Remarks on the secant method for the solution of nonlinear operator equations

9 years ago

Abstract

Let $X_{1},X_{2}$ be two Banach spaces and $f:X_{1}\rightarrow X_{2}$ a nonlinear equation. We study the chord method for solving the equation $f\left( x\right) =0$. Assuming the first order divided differences of $f$ satisfy a Holder type condition, we obtain sufficient convergence conditions and error estimations at each step.

Authors

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)

Keywords

chord method; divided differences; Holder condition

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Cite this paper as:

I. Păvăloiu, 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.

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] Dennis, J.E., Toward a unified convergence theory for Newton like methods, Nonlinear Functional analysis and Applications (Ed. by L.B. Rall), pp. 425–472, New York, John Wiley (1986).

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

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"Babeş-Bolyai" University

Faculty of Mathematics and Physics

Research Seminars

Seminar on Mathematical Analysis

Preprint Nr.7, 1991, pp.127-132

Remarks on the secant method for the solution of nonlinear operatorial equations

Ion Păvăloiu

This note has for purpose some refinements of the convergence conditions and error delimitations obtained by I.K. Argyros in [2] with respect to the secant method for the solution of the equation:

(1)

f (x) = 0,

where $f : X_{1} \to X_{2}$ is a nonlinear operator, while $X_{1}$ and $X_{2}$ are Banach spaces.

If we denote by $[x, y; f]$ the divided difference of the mapping $f$ on the point $x$ and $y$ , then for fixed $x, y$ we have $[x, y; f] \in ℒ (X_{1}, X_{2}) .$ It is known that in certain conditions the sequence ${(x_{n})}_{n \geq 0}$ generated by the secant method:

(2)

x_{n + 1} = x_{n} - {[x_{n - 1}, x_{n}; f]}^{- 1} f (x_{n}), x_{0}, x_{1} \in X_{1}, n = 1, 2, \dots

converges to the solution $x^{*}$ of equation (1).

1. Generalizing a result on J.E. Dennis [3], I.K. Argyros [2] studies the convergence of the method (2) with the assumptions that the operator $f$ is Fréchet differentiable, while the derivative $f^{'} (x)$ fulfils a Hölder-like condition on a set $D \subset X_{1},$ namely there exist a constant $C > 0$ and number $p \in (0, 1]$ such that the inequality:

(3)

‖ f^{'} (x) - f^{'} (y) ‖ \leq C {‖ x - y ‖}^{p}

holds for every $x, y \in D$ . In this case we shall say that $f^{'} (\cdot) \in H_{D} (C, p) .$

In the quoted paper I.K. Argyros defines the divided difference operator $[x, y; f]$ as a linear operator which fulfils the conditions:

(4)

[x, y; f] (y - x) = f (y) - f (x), \forall x, y \in D,

and, in addition, for every $x, y, u \in D$ the following inequality holds:

(5)

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

where $l_{1} \geq 0, l_{2} \geq 0$ are constants which do not depend on $x, y$ and $u,$ while $p \in (0, 1] .$

Let $x^{*}$ be a simple solution of (1). We mean by that the mapping $f^{'} (x^{*})$ admits a bounded inverse mapping, and if $[x^{*}, x^{*}; f] = f^{'} (x^{*})$ then $[x^{*}, x^{*}; f]$ admits a bounded inverse mapping. Thus the continuity of the mapping $[x, y; f]$ with respect to the variable $x$ and $y$ ensures the existence of a number $ε > 0$ such that the mapping $[x, y; f]$ admits a bounded inverse mapping for every $x, y \in U (x^{*}, ε),$ where $U (x^{*}, ε) = {x \in X_{1} : ‖ x - x^{*} ‖ < ε}$ that is, the set $B (x, y) = ‖ {[x, y; f]}^{- 1} ‖$ is uniformly bounded in $U (x^{*}, ε) = {x \in X_{1} : ‖ x - x^{*} ‖ \leq ε}$ .

Theorem 1.

[2] Let $f : X_{1} \to X_{2}$ and let $D \subset X_{1}$ an open set. The following conditions are fulfilled:

(a)

$x^{*} \in D$ is a simple solution of the equation (1);
(b)

there exist $ε \in 0, b > 0$ such that $‖ {[x, y; f]}^{- 1} ‖ \leq b$ for every $x, y \in U (x^{*}, ε);$
(c)

there exists a convex set $D_{0} \subset D$ such that $x^{*} \in D_{0},$ and there exists $ε_{1} > 0,$ with $0 < ε_{1} < ε$ such that $f^{'} (\cdot) \in H_{D_{0}} (C, p)$ for every $x, y \in D_{0}$ and $U (x^{*}, ε_{1}) \subset D_{0}$ .

Let $r > 0$ such that:

(6)

0 < r < \min {ε_{1}, {(q (p))}^{- 1 / p}}

where:

(7)

q (p) = \frac{b}{p + 1} [2^{p} (l_{1} + l_{2}) (1 + p) + C] .

Then, if $x_{0} x_{1} \in \bar{U} (x^{*}, r),$ the iterates $x_{n}, n = 2, 3, \dots,$ generated by (2) are well defined and belong to the set $\bar{U} (x^{*}, r),$ while the sequence ${(x_{n})}_{n \geq 0}$ converges to the unique solution $x^{*}$ of equation (1).

Moreover, the following estimation:

(8)

‖ x_{n + 1} - x^{*} ‖ \leq γ_{1} {‖ x_{n - 1} - x^{*} ‖}^{p} \cdot ‖ x_{n} - x^{*} ‖ + γ_{2} {‖ x_{n} - x^{*} ‖}^{p + 1}

holds for sufficiently great $n$ , where:

(9)

γ_{1} = b (l_{1} + l_{2}) 2^{p},

(10)

γ_{2} = \frac{b C}{1 + p}

while $l_{1}, l_{2}$ and $p$ were precised by the relation (5).

In order to prove this theorem the author uses the following two lemmas:

Lemma 1.

[2]. Let $f : X_{1} \to X_{2}$ and $D \subset X_{1}$ . Suppose that $D$ is an open set and $f^{'} (\cdot)$ does exist in every point of $D$ . If, for a convex set $D_{0} \subseteq D, f^{'} (\cdot) \in H_{D_{0}} (C, p),$ then for every $x, y \in D_{0}$ the following inequality holds:

‖ f (x) - f (y) - f^{'} (x) (y - x) ‖ \leq \frac{C}{1 + p} {‖ x - y ‖}^{1 + p} .

Lemma 2.

[2]. If $[x, y; f]$ fulfils the conditions (4) and (5), the following relations hold:

(a)

$[x, x; f] = f^{'} (x)$ for every $x \in D_{0};$
(b)

$f^{'} (\cdot) \in H_{D_{0}} (2 (l_{1} + l_{2}), p) .$

From the proof of Theorem 1 follows, for the error estimation and for the convergence speeds of the sequence ${(x_{n})}_{n \geq 0},$ the inequality:

(11)

‖ x_{n + 1} - x^{*} ‖ \leq {(M (r))}^{n + 1} ‖ x_{0} - x^{*} ‖

where one shows that $M (r) \in (0, 1) .$

2. We shall make further down some remarks upon the above exposed results, showing that the hypotheses imposed in [2] can lead to more rich conclusions with respect to both the convergency order of the secant method and the error estimation.

Suppose that $x_{0}$ and $x_{1}$ fulfil the conditions:

(a’)

$‖ x^{*} - x_{0} ‖ \leq α d_{0};$
(b’)

$‖ x^{*} - x_{1} ‖ \leq \min {α d_{0}^{t_{1}}, ‖ x^{*} - x_{0} ‖}$

where $0 < d_{0} < 1, α = {(q (p))}^{- 1},$ while $t_{1}$ is the positive root of the equation:

(12)		$t^{2} - t - p$	$= 0$
	$namely t_{1}$	$= \frac{1 + {(1 + 4 p)}^{1 / 2}}{2} .$

Using the condition (4) and (5), Lemmas 1 and 2, and the hypotheses of 1, it results easily from (2),for $n = 1$ , the inequality [2]:

(13)

‖ x_{2} - x^{*} ‖ \leq γ_{1} {‖ x_{0} - x^{*} ‖}^{p} ‖ x_{1} - x^{*} ‖ + γ_{2} {‖ x_{1} - x^{*} ‖}^{p + 1}

from which, using (a’) and (b’) and the fact that $t_{1}$ is a root of equation (12), we obtain:

	$‖ x_{2} - x^{*} ‖$	$\leq γ_{1} α^{p} d_{0}^{p} α d_{0}^{t_{1}} + γ_{2} α^{1 + p} d_{0}^{t_{1} (1 + p)}$
		$= α^{1 + p} (γ_{1} d_{0}^{t_{1} + p} + γ_{2} d_{0}^{t_{1} (1 + p)})$
		$= α^{1 + p} d_{0}^{t_{1} + p} (γ_{1} + γ_{2} d_{0}^{p (t_{1} - 1)})$
		$= α d_{0}^{t_{1}^{2}} (γ_{1} + γ_{2} d_{0}^{p (t_{1} - 1)}) α^{p} .$

But

(γ_{1} + γ_{2} d_{0}^{p (t_{1} - 1)}) α^{p} = \frac{γ_{1} + γ_{2} d_{0}^{p (t_{1} - 1)}}{γ_{1} + γ_{2}} < 1,

then the following inequality holds

‖ x_{2} - x^{*} ‖ \leq α d_{0}^{t_{1}^{2}} .

We prove now that $‖ x_{2} - x^{*} ‖ \leq ‖ x_{1} - x^{*} ‖ .$ From the inequality (13) we obtain:

		$‖ x_{2} - x^{} ‖ (γ_{1} α^{p} d_{0}^{p} + γ_{2} α^{p} d_{0}^{t_{1} p}) ‖ x_{1} - x^{} ‖ \leq$
		$\leq α^{p} d_{0}^{p} (γ_{1} + γ_{2} d_{0}^{p (t_{1} - 1)}) ‖ x_{1} - x^{} ‖ < ‖ x_{1} - x^{} ‖$

since $d_{0}^{p} < 1$ and, as we saw above, $α^{p} (γ_{1} + γ_{2} d_{0}^{p (t_{1} - 1)}) < 1$ .

Assume now that for $n \in ℕ,$ $n \geq 2,$ the following relations hold:

(a”)

$‖ x_{n - 1} - x^{*} ‖ \leq α d_{0}^{t_{1}^{n - 1}};$
(b”)

$‖ x_{n} - x^{*} ‖ \leq \min {α d_{0}^{t_{1}^{n}}, ‖ x_{n - 1} - x^{*} ‖}$

Proceeding as in the case of $x_{2},$ and taking into account (a”), (b”) and (8), we obtain:

	$‖ x_{n + 1} - x^{*} ‖$	$\leq α^{1 + p} d_{0}^{t_{1}^{n + 1}} \cdot (γ_{1} + γ_{2} d_{0}^{p t_{1}^{n - 1} (t_{1} + 1)}) =$
		$= α d_{0}^{t_{1}^{n + 1}} \cdot α^{p} (γ_{1} + γ_{2} d_{0}^{p t_{1}^{n - 1} (t_{1} - 1)}) \leq α d_{0}^{t_{1}^{n + 1}},$

since, as previously, it is easy to show that:

α^{p} (γ_{1} + γ_{2} d_{0}^{p t_{1}^{n - 1} (t_{1} - 1)}) < 1

In order to complete the proof, we shall show that:

‖ x_{n + 1} - x^{*} ‖ \leq ‖ x_{n} - x^{*} ‖

Indeed, form (8) we deduce:

‖ x_{n + 1} - x^{*} ‖ \leq (γ_{1} α^{p} d_{0}^{p t_{1}^{n - 1}} + γ_{2} α^{p} d_{0}^{p t_{1}^{n}}) ‖ x_{n} - x^{*} ‖ .

But $d_{0} < 1$ and $α^{p} (γ_{1} + γ_{2} d_{0}^{p (t_{1} - 1)}) < 1,$ therefore:

‖ x_{n + 1} - x^{*} ‖ \leq ‖ x_{n} - x^{*} ‖ .

We proved in this way the following theorem:

Theorem 2.

If the conditions of Theorem 1 are fulfilled, with the difference that $x_{0}$ and $x_{1}$ are chosen in such a manner to verify the relations (a’) and (b’), where $α = {(q (p))}^{- 1 / p}$ and $d_{0} \in (0, 1),$ then, for every $n \in ℕ, x_{n} \in U = {x \in X_{1} | ‖ x - x^{*} ‖ < α}$ and the following inequality holds:

(14)

‖ x_{n + 1} - x^{*} ‖ \leq α d_{0}^{t_{1}^{n + 1}}, n = 0, 1, \dots

Remark.

The inequality (14) contains in its right-hand side a number substantially smaller than that yielded by relation (11).

References

[1]
[2] Argyros, I.K., The secant method and fixed points of nonlinear operators, Mh. Math. 106, 85–94 (1988).
[3] Dennis, J.E., Toward a unified convergence theory for Newton like methods, Nonlinear Functional analysis and Applications (Ed. by L.B. Rall), pp. 425–472, New York, John Wiley (1986).
[4] ^†^†margin: clickable $\to$ Păvăloiu, I., Introduction to the Theory of Approximation of Equations Solutions, Dacia Ed., Cluj-Napoca, 1976 (in Romanian).

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Remarks on the secant method for the solution of nonlinear operator equations

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Remarks on the secant method for the solution of nonlinear operatorial equations

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