Some Families of Two-Step Simultaneous Methods for Determining Zeros of Nonlinear Equations

Abstract

We construct two families of two-step simultaneous methods, one of order four and the other of order six, for determining all the distinct zeros of single variable nonlinear equations. The convergence analysis of both the families of methods and the numerical results are also given in order to demonstrate the efficiency and the performance of the new iterative simultaneous methods.

1. Introduction

Determining the zeros of non-linear equations is among the oldest problems in mathematics, whereas the non-linear equations have wide range of applications in science and engineering.

There are numerical methods which find one root at a time, such as Newton's method, and the methods which find all the roots at a time, namely, simultaneous methods, such as Weierstrass method.

The methods for simultaneous finding of all roots of the non-linear equations are very popular as compared to the methods for individual finding of the roots. These methods have a wider region of convergence, are more stable, and can be implemented for parallel computing. More details on simultaneous methods, their convergence properties, computational efficiency, and parallel implementation may be found in [1–5] and references cited therein.

The main goal of this paper is to develop simultaneous methods which have high convergence order and computational efficiency.

2. A Family of Two-Step Fourth-Order Simultaneous Methods for Distinct Roots

In this section, we develop a family of new two-step iterative methods for the simultaneous approximation of all zeros of a non-linear equation using Weierstrass’ correction.

There exist many one-step iterative methods for extracting roots of non-linear equations. We consider the following two-step alpha family: 𝑦_𝑖=𝑥_𝑖−𝑓𝑥^𝑖𝑧_𝑖𝑓^𝑥_𝑖,=𝑦_𝑖−𝑓𝑦^𝑖𝑓^𝑦_𝑖−𝛼𝑓𝑦_𝑖,(2.1) where 𝛼 is any arbitrary real parameter. This family was proposed by Li et al. [6] and has fourth-order convergence. If 𝛼=0, then (2.1) reduces to the following iterative method: 𝑦_𝑖=𝑥_𝑖−𝑓𝑥^𝑖𝑧_𝑖𝑓^𝑥_𝑖,=𝑦_𝑖−𝑓𝑦^𝑖𝑓^𝑦_𝑖.(2.2) This is a well-known two-step Newton's method presented and considered by Traub [7] and has a fourth-order convergence.

Let 𝑊_𝑖𝑥_𝑖=𝑓𝑥^𝑖∏^{𝑛𝑗≠𝑖𝑗=1}𝑥_𝑖−𝑥_𝑗Weierstrass^correction.(2.3) Equation (2.1) can be written as 𝑦_𝑖=𝑥_𝑖−𝑓𝑥^𝑖𝑧_𝑖𝑓^𝑥_𝑖,=𝑦_𝑖−𝑓𝑦^𝑖/𝑓^𝑦_𝑖1−𝛼𝑓𝑦_𝑖/𝑓^𝑦_𝑖.(2.4) Replacing 𝑓/𝑓′ by 𝑊_𝑖 in (2.4), we get 𝑧_𝑖𝑦_𝑖=𝑥_𝑖−𝑊_𝑖𝑥_𝑖,=𝑦_𝑖−𝑊^𝑖𝑦_𝑖1−𝛼𝑊_𝑖𝑦_𝑖,(2.5) where 𝑊_𝑖(𝑥_𝑖) and 𝑊_𝑖(𝑦_𝑖) are given by (2.3).

Hence, we get a family of new two-step iterative methods (2.5) abbreviated as MR1, which depends upon a real parameter alpha for extracting all roots of a non-linear equation.

Remark 2.1. Two-step Weierstrass’ method is a special case of our family of methods if 𝛼=0, in (2.5).

3. Convergence Analysis

In this section, we prove the following theorem on the convergence order of family of two-step simultaneous methods (2.5).

Theorem 3.1. Let 𝑓(𝑥)=0 be a non-linear equation with n number of simple roots 𝜉₁,𝜉₂,…,𝜉_𝑛. If 𝑥₁,𝑥₂,…,𝑥_𝑛 are the initial approximations of the roots, respectively, then, for arbitrary 𝛼 and sufficiently close initial approximations, the order of convergence of (2.5) equals four.

Proof. We denote, 𝜖_𝑖=𝑥_𝑖−𝜉_𝑖, 𝜖_𝑖′=𝑦_𝑖−𝜉_𝑖, and 𝜖̂_𝑖=𝑧_𝑖−𝜉_𝑖.
Considering the first equation of (2.5), 𝑦_𝑖=𝑥_𝑖−𝑊_𝑖𝑥_𝑖,(3.1) we have that 𝜖^𝑖==𝜖𝜖_𝑖𝑖−𝑊_𝑖𝑥_𝑖,1−𝐴_𝑖,(3.2) where 𝐴_𝑖=𝑊^𝑖𝑥_𝑖𝜖_𝑖=^𝑛_{𝑗≠𝑖𝑗=1}𝑥^𝑖−𝜉_𝑗𝑥_𝑖−𝑥_𝑗.(3.3) Now, if 𝜉_𝑖 is a simple root, then, for small enough 𝜖, |𝑥_𝑖−𝑥_𝑗| is bounded away from zero, and so 𝑥^𝑖−𝜉_𝑗𝑥_𝑖−𝑥_𝑗=1+𝑥^𝑗−𝜉_𝑗𝑥_𝑖−𝑥_𝑗=1+𝑂(𝜖),_𝑛_{𝑗≠𝑖𝑗=1}𝑥_𝑖−𝜉_𝑗𝑥_𝑖−𝑥_𝑗=(1+𝑂(𝜖))^𝑛−1=1+(𝑛−1)𝑂(𝜖)+⋯=1+𝑂(𝜖).(3.4) This implies that 𝐴_𝑖=1+𝑂(𝜖).(3.5) Hence, 𝐴_𝑖−1=𝑂(𝜖).(3.6) Thus, (3.2) gives 𝜖_𝑖′=𝑂𝜖².(3.7) Now, considering second equation of (2.5), we have that 𝜖̂_𝑖=𝜖_𝑖′−𝑊^𝑖𝑦_𝑖=𝜖_𝑖1−𝛼𝑊_𝑖𝑦_𝑖,=𝜖_𝑖1−1−𝛼𝐵𝑊^𝑖_𝑖𝑦_𝑖,=𝜖_𝑖1−𝐵^𝑖1−𝛼𝑊_𝑖𝑦_𝑖⁻¹,=𝜖_𝑖1−𝐵_𝑖1+𝛼𝑊_𝑖𝑦_𝑖+⋯,1−𝐵_𝑖−𝛼𝑊_𝑖𝑦_𝑖𝐵_𝑖−⋯,(3.8) where 𝐵_𝑖=𝑊^𝑖𝑦_𝑖𝜖^𝑖=^𝑛_{𝑗≠𝑖𝑗=1}𝑦^𝑖−𝜉_𝑗𝑦_𝑖−𝑦_𝑗.(3.9) With the same arguments as in (3.2), we have that 𝐵_𝑖=1+𝑂𝜖².(3.10) Now, 𝑊_𝑖=𝑓𝑦^𝑖∏^{𝑛𝑗≠𝑖}𝑦_𝑖−𝑦_𝑗=∏^𝑛𝑗=1𝑦_𝑖−𝜉_𝑗∏^{𝑛𝑗=1𝑗≠𝑖}𝑦_𝑖−𝑦_𝑗=𝜖^𝑖𝑛_{𝑗=1𝑗≠𝑖}𝑦^𝑖−𝜉_𝑗𝑦_𝑖−𝑦_𝑗.(3.11) Using (3.9), this implies that 𝑊_𝑖=𝜖^𝑖𝐵_𝑖.(3.12) Hence, (3.8) gives 𝜖̂_𝑖=𝜖^𝑖1−𝐵_𝑖−𝛼𝜖^𝑖𝐵^2𝑖−⋯.(3.13) Now, 𝜖^𝑖=𝑂(𝜖²)and 𝐵_𝑖=1+𝑂(𝜖²). Thus 𝜖̂_𝑖=𝑂𝜖⁴,(3.14) which proves the theorem.

4. A Family of Two-Step Sixth-Order Simultaneous Methods for Distinct Roots

Here, we develop a new family of two-step simultaneous iterative methods of order six using the same alpha family (2.1).

Consider 𝑁_𝑖𝑥_𝑖=𝑓𝑥^𝑖𝑓^𝑥_𝑖Newton^𝑊scorrection,(4.1)_𝑖𝑥_𝑖=𝑓𝑥^𝑖∏^{𝑛𝑗≠𝑖𝑗=1}𝑥_𝑖−𝑥_𝑗Weierstrass^correction.(4.2) Taking logarithmic derivatives of (4.2), we have that 𝑊^𝑖𝑥_𝑖𝑊_𝑖𝑥_𝑖=𝑓^𝑥_𝑖𝑓𝑥_𝑖−^𝑛_{𝑗≠𝑖𝑗=1}1𝑥_𝑖−𝑥_𝑗.(4.3) Replacing 𝑓′/𝑓 by 𝑊^𝑖/𝑊_𝑖 in (2.4), we have that 𝑦_𝑖=𝑥_𝑖−1𝑧_𝑖𝑊^𝑖𝑥_𝑖/𝑊_𝑖𝑥_𝑖,=𝑦_𝑖−1𝑊^𝑖𝑦_𝑖/𝑊_𝑖𝑦_𝑖−𝛼.(4.4) Using (4.1) and (4.3) in the above, we have that 𝑦_𝑖=𝑥_𝑖−1𝑧_𝑖1/𝑁_𝑖𝑥_𝑖−∑^{𝑛𝑗≠𝑖𝑗=1}1/𝑥_𝑖−𝑥_𝑗,=𝑦_𝑖−11/𝑁_𝑖𝑦_𝑖∑^{𝑛𝑗≠𝑖𝑗=1}1/𝑦_𝑖−𝑦_𝑗−𝛼.(4.5) Thus, we get a family of new simultaneous iterative methods (4.5), abbreviated as MR2.

Remark 4.1. Two-step Ehrlich-Alberth’s method is a special case of our method, if 𝛼=0,in (4.5).

5. Convergence Analysis

Here, we prove the following theorem on the convergence order of a family of two-step simultaneous methods (4.5).

Theorem 5.1. Let  𝑓(𝑥)=0 be a non-linear equation with n number of simple roots 𝜉₁,𝜉₂,…,𝜉_𝑛. If 𝑥₁,𝑥₂,…,𝑥_𝑛 are the initial approximations of the roots, respectively, then, for arbitrary 𝛼 and sufficiently close initial approximations, the order of convergence of family (4.5) is six.

Proof. Consider the Newton correction 𝑁_𝑖𝑥_𝑖=𝑓𝑥^𝑖𝑓′𝑥_𝑖.(5.1) This implies that 1𝑁_𝑖𝑥_𝑖=𝑓^𝑥_𝑖𝑓𝑥_𝑖=^𝑛_𝑗=11𝑥_𝑖−𝜉_𝑗,(5.2) where 𝜉_𝑖 is the exact root and 𝑥_𝑖 is its approximation. This gives 1𝑁_𝑖𝑥_𝑖=𝑥_𝑖−1𝜉_𝑖+^𝑛_{𝑗≠𝑖𝑗=1}1𝑥_𝑖−𝜉_𝑗.(5.3) Using (5.3) in (4.5), we have that 𝑦_𝑖=𝑥_𝑖−11/𝑥_𝑖−𝜉_𝑖+∑^{𝑛𝑗≠𝑖𝑗=1}1/𝑥_𝑖−𝜉_𝑗−∑^{𝑛𝑗≠𝑖𝑗=1}1/𝑥_𝑖−𝑥_𝑗.(5.4) Let 𝜖_𝑖=𝑥_𝑖−𝜉_𝑖,𝜖_𝑖′=𝑦_𝑖−𝜉_𝑖.(5.5) Thus, from (5.4), we have that 𝜖^𝑖=𝜖_𝑖−1=𝜖_𝑖1/𝜖_𝑖+∑^{𝑛𝑗≠𝑖𝑗=1}1/𝑥_𝑖−𝜉_𝑗−∑^{𝑛𝑗≠𝑖𝑗=1}1/𝑥_𝑖−𝑥_𝑗,−𝜖^𝑖=𝜖_𝑖1+𝜖_𝑖∑^{𝑛𝑗≠𝑖𝑗=1}𝑥_𝑖−𝑥_𝑗−𝑥_𝑖−𝜉_𝑗/𝑥_𝑖−𝜉_𝑗𝑥_𝑖−𝑥_𝑗−𝜖^𝑖=𝜖_𝑖1+𝜖_𝑖∑^{𝑛𝑗≠𝑖𝑗=1}−𝑥_𝑗−𝜉_𝑗/𝑥_𝑖−𝜉_𝑗𝑥_𝑖−𝑥_𝑗−𝜖^𝑖1+𝜖_𝑖∑^{𝑛𝑗≠𝑖𝑗=1}𝜖_𝑗𝐴_𝑖𝑗,(5.6) where 𝐴_𝑖𝑗=−1𝑥_𝑖−𝜉_𝑗𝑥_𝑖−𝑥_𝑗.(5.7) On simplification, we have that 𝜖^𝑖=𝜖^2𝑖∑^{𝑛𝑗≠𝑖𝑗=1}𝜖_𝑗𝐴_𝑖𝑗1+𝜖_𝑖∑^{𝑛𝑗≠𝑖𝑗=1}𝜖_𝑗𝐴_𝑖𝑗.(5.8) If we assume that absolute values of all errors 𝜖_𝑗(𝑗=1,…,𝑛) are of the same order as, say |𝜖_𝑗|=𝑂(|𝜖|), then 𝜖^𝑖=𝜖^2𝑖∑^{𝑛𝑗≠𝑖𝑗=1}𝜖_𝑗𝐴_𝑖𝑗1+𝜖_𝑖∑^{𝑛𝑗≠𝑖𝑗=1}𝜖_𝑗𝐴_𝑖𝑗=𝑂𝜖³.(5.9) Now considering the second equation of (4.5), we have that 𝑧_𝑖=𝑦_𝑖−11/𝑁_𝑖−∑^{𝑛𝑗≠𝑖𝑗=1}1/𝑦_𝑖−𝑦_𝑗−𝛼.(5.10) This gives 𝑧_𝑖=𝑦_𝑖−11/𝑦_𝑖−𝜉_𝑖+∑^{𝑛𝑗≠𝑖𝑗=1}1/𝑦_𝑖−𝜉_𝑗−∑^{𝑛𝑗≠𝑖𝑗=1}1/𝑦_𝑖−𝑦_𝑗−𝛼.(5.11) This implies that 𝜖̂_𝑖=𝜖_𝑖′−1=𝜖_𝑖1/𝜖^𝑖+∑^{𝑛𝑗≠𝑖𝑗=1}1/𝑦_𝑖−𝜉_𝑗−∑^{𝑛𝑗≠𝑖𝑗=1}1/𝑦_𝑖−𝑦_𝑗−𝛼′−𝜖^𝑖1+𝜖^𝑖∑^{𝑛𝑗≠𝑖𝑗=1}𝑦_𝑖−𝑦_𝑗−𝑦_𝑖+𝜉_𝑗/𝑦_𝑖−𝜉_𝑗𝑦_𝑖−𝑦_𝑗−𝛼𝜖^𝑖=𝜖^𝑖−𝜖^𝑖1+𝜖^𝑖∑^{𝑛𝑗≠𝑖𝑗=1}−𝑦_𝑗−𝜉_𝑗/𝑦_𝑖−𝜉_𝑗𝑦_𝑖−𝑦_𝑗−𝛼𝜖^𝑖.(5.12) Let 𝐵_𝑖𝑗=−1𝑦_𝑖−𝜉_𝑗𝑦_𝑖−𝑦_𝑗.(5.13) Hence, the above equation becomes 𝜖̂_𝑖=𝜖^𝑖−𝜖^𝑖1+𝜖^𝑖∑^{𝑛𝑗≠𝑖𝑗=1}𝜖^𝑗𝐵_𝑖𝑗−𝛼𝜖^𝑖.(5.14) On simplification, we have that 𝜖̂_𝑖=𝜖^𝑖²∑^{𝑛𝑗≠𝑖𝑗=1}𝜖^𝑗𝐵_𝑖𝑗−𝛼1+𝜖^𝑖∑^{𝑛𝑗≠𝑖𝑗=1}𝜖^𝑗𝐵_𝑖𝑗−𝛼.(5.15) Since 𝜖^𝑖=𝑂(𝜖³), from (5.9), 𝜖̂_𝑖=𝑂𝜖³²∑^{𝑛𝑗≠𝑖𝑗=1}𝜖^𝑗𝐵_𝑖𝑗−𝛼1+𝜖^𝑖∑^{𝑛𝑗≠𝑖𝑗=1}𝜖^𝑗𝐵_𝑖𝑗−𝛼=𝑂𝜖⁶.(5.16) This shows that our family of two-step simultaneous methods (4.5) has sixth-order convergence.

6. Numerical Results

We consider here some numerical examples in order to demonstrate the performance of our family of fourth- and sixth-order two-step simultaneous methods, namely MR1 (2.5) and MR2 (4.5). We compare our family of methods with Zhang et al. method of fifth-order convergence and use the abbreviations as ZPH [8].

All the computations are performed using Maple 7.0, using 64 digits floating point arithmetic. We take 𝜖_𝑖=10⁻³⁰ as tolerance and use the following stopping criteria for estimating the zeros: 𝑒_𝑖=||𝑧^𝑖(𝑛+1)−𝑧^𝑖(𝑛)||<𝜖_𝑖,foreach𝑖,(6.1) and 𝑒_𝑖 represents the absolute error.

In all the examples for MR1 and MR2, we have taken 𝛼=0.5.

Example 6.1. Consider 𝑓(𝑧)=𝑧⁴−1(6.2) with the exact zeros 𝜉₁=1,𝜉₂=−1,𝜉₃=𝑖,𝜉₄=−𝑖.(6.3) The initial approximations have been taken as: 𝑧¹⁽⁰⁾=0.5+0.5𝑖,𝑧²⁽⁰⁾=−1.36+0.42𝑖,𝑧³⁽⁰⁾=−0.25+1.28𝑖,𝑧⁴⁽⁰⁾=0.46−1.37𝑖.(6.4) The numerical comparison is given in Table 1.

Table 1 

Example 6.2. Consider 𝑓(𝑧)=𝑧⁷+𝑧⁵−10𝑧⁴−𝑧³−𝑧+10(6.5) with the exact zeros 𝜉₁=2,𝜉₂=1,𝜉₃=−1,𝜉₄=𝑖,𝜉₅=−𝑖,𝜉₆=−1+2𝑖,𝜉₇=−1−2𝑖.(6.6) The initial approximations have been taken as 𝑧¹⁽⁰⁾=1.66𝑧+0.23𝑖,𝑧²⁽⁰⁾=1.36−0.31𝑖,𝑧³⁽⁰⁾=−0.76+0.18𝑖,𝑧⁴⁽⁰⁾=−0.35+1.17𝑖,₅₍₀₎=0.29−1.37𝑖,𝑧⁶⁽⁰⁾=−0.75+2.36𝑖,𝑧⁷⁽⁰⁾=−1.27−1.62𝑖.(6.7) The numerical comparison is shown in Table 2.

Table 2 

Example 6.3. Consider 𝑓(𝑧)=𝑧³+5𝑧²−4𝑧−20+cos𝑧³+5𝑧²−4𝑧−20−1(6.8) with the exact zeros 𝜉₁=−5,𝜉₂=−2,𝜉₃=2.(6.9) The initial approximations have been taken as: 𝑧¹⁽⁰⁾=−5.1,𝑧²⁽⁰⁾=−1.8,𝑧³⁽⁰⁾=1.9.(6.10) The numerical comparision is shown in Table 3.

Table 3 

7. Conclusions

We have developed here two families of two-step simultaneous iterative methods of order four and six for determination of all the distinct zeros of non-linear equations. Weierstrass’ two-step and Ehrlich-Alberth’s two-step simultaneous methods are special cases of our family of methods. From Tables 1–3, we observe that our methods are very effective and efficient as compared to fifth-order simultaneous method of Zhang et al. [8]. Our results can be considered as an improvement and generalization of the previously known results in the existing literature.