	$\leq ‖ x_{n + s} - x_{n + s - 1} ‖ + ‖ x_{n + s - 1} - x_{n + s - 2} ‖ + \dots + ‖ x_{n + 1} - x_{n} ‖$
		$\leq B (‖ f (x_{n}) ‖ + ‖ f (x_{n + 1}) ‖ + \dots + ‖ f (x_{n + s - 1}) ‖)$
		$\leq \frac{B}{ρ_{0}^{1 / (p + q - 1)}} (ε_{0}^{{(p + q)}^{n}} + ε_{0}^{{(p + q)}^{n + 1}} + \dots + ε_{0}^{{(p + q)}^{n + s - 1}})$
		$\leq \frac{B ε_{0}^{{(p + q)}^{n}}}{ρ_{0}^{1 / (p + q - 1)}} (1 + ε^{p + q - 1} + ε^{{(p + q)}^{2} - 1} + ε^{{(p + q)}^{s - 1} - 1})$
		$\leq \frac{B ε_{0}^{{(p + q)}^{n}}}{ρ_{0}^{1 / (p + q - 1)} (1 - ε_{0}^{p + q - 1})},$

that is, for every $s, n \in ℕ$ the following inequality holds:

‖ x_{n + s} - x_{n} ‖ \leq \frac{B ε_{0}^{{(p + q)}^{n}}}{ρ_{0}^{1 / (p + q - 1)} (1 - ε_{0}^{p + q - 1})},

from which, since $ε_{0} < 1$ , it results that the sequence ${(x_{n})}_{n \geq 0}$ is fundamental. Since $X_{1}$ is a Banach space, there exists $lim_{n \to \infty} x_{n} = \bar{x}$ ,

and

‖ \bar{x} - x_{n} ‖ \leq \frac{B ε_{0}^{{(p + q)}^{n}}}{ρ_{0}^{1 / (p + q - 1)} (1 - ε_{0}^{p + q - 1})},

which leads, for $n = 0,$ to $\bar{x} \in S$ .

From the inequality $ε_{n} \leq ε_{0}^{{(p + q)}^{n}},$ for $n \to \infty,$ we obtain:

f (\bar{x}) = lim_{n \to \infty} f (x_{n}) = 0,

and one sees that $\bar{x}$ is the solution of the equation (1). ∎

References

[1]
[2] Argyros, I.K., The secant method and fixed points of nonlinear operators, Mh. Math. 106, 85–94 (1988).
[3] ^†^†margin: clickable $\to$ Păvăloiu, I., Sur la méthode de Steffensen pour la résolution des équations operationnelles non linéaires, Revue Roumaine des Mathematiques pures et appliquees, 1, XIII, 149–158 (1968).
[4] Păvăloiu, I., ^†^†margin: clickable $\to$ Introduction in the Theory of Approximation of Equations Solutions, Dacia Ed., Cluj-Napoca 1976 (in Romanian).
[5] ^†^†margin: clickable $\to$ Păvăloiu, I., Remarks on the secant method for the solution of nonlinear operatorial equations, Research Seminars, Seminar on Mathematical Analysis, Preprint no. 7, (1991), pp. 127–132.
[6] Ul’m, S., Ob obobscenie metod Steffensen dlea resenia nelineinîh operatornîh urnavnenii, Journal Vîcisl., mat. i mat.-fiz. 4, 6 (1964).

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On the convergency of a Steffensen-type method

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On the Convergency of a Steffensen-type Method

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