^{1}

^{*}

^{2}

^{*}

^{2}

^{*}

^{2}

^{*}

^{3}

^{*}

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.

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) There exist

2) There exist

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

Definition 1.1 [

then p is order of convergence.

Theorem 1.2 (see [

Consider the nonlinear equation

we can rewrite the above equation as

We suppose that

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

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

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

Theorem 3.1 Let

Proof. Let ^{th} and (n + 1)^{th} iterations respectively. Then expanding

and

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

Proof. Let

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.

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

1)

2)

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

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.

MuhammadSaqib,MuhammadIqbal,ShahzadAhmed,ShahidAli,TariqIsmaeel, (2015) New Modification of Fixed Point Iterative Method for Solving Nonlinear Equations. Applied Mathematics,06,1857-1863. doi: 10.4236/am.2015.611163