This paper deals with a new numerical iterative method for finding the approximate solutions associated with both scalar and vector nonlinear equations. The iterative method proposed here is an extended version of the numerical procedure originally developed in previous works. The present study proposes to show that this new root-finding algorithm combined with a stationary-type iterative method (e.g., Gauss-Seidel or Jacobi) is able to provide a longer accurate solution than classical Newton-Raphson method. A numerical analysis of the developed iterative method is addressed and discussed on some specific equations and systems.

Solving both nonlinear equations and systems is a situation very often encountered in various fields of formal or physical sciences. For instance, solid mechanics is a branch of physics where the resolution of problems governed by nonlinear equations and systems occurs frequently [

We consider vector-valued function

Equation (

With the aim of numerically solving system

In order to solve a system of nonlinear equations, any RFA can be used if it is combined with a SIP (i.e., Jacobi or Gauss-Seidel) [

in the case of Jacobi procedure:

in the case of Gauss-Seidel procedure:

In previous works [

The used RFA is based on the following main steps (see [

In the first step, we consider the iterative tangent and normal straight lines

where

In the second step, we introduce the iterative exact and inexact local curvature associated with the curve representing nonlinear function

where

In the third step, we define the iterative center

By taking (

where

In the fourth step, we introduce the iterative point

where

In the fifth step, we define the iterative straight line

where

In the sixth step, we introduce the iterative straight line

where

In the last step, we define the iterative point

with

where

In line with (

Schematic diagram with the specific entities used by the new RFA applied on

The new iterative method that we will thereafter appoint as “Adaptative Geometric-based Algorithm” (AGA) enables providing a more convenient approximate solution associated with a system of nonlinear equations of type

Geometric interpretation of the new RFA (i.e., AGA) applied on

The different conditions

First condition [BC1] is

Second condition [BC2] is

Third condition [BC3] is

Fourth condition [BC4] is

Fifth condition [BC5] is

Sixth condition [BC6] is

In this section, we propose to evaluate the predictive abilities associated with the numerical iterative method developed in Section

The used iterative methods for the different examples are as follows (see [

Newton-Raphson Algorithm (NRA):

where

Standard Newton’s Algorithm (SNA):

Third-order Modified Newton Method (TMNM):

In order to stop the iterative process associated with each considered algorithm, we consider three coupled types of criteria for dealing with nonlinear equations:

For scalar-valued equations

(C1S) on the iteration number,

where

(C2S) on the residue error,

where

(C3S) on the approximation error,

where

For vector-valued equations

(C1V) on the iteration number, we adopt the same condition that (C1S), that is,

(C2V) on the residue error,

where

(C3V) on the approximation error,

where

Here, for the stopping criteria (C1S, C1V), (C2S, C2V), and (C3S, C3V) associated with the iterative process, we consider: (i) the maximum number of iterations

We consider the following nonlinear equations.

In case

Consider the following:

Consider the following:

In case

Consider the following:

Consider the following:

All the numerical results of Examples

Evolution of approximate solutions

Evolution of approximate solutions

Evolution of residue error (C2S) and approximation error (C3S) associated with (

Evolution of residue error (C2S) and approximation error (C3S) associated with (

Evolution of approximate solutions

Evolution of approximate solutions

Evolution of residue error (C2S) and approximation error (C2S) associated with (

Evolution of residue error (C2S) and approximation error (C3S) associated with (

Evolution of approximate solutions (

Evolution of approximate solutions (

Evolution of residue error using (C2V1) and (C2V2) conditions associated with (

Evolution of residue error using (C2V1) and (C2V2) conditions associated with (

Evolution of the approximation error using (C3V1) and (C3V2) conditions associated with (

Evolution of the approximation error using (C3V1) and (C3V2) conditions associated with (

Evolution of approximate solutions (

Evolution of approximate solutions (

Evolution of residue error using (C2V1) and (C2V2) conditions associated with (

Evolution of approximation error for (C3V1) and (C3V2) conditions associated with (

Evolution of approximate solutions (

Evolution of approximate solutions (

Evolution of residue error using (C2V1) and (C2V2) conditions associated with (

Evolution of residue error using (C2V1) and (C2V2) conditions associated with (

Evolution of the approximation error using (C3V1) and (C3V2) associated with (

Evolution of the approximation error using (C3V1) and (C3V2) conditions associated with (

Evolution of approximate solutions (

Evolution of approximate solutions (

Evolution of the residue error using (C2V1) and (C2V2) conditions associated with (

Evolution of residue error using (C2V1) and (C2V2) conditions associated with (

Evolution of approximation error using (C3V1) and (C3V2) conditions associated with (

Evolution of the approximation error using (C3V1) and (C3V2) conditions associated with (

The present work concerns a new numerical iterative method for approximating the solutions of both scalar and vector nonlinear equations. Based on an iterative procedure previously developed in a study, we propose here an extended form of this numerical algorithm including the use of a stationary-type iterative procedure in order to solve systems of nonlinear equations. A predictive numerical analysis associated with this proposed method for providing a more accurate approximate solution in regard to nonlinear equations and systems is tested, assessed and discussed on some specific examples.

The author declares that there are no competing interests regarding the publication of this paper.