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

Mohammad Hasan Khani¹, Jalil Rashidinia², Sajjad Zia Borujeni^1,2*

¹Department of Mathematics, Islamic Azad University, Shahin Shahr Branch, Isfahan, Iran.
²Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran.

DOI: 10.4236/alamt.2015.53012                

Abstract

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.

Keywords

HAM, HPM, Linear System

1. 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. In this article we used homotopy analysis method [14] [15] with different H(x) to solve linear system and showed that our results were better than the HPM results; then convergence of the method was considered.

Consider a linear system

(1)

where is nonsingular and is a vector.

First of all, the basic ideas of the homotopy analysis method are being discussed.

Let be an initial guess of x, and be called the embedding parameter. The homotopy analysis method is based on a kind of continuous mapping such that, as the embedding parameter q increases from 0 to 1, varies from the initial guess to the exact solution x. To ensure this, choose such an auxiliary linear operator as

(2)

and we define the operator

(3)

Let and denote the so-called auxiliary parameter and auxiliary matrix, respectively. Using the embedding parameter, we construct a family of equations

from (2) and (3) we have

(4)

Obviously, at q = 0 and q = 1, one has and respectively. Thus, as q increases from 0 to 1, varies continuously from to x. Such kind of continuous variation is called deformation in topology [16] . We call the family of equations like (4) the zeroth-order deformation equation. Now we define mth-order deformation derivative

(5)

where Because is now a function of the embedding parameter q, by Taylors Theorem, we expand in a power series of the embedding parameter q as follows:

By using (5) we have

(6)

If the series (6) is convergent at q = 1, then using the relationship one has the series solution

(7)

Now we have the so-called mth-order deformation equation

(8)

where

(9)

and

(10)

By using (2) we obtain

(11)

Also by using (3) and (9) we have

and then

Finally by using (11) we obtain

(12)

Now with the initial guess and we have

(13)

hence, by substituting (13) in (7) we obtain

(14)

and by factor of we have

(15)

Now we have to prove the convergence of (15).

Theorem 1. The sequence is a Cauchy sequence if

Proof: Following ([2] , Theorem 1) we have to show that

Now considering

then

let then

so we have

since then we obtain

which completes the proof.

2. 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 Dividing (1) by and without loss of generality we can obtain

(16)

where, such that

(17)

and

Now we apply different H(x) and the convergence of the method is tested.

1) we propose with

(18)

and show that

Theorem 2. If A is diagonally dominated and, where is defined in (17) then

Proof: By direct calculation we have

and first row is satisfied:

Since A is diagonally dominated, B is diagonally dominated and we have

(19)

Now by using (19) we obtain

This relation satisfis for other rows also and

2) We propose with

(20)

and show that

Theorem 3. If A is diagonally dominated and, where is defined in (17) then

Proof: Following Theorem (2)

such that

and last row is satisfied:

This relation satisfis for other rows also

3) We propose such that S and R was explained in (18) and (20) respectively and show that

Theorem 4. If A is diagonally dominated and, where is defined in (17) then

Proof: Similar to proof of Theorems (2) and (3).

4) We propose with

(21)

and show that

Theorem 5. If A is diagonally dominated and, where is defined in (17) then

Proof: Following Theorem (2) after expanding according to the first row we have

This relation satisfis for other rows also

5) We propose such that and was explained in (21) and (20) respectively and show that

Theorem 6. If A is diagonally dominated and, where is defined in (17) then

Proof: Similar to proof of Theorems (3) and (5).

6) We propose such that U is the strictly upper triangular part of A and show that

Theorem 7. If A is diagonally dominated and, where is defined in (17) then

Proof: Following Theorem (2) after expanding according to the first row we have

This relation satisfis for other rows also

7) We propose such that U is the strictly upper triangular part of A and R was explained in (20) and show that

Theorem 8. If A is diagonally dominated and, where is defined in (17) then

Proof: Similar to proof of Theorems (3) and (7).

Now in the next section we apply for solving numerical examples.

3. 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.

Example 1. Consider the linear system, that 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 are same and the results are same too.

Example 2. In this example we apply HAM method for solving the linear system

where A is a matrix, b is a vector that its components are sum of the row components of the corresponding matrix and the exact solution is. The numerical results are in Table 2.

Table 1. Camparision between HPM and HAM for 3 ´ 3 system.

Table 2. Camparision between HPM and HAM for 1000 ´ 1000 system.

4. Conclusion

From the numerical results, we have seen that the HAM method with different produces a spectral radius smaller than the HPM and with the less iteration we obtain the desirable result.

Acknowledgements

We thank Islamic Azad University for support researcher plan entitled: “Combination of Iterative methods and semi analytic methods for solving linear systems” and the Editor and the referee for their comments.

NOTES

^*Corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

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