On the chord method

9 years ago

Abstract

Let $X_{1},X_{2}$ be two Banach spaces, $f:X_{1}\rightarrow X_{2}$ a nonlinear mapping and consider the chord method for solving the equation $f\left(x\right) =0$: \[x_{n+1}=x_n-[x_{n-1},x_{n};f]^{-1}f(x_n), \quad n=1,…\] Under some simple 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 show that the chord method converge to the solution. We obtain error estimations and determine the convergence order.

Authors

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)

Keywords

chord method; equations in Banach spaces; error estimation; convergence order

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I. Păvăloiu, On the chord method, Bul. Ştiinţ. Univ. Baia Mare, Seria B. Fasc. Mat.-Fiz., 7 (1991), pp. 61-66.

About this paper

Journal

Bul. Ştiinţ. Univ. Baia Mare, Seria B. Fasc. Mat.-Fiz., 7 (1991)

Publisher Name

Bul. Ştiinţ. Univ. Baia Mare

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http://www.jstor.org/stable/44002517

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References

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

[2] Balazs, M. ¸si Goldner, G., Observatii asupra diferentelor divizate si asupra metodei coardei. Revista de analiza numerica si teoria aproximatiei vol. 3 (1974) fasc. 1, 19–30.

[3] 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.

[4] Pavaloiu, I., Introducere in teoria aproximarii solutiilor ecuatiilor. Ed. Dacia, ClujNapoca, 1976.

[5] Schmidt, I. W., Eine Ubertagungen der Regula Falsi auf Gleichungen, in ”Banachraumen” I ZAMM, 48, 1–8 (1963).

[6] Schmidt, I. W., Eine Ubertagungen der Regula Falsi auf Gleichungen, in ”Banachraumen” II 97–110 (1963).

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Bul.St.Univ.Baia Mare

Seria B, Matematică-Informatică, vol.VII (1991) Nr.1-2, 61-66

On the chord method

I. Păvăloiu (Cluj-Napoca)

In the paper [1], I.K. Argyros considers as divided difference of the mapping $f : X_{1} \to X_{2}$ , where $X_{1}$ and $X_{2}$ are Banach spaces, a linear mapping $[x, y; f] \in ℒ (X_{1}, X_{2})$ which fulfils the following conditions:

(a)

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

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

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

In [1] the hypothesis that the equation:

(1)

f (x) = 0

admits a simple solution $x^{*}$ in adopted, and conditions for the convergence of the sequence ${(x_{n})}_{n \geq 0}$ generated by the chord method:

(2)

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

are given.

In a recent paper [2] there is shown that, with the hypotheses considered in [1], the convergence speed of the sequence generated by (2) and the error estimation are featured by the inequality:

(3)

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

where $α$ is a precised constant, $0 < d_{0} < 1$ and $t_{1}$ is the positive root of the equation $t^{2} - t - p .$

We shall admit further down that the divided difference operator fulfils the conditions (a) and (b), and search for supplementary conditions in order to make equation (1) admit a solution $x^{*}$ into a precised domain $D_{0}$ and the sequence ${(x_{n})}_{n \geq 0}$ generated by (2) converge to this solution.

Observe firstly that the identity:

(4)

x_{n} - {[x_{n - 1}, x_{n}; f]}^{- 1} f (x_{n}) = x_{n - 1} - {[x_{n - 1}, x_{n}; f]}^{- 1} f (x_{n - 1})

holds for every $n = 1, 2, \dots$ with the hypothesis that the linear mapping $[x_{n - 1}, x_{n}; f]$ admits an inverse mapping.

The following identity

(5)		$f (x_{n + 1}) =$	$f (x_{n}) + [x_{n - 1}, x_{n}; f] (x_{n + 1} - x_{n})$
		$+ ([x_{n}, x_{n + 1}; f] - [x_{n - 1}, x_{n}; f]) (x_{n + 1} - x_{n}), n = 1, 2, \dots$

Let $B > 0, α > 0, 0 < d_{0} < 1,$ and $x_{0}, x \in X_{1} .$ Consider the sphere

(6)

U = {x \in X_{1} : ‖ x - x_{0} ‖ \leq \frac{B α d_{0}}{1 - d_{0}^{t_{1} - 1}}}

where $t_{1} = \frac{1 + \sqrt{1 + 4 p}}{2}$ that is, the positive root, the equation:

(7)

t^{2} - t - p = 0

The following theorem holds:

Theorem 1.

If the divided difference $[x, y; f]$ fulfils the conditions (a) and (b) for every $x, y \in U$ and the following hypotheses:

(1)

the mapping $[x, y; f]$ admits a bounded inverse mapping for every $x, y \in U,$ namely there exists a constant $B > 0$ such that $‖ {[x, y; f]}^{- 1} ‖ \leq B$
(ii)

$α = \frac{1}{B^{(1 + p) / p} {(l_{1} + l_{2} + l_{3})}^{1 / p}};$
(iii)

$‖ x_{1} - x_{0} ‖ \leq B α d_{0}, ‖ f (x_{0}) ‖ \leq α d_{0}, ‖ f (x_{1}) ‖ \leq α d_{0}^{t_{1}}$

are also fulfilled, then equation (1) has at least one solution $x^{*} \in U$ and the sequence ${(x_{n})}_{n \geq 0}$ generated by (2) converges to $x^{*},$ the convergence speed and the error estimation being featured by the inequality:

$‖ x^{*} - x_{n} ‖ \leq \frac{B α d_{0}^{t_{1}^{n}}}{1 - d_{0}^{t_{1}^{n} (t_{1} - 1)}} .$

Proof.

From (2) for $n = 1$ we deduce:

‖ x_{2} - x_{1} ‖ \leq B ‖ f (x_{1}) ‖ \leq B α d_{0}^{t_{1}}

from which, taking also into account iii. it follows

	$‖ x_{2} - x_{0} ‖$	$\leq ‖ x_{2} - x_{1} ‖ + ‖ x_{1} - x_{0} ‖$
		$\leq B α d_{0}^{t_{1}} + B α d_{0}$
		$\leq B α d_{0} (1 + d_{0}^{t_{1} - 1})$
		$< \frac{B α d_{0}}{1 - d_{0}^{t_{1} - 1}}$

from which it results that $x_{2} \in U$ .

Using the fact that $x_{2} \in U,$ the identities (4) and (5), and the inequality a), we obtain

	$‖ f (x_{2}) ‖$	$\leq ‖ x_{2} - x_{1} ‖ (l_{1} {‖ x_{2} - x_{0} ‖}^{p} + l_{2} {‖ x_{1} - x_{0} ‖}^{p} + l_{3} {‖ x_{2} - x_{1} ‖}^{p})$
		$\leq B ‖ f (x_{1}) ‖ (l_{1} B^{p} {‖ f (x_{0}) ‖}^{p} + l_{2} {‖ x_{1} - x_{0} ‖}^{p} + l_{3} B^{p} {‖ f (x_{1}) ‖}^{p})$
		$\leq B α d_{0}^{t_{1}} (l_{1} B^{p} α^{p} d_{0}^{p} + l_{2} B^{p} α^{p} d_{0}^{p} + l_{3} B^{p} α^{p} d_{0}^{t_{1}})$
		$\leq B^{p + 1} α^{p + 1} d_{0}^{t_{1} + p} (l_{1} + l_{2} + l_{3} d_{0}^{p (t_{1} - 1)})$
		$= B^{p + 1} α^{p + 1} (l_{1} + l_{2} + l_{3} d_{0}^{p (t_{1} - 1)}) d_{0}^{t_{1}^{2} - p} \leq α d_{0}^{t_{1}^{2}}$

since $α^{p} B^{p + 1} (l_{1} + l_{2} + l_{3} d_{0}^{p (t_{1} - 1)}) \leq α^{p} B^{p + 1} (l_{1} + l_{2} + l_{3}) < 1 .$

From the above inequality follows therefore:

‖ f (x_{2}) ‖ \leq α d_{0}^{t_{1}^{2}}

Suppose by induction that:

(a’)

$x_{i} \in U, i = 0, 1, \dots, k;$
(b’)

$‖ f (x_{i}) ‖ \leq α d_{0}^{t_{1}^{i}}, i = 1, 2, \dots, k .$

Then, for $x_{k + 1}$ we have:

	$‖ x_{k + 1} - x_{0} ‖ \leq ‖ x_{k + 1} - x_{k} ‖ + ‖ x_{k} - x_{k - 1} ‖ + \dots + ‖ x_{1} - x_{0} ‖$
	$\leq B ‖ f (x_{k}) ‖ + B ‖ f (x_{k - 1}) ‖ + \dots + B α d_{0}$
	$\leq B α d_{0}^{t_{1}^{k}} + B α d_{0}^{t_{1}^{k - 1}} + \dots + B α d_{0}$
	$= B α d_{0} (1 + d_{0}^{t_{1} - 1} + d_{0}^{t_{1}^{2} - 1} + \dots + d_{0}^{t_{1}^{k} - 1})$
	$\leq B α d_{0} (1 + d_{0}^{t_{1} - 1} + d_{0}^{2 (t_{1} - 1)} + \dots + d_{0}^{k (t_{1} - 1)}) \leq \frac{B α d_{0}}{1 - d_{0}^{t_{1} - 1}}$

from which follows that $x_{k + 1} \in U .$ Proceeding now for $x_{k + 1},$ as in the case of $x_{2},$ we obtain:

	$‖ f (x_{k + 1}) ‖$	$\leq B^{p + 1} α^{p + 1} (l_{1} + l_{2} + l_{3} d_{o}^{p t_{1}^{k - 1} (t_{1} - 1)}) d_{0}^{t_{1}^{k - 1} (t_{1} + p)}$
		$\leq B^{p + 1} α^{p + 1} (l_{1} + l_{2} + l_{3}) d_{0}^{t_{1}^{k + 1}} \leq α d_{0}^{t_{1}^{k + 1}}$

It results therefore that the relations (a’) and (b’) hold for every $i \in ℕ .$

Now we shall show that the sequence ${(x_{n})}_{n \geq 0}$ is fundamental.

Indeed, for every $n, s \in ℕ$ we have:

	$‖ x_{n + s} - x_{n} ‖$	$\leq \sum_{k = n}^{n + s - 1} ‖ x_{k + 1} - x_{k} ‖ \leq \sum_{k = n}^{n + s - 1} B ‖ f (x_{k}) ‖ \leq B α \sum_{k = n}^{n + s - 1} d_{0}^{t_{1}^{k}}$
		$= B α d_{0}^{t_{1}^{n}} \sum_{k = n}^{n + s - 1} d_{0}^{t_{1}^{k} - t_{1}^{n}} = B α d_{0}^{t_{1}^{n}} \sum_{k = n}^{n - s - 1} d_{0}^{t_{1}^{n} (t_{1}^{k - n} - 1)}$
		$\leq B α d_{0}^{t_{1}^{n}} \sum_{k = n}^{n + s - 1} d_{0}^{t_{1}^{n} (k - n) (t_{1} - 1)} = B α d_{0}^{t_{1}^{n}} \sum_{k = n}^{n + s - 1} {(d_{0}^{t_{1}^{n} (t_{1} - 1)})}^{k - n}$
		$\leq \frac{B α d_{0}^{t_{1}^{n}}}{1 - d_{0}^{t_{1}^{n} (t_{1} - 1)}} .$

By the last inequality and the fact that $0 < d_{0} < 1$ and $t_{1} > 1$ follows that the sequence ${(x_{n})}_{n \geq 2}$ is fundamental. For $s \to \infty,$ from the inequality:

‖ x_{n + s} - x_{n} ‖ \leq \frac{B α d_{0}^{t_{1}^{n}}}{1 - d_{0}^{t_{1}^{n} (t_{1} - 1)}}

follows the inequality:

‖ x^{*} - x_{n} ‖ \leq \frac{B α d_{0}^{t_{1}^{n}}}{1 - d_{0}^{t_{1}^{n} (t_{1} - 1)}} .

In [1] Argyros showed that if the divided difference $[x, y; f]$ fulfils the conditions (a) and (b) then $f$ is Fréchet differentiable and $[x, x; f] = f^{'} (x) .$ From this fact follows that the mapping $f$ is continuous on $B$ : hence at limit for $n \to \infty$ in the inequality:

‖ f (x_{n}) ‖ \leq α d_{0}^{t_{1}^{n}},

one obtains:

‖ f (x^{*}) ‖ \leq 0

from which results $f (x^{*}) = 0 .$ With this the theorem is entirely proved ∎

Remark 2.

In [5], [6] Schmidt imposes in the divided difference conditions similar to the conditions (a) and (b) given by Argyros in [1], but for $p = 1$ . The same conditions are reproduced in [2], too.

References

[1] Argyros, I. K., The secant method and fixed points on nonlinear operators, Mh. Math., 106 (1988), 85–94.
[2] Balázs, M. şi Goldner, G., ^†^†margin: clickable $\to$ Observaţii asupra diferenţelor divizate şi asupra metodei coardei. Revista de analiză numerică şi teoria aproximaţiei vol. 3 (1974) fasc. 1, 19–30.
[3] ^†^†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.
[4] Păvăloiu, I., ^†^†margin: clickable $\to$ Introducere în teoria aproximării soluţiilor ecuaţiilor. Ed. Dacia, Cluj-Napoca, 1976.
[5] Schmidt, I. W., Eine Ubertagungen der Regula Falsi auf Gleichungen, in ”Banachräumen” I ZAMM, 48, 1–8 (1963).
[6] Schmidt, I. W., Eine Ubertagungen der Regula Falsi auf Gleichungen, in ”Banachräumen” II 97–110 (1963).

Institutul de Calcul

Academia Română

Cluj-Napoca

1991

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On the chord method

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On the chord method

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