1. Introduction
Let j denote a fixed natural number, the set be open and convex, and stand for a continuously differentiable mapping with a Jacobian denoted by .
Numerous problems from applied mathematics, scientific computing, and engineering can be written using mathematical modeling [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10] as a system of equations in the form
A solution of the system of equations is attainable in analytical form only in special cases. That explains why most solution schemes for such a system of equations are iterative.
Newton’s is undoubtedly the most popular method and it is defined for
and each
by
Newton’s method is of convergence order two and has served as the first substep of higher convergence order schemes due to its computational efficiency (CE). In particular, Newton’s method is the first optimal (vectorial) scheme in the sense of an assumption which is given next.
Conjecture 1 ([
5]).
The convergence order of any Newton-type method (without memory), which is defined on cannot exceed the bound , , where is the number of function evaluations in the entries of per iteration and is the number of function evaluations of F. Moreover, the iterative method is called optimal if it reaches . Two-step optimal Newton-type methods with accelerators of order four have already been studied in Refs. [
5,
11], respectively, and are defined for
by
where
is a function such that
or
where, for
,
We shall also use the equivalent versions of
given for
by
The local convergence order four is established in [
11] for method (
4) using Taylor series expansions and by assuming the existence of at least the fifth derivative of the mapping
F, which is not on method (
4) (or method (
3)). However, there are several issues limiting the applicability of these methods.
1.1. Motivational Issues
- ()
The convergence order four is shown in [
11] by utilizing Taylor series expansions and assuming the existence of at least the fifth derivative of
F, which is not in method (
4). In particular, the following local convergence result is shown in Ref. [
11] for method (
4).
Theorem 1. Suppose that is sufficiently many times differentiable in a neighborhood of a simple solution . Then, the sequence generated by method (
4)
is convergent to . Moreover, the following error equation holdswhere . It is worth noting that the proof of this result uses Taylor series expansions and requires the existence of at least the fifth derivative of the mapping F, which is not in the method.
We look at a toy example where the results of Ref. [
11] cannot apply.
Let
,
. Define
by
where
and
. It follows by this definition that the fourth derivative of
F does not exist, since for
,
is discontinuous at zero. Notice that
solves the equation
. Moreover, if
, both methods (
3) and (
4) converge to
. So, this observation suggests that the sufficient convergence conditions in [
11] or other studies using Taylor series can be replaced by weaker ones.
- ()
There are no computable a priori estimates on the error distances . Hence, we cannot tell in advance how many iterations are required to achieve a desired error tolerance .
- ()
There is no information on the uniqueness of the solution in a neighborhood of it.
- ()
The more challenging and important semi-local convergence analysis has not been studied previously.
1.2. Innovation
These problems constitute our motivation for this paper. Problems – are addressed as follows.
The local convergence analysis is presented using only conditions on the mappings which appear in method (
4), i.e.,
F and
.
A natural number k is determined in advance such that for each . Moreover, the radius of convergence is given. Consequently, the initial points are picked from a specific neighborhood of such that .
A domain is specified containing only one solution of the system of equations .
The semi-local convergence analysis is provided using majorizing sequences [
1].
Both types of convergence analysis relying on generalized continuity are used to control and sharpen the error estimates and .
Since, we will refer to neighborhoods of points in , we use the standard notation for open and closed balls. Given a point and a radius , we obtain the following:
The
open ball of radius
r around
x is as follows:
which includes all points strictly within distance
r from
x.
The
closed ball of radius
r around
x is as follows:
which also contains the boundary points exactly at distance
r.
The remainder of the paper is organized as follows. In
Section 2, we introduce our notation and establish the local convergence theorems, culminating in the error bounds and uniqueness results.
Section 3 contains the semi-local convergence analysis via majorizing sequences.
Section 4 discusses the implementation details, including the computational cost and comparisons with classical schemes. Finally, in
Section 5, we conclude with numerical experiments that showcase the benefits of our approach, highlighting cases where existing fourth-order methods struggle or require more stringent assumptions.
2. Local Area Convergence
Some convergence conditions are needed. Let .
Suppose the following hold:
- ()
There exists a continuous, nondecreasing function such that the function has a minimal positive zero. We shall call such zero by and set .
- ()
There exists a continuous, nondecreasing function
such that for
defined by
the function
has a minimal positive zero in
. We shall denote such a zero by
.
- ()
For
, the function
defined as
is such that
has a minimal positive zero in
, which is called
.
Set , where .
- ()
For functions
,
, and
and
given by
and
the function
has a minimal positive zero in
, which is denoted by
.
By the conditions (
)–(
) and (
5), we have the following for each
:
and
We show in Theorem 2 that the number
is a convergence radius for method (
4).
Next, the parameter and functions and w are associated with .
- ()
There exists an invertible linear operator
, where
k is a natural number and a solution
of the nonlinear system of equations
such that for each
,
Set .
- ()
for each .
- ()
.
- ()
There exists a parameter
such that
Remark 1. One can choose , the identity operator, or , where is a convenient point other than , or . The last selection implies that is a simple solution. However, no such assumption is made or implied here. Thus, the method (
4)
can be used to approximate solutions of multiplicity 2, 3, …. Other choices for S are also possible [1,2,7,10]. The main local area convergence is based on the conditions ()–(). Set .
Theorem 2. Suppose that conditions ()–() hold. Then, the following items hold for the sequence , provided : and .
Proof. Notice that item (
9) holds if
. Induction is used to show items (
9)–(
11). Let
. The application of the conditions (
) and (
) and definitions (
5) and (
6) can, in turn, give
The Banach lemma on linear invertible operators [
1,
7] and (
12) assure the existence of
and the following estimate:
Notice now that for
, the iterate
is well defined by the first substep of method (
4) if
and
In view of condition (
), definition (
5), and estimate (
13), we obtain the following by (
14):
It follows that, by estimate (
15), item (
10) holds for
and the iterate
. Notice that by (
13), for
and the condition (
), the accelerators
, and
and the iterate
are well defined. Moreover, we can, in turn, write the following:
Some estimates are needed before we revisit (
16).
By condition (
), (
15), and the induction hypotheses, we obtain, in turn,
Thus, by (
18) and the condition (
), we have
In view of (
15), (
18)–(
21), (
5) and (
8), for
, estimate (
16) gives
Hence, item (
11) holds if
and the iterate
. We also have for
and (
22) that
Finally, by (
23), we deduce that
. □
A domain is defined with only as a solution of the system of equations .
Proposition 1. Suppose the following:
There exists solving system of equations for some , and there exists such that the condition () holds in , Set . Then, the only solution of the system of equations in the domain is .
Proof. Let us consider
, provided that
. In view of the condition (
) and (
24), we have
Therefore,
. Finally, from the identity
we conclude that . □
Remark 2. - (i)
The function has two versions. So, in practice, we shall use the smaller of the two. Notice that if the two versions cross on , then is chosen as the smallest on each interval.
- (ii)
A choice for and is provided if all conditions ()–() hold in Proposition 1.
3. Semi-Local Area Convergence
In this section, the formulas and calculations are similar to the local area convergence. But the terms are switched by and , respectively.
Suppose the following:
- ()
There exists a continuous, nondecreasing function so that the function has a minimal positive zero. Let us denote such a zero by s. Set .
- ()
There exists a continuous and nondecreasing function .
Define the sequences
, and
for
, some
and each
, by
and
A general convergence condition for the sequence is needed since it is shown to be majorizing for in Theorem 3.
- ()
There exists
such that for each
,
In view of the condition (
) and (
26), it follows that
and there exists
such that
It is known that is the unique least upper bound of the sequence .
There exists a relationship between the functions
and
and the mappings in the method (
4).
- ()
There exist
and an invertible mapping
S so that for each
,
Set .
Notice that if
, the condition (
) gives
Thus, the linear mapping is invertible. Consequently, the iterate is well defined by the first substep of the method. Thus, we can choose .
- ()
for each .
- ()
There exists
so that for each
,
and
- ()
.
Remark 3. Some possible selections for S are the following: , , or , where is an auxiliary point other than . Other choices for S are also possible.
The semi-local area convergence for method (
4) can now follow.
Theorem 3. Suppose that conditions ()–() hold. Then, the sequence generated by method (
4)
satisfies the assertions Moreover, the point is well-defined and solves the system of equations .
Proof. The assertions (
27)–(
29) are shown by induction. Notice that by the definition of
, (
26) and the method (
4) we have that (
27) holds, if
and
. So, the assertions (
27) and (
28) hold if
and the iterate
.
Then, by (
)–(
) and the induction hypotheses, we obtain that
exists and
As in the local case, we need some estimates.
Then, by the second substep of (
4), (
30)–(
34), and the triangle inequality, we have from
and
Thus, assertion (
29) holds, and the iterate
.
Then, in view of the first substep of the method (
4), we can write, in turn, that
where (
40) is obtained as (
31) with
replaced by
.
Hence, in view of (
40) and (
41), (
39) gives
By the first substep of method (
4), for
, (
26), (
42), and (
), we have
and
The induction for assertions (
27)–(
29) is completed. Notice that by condition (
), the sequence
is Cauchy as convergent. But all the iterates are such that
and (
27)–(
30) hold. It follows that the sequence
is also Cauchy in
and, as such, it has a limit denoted by
.
Let
in estimate (
42) to obtain
. Furthermore, by assertions (
28) and (
29), and the triangle inequality, one can have
So, for
, and using the triangle inequality,
Therefore, we conclude by (
46) that, if
,
□
Next, a domain is given with only one solution for the system of equations .
Proposition 2. Suppose that there exists a solution for some , the condition () holds on , and there exists so that Set .
Then, the only solution of system of equations in the domain is .
Proof. Suppose that there exists solver
for the system of equations
in the domain
satisfying
. Let us define the linear mapping by
. Then, by condition (
) and (
48), we have, in turn, that
It follows from (
49) that the linear mapping
L is invertible. Thus, by the identity
we deduce that . □
Remark 4. - (i)
The limit point in the condition () can be switched by s, given in ().
- (ii)
If all conditions, ()–(), hold in Proposition 2, then one can take and .
4. Numerical Work
In this section, we first consider some alternatives to conditions () and ().
Case: and . Local area convergence. In view of the estimate
Thus, we have from (
50) that
Consequently, we can drop the condition (
) and replace the function
h by
Semi-local area convergence
General Case:
Local area convergence
Let us introduce the following conditions:
for some and each , or
for each .
In this case, notice that
or equivalently
. Thus, we can set
Semi-local area convergence
Suppose that
for some finite and each .
Notice that in view of (
55), one can also consider the following hybrid method:
but
is replaced by
, defined for some finite
i by the following:
To further validate the efficiency and accuracy of the proposed method, we present three numerical examples of varying complexity. These examples serve as benchmarks to compare the convergence behavior, computational efficiency, and numerical stability of different iterative methods.
In all calculations, the default tolerance was set to to ensure high precision in the numerical results. The maximum number of iterations for each method was limited to 50 to prevent excessive computational overhead while maintaining convergence efficiency. This limitation was chosen based on empirical observations that most efficient methods reach convergence well within this range, making further iterations redundant and computationally wasteful. Additionally, the reported CPU timing was obtained as the average over 50 independent runs, providing a more stable and representative measure of computational performance by reducing the impact of potential fluctuations in execution time. This approach is valuable in ensuring that the results are not unduly influenced by temporary variations in computational load, background processes, or system-specific execution conditions. Such fluctuations can distort comparative performance assessments, leading to misleading conclusions about the relative efficiency of the evaluated methods. All numerical experiments were conducted using Google Colab’s cloud computing resources. The runtime used for computations was equipped with an Intel Xeon CPU @2.20 GHz, 13 GB RAM, and a Tesla K80 accelerator with 12 GB of GDDR5 VRAM. This environment ensured that performance benchmarks were consistent and comparable across different test cases.
The first example focuses on a small system to illustrate the fundamental properties of the methods, while the second example scales up the problem size to assess medium-sized system performance. The third example examines a large-scale nonlinear system to demonstrate how well the methods handle computational challenges at an increased scale.
In this section, we compare the performance of Method (
4) with two established iterative methods for solving systems of nonlinear equations. The first is Method (8) from [
12]. The second is a sixth-order method without memory [
13]. For completeness, we recall their definitions below as used in the numerical experiments.
- 2.
A sixth-order method of convergence without memory [
13]
Example 1. Let 3 and Ω
= B, and define the mapping for , by From this definition, it follows that the Jacobian of mapping F is given by Notice that solves the system of equations and . Then, for , the conditions hold if and , since .
Next, we compute ρ using (
5)
, yielding the following radii: , , , and . Among these values, the minimum radius of convergence is . The chosen parameters are and . For the numerical experiment, the initial guess was set to . We compare the methods’ performance for this example in Table 1. Example 2. Consider the nonlinear system of equations of size 200 as follows: We set the initial estimate , , and to obtain the solution The results are summarized in Table 2. Example 3. We analyze a large-scale nonlinear equation system of order 300 with 300 variables to demonstrate the method’s efficiency and scalability in handling complex computational problems. This study highlights the method’s capability to address significant computational challenges in large-scale systems.
To demonstrate its broad applicability to real-world large-scale nonlinear problems, we consider the following system: The required solution for this system is given by . We used as the initial guess, and the parameters were set to and . Given the complexity of this system, computational efficiency becomes a critical factor. We compare the performance of various methods in Table 3. The numerical experiments presented demonstrate the efficiency and robustness of the tested methods across different problem sizes. The proposed approach consistently showed reduced computation time while maintaining high accuracy, particularly in large-scale systems. These results confirm the method’s potential for practical applications in solving complex systems of nonlinear equations efficiently.
5. Conclusions
A finer local convergence analysis for method (
4) is presented without Taylor series expansions, as used in Ref. [
11], which in turn brings the drawbacks
. The new analysis uses generalized continuity assumptions to control the derivative and sharpen the bounds on the error distances
. Moreover, the rest of the assumptions rely only on the mapping of method (
4), i.e.,
F and
. Furthermore, the more challenging and important semi-local convergence analysis, which has not been studied previously, is also provided by relying on majorizing sequences. The same technique can be used to extend the applicability of other methods, such as (
3), or other methods along the same lines [
2,
3,
4,
5,
6,
7,
12,
14,
15,
16,
17,
18,
19,
20]. This is the direction of our future work.
Numerical experimentations complete this paper.