Muhammad Saqib¹, Muhammad Iqbal², Shahzad Ahmed², Shahid Ali², Tariq Ismaeel^3
1Department of Mathematics, Govt. Degree College, Kharian, Pakistan.
²Department of Mathematics, Lahore Leads University, Lahore, Pakistan.
³Department of Mathematics, GC University, Lahore, Pakistan.
DOI: 10.4236/am.2015.611163   PDF    HTML   XML   4,831 Downloads   7,349 Views   Citations

Abstract

In this paper, we have modified fixed point method and have established two new iterative methods of order two and three. We have discussed their convergence analysis and comparison with some other existing iterative methods for solving nonlinear equations.

Keywords

Modifications, Fixed Point Method, Nonlinear Equations

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1. Introduction

In recent much attention has been given to establish new higher order iteration schemes for solving nonlinear equations. Many iteration schemes have been established by using Taylor series, Adomain decomposition, Homotopy pertrubation technique and other decomposition techniques [1] -[6] . We shall modify the fixed point method using taylor series on the functional equation of nonlinear equation. Initially, we do not put any restrictions on the original function f. In fixed point method, we rewrite as where

1) There exist such that for all

2) There exist such that for all

The order of convergence of a sequence of approximation is defined as:

Definition 1.1 [7] Let the sequence converges to. If there is a positive integer p and real number C such that

then p is order of convergence.

Theorem 1.2 (see [6] ). Suppose that. If, for and , then the sequence is of order m.

2. New Iteration Scheme

Consider the nonlinear equation

(2.1)

we can rewrite the above equation as

(2.2)

We suppose that is a root of (2.1) and is initial guess close to. We can rewrite Equation (2.2) by using Taylor’s expansion as:

(2.3)

if we truncate Equation (2.3) after second term then, we obtained

From above formulation we suggest the following algorithm for solving nonlinear Equation (2.1).

In algorithem form, we can write

we approximate

Thus

if we take

then we have the following algorithem;

Algorithm 2.1 For a given, we approximation solution by the iteration scheme:

If we truncate Equation (2.3) after third term then we have

In algorithem form, we can write

we approximate

By substituting in above, we have

Thus, we have the following algorithem;

Algorithm 2.2 For a given, we approximation solution by the iteration scheme:

3. Convegence Analysis

In this section, we discuss the convergence of Algorithm (2.1) and (2.2).

Theorem 3.1 Let for an open interval I and consider that the nonlinear equation (or) has simple root, where be sufficiently smooth in the neighbourhood of the root. If is sufficiently close to then iteration scheme defined by Algorithm 2.1 has at least second order convergence.

Proof. Let be simple zero of and be its functional equation. Let and be errors at n^th and (n + 1)^th iterations respectively. Then expanding and about, we have

(3.1)

and

(3.2)

Algorithem (2.1) is given by

By substituting values from Equations (3.1) and (3.2) in above, we get

Hence algorithem (2.1) has second order convergence.

Theorem 3.2 Let for an open interval I and consider that the nonlinear equation (or) has simple root, where be sufficiently smooth in the neighbourhood of the root. If is sufficiently close to then iteration scheme defined by Algorithm 2.2 has at least third order convergence.

Proof. Let be simple zero of and be its functional equation. Let and be errors at and iterations respectively. Then expanding, and about, we have

(3.3)

(3.4)

(3.5)

Algorithem (2.2) is given by

By substituting values from Equations (3.3), (3.4) and (3.5) in above, we get

Hence the order of convergence fo algorithm 2.2 is least 3.

4. Numerical Results

In this section, we present some example to make the comparitive study of fixed point method (FPM), Newton method (NM), Abbasbandy method (AM), Homeier method (HM), Chun method (CM), Householder method (HHM), Algorithem 2.1 and Algorithm 2.2 developed in this paper. We use. The following criterias are used for computer programs:

1)

2)

We consider the following examples to illustarate the performance of our newly established iteration scheme.

Comparison Table

5. Conclusion

We have modified the fixed point method for solving nonlinear equations. We have established two new algorithems of convergence order two and three. We have solved some nonlinear equations to show the performance and efficiency of our newly developed iteration schemes. From comparison table, we conclude that these schemes perform much better than Newton method, Abbasbandy method, Chun method, Homeier method, Householder method etc.

Conflicts of Interest

The authors declare no conflicts of interest.

References

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