An Iterative Scheme for Evaluating Simultaneous Roots of Nonlinear Problems with Applications

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An Iterative Scheme for Evaluating Simultaneous Roots of Nonlinear Problems with Applications

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Submitted:

29 September 2023

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30 September 2023

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Abstract

In this article, an iterative method of two step is proposed for finding all roots simultaneously of polynomial equations. The order of convergence of the proposed algorithm is 2m, by using any iterative scheme of order m. Numerical tests are performed to confirm the theoretical results and to compare the proposed scheme with existing methods for finding all roots simultaneously.

Keywords:

Subject: Computer Science and Mathematics - Mathematics

1. Introduction

Let us consider the problem of solving any polynomial equation

f (x) = 0

, where

f : D \subseteq C \to C

a is nonlinear, continuous function on any domain D. Nonlinear equations have wide range of applications in various fields of science and engineering like in the theory of control systems, nonlinear circuits, analysis of transfer functions, various mathematical models, differential and difference equations, eigenvalue problems etc. [1]. This arises the development of large number of numerical methods which take the form of iterative procedures. These algorithms are distributed into single roots, multiple roots and simultaneous root finding methods.

In recent years, methods of finding simultaneous roots are very popular as these methods are more stable and can be applied for parallel computing (for example [2,3,4,5]). One of the best known schemes, for approximating the n simple roots

α_{i}

i = 1, 2, \dots, n

, of a polynomial f of degree n, is the Weierstrass method [6] defined by the expression

x_{i}^{(k + 1)} = x_{i}^{(k)} - W (x_{i}^{(k)}) = x_{i}^{(k)} - \frac{f (x_{i}^{(k)})}{\prod_{j = 1, j \neq i}^{n} (x_{i}^{k} - x_{j}^{k})}, k = 0, 1, \dots

In the 1960s, the Weierstrass method was rediscovered by Durand [7], Dochev [8], Kerner [9] and Prešić [10].

The main objective of this paper is to construct iterative schemes, of high order of convergence, for finding all roots of a polynomial equation, simultaneously. For any fixed point iterative method of order m, we construct a new root finding method of order of convergence

2 m

, in Section 2. The convergence of the proposed technique is proved in Section 3. We compare the results obtained with the proposed methods and with other known schemes like Newton’s method, Ostrowki method [11], Jarratt’s method [12], Cordero et al. methods [13,14], Mir et al. schemes [15,16], Shams et al. method [17] and Petkovic et al. scheme [2]. Several numerical experiments are performed in Section 4 by using the mentioned algorithms. With some conclusion in Section 5, we finish the manuscript.

2. Simultaneous methods construction

Let us consider a polynomial equation

f (x) = 0

having n simple roots, denoted by

ξ_{i}

i = 1, 2, \dots, n

. Consider well-known Newton’s scheme, with quadratic convergence:

x_{i}^{(k + 1)} = x_{i}^{(k)} - \frac{f (x_{i}^{(k)})}{f^{'} (x_{i}^{(k)})}, k = 0, 1, \dots

(1)

We take Weierstrass’s correction as follows

\frac{f (x_{i}^{(k)})}{f^{'} (x_{i}^{(k)})} = W (x_{i}^{(k)}) = \frac{f (x_{i}^{(k)})}{\prod_{j = 1, j \neq i}^{n} (x_{i}^{k} - x_{j}^{k})},

(2)

Using (2.2) in (2.1), we have

x_{i}^{(k + 1)} = x_{i}^{(k)} - \frac{f (x_{i}^{(k)})}{\prod_{j = 1, j \neq i}^{n} (x_{i}^{k} - x_{j}^{k})},

(3)

Replacing

x_{i}^{k}

{\bar{x_{i}}}^{k}

, kth-iteration obtained by any scheme, in (2.3), we obtain

x_{i}^{(k + 1)} = {\bar{x}}_{i}^{(k)} - \frac{f ({\bar{x}}_{i}^{(k)})}{\prod_{j = 1, j \neq i}^{n} ({\bar{x}}_{i}^{k} - {\bar{x}}_{j}^{k})}, k = 0, 1, \dots

(4)

Thus, a new two-step iterative method is constructed for finding all roots simultaneously by taking first step of any existing fixed point iterative method with order of convergence m, having the form

{\bar{x}}^{k + 1} = ψ (x^{k})

, and by taking second step as equation (2.4), which is denoted by

ϕ_{t}

and having order of convergence

2 m

as follows:

\begin{matrix} {\bar{x}}_{i}^{k} = & ψ (x_{i}^{k}), \\ x_{i}^{(k + 1)} = & {\bar{x}}_{i}^{(k)} - \frac{f ({\bar{x}}_{i}^{(k)})}{\prod_{j = 1, j \neq i}^{n} ({\bar{x}}_{i}^{k} - {\bar{x}}_{j}^{k})}, i = 1, 2, . . ., n . \end{matrix}

(5)

2.1. Some special cases

We are going to describe some special cases that we will use in the numerical section.

(1): Taking as first step Newton’s method we obtain the following iterative scheme of fourth-order of convergence, denoted by $N_{2 t}$ , with the following expression

$\begin{matrix} {\bar{x}}_{i}^{(k)} = & x_{i}^{(k)} - \frac{f (x_{i}^{(k)})}{f^{'} (x_{i}^{(k)})}, \\ x_{i}^{(k + 1)} = & {\bar{x}}_{i}^{(k)} - \frac{f ({\bar{x}}_{i}^{(k)})}{\prod_{j = 1, j \neq i}^{n} ({\bar{x}}_{i}^{k} - {\bar{x}}_{j}^{k})}, k = 0, 1, \dots \end{matrix}$

(6)
(2): Now, we consider as first step Ostrowski’s method [11], having order of convergence four, then by applying the proposed idea, we obtain an eighth-order of convergence, denoted by $T_{4 t}$ , as follows

$\begin{matrix} y_{i}^{(k)} = & x_{i}^{(k)} - \frac{f (x_{i}^{(k)})}{f^{'} (x_{i}^{(k)})}, \\ {\bar{x}}_{i}^{(k)} = & x_{i}^{(k)} - (\frac{f (y_{i}^{(k)}) - f (x_{i}^{(k)})}{2 f (y_{i}^{(k)}) - f (x_{i}^{(k)})}) (\frac{f (x_{i}^{(k)})}{f^{'} (x_{i}^{(k)})}), \\ x_{i}^{(k + 1)} = & {\bar{x}}_{i}^{(k)} - \frac{f ({\bar{x}}_{i}^{(k)})}{\prod_{j = 1, j \neq i}^{n} ({\bar{x}}_{i}^{k} - {\bar{x}}_{j}^{k})}, k = 0, 1, \dots \end{matrix}$

(7)
(3): Consider as first step of Jarratt’s method [12], which has order of convergence four, then applying the second step of (5) we obtain an iterative method of eighth-order convergence, denoted by $J_{4 t}$ , as follows

$\begin{matrix} y_{i}^{(k)} = & x_{i}^{(k)} - \frac{2 f (x_{i}^{(k)})}{3 f^{'} (x_{i}^{(k)})}, \\ {\bar{x}}_{i}^{(k)} = & x_{i}^{(k)} - (1 - \frac{3}{2} \frac{f^{'} (y_{i}^{(k)}) - f^{'} (x_{i}^{(k)})}{3 f^{'} (y_{i}^{(k)}) - f^{'} (x_{i}^{(k)})}) (\frac{f (x_{i}^{(k)})}{f^{'} (x_{i}^{(k)})}), \\ x_{i}^{(k + 1)} = & {\bar{x}}_{i}^{(k)} - \frac{f ({\bar{x}}_{i}^{(k)})}{\prod_{j = 1, j \neq i}^{n} ({\bar{x}}_{i}^{k} - {\bar{x}}_{j}^{k})}, k = 0, 1, \dots \end{matrix}$

(8)
(4): Considering as first step the Cordero et al. scheme [13], which have four order of convergence and applying the proposed scheme, result an iterative scheme on eighth-order convergence, denoted by $M_{4 t}$ , as follows

$\begin{matrix} y_{i}^{(k)} = & x_{i}^{(k)} - \frac{f (x_{i}^{(k)})}{f [w_{i}^{(k)}, x_{i}^{(k)}]}, \\ {\bar{x}}_{i}^{(k)} = & y_{i}^{(k)} - H (μ_{i}^{(k)}) \frac{f (y_{i}^{(k)})}{f [y_{i}^{(k)}, x_{i}^{(k)}] - f^{'} (x_{i}^{(k)})}, \\ x_{i}^{(k + 1)} = & {\bar{x}}_{i}^{(k)} - \frac{f ({\bar{x}}_{i}^{(k)})}{\prod_{j = 1, j \neq i}^{n} ({\bar{x}}_{i}^{k} - {\bar{x}}_{j}^{k})}, k = 0, 1, \dots, \end{matrix}$

(9)

where $w_{i}^{(k)} = x_{i}^{(k)} + β f (x_{i}^{(k)})$ and $μ_{i}^{(k)} = \frac{f (y_{i}^{(k)})}{f^{'} (w_{i}^{(k)})}$ , being $β$ an arbitrary parameter.

3. Convergence Analysis

The convergence analysis of the family of simultaneous method (5) is presented in the following result.

Theorem 1.

Let

ξ_{1}, ξ_{2}, \dots, ξ_{n}

be all simple roots of the polynomial equation

f (x) = 0

. If

x_{1}^{(0)}, x_{2}^{(0)}, \dots, x_{n}^{(0)}

are the initial approximations of the roots

ξ_{1}, ξ_{2}, \dots, ξ_{n}

, respectively, and they are sufficiently close of them, then the order of convergence of method (5) is

2 m

Proof.

Let

ϵ_{i} = x_{i}^{(k)} - ξ_{i},

(10)

ϵ_{i}^{'} = {\bar{x}}_{i}^{(k)} - ξ_{i},

(11)

ϵ_{i}^{''} = x_{i}^{(k + 1)} - ξ_{i},

(12)

be the errors of approximations

x_{i}^{(k)}

{\bar{x}}_{i}^{(k)}

and

x_{i}^{(k + 1)}

, respectively.

Now, we consider the second step of proposed scheme

x_{i}^{(k + 1)} = {\bar{x}}_{i}^{(k)} - \frac{f ({\bar{x}}_{i}^{(k)})}{\prod_{j = 1, j \neq i}^{n} ({\bar{x}}_{i}^{k} - {\bar{x}}_{j}^{k})},

(13)

and subtract

ξ_{i}

from both sides of (13)

x_{i}^{(k + 1)} - ξ_{i} = {\bar{x}}_{i}^{(k)} - ξ_{i} - \frac{f ({\bar{x}}_{i}^{(k)})}{\prod_{j = 1, j \neq i}^{n} ({\bar{x}}_{i}^{k} - {\bar{x}}_{j}^{k})},

(14)

Then using the errors defined before, we have

ϵ_{i}^{''} = ϵ_{i}^{'} - \frac{f ({\bar{x}}_{i}^{(k)})}{\prod_{j = 1, j \neq i}^{n} ({\bar{x}}_{i}^{k} - {\bar{x}}_{j}^{k})},

(15)

ϵ_{i}^{''} = ϵ_{i}^{'} (1 - \frac{1}{ϵ_{i}^{'}} \frac{f ({\bar{x}}_{i}^{(k)})}{\prod_{j = 1, j \neq i}^{n} ({\bar{x}}_{i}^{k} - {\bar{x}}_{j}^{k})}),

(16)

ϵ_{i}^{''} = ϵ_{i}^{'} (1 - E_{i}),

(17)

where

E_{i} = \frac{1}{ϵ_{i}^{'}} \frac{f ({\bar{x}}_{i}^{(k)})}{\prod_{j = 1, j \neq i}^{n} ({\bar{x}}_{i}^{k} - {\bar{x}}_{j}^{k})} .

(18)

Now, as

f (x)

can be described in the form

f (x) = \prod_{j = 1}^{n} (x - ξ_{j}),

(19)

equation (18), becomes

E_{i} = \prod_{j = 1, j \neq i}^{n} \frac{({\bar{x}}_{i}^{k} - ξ_{j})}{({\bar{x}}_{i}^{k} - {\bar{x}}_{j}^{k})},

(20)

\frac{({\bar{x}}_{i}^{k} - ξ_{j})}{({\bar{x}}_{i}^{k} - {\bar{x}}_{j}^{k})} = 1 + \frac{({\bar{x}}_{j}^{k} - ξ_{j})}{({\bar{x}}_{i}^{k} - {\bar{x}}_{j}^{k})},

(21)

and from (3.2)

({\bar{x}}_{j}^{k} - ξ_{j}) = O (ϵ^{'})

. For a simple root

ξ

and small enough

ϵ

| ({\bar{x}}_{i}^{k} - {\bar{x}}_{j}^{k}) |

is bounded and away from zero, so equation (3.12) becomes,

\frac{({\bar{x}}_{i}^{k} - ξ_{j})}{({\bar{x}}_{i}^{k} - {\bar{x}}_{j}^{k})} = 1 + \frac{({\bar{x}}_{j}^{k} - ξ_{j})}{({\bar{x}}_{i}^{k} - {\bar{x}}_{j}^{k})} = 1 + O (ϵ^{'}),

(22)

Using (3.13) in (3.11)

\prod_{j \neq i j = 1}^{n} \frac{({\bar{x}}_{i}^{(k)} - ξ_{j})}{({\bar{x}}_{i}^{k} - {\bar{x}}_{j}^{k})} = {(1 + O (ϵ^{'}))}^{n - 1} = 1 + (n - 1) O (ϵ^{'}) = 1 + O (ϵ^{'}),

E_{i} = 1 + O (ϵ^{'}),

E_{i} - 1 = O (ϵ^{'}) .

Thus, (3.8) gives

ϵ_{i}^{''} = O {(ϵ^{'})}^{2} .

(23)

Since the first step has order m, we have

ϵ_{i}^{'} = O (ϵ^{m})

. Thus

ϵ_{i}^{''} = O {(ϵ^{m})}^{2} = O {(ϵ)}^{2 m} .

(24)

Therefore, the order of convergence

2 m

is proved. □

4. Numerical experiments

In this section, different numerical experiments are performed in order to analyze the behavior of the proposed scheme. As discussed in the previous section, we consider Newton’s method (denoted by

N_{2 t}

), Ostrowski’s scheme (denoted by

T_{4 t}

), Jarratt’s method (denoted by

J_{4 t}

), Cordero et al. method (denoted by

M_{4 t}

) modified by applying the second step of our proposal. We compare the results obtained from these modified methods with the known schemes for finding simultaneous roots given by Mir et al. (denoted by

M M N_{8}

M R_{6}

M R_{4}

), Shams (denoted by

S_{5}

) and Petkovic et al. (denoted by

P_{10}

). The iterative expressions of these schemes for finding all roots simultaneously are:

Mir et al. scheme [15], $(M M N_{8})$ :

$\begin{matrix} y_{i}^{(k)} = & x_{i}^{(k)} - \frac{f (x_{i}^{(k)})}{f^{'} (x_{i}^{(k)})}, \\ z_{i}^{(k)} = & x_{i}^{(k)} - \frac{1}{\frac{f^{'} (x_{i}^{(k)})}{f (x_{i}^{(k)})} - \sum_{j = 1, j \neq i}^{n} \frac{1}{(x_{i}^{(k)} - y_{j}^{(k)})}}, \\ x_{i}^{(k + 1)} = & z_{i}^{(k)} - \frac{1}{\frac{f^{'} (z_{i}^{(k)})}{f (z_{i}^{(k)})} - \sum_{j = 1, j \neq i}^{n} \frac{1}{(z_{i}^{(k)} - z_{j}^{(k)})} - α} . \end{matrix}$
Mir et al. scheme [16], $(M R_{6})$ :

$\begin{matrix} z_{i}^{(k)} = & x_{i}^{(k)} - \frac{1}{\frac{f^{'} (x_{i}^{(k)})}{f (x_{i}^{(k)})} - \sum_{j = 1, j \neq i}^{n} \frac{1}{(x_{i}^{(k)} - x_{j}^{(k)})}}, \\ x_{i}^{(k + 1)} = & z_{i}^{(k)} - \frac{1}{\frac{f^{'} (z_{i}^{(k)})}{f (z_{i}^{(k)})} - \sum_{j = 1, j \neq i}^{n} \frac{1}{(z_{i}^{(k)} - z_{j}^{(k)})} - α} . \end{matrix}$
Mir et al. scheme [16], $(M R_{4})$ :

$\begin{matrix} w_{i} (x_{i}) = & \frac{f (x_{i})}{\prod_{j = 1, j \neq i}^{n} (x_{i} - y_{j})}, \\ y_{i} = & x_{i} - w_{i} (x_{i}), \\ z_{i} = & y_{i} - \frac{W_{i} (y_{i})}{1 - α W_{i} (y_{i})} . \end{matrix}$
Shams et al. scheme [17], $(S_{5})$ :

$\begin{matrix} y_{i}^{(k)} = & x_{i}^{(k)} - \frac{f (x_{i}^{(k)})}{f^{'} (x_{i}^{(k)})}, \\ z_{i}^{(k)} = & y_{i}^{(k)} - \frac{f^{'} (x_{i}^{(k)}) - f^{'} (y_{i}^{(k)})}{α f^{'} (y_{i}^{(k)}) + (2 - α) f^{'} (x_{i}^{(k)})} \frac{f (x_{i}^{(k)})}{f^{'} (x_{i}^{(k)})}, \\ x_{i}^{(k + 1)} = & x_{i}^{(k)} - \frac{1}{\frac{f^{'} (x_{i}^{(k)})}{f (x_{i}^{(k)})} - \sum_{j = 1, j \neq i}^{n} \frac{1}{(x_{i}^{(k)} - z_{j}^{(k)})}} . \end{matrix}$
Petkovic et al. scheme [2], $(P_{10})$ :

$\begin{matrix} y_{i}^{(k)} & = & x_{i}^{(k)} - \frac{f (x_{i}^{(k)})}{f^{'} (x_{i}^{(k)})}, \\ z_{i}^{(k)} & = & y_{i}^{(k)} - \frac{f (y_{i}^{(k)}) f (x_{i}^{(k)})}{{(f (x_{i}^{(k)}) - f (y_{i}^{(k)}))}^{2}} \frac{f (x_{i}^{(k)})}{f^{'} (x_{i}^{(k)})}, \\ t_{i}^{(k)} & = & z_{i}^{(k)} - (\frac{f (y_{i}^{(k)}) {(f (x_{i}^{(k)}))}^{2}}{{(f (y_{i}^{(k)}) - f (z_{i}^{(k)}))}^{2}}) (\frac{x_{i}^{(k)} - z_{i}^{(k)}}{{(f (x_{i}^{(k)}) - f (z_{i}^{(k)}))}^{2}} - \\ \frac{1}{(f (x_{i}^{(k)}) - f (z_{i}^{(k)})) f^{'} (x_{i}^{(k)})} - \frac{f (y_{i}^{(k)})}{f^{'} (x_{i}^{(k)}) {(f (x_{i}^{(k)}) - f (y_{i}^{(k)}))}^{2}}), \\ x_{i}^{(k + 1)} & = & x_{i}^{(k)} - \frac{1}{\frac{f^{'} (x_{i}^{(k)})}{f (x_{i}^{(k)})} - \sum_{j = 1, j \neq i}^{n} \frac{1}{(x_{i}^{(k)} - t_{j}^{(k)})}} . \end{matrix}$

In the Mir et al. scheme and Shames et al. scheme, we consider the value of

α = 30

. The outcomes of the experiments have been completed in Mathematica version 11.1 software. We choose

∥ x^{(k + 1)} - x^{(k)} ∥_{2} + {∥ F (x^{(k + 1)}) ∥}_{2} < T o l

as stopping criterion, with a tolerance,

T o l = 10^{- 300}

, where

F (x^{(k + 1)}) = (f (x_{1}^{(k + 1)}), . . . . . ., f (x_{n}^{(k + 1)})) .

The norm of the distance between two approximations is given by

∥ x^{(k + 1)} - x^{(k)} ∥_{2}

and the approximated computational order of convergence (ACOC), [18], which is an estimation of the theoretical order of convergence

ρ

whose expression is

ρ \sim A C O C = \frac{ln (\frac{∥ x^{(k + 2)} - x^{(k + 1)} ∥_{2}}{∥ x^{(k + 1)} - x^{(k)} ∥_{2}})}{ln (\frac{∥ x^{(k + 1)} - x^{(k)} ∥_{2}}{∥ x^{(k)} - x^{(k - 1)} ∥_{2}})} .

Example 1.

Consider the polynomial

f (x) = (x + 1) (x + 3) (x^{2} - 2 x + 2) (x - 1) (x^{2} - 4 x + 5) (x^{2} + 4 x + 5),

whose exact roots are

ξ_{1} = - 1, ξ_{2} = - 3, ξ_{3} = 1 + \dot{ι}, ξ_{4} = 1 - \dot{ι}, ξ_{5} = 1, ξ_{6} = - 2 + \dot{ι}, ξ_{7} = - 2 - \dot{ι}, ξ_{8} = 2 + \dot{ι}, ξ_{9} = 2 - \dot{ι} .

The initial approximation has been taken as:

x_{1}^{(0)} = - 1.3 + 0.2 \dot{ι}, x_{2}^{(0)} = - 2.8 - 0.2 \dot{ι}, x_{3}^{(0)} = 1.2 + 1.3 \dot{ι}, x_{4}^{(0)} = 0.8 - 1.2 \dot{ι}, x_{5}^{(0)} = 0.8 - 0.3 \dot{ι}, x_{6}^{(0)} = - 1.8 + 1.2 \dot{ι},

x_{7}^{(0)} = - 1.8 - 1.2 \dot{ι}, x_{8}^{(0)} = 1.8 + 0.8 \dot{ι}, x_{9}^{(0)} = 1.8 - 0.8 \dot{ι} .

Table 1 represents the results obtained from the different iterative methods. We can observed that the proposed methods

T_{4 t}

J_{4 t}

N_{2 t}

and

M_{4 t}

have less absolute error than other methods and are faster than existing schemes.

Example 2.

In this example, let us consider the polynomial

\begin{matrix} f (x) & = & (x - 4) (x^{2} - 1) (x^{4} - 16) (x^{2} + 9) (x^{2} + 16) (x^{2} + 2 x + 5) (x^{2} + 2 x + 2) (x^{2} - 2 x + 2) \\ (x^{2} - 4 x + 5) (x^{2} - 2 x + 10), \end{matrix}

with the exact roots

\begin{array}{l} ξ_{1} = 4, ξ_{2} = - 1, ξ_{3} = 2, x i_{4} = - 2, ξ_{5} = 2 \dot{ι}, ξ_{6} = - 2 \dot{ι}, ξ_{7} = 3 \dot{ι}, ξ_{8} = - 3 \dot{ι}, ξ_{9} = - 1 + 2 \dot{ι}, ξ_{10} = - 1 - 2 \dot{ι}, \\ ξ_{11} = - 1 + \dot{ι}, ξ_{12} = - 1 - \dot{ι}, ξ_{13} = 1 + \dot{ι}, ξ_{14} = 1 - \dot{ι}, ξ_{15} = 2 + \dot{ι}, ξ_{16} = 2 - \dot{ι}, ξ_{17} = 1 + 3 \dot{ι}, ξ_{18} = 1 - 3 \dot{ι}, ξ_{19} = 4 \dot{ι}, \\ ξ_{20} = - 4 \dot{ι}, ξ_{21} = 1 \end{array}

The initial approximations we have taken are:

\begin{array}{l} x_{1}^{(0)} = 4.2 + 0.1 \dot{ι}, x_{2}^{(0)} = - 1.2 + 0.1 \dot{ι}, x_{3}^{(0)} = 2.2 + 0.1 \dot{ι}, x_{4}^{(0)} = - 2.2 - 0.1 \dot{ι}, x_{5}^{(0)} = 0.2 + 2.1 \dot{ι}, x_{6}^{(0)} = 0.2 - 2.1 \dot{ι}, \\ x_{7}^{(0)} = 0.2 + 3.1 \dot{ι}, x_{8}^{(0)} = 0.2 - 3.1 \dot{ι}, x_{9}^{(0)} = - 1.2 + 2.1 \dot{ι}, x_{10}^{(0)} = - 1.2 - 2.1 \dot{ι}, x_{11}^{(0)} = - 1.2 + 1.1 \dot{ι}, x_{12}^{(0)} = - 1.2 - 1.1 \dot{ι}, \\ x_{13}^{(0)} = 1.2 + 1.1 \dot{ι}, x_{14}^{(0)} = 1.2 - 1.1 \dot{ι}, x_{15}^{(0)} = 2.2 + 1.1 \dot{ι}, x_{16}^{(0)} = 2.2 - 1.1 \dot{ι}, x_{17}^{(0)} = 1.2 + 3.1 \dot{ι}, x_{18}^{(0)} = 1.2 - 3.1 \dot{ι}, x_{19}^{(0)} = 0.2 + 4.1 \dot{ι}, x_{20}^{(0)} = 0.2 - 4.1 \dot{ι}, x_{21}^{(0)} = 1.1 + 0.2 \dot{ι} . \end{array}

Table 2 represents the results obtained from different iterative methods, observing that the number of iterations in most of the proposed method is less than the number of iterations of the known schemes. We can observe that

T_{4 t}

and

M_{4 t}

obtain the best results as compared to existing schemes.

Example 3.

Consider the polynomial of degree seven

f (x) = x^{7} + x^{5} - 10 x^{4} - x^{3} - x + 10,

with exact roots

ξ_{1} = 2, ξ_{2} = 1, ξ_{3} = - 1, ξ_{4} = \dot{ι}, ξ_{5} = - \dot{ι}, ξ_{6} = - 1 + 2 \dot{ι}, ξ_{7} = - 1 - 2 \dot{ι} .

The initial approximations taken are:

\begin{array}{l} x_{1}^{(0)} = 1.66 + 0.23 \dot{ι}, x_{2}^{(0)} = 1.36 - 0.31 \dot{ι}, x_{3}^{(0)} = - 0.76 + 0.18 \dot{ι}, x_{4}^{(0)} = - 0.35 + 1.17 \dot{ι}, x_{5}^{(0)} = 0.29 - 1.37 \dot{ι}, \\ x_{6}^{(0)} = - 0.75 + 2.36 \dot{ι}, x_{7}^{(0)} = - 1.27 - 1.62 \dot{ι} . \end{array}

Table 3 represents the results obtained from the different iterative methods used until now.

Example 4.

Now, let us consider the polynomial

f (x) = (x + 1) (x + 2) (x^{2} - 2 x + 2) (x^{2} + 1) (x - 2) (x + 2 - \dot{ι}),

whose exact roots are

ξ_{1} = - 1, ξ_{2} = - 2, ξ_{3} = 1 + \dot{ι}, ξ_{4} = 1 - \dot{ι}, ξ_{5} = + \dot{ι}, ξ_{6} = - \dot{ι}, ξ_{7} = 2, ξ_{8} = - 2 + \dot{ι} .

In this example, we take as initial approximations:

\begin{array}{l} x_{1}^{(0)} = - 1.3 + 0.2 \dot{ι}, x_{2}^{(0)} = - 2.2 - 0.3 \dot{ι}, x_{3}^{(0)} = 1.3 + 1.2 \dot{ι}, x_{4}^{(0)} = 0.7 - 1.2 \dot{ι}, x_{5}^{(0)} = - 0.2 + 0.8 \dot{ι}, x_{6}^{(0)} = 0.2 - 1.3 \dot{ι}, \\ \begin{matrix} x_{7}^{(0)} = 2.2 - 0.3 \dot{ι}, x_{8}^{(0)} = - 2.2 + 0.7 \dot{ι} . \end{matrix} \end{array}

In Table 4 we can observe the results for this example. Some proposed methods show much better results than the other ones.

Example 5.

In this example, we consider

f (x) = (x + 3) (x - 2 \dot{ι}) (x^{2} + 4 x + 5) (x^{2} - 4 x + 5),

whose roots are

ξ_{1} = - 3, ξ_{2} = 2 \dot{ι}, ξ_{3} = - 2 + \dot{ι}, ξ_{4} = - 2 - \dot{ι}, ξ_{5} = 2 + \dot{ι}, ξ_{6} = 2 - \dot{ι} .

In this case, we use the initial approximations:

\begin{array}{l} x_{1}^{(0)} = - 0.33 + 0.2 \dot{ι}, x_{2}^{(0)} = 0.3 + 2.3 \dot{ι}, x_{3}^{(0)} = - 2.3 + 1.2 \dot{ι}, x_{4}^{(0)} = - 2.3 - 1.2 \dot{ι}, x_{5}^{(0)} = 2.3 + \\ 1.2 \dot{ι}, x_{6}^{(0)} = 2.3 - 1.2 \dot{ι} . \end{array}

Based on the results obtained, shown in Table 5, it is clear that proposed methods are, in general, faster than the existing ones.

Example 6.

To show the real life problems of nonlinear equations, we apply the proposed methods, for Classical Dynamics of Blocks on an inclined planes [19]. The role of eigenvalue/eigenvector pairs is presented here. It is shown that the lower frequencies of eigenvalue/eigenvector pairs dominant the solution of this system. The dynamical problem is to determine block acceleration

a_{B}

and cable tensions

T_{A}

T_{B}

and

T_{C}

just after the system is released from rest. The fourth order matrix from Newton’s laws are given by:

[\begin{matrix} 0.5 m_{A} & - 1 & 0 & 0 \\ m_{B} & 0 & 1 & 0 \\ 0.5 \frac{J}{R} & 0 & - R & R \\ 0.5 m_{p} & 1 & - 1 & - 1 \end{matrix}] [\begin{matrix} a_{B} \\ T_{A} \\ T_{B} \\ T_{C} \end{matrix}] = [\begin{matrix} - μ m_{A} g \\ m_{B} g (s i n (45) - μ c o s (45)) \\ 0 \\ 0 \end{matrix}]

The following numerical values are used:

Block & Pulley Weights:

$W_{A} = 200 l b, W_{B} = 300 l b, W_{P} = 50 l b,$
Surface coefficient of friction μ = 0.2,
Pulley Mass Moment of Inertia

$J = \frac{1}{2} m_{P} R^{2}, A_{B} = \frac{a_{B}}{g}, F_{A} = \frac{T_{A}}{m_{A} g}, F_{A} = \frac{T_{B}}{m_{A} g}, F_{C} = \frac{T_{C}}{m_{A} g}, K_{B} = \frac{m_{B}}{m_{A}}, K_{B} = \frac{T_{B}}{m_{A}}$

with K^B = 0.67, K_P = 0.25.

The dimensionless numerical matrix A is as follows:

A = [\begin{matrix} 0.5 & - 1 & 0 & 0 \\ 0.67 & 0 & 1 & 0 \\ 0.063 & 0 & - 1 & 1 \\ 0.13 & 1 & - 1 & - 1 \end{matrix}]

The characteristic equation of matrix A is given by:

f(x) = (x² - 0.807213 x + 0.072386)(x² + 0.230721x + 0.280855),

whose roots are

ζ₁ ≈ − 1.153605 + 1.215626ι, ζ₂ ≈ − 1.153605 − 1.215626ι, ζ₃ ≈ 0.4036065 + 0.7489738ι, ζ₄ ≈ 0.4036065 − 0.7489738ι.

The initial approximation used are:

x_{1}^{(0)} = - 3 + 3 \dot{ι}, x_{2}^{(0)} = - 3 - 3 \dot{ι}, x_{3}^{(0)} = 2 + 2 \dot{ι}, x_{4}^{(0)} = 2 - 2 \dot{ι} .

The results obtained by the different methods appear in Table 6. The characteristics of these numerical results are similar to those in the other examples.

Example 7.

In this example, we apply the proposed methods for a mechanism composed of a block and disc connected by a pulley system [19]. The role of eigenvalue/eigenvector pairs is shown that the lower frequencies of eigenvalue/eigenvector pairs dominant the solution of this system. The dynamical problem is to determine block acceleration

a_{B}

and disc acceleration

a_{A}

disc friction force

F_{A}

, cable tensions

T_{A}

T_{B}

T_{C}

and

T_{D}

just after the system is released from rest. The seventh order matrix from Newton’s laws are given by:

[\begin{matrix} m_{A} & 0 & - 1 & 1 & 0 & 0 & 0 \\ 5 J_{A} & 0 & 0.2 & 0 & 0 & 0 & 0 \\ 10 J_{C} & 0 & 0 & - 0.1 & 0 & 0.1 & 0 \\ 0 & m_{D} & 0 & 0 & 1 & - 1 & - 1 \\ 0 & 10 J_{D} & 0 & 0 & 0 & - 0.1 & 0.1 \\ 0 & m_{B} & 0 & 0 & - 1 & 0 & 0 \\ - 0.5 & 1 & 0 & 0 & 0 & 0 & 0 \end{matrix}] [\begin{matrix} a_{A} \\ a_{B} \\ F_{A} \\ T_{A} \\ T_{B} \\ T_{C} \\ T_{D} \end{matrix}] = [\begin{matrix} 0 \\ τ_{A} \\ 0 \\ - m_{D} g \\ 0 \\ - m_{B} g \\ 0 \end{matrix}] .

The following numerical values are used:

Disk & Block Masses: $m_{A} = 20 k g$ , $m_{B} = 10 k g$ ,
Pulley Masses: $m_{C} = m_{D} = 15 k g$ ,
Disk Mass Moment of Inertia: $J_{A} = 0.4 k g - m^{2}$ ,
Pulleys Mass Moment of Inertia: $J_{C} = J_{D} = 0.075 k g - m^{2}$ ,
External Torque: $τ_{A} = 10 N - m$ .

The dimensionless numerical matrix A is as follows:

A = [\begin{matrix} 20 & 0 & - 1 & 1 & 0 & 0 & 0 \\ 2 & 0 & 0.2 & 0 & 0 & 0 & 0 \\ 0.75 & 0 & 0 & - 0.1 & 0 & 0.1 & 0 \\ 0 & 15 & 0 & 0 & 1 & - 1 & - 1 \\ 0 & 0.75 & 0 & 0 & 0 & - 0.1 & 0.1 \\ 0 & 10 & 0 & 0 & - 1 & 0 & 0 \\ - 0.5 & 1 & 0 & 0 & 0 & 0 & 0 \end{matrix}] .

The characteristic equation of matrix A is given by

\begin{matrix} f (x) & = & (x - 20.0364) (x - 0.765264) (x - 0.0923177) (x^{2} + 0.388177 x + 0.180708) \\ (x^{2} + 0.505777 x + 0.071346), \end{matrix}

whose roots are

\begin{array}{l} ξ_{1} \approx 20.0364, ξ_{2} \approx 0.765264, ξ_{3} \approx 0.0923177, ξ_{4} \approx - 0.194088 + 1.33019 \dot{ι}, ξ_{5} \approx - 0.194088 - \\ 1.33019 \dot{ι}, ξ_{6} \approx - 0.252888 + 0.085985 \dot{ι}, ξ_{7} \approx - 0.252888 - 0.085985 \dot{ι} . \end{array}

The initial approximation has been taken as:

\begin{array}{l} x_{1}^{(0)} = 21.5 + 0.01 \dot{ι}, x_{2}^{(0)} = 1.7 + 0.02 \dot{ι}, x_{3}^{(0)} = 0.5 + 0.03 \dot{ι}, x_{4}^{(0)} = - 0.55 + 2.3 \dot{ι}, x_{5}^{(0)} = - 0.55 - \\ 2.3 \dot{ι}, x_{6}^{(0)} = - 0.55 + 0.02 \dot{ι}, x_{7}^{(0)} = - 0.55 - 0.02 \dot{ι} . \end{array}

The obtained numerical results for this example appear in Table 7.

In the last table, we show the number of iterations needed in each example for satisfying the stopping criterion.

Table 8. Comparison of the number of iterations with different examples and methods.

Method	$N I_{e x 1}$	$N I_{e x 2}$	$N I_{e x 3}$	$N I_{e x 4}$	$N I_{e x 5}$	$N I_{e x 6}$	$N I_{e x 7}$	Total NI	Average NI
$T_{4 t}$	5	5	6	5	5	5	6	37	5.29
$J_{4 t}$	5	5	6	5	5	5	6	37	5.29
$N_{2 t}$	7	7	7	7	6	8	9	51	7.29
$M_{4 t}$	7	6	7	6	6	7	7	46	6.57
$M M N_{8}$	6	5	6	5	5	6	8	41	5.86
$M R_{6}$	6	6	6	6	6	7	8	45	6.43
$M R_{4}$	9	8	8	8	7	10	11	61	8.71
$S_{5}$	6	6	7	6	6	7	9	47	6.71
$P_{10}$	5	5	5	5	4	5	7	36	5.14

14.1cm NI = Number of iterations.

5. Conclusions

In this manuscript, we combine the Weierstrass method with any iterative scheme for designing new iterative algorithms for approximating simultaneously all the roots of a real or complex polynomial. The order of convergence of the proposed schemes is

2 m

, being m the order of the iterative method that combine with Weierstrass’s scheme.

Several numerical tests confirm the theoretical results and allow us to compare the proposed schemes with other known in the literature.

Author Contributions

Conceptualization, R.B. and A.C.; methodology, J.R.T. and A.C.; software, M.P.; validation, R.B. and J.R.T.; formal analysis, P.T.-N.; investigation, S.B.; resources, S.B.; writing—original draft preparation, M.P.; writing—review and editing, P.T.-N.; visualization, R.B.; supervision, A.C. and J.R.T. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Not applicable

Conflicts of Interest

The authors declare no conflict of interest.

References

Sendov, B., Andreev, A., and Kjurkchiev, N. (1994). Numerical solution of polynomial equations. Handbook of numerical analysis, 3, 625–778. [CrossRef]
Petković, M. S., Petković, L. D., and Džunić, J. (2014). On an efficient simultaneous method for finding polynomial zeros. Appl. Math. Letters, 28, 60–65. [CrossRef]
Cordero, A., Garrido, N., Torregrosa, J.R., and Triguero-Navarro, P. (2022). Iterative schemes for finding all roots simultaneously of nonlinear equations. Appl. Math. Letters, 134, 108325. [CrossRef]
Chinesta, F., Cordero, A., Garrido, N., Torregrosa, J., and Triguero-Navarro, P. (2023). Simultaneous roots for vectorial problems. Comput. Appl. Math., 42:227. [CrossRef]
Proinov, P., and Ivanov, S. (2019). Convergence analysis of Sakurai-Torii-Sugiura iterative method for simultaneous approximation of polynomial zeros. J. Comput. Appl. Math., 357, 56–70. [CrossRef]
Weierstrass, K. (1891). Neuer beweis des satzes, dass jede ganze rationale function einer Veranderlichen dargestellt werden kann als ein product aus linearen functionen derselben Veranderlichen, Sitzungsber. K. Preuss. Akad. Wiss. (Berlin), 1085–1101.
Durand, E. (1960). Numériques des Equations Algébriques, Tome 1: Equations du tipe F(x)=0, Racines d’un Polynome, Masson, Paris.
Dochev, K. (1962). Modified Newton method for the simultaneous computation of all roots of a given algebraic equation. Phys. Math. J. Bulg. Acad. Sci. 5, 136–139.
Kerner, I.O. (1966). Ein gesamtschrittverfahren zur Berechnung der Nullstellen van Polynomen. Numer. Math. 8, 290–294. [CrossRef]
Prešsić, S.B. (1966). Un proced e iteratif pour la factorisation des polynomes. C.R. Acad. Sci. Paris Ser. A, 262, 862–863.
King, R. F. (1973). A family of fourth order methods for nonlinear equations. SIAM J. on Numer Anal., 10(5), 876–879. [CrossRef]
Jarratt, P. (1966). Some fourth order multipoint iterative methods for solving equations. Math. of Comput., 20(95), 434–437. [CrossRef]
Cordero, A., Garrido, N., Torregrosa, J. R., and Triguero-Navarro, P. (2021). Memory in the iterative processes for nonlinear problems. Math. Meth. in the Appl. Sci., 46, 4145–4158. [CrossRef]
Cordero, A., Torregrosa, J. R., and Triguero-Navarro, P. (2021). A general optimal iterative scheme with arbitrary order of convergence. Symmetry, 13(5), 884. [CrossRef]
Mir, N. A., Shams, M., Rafiq, N., Akram, S., and Ahmed, R. (2020). On Family of Simultaneous Method for Finding Distinct as Well as Multiple Roots of Non-linear Equation. Punjab University J. of Math., 52(6).
Mir, N. A., Muneer, R., and Jabeen, I. (2011). Some families of two-step simultaneous methods for determining zeros of nonlinear equations. ISRN Appl. Math., 1–11. [CrossRef]
Shams, M., Rafiq, N., Kausar, N., Agarwal, P., Park, C., and Mir, N. A. (2021). On iterative techniques for estimating all roots of nonlinear equation and its system with application in differential equation. Adv. Diff. Equations, 1, 1–18. [CrossRef]
Cordero, A., and Torregrosa, J. R. (2007). Variants of Newton’s method using fifth-order quadrature formulas. Applied Mathematics and Computation, 190(1), 686-698. [CrossRef]
Panza, M. J. (2018). Application of power method and dominant eigenvector/eigenvalue concept for approximate eigenspace solutions to mechanical engineering algebraic systems. American Journal of Mechanical Engineering, 6(3), 98–113. [CrossRef]

Table 1. Numerical results for Example 1 based on various methods.

Method	$\| \| x^{(3)} - x^{(2)} \| \|$	$\| \| x^{(4)} - x^{(3)} \| \|$	$\| \| x^{(5)} - x^{(4)} \| \|$	$\| \| f (x^{(5)}) \| \|$	$ρ$	Time (s)
$T_{4 t}$	$2.2 (- 7)$	$5.7 (- 53)$	$1.3 (- 417)$	$1.9 (- 3331)$	8.0232	0.671
$J_{4 t}$	$1.9 (- 7)$	$2.2 (- 53)$	$7.2 (- 421)$	$1.7 (- 3357)$	8.0230	1.017
$N_{2 t}$	$1.0 (- 2)$	$2.3 (- 8)$	$6.8 (- 31)$	$7.0 (- 118)$	4.1001	0.532
$M_{4 t}$	$1.6 (- 2)$	$3.2 (- 5)$	$1.4 (- 24)$	$4.4 (- 175)$	9.8547	0.657
$M M N_{8}$	$2.3 (- 5)$	$1.2 (- 37)$	$2.5 (- 297)$	$4.9 (- 2375)$	8.1617	0.734
$M R_{6}$	$9.1 (- 5)$	$2.5 (- 26)$	$6.4 (- 158)$	$1.0 (- 948)$	6.1057	0.812
$M R_{4}$	$3.5 (- 1)$	$8.1 (- 2)$	$2.3 (- 4)$	$4.0 (- 11)$	4.3266	0.531
$S_{5}$	$5.4 (- 4)$	$9.8 (- 16)$	$8.0 (- 77)$	$2.5 (- 379)$	5.2340	1.390
$P_{10}$	$2.5 (- 7)$	$2.2 (- 68)$	$8.6 (- 681)$	$4.9 (- 6800)$	10.020	3.313

Table 2. Numerical results for Example 2 based on various methods.

Method	$\| \| x^{(3)} - x^{(2)} \| \|$	$\| \| x^{(4)} - x^{(3)} \| \|$	$\| \| x^{(5)} - x^{(4)} \| \|$	$\| \| f (x^{(5)}) \| \|$	$A C O C$	Time (s)
$T_{4 t}$	$5.6 (- 15)$	$4.8 (- 113)$	$8.4 (- 897)$	$5.8 (- 7155)$	8.0159	3.203
$J_{4 t}$	$9.5 (- 15)$	$4.1 (- 111)$	$3.0 (- 881)$	$1.5 (- 7030)$	8.0145	6.812
$N_{2 t}$	$2.7 (- 4)$	$1.3 (- 14)$	$2.0 (- 55)$	$6.7 (- 207)$	4.0848	2.359
$M_{4 t}$	$2.7 (- 4)$	$1.3 (- 14)$	$2.4 (- 64)$	$1.9 (- 448)$	8.0343	1.703
$M M N_{8}$	$3.0 (- 14)$	$1.9 (- 108)$	$3.1 (- 862)$	$5.5 (- 6883)$	8.0215	4.345
$M R_{6}$	$9.7 (- 7)$	$7.2 (- 38)$	$6.3 (- 227)$	$4.9 (- 1358)$	6.0741	3.734
$M R_{4}$	$2.6 (- 1)$	$1.3 (- 2)$	$1.0 (- 7)$	$9.7 (- 19)$	5.7548	1.328
$S_{5}$	$8.8 (- 7)$	$4.2 (- 30)$	$6.0 (- 146)$	$2.9 (- 716)$	5.0487	9.390
$P_{10}$	$1.7 (- 17)$	$7.1 (- 166)$	$3.7 (- 1652)$	$7.8 (- 16500)$	10.018	31.219

Table 3. Numerical results for Example 3 based on various methods.

Method	$\| \| x^{(3)} - x^{(2)} \| \|$	$\| \| x^{(4)} - x^{(3)} \| \|$	$\| \| x^{(5)} - x^{(4)} \| \|$	$\| \| f (x^{(5)}) \| \|$	$A C O C$	Time (s)
$T_{4 t}$	$6.8 (- 5)$	$7.7 (- 34)$	$2.2 (- 265)$	$2.3 (- 2115)$	8.0366	0.985
$J_{4 t}$	$1.2 (- 4)$	$8.5 (- 32)$	$4.9 (- 249)$	$1.7 (- 1989)$	8.0390	0.927
$N_{2 t}$	$5.5 (- 2)$	$6.9 (- 6)$	$2.0 (- 21)$	$2.1 (- 81)$	4.1600	0.311
$M_{4 t}$	$5.7 (- 2)$	$4.6 (- 6)$	$2.1 (- 35)$	$2.6 (- 266)$	8.0321	1.109
$M M N_{8}$	$3.6 (- 3)$	$1.9 (- 23)$	$1.5 (- 183)$	$1.3 (- 1462)$	8.1090	0.735
$M R_{6}$	$8.5 (- 3)$	$4.8 (- 13)$	$7.5 (- 76)$	$4.7 (- 453)$	6.1383	0.422
$M R_{4}$	$1.9 (- 1)$	$5.8 (- 3)$	$9.0 (- 9)$	$9.2 (- 30)$	4.4467	0.376
$S_{5}$	$6.5 (- 3)$	$2.5 (- 11)$	$3.1 (- 52)$	$4.6 (- 256)$	5.1550	0.530
$P_{10}$	$7.7 (- 5)$	$6.8 (- 42)$	$3.7 (- 411)$	$3.9 (- 4103)$	10.121	2.219

Table 4. Numerical results for Example 4 based on various methods.

Method	$\| \| x^{(3)} - x^{(2)} \| \|$	$\| \| x^{(4)} - x^{(3)} \| \|$	$\| \| x^{(5)} - x^{(4)} \| \|$	$\| \| f (x^{(5)}) \| \|$	$A C O C$	Time (s)
$T_{4 t}$	$6.0 (- 11)$	$3.2 (- 82)$	$2.7 (- 652)$	$1.9 (- 5210)$	8.0151	1.047
$J_{4 t}$	$1.2 (- 10)$	$1.3 (- 79)$	$1.7 (- 631)$	$4.4 (- 5044)$	8.0154	1.738
$N_{2 t}$	$1.3 (- 3)$	$3.2 (- 12)$	$1.7 (- 46)$	$2.1 (- 181)$	4.0672	0.511
$M_{4 t}$	$1.2 (- 3)$	$1.1 (- 15)$	$3.9 (- 111)$	$4.3 (- 872)$	8.1326	0.500
$M M N_{8}$	$3.0 (- 9)$	$7.4 (- 69)$	$5.2 (- 545)$	$3.8 (- 4357)$	8.0358	1.113
$M R_{6}$	$2.0 (- 4)$	$3.4 (- 22)$	$8.4 (- 129)$	$2.8 (- 766)$	6.0439	1.578
$M R_{4}$	$8.9 (- 2)$	$2.5 (- 4)$	$2.2 (- 14)$	$1.9 (- 52)$	4.4202	0.390
$S_{5}$	$3.2 (- 5)$	$9.6 (- 22)$	$2.4 (- 104)$	$3.5 (- 515)$	5.0449	1.828
$P_{10}$	$9.1 (- 5)$	$1.1 (- 42)$	$6.4 (- 420)$	$1.2 (- 4192)$	10.139	2.593

Table 5. Numerical results for Example 5 based on various methods.

Method	$\| \| x^{(3)} - x^{(2)} \| \|$	$\| \| x^{(4)} - x^{(3)} \| \|$	$\| \| x^{(5)} - x^{(4)} \| \|$	$\| \| f (x^{(5)}) \| \|$	$A C O C$	Time (s)
$T_{4 t}$	$4.4 (- 30)$	$2.4 (- 236)$	$1.7 (- 1886)$	$2.9 (- 15085)$	8.0051	0.735
$J_{4 t}$	$6.3 (- 30)$	$4.4 (- 235)$	$2.9 (- 1876)$	$1.6 (- 15003)$	8.0051	0.875
$N_{2 t}$	$1.7 (- 7)$	$1.2 (- 28)$	$3.7 (- 113)$	$6.7 (- 449)$	4.0305	0.36
$M_{4 t}$	$1.6 (- 5)$	$4.3 (- 32)$	$1.2 (- 244)$	$1.3 (- 1942)$	8.0400	0.360
$M M N_{8}$	$1.0 (- 18)$	$8.8 (- 146)$	$3.7 (- 1162)$	$5.9 (- 9291)$	8.0086	0.422
$M R_{6}$	$1.6 (- 7)$	$9.2 (- 43)$	$2.6 (- 256)$	$1.4 (- 1538)$	6.0637	0.468
$M R_{4}$	$2.2 (- 2)$	$1.5 (- 7)$	$1.5 (- 27)$	$1.2 (- 105)$	4.0698	0.344
$S_{5}$	$8.9 (- 9)$	$6.3 (- 41)$	$8.6 (- 202)$	$9.9 (- 1004)$	5.0354	0.937
$P_{10}$	$9.4 (- 37)$	$2.1 (- 362)$	$1.9 (- 3617)$	$1.9 (- 36170)$	10.014	0.891

Table 6. Numerical results for Example 6 based on various methods.

Method	$\| \| x^{(3)} - x^{(2)} \| \|$	$\| \| x^{(4)} - x^{(3)} \| \|$	$\| \| x^{(5)} - x^{(4)} \| \|$	$\| \| f (x^{(5)}) \| \|$	$A C O C$	Time (s)
$T_{4 t}$	$2.7 (- 5)$	$1.9 (- 39)$	$1.9 (- 312)$	$2.7 (- 2495)$	8.0544	0.359
$J_{4 t}$	$2.8 (- 5)$	$3.3 (- 39)$	$1.8 (- 3101)$	$2.0 (- 2479)$	8.0541	0.359
$N_{2 t}$	$1.9 (- 1)$	$2.5 (- 4)$	$8.7 (- 16)$	$1.1 (- 60)$	4.3190	0.375
$M_{4 t}$	$1.9 (- 1)$	$1.6 (- 5)$	$7.9 (- 38)$	$4.2 (- 295)$	8.4883	0.515
$M M N_{8}$	$1.2 (- 1)$	$2.4 (- 8)$	$6.9 (- 62)$	$2.3 (- 489)$	8.1543	0.406
$M R_{6}$	$3.9 (- 1)$	$2.6 (- 3)$	$3.0 (- 16)$	$4.6 (- 93)$	6.4975	0.406
$M R_{4}$	$8.7 (- 1)$	$5.0 (- 1)$	$2.1 (- 1)$	$7.2 (- 2)$	-1.4440	0.327
$S_{5}$	$1.2 (- 1)$	$4.0 (- 5)$	$2.8 (- 22)$	$4.4 (- 107)$	5.2972	0.407
$P_{10}$	$5.4 (- 1)$	$6.6 (- 6)$	$5.5 (- 54)$	$1.1 (- 533)$	10.047	0.641

Table 7. Numerical results for Example 7 based on various methods.

Method	$\| \| x^{(3)} - x^{(2)} \| \|$	$\| \| x^{(4)} - x^{(3)} \| \|$	$\| \| x^{(5)} - x^{(4)} \| \|$	$\| \| f (x^{(5)}) \| \|$	$A C O C$	Time (s)
$T_{4 t}$	$2.3 (- 2)$	$1.9 (- 9)$	$6.3 (- 66)$	$2.3 (- 517)$	8.0187	0.953
$J_{4 t}$	$2.4 (- 2)$	$3.3 (- 9)$	$6.1 (- 64)$	$1.8 (- 501)$	8.1425	2.125
$N_{2 t}$	$1.6 (- 1)$	$1.5 (- 2)$	$5.4 (- 6)$	$2.3 (- 19)$	3.9359	0.766
$M_{4 t}$	$9.0 (- 2)$	$3.4 (- 4)$	$3.3 (- 22)$	$6.5 (- 166)$	8.0586	0.812
$M M N_{8}$	$5.5 (- 1)$	$8.7 (- 2)$	$1.2 (- 5)$	$1.2 (- 35)$	6.8550	5.359
$M R_{6}$	$4.7 (- 1)$	$1.7 (- 1)$	$2.5 (- 3)$	$1.9 (- 14)$	8.0862	5.435
$M R_{4}$	$3.0 (- 1)$	$4.3 (- 1)$	$1.6 (- 1)$	$3.6 (- 2)$	0.16920	2.735
$S_{5}$	$5.5 (- 1)$	$1.3 (- 1)$	$2.9 (- 3)$	$1.1 (- 9)$	4.8264	1.172
$P_{10}$	$1.4 (- 1)$	$3.0 (- 4)$	$1.1 (- 28)$	$9.0 (- 276)$	10.067	5.688

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An Iterative Scheme for Evaluating Simultaneous Roots of Nonlinear Problems with Applications

Abstract

1. Introduction

2. Simultaneous methods construction

2.1. Some special cases

3. Convergence Analysis

4. Numerical experiments

5. Conclusions

Author Contributions

Data Availability Statement

Conflicts of Interest

References

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