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.
Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)
chord method; divided differences; Holder condition
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.
[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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Remarks on the secant method for the solution of nonlinear operatorial equations
"Babeş-Bolyai" University
Faculty of Mathematics and Physics
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:
where is a nonlinear operator, while and
are Banach spaces.
If we denote by the divided difference of the mapping
on the point and , then for fixed we have It is known that in
certain conditions the sequence generated by
the secant method:
| (2) |
|
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converges to the solution 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 is Fréchet differentiable, while the
derivative fulfils a Hölder-like condition
on a set namely there exist a constant and number
such that the inequality:
| (3) |
|
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holds for every . In this case we shall say that
In the quoted paper I.K. Argyros defines the divided difference operator
as a linear operator which fulfils the conditions:
| (4) |
|
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and, in addition, for every the following inequality
holds:
| (5) |
|
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where are constants which do not depend on and
while
Let be a simple solution of (1). We mean by that the
mapping admits a bounded inverse mapping, and if
then admits a bounded inverse mapping.
Thus the continuity of the mapping with respect
to the variable and ensures the existence of a number
such that the mapping admits a bounded inverse
mapping for every where
that is, the set is uniformly bounded in
.
Theorem 1.
[2] Let and let
an open set. The following conditions are fulfilled:
-
(a)
is a simple solution of the equation
(1);
-
(b)
there exist such that for every
-
(c)
there exists a convex set such that and there exists with such that for every and .
Let such that:
| (6) |
|
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where:
| (7) |
|
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Then, if the iterates
generated by (2) are well defined and
belong to the set while the sequence
converges to the unique solution
of equation (1).
Moreover, the following estimation:
| (8) |
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holds for sufficiently great , where:
while and were precised by the relation (5).
In order to prove this theorem the author uses the following two lemmas:
Lemma 1.
[2]. Let and
. Suppose that is an open set and does exist in every point of . If, for a convex set
then for every the following inequality holds:
|
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Lemma 2.
[2]. If fulfils the
conditions (4) and (5), the following relations hold:
-
(a)
for every
-
(b)
From the proof of Theorem 1 follows, for the error estimation
and for the convergence speeds of the sequence the inequality:
| (11) |
|
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where one shows that
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 and fulfil the conditions:
-
(a’)
-
(b’)
where while is the
positive root of the equation:
| (12) |
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Using the condition (4) and (5), Lemmas 1 and
2, and the hypotheses of 1, it results easily from
(2),for , the inequality [2]:
| (13) |
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from which, using (a’) and (b’) and the fact that is a root of
equation (12), we obtain:
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But
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then the following inequality holds
We prove now that From the inequality (13) we obtain:
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since and, as we saw above, .
Assume now that for the following relations hold:
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(a”)
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(b”)
Proceeding as in the case of and taking into account
(a”), (b”) and (8), we obtain:
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since, as previously, it is easy to show that:
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In order to complete the proof, we shall show that:
Indeed, form (8) we deduce:
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But and therefore:
We proved in this way the following theorem:
Theorem 2.
If the conditions of Theorem 1 are
fulfilled, with the difference that and are chosen in such a
manner to verify the relations (a’) and (b’), where
and then, for every and the following inequality holds:
| (14) |
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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
Păvăloiu, I.,
Introduction to the Theory of
Approximation of Equations Solutions, Dacia Ed., Cluj-Napoca, 1976 (in Romanian).
This paper is in final form and no version of it is or will be
submitted for publication elsewhere.
