Application of Different H ( x ) in Homotopy Analysis Methods for Solving Systems of Linear Equations

In this paper, we present homotopy analysis method (HAM) for solving system of linear equations and use of different H(x) in this method. The numerical results indicate that this method performs better than the homotopy perturbation method (HPM) for solving linear systems.


Introduction
Approximating the solutions of the system of linear and nonlinear equations has widespread applications in applied mathematics [1]- [11].Many techniques including homotopy perturbation method (HPM) [12] and iterative methods [13] were suggested to search for the solution of linear systems.In 2009 Keramati [2] and in 2011 Liu [3] in their articles applied HPM to the solution of the system Ax b = .In this article we used homotopy analysis method [14] [15] with different H(x) to solve linear system Ax b = and showed that our results were better than the HPM results; then convergence of the method was considered.Consider a linear system , Ax b = where is nonsingular and , n x b R ∈ is a vector.First of all, the basic ideas of the homotopy analysis method are being discussed.Let 0 x be an initial guess of x, and [ ] 0,1 q ∈ be called the embedding parameter.The homotopy analysis method is based on a kind of continuous mapping ( ) x q φ → such that, as the embedding parameter q increases from 0 to 1, ( ) x q φ varies from the initial guess 0 x to the exact solution x.To ensure this, choose such an auxiliary linear operator as and we define the operator denote the so-called auxiliary parameter and auxiliary matrix, respectively.Using the embedding parameter , we construct a family of equations 2) and (3) we have Obviously, at q = 0 and q = 1, one has ( ) 0 , 0 = respectively.Thus, as q increases from 0 to 1, ( ) varies continuously from 0 x to x.Such kind of continuous variation is called deforma- tion in topology [16].We call the family of equations like (4) the zeroth-order deformation equation.Now we define mth-order deformation derivative is now a function of the embedding parameter q, by Taylors Theorem, we expand ( ) x q φ in a power series of the embedding parameter q as follows: ( ) ( ) ( ) x q x q x q m q φ φ φ x q x x q If the series ( 6) is convergent at q = 1, then using the relationship ( ) Now we have the so-called mth-order deformation equation where and 0 when 1, 1 otherwise.
Now we have to prove the convergence of (15).
which completes the proof.

Main Results
In this section For solving the linear system (1) we apply different H(x) and the convergence of the method is checked.At first assume that A is a nonsingular diagonally dominate matrix and 0, 1, 2, , .
ii a i n ≠ = Dividing (1) by ii a and without loss of generality we can obtain where , , .
Now we apply different H(x) and the convergence of the method is tested.1) we propose ( ) and show that ( ) and first row is satisfied: .
This relation satisfis for other rows also and ( ) We propose ( ) and show that ( ) , and last row is satisfied: This relation satisfis for other rows also ( ) such that S and R was explained in (18) and (20) respectively and show that ( ) Proof: Similar to proof of Theorems ( 2) and (3).4) We propose ( ) ( ) and show that ( ) Proof: Following Theorem (2) after expanding ( ) according to the first row we have This relation satisfis for other rows also ( )

Numerical Results
In this section, we present some numerical examples to apply HAM and HPM methods for solving linear system.We used of Matlab 2013 for numerical results.and the exact solution is Table 1 shows the iteration number,error,spectral radius of iteration matrix and computation time.According to Table 1 we obtain the desirable result for solving this system by seven iterations with HAM and ( ) while by HPM method we used of fourteen iteration.
In this example the matrices S and ( ) S m are same and the results are same too.Example 2. In this example we apply HAM method for solving the linear system

Ax b =
where A is a 1000 1000 × matrix, b is a 1000 1 × vector that its components are sum of the row components of the corresponding matrix and the exact solution is [ ] T 1 1 1 .The numerical results are in Table 2.
Similar to proof of Theorems (3) and (5).6) We propose ( ) H x I U = − such that U is the strictly upper triangular part of A and show that ( ) the strictly upper triangular part of A and R was explained in (20) and show that ( ) Similar to proof of Theorems (3) and(7).Now in the next section we apply ( ) H x for solving numerical examples.

Example 1 .
Consider the linear system Ax b By direct calculation we have