1. Introduction
Driven by the needs of applications in applied mathematics, Refs. [
1,
2,
3,
4,
5,
6] finding a solution
of the equation
is considered significant, where
is a nonlinear Fréchet differentiable operator. Here and below,
T and
are Banach spaces and
is an open convex set in
Due to the complexity in finding closed form solutions to (
1), it is often advised to adopt iterative methods to find
. Newton’s method has been modified in numerous ways in the studies found in [
7,
8,
9,
10,
11,
12,
13,
14]. This paper is based on the Newton–Gauss iterative method studied by Zhongli Liu et al. in [
15]. Precisely, in [
15], Zhongli Liu et al. constructed the following method (see (
2) below) by employing the two-point Gauss quadrature formula, given for each
as
where
,
This was further extended to a method of order five given by
Recall [
1] that a sequence
converges to
with convergence order
if for
where
c is called the rate of convergence.
The current level of research is established in [
7,
8,
9,
10,
11,
12,
13,
14,
15]. In these references, the convergence order was established using higher order derivatives. Thus, these results cannot be applied to solve equations involving operators that are not at least five (if the order of the method is four) times differentiable. This limits their applicability, and other problems also exist:
- (a)
No computable error bounds are provided;
- (b)
There is no information on the uniqueness domain of the solution;
- (c)
The local convergence analysis is provided only when ;
- (d)
The more interesting (than the local) semilocal convergence analysis is not provided.
We address all of these concerns in the more general setting of a Banach space using generalized conditions and majorizing sequences. Moreover, our extended method (see (
4)) is of order six, not five.
The study on the order of convergence of (
2) and (
3) in [
15] involves Taylor expansion. The major concern in [
15] is the necessity of assumptions on the derivatives of
up to order five, reducing the utility of the above methods. As an example, consider
given by
It follows that
Observe that
is not bounded.
Our study is solely based on obtaining the required order of convergence of the above methods without using Taylor expansion and assumptions only on
and
This enhances the applicability of the considered iterative method to a wider range of practical problems. Our approach can be used to study other similar methods [
5,
6,
16,
17,
18].
Furthermore, we have extended this to a method of order six with the ideas in [
3,
4] defined for
given by
The outline of the article is as follows. We discuss the convergence of the methods (
2), (
3) and (
4) in
Section 2,
Section 3 and
Section 4 respectively. Semilocal convergence of the methods is developed in
Section 5.
Section 6 deals with examples.
Section 7 is dedicated to the dynamics and the basins of attraction of the methods (
2), (
3) and (
4).
Section 8 gives the conclusion.
2. Convergence Analysis of (2)
Hereafter, and for some
The following assumptions are made in our study:
- (A1)
a simple solution
of (
1) exists and
;
- (A2)
- (A3)
;
- (A4)
;
- (A5)
- (A6)
for parameter to be specified in what follows and and are scalars.
We define the functions
by
and let
Observe that
is a continuous nondecreasing function. Furthermore,
and
Therefore, there exists a smallest zero
for
Let the functions
be defined by
and
Observe that
is a nondecreasing and continuous function,
and
Therefore,
has a smallest zero
Let
Then,
and
∀
For convenience, we use the notation and
Theorem 1. Assume (A1)–(A6) hold, and let r be as in (5). Then, the sequence given by the formula (2), for exists, and Furthermore,and Proof. This is proved utilizing induction. Let
Using (A2) gives
By the Banach result on invertible operators [
1,
2], it follows that
The Mean Value Theorem (MVT) leads to
for any
Next, because
by (
9) (with
), we have
Thus, by our assumptions,
So, the iterate
and the result holds for
In addition,
is well defined. In fact,
where we also used
and
That is
Therefore, by Banach lemma on invertible operators [
1]
exists and
From (
2) and (
9), we obtain
By (MVT) for second derivatives, (
12) gives
where
and
Then, by (
12) and assumption (A4),
Similarly, one can observe
By the assumptions(A1)–(A5) and (
12), we get
Similarly,
Note that
where
Then
So, by (
12) and assumption (A3),
Combining (
14)–(
19) gives
Therefore, because
we have
, so the iterate
.
The induction for (
6) and (
7) is completed, if one replaces
in the above conclusions with
respectively. □
5. Semilocal Convergence
Generalized
continuity conditions, as well as real majorizing sequences, are utilized to show the semilocal convergence of the three methods under the same set of conditions [
1,
2,
5].
Suppose: ∃ a nondecreasing and continuous real function defined on such that the function has a minimal root Let be a nondecreasing and continuous real function on
We shall show that the following scalar functions are majorizing for Method (
2), Method (
3) and Method (
4), respectively, for
and each
and
and
and
We choose in practice the smallest version of the possible sequences
Next, the convergence is developed for these sequences.
Lemma 1. Let be such that for each Then, the scalar sequences generated by the formulas (30), (31) and (32) are convergent to some Proof. It follows from the definitions and the condition (
33) that these sequences are nondecreasing and bounded from above by
Hence, they are convergent to some
□
Remark 1. The parameter is an upper bound of each of these scalar sequences, and it is not the same in general. Moreover, it is the least and unique.
The functions the parameter and the scalar sequences are associated with the operator as follows:
- (E1)
and a parameter such that the operator is well defined and
- (E2)
for each
Set
- (E3)
for each
- (E4)
The conditions of Lemma 1 hold and
- (E5)
Next, we first present the convergence of Method (
2).
Theorem 4. Under the conditions (E1)–(E5), the sequence generated by the formula (2) is convergent to some satisfying and Proof. The assertions
and
shall be established by induction. The assertion (
35) holds for
by (E1) and the first substep of Method (
2). Then, we also have that
Then, as in (
11), but using
for
respectively, we obtain
thus
where we also used
and
so
Similarly, for
thus
It follows from (
37) and (
38) that
exist. Moreover, we get
We also need the estimate
or
By using (
20) and (
37)–(
41), we get
and
Thus, the iterate
and the assertions (
36) and (
38) (for
) hold. Furthermore, by the first substep of Method (
2), the iterate
is well defined and
Notice that
Then, by (
20), (
43), (
44) and (
38) (for
),
and
Hence, the induction for the assertions (
35), (
36) is completed and the iterates
By the condition (E4), the sequence
is Cauchy. It follows from (
42) and (
45) that the sequence
is also fundamental in
so
In view of (
44) and the continuity of the operator
it follows that
(if
). Let
be an integer. Then, we have the estimate
By letting
in (
48), we show the assertion (
36). □
Similarly, we show the convergence for Method (
3) and Method (
4), respectively.
Theorem 5. Under the conditions (E1)–(E5), the sequence given by (3) is convergent to some satisfying andand Proof. The assertions (
49), (
50) and (
52) are given in Theorem 4. We have from the last substep of Method (
3)
However,
so we get
□
Theorem 6. Under the conditions (E1)–(E5), the sequence given by Method (4) is convergent to some satisfying andwhere the sequences are given by the formula (32). Proof. The proof is as in the proof of Theorem 4, but we use Method (
4) to obtain instead
□
Next, a region is specified for the solution.
Proposition 2. Suppose:
(1) such that for some ;
(2) The condition (E2) holds on the ball ;
(3) There exists such thatDefine the region . Then the unique solution of the equation in the region is . Proof. Define the linear operator
for some
with
It follows, by the conditions (E2) and (
59) in turn, that
Therefore, we get
and consequently,
□
Remark 2. (1) Under all the conditions (E1)–(E5), we can choose and
(2) The limit point in the condition (E5) can be replaced by λ or μ given in the Lemma 1.
7. Basins of Attractions
To obtain the convergence region of Methods (
2), (
3) and (
4), we study the Basins of Attraction (BA) (i.e., the collection of all initial points from which the iterative method converges to a solution of a given equation) and Julia sets (JS) (i.e., the complement of basins of attraction) [
18]. In fact, we study the BA associated with the roots of the three systems of equations given in Examples 5–7.
Example 5.
The solutions are and
Example 6.
The solutions are and
Example 7.
The solutions are and
The region
which contains all the roots of Examples 5–7 is used to plot BA and JS. We choose an equidistant grid of
points in
as the initial guess
, for Methods (
2), (
3) and (
4). A tolerance level of
and a maximum of 50 iterations are used. A color is assigned to each attracting basin corresponding to each root, and if we do not obtain the desired tolerance with the fixed iterations, we assign the color black (i.e., we decide that the iterative method starting at
does not converge to any of the roots). In this way, we distinguish each BA by their respective colors for the distinct roots of each method.
Figure 1,
Figure 2 and
Figure 3 demonstrate the BA corresponding to each root of the above examples (Examples 5–7) for Methods (
2), (
3) and (
4). The JS (black region), which contains all the initial points from which the iterative method does not converge to any of the roots, can easily be observed in the figures.
All the calculations in this paper were performed on a 16-core 64-bit Windows machine with Intel Core i7-10700 CPU @ 2.90GHz, using MATLAB.
In
Figure 1 (corresponding to Example 5), the red region is the set of all initial points from which the iterates (
2), (
3) and (
4) converge to
the blue region is the set of all initial points from which the iterates (
2), (
3) and (
4) converge to
and the green region is the set of all initial points from which the iterates (
2), (
3) and (
4) converge to
The black region represents the Julia set.
In
Figure 2 (corresponding to Example 6), the red region is the set of all initial points from which the iterates (
2), (
3) and (
4) converge to
the blue region is the set of all initial points from which the iterates (
2), (
3) and (
4) converge to
and the green region is the set of all initial points from which the iterates (
2), (
3) and (
4) converge to
The black region represents the Julia set.
In
Figure 3 (corresponding to Example 7), the red region is the set of all initial points from which the iterates (
2), (
3) and (
4) converge to
the blue region is the set of all initial points from which the iterates (
2), (
3) and (
4) converge to
the green region is the set of all initial points from which the iterates (
2), (
3) and (
4) converge to
and the yellow region is the set of all initial points from which the iterates (
2), (
3) and (
4) converge to
The black region represents the Julia set.