1. Introduction
Many problems in computational sciences and other areas are related with the problem of approximating a locally solution
using mathematical modeling [
1] of a general nonlinear equation or a system of equations in the form
with
G being a continuous operator mapping a convex subset
D of a Banach space
into a Banach space
.
Solutions of such equations can hardly be found in closed form. So, most solution methods for these equations are iterative. The study about convergence matter of iterative procedures is usually based on two types: semi-local and local convergence analysis. The semi-local convergence (SL) is using data around an initial point, to give criteria assuring the convergence of the iterative procedure. The local one (LC) uses information around a solution, to find the radii of convergence balls. Note that in computational sciences, the practice of numerical analysis for finding such solutions is connected to Newton’s method (NM),
NM is undoubtedly the most popular method for generating a sequence
quadratically (under certain hypotheses [
1]) converging to
. Here,
the space of bounded linear operators from
into
. There is extensive literature on local as well as semi-local convergence results for NM under Lipschitz-type conditions.
There are several iterative processes that consider the use of divided differences instead of derivatives because the operator
G can be not differentiable. The operator
,
, with
, is the first divided difference when
We can consider the approximation
, where
and
are known data at the point
. Depending on the data, this approximation will improve and we can show the Secant-like method [
2,
3,
4,
5]
In general, symmetric divided differences perform better. We can see this in the Center–Steffensen (
and
) [
2,
6] and Kurchatov methods (
and
) [
7]; they both maintain the quadratic convergence of Newton’s method by approximating the derivative through symmetric divided differences with respect to the
[
1,
3,
8,
9,
10,
11]. Following this idea, in this work we consider the derivative-free point-to-point iterative process of the Steffensen-like method given by
where
for a real number
. Thus, we are considering a symmetric divided difference to approximate the derivative of NM. Furthermore, by varying the parameter
, we can approach the value
.
In a similar way, we can consider the Inexact Steffensen-like method
Our objective in this work focuses on verifying that theses are iterative processes, methods (
4) and (
5), and have a behavior like Newton’s method in differentiable situations and maintains this behavior for non-differentiable situations, where Newton’s method is not applicable.
The rest of the work is organized as follows: The SL of methods (
4) and (
5) are given in
Section 2 and
Section 3, respectively, whereas the numerical examples are given in
Section 4. A discussion is presented in
Section 5, whereas the conclusions can be found in
Section 6.
2. Convergence for Steffensen-like Method
Along the work, we denote and
, respectively, for the open and closed balls with center and of radius .
We start by presenting our extension of the celebrated Newton–Kantorovich theorem for solving nonlinear equations given in [
1] under the following conditions:
There exist and such that and
.
There exists such that for that
.
Set .
There exists such that for that
.
Theorem 1 (Extended Newton–Kantorovich theorem)
. Let be a continously Fréchet differentiable operator. Assume that conditions – and are satisfied. Further, assumeThen, sequence generated by Newton’s method (2) is well defined in , remains in and converges to a solution of equation , where is the smallest positive zero of polynomial . Moreover, the following estimates holdandwhereandFurthermore, if there exists , such thatthe limit point is the only solution of equation in the set . Remark 1. The following Lipschitz condition has been used for someand all:
However, thenandsinceThe sufficient SL convergence condition given by Kantorovich [1] (see also [3,6,8,9,10,11]) under and (in non-affine invariant form) isThen, by (6)–(10), we havebut not vice versa, unless if . The error estimates under (10) are less precise as well as the uniqueness results, since replaces M (and ). Similar extensions hold in the case of the Steffensen-like method (
4). Indeed, let us consider
and present the semi-local convergence result in Theorem 1 of [
2], but given in affine form.
Theorem 2. Let be a continuously differentiable operator. Assume conditions and withare satisfied. Moremover, assumeandwhere is the smallest positive root of polinomialThen, sequence generated by Steffensen-like method (4) is well defined in , remains in and converges to a solution of equation . Moreover, the following error estimates holdandwhereandFurthermore, the limit point is unique in , where Remark 2. In order to compare Theorem 2 with its extension that follows as in [2], we introduce the divided difference of order one [2,3,7] given bywith Using
) (instead
used in [
2]) and (
19), we get the following extension of Theorem 2.
Assume
Then, we have
from which it follows by the Banach lemma on invertible operators [
1,
5,
8] that
Set
and
Hence, we get:
Theorem 3. Let be a continuously differentiable operator. Assume conditions – and (20) are satisfied. Moreover, assumeandThen, sequence generated by Steffensen-like method (4) is well defined in , remains in and converges to a solution of equation . Moreover, the following error estimates holdandFurthermore, the limit point is unique in . Proof. Simply repeat proof in Theorem 2 in [
2] but in affine invariant form, and use
instead of
for the upper bounds on the inverse of the operators involved. □
Remark 3. In view of (7)–(9), we haveandwhich justify the advantages as stated in the introduction. The computation of parameter requires that of and M. Hence, advantages are obtained under the same computational cost as before. A further improvement can be obtained if in and is replaced by , since in this case, tighter can replace M in all previous results with since . 3. Convergence of Inexact Steffensen-like Method
We shall first develop an auxiliary result conerning a majorizing sequence for method (
5). Let
a and
b be real numbers. Define parameters and functions
where,
Notice that is the unique positive root of equation .
Sequence shall be shown to be majorizing for sequence in Theorem 4. However, first two convergence results are presented for sequence .
Lemma 1. Assume that for Then, the following items holdand Proof. It follows immediately by the definition of sequence
and condition (
24). □
Next, a second convergent result follows for sequence .
Lemma 2. Let , and be real numbers. Assume thatThen, sequence generated by (23) is well defined, non-decreasing, bounded from above by and converges to its unique least upper bound , which satisfies Proof. We shall show, using mathematical induction, that for
Estimate (
27) is satisfied for
by condition (
25). Then, by (
23), we have
Then, by the induction hypothesis (
28), instead of showing (
27), it suffices to show
or
Estimate (
29) motivates us to introduce the sequence of functions
.
We need a relationship between consecutive functions
. We can write
In particular, by the definition of
and Equation (
30), we get
. However, then, (
29) holds if
where
So, we must show instead of (
31), that
, which is true by condition (
25). Hence, the induction for (
27) is completed.
It follows that the sequence
is non-decreasing, bounded from above by
, and as such it converges to its unique least upper bound
, which satisfies (
26). □
Next, we present the semi-local convergence analysis of method (
5) using conditions
,
,
, and the preceding Lemma and notations.
Theorem 4. Assume conditions , , , with replaced by and (24) or (25) are satisfied. Then, sequence generated by method (5) is well defined in , remains in for and converges to a solution of equation . Moreover, the following estimates holdand Furthermore, the limit point is the only solution of equation in the set
, where .
Proof. It follows from the estimates
Therefore,
leading to
so, we conclude
The rest follows as in Theorem 2 in [
2]. □
Remark 4. - (a)
Comments similar to the one given in Remark 3 can follow.
- (b)
As in the case of Lemma 1, convergence criterion (6) can be replaced byin the case of NM. Finally, it is worth noticing that conditions (24) and (32) are weaker than (25) and (6) (or (10)), respectively.
5. Discussion
Note that if all methods are convergent, the new error bounds are at least as tight, since the Lipschitz constants are at least as small. For instance, in the Example 1, the Lipschitz constant
(see (
)) used before is
, so
, where
and
M are the constants used. Then, the new majorizing sequence (see Theorem 1) is tighter than the one used by Kantorovich where
(for Newton’s method). The same is true for Steffensen-like methods (
4) and (
5) (see Theorem 2 and Remark 3). Besides, if all Newton and Steffensen methods are compared at the same time, then if the error estimated are obtained using majorizing sequences which in turn depend on the “
M” and “
K” constants, respectively, then the tighter error bounds will be given by those with the smallest constants. Notice also that such a comparison was the main topic in the motivational paper [
2]. Observe that the methods (
4) and (
5) are derivative free. They should be used when the derivative is hard to find or it does not exist. It is clear that for sufficiently small
, these methods will be similar to Newton’s (see also [
2]).