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It is well known that the matrix equations play a significant role in engineering and applicable sciences. In this research article, a new modification of the homotopy perturbation method (HPM) will be proposed to obtain the approximated solution of the matrix equation in the form AX = B. Moreover, the conditions are deduced to check the convergence of the homotopy series. Numerical implementations are adapted to illustrate the properties of the modified method.

Let

Matrix equations are arisen in control theory, signal processing, model reduction, image restoration, ordinary and partial differential equations and several applications in science and engineering. There are various approaches either direct methods or iterative methods to evaluate the solution of these equations [

The HPM that was proposed first time by Doctor He [

In terms of linear algebra, Keramati [

According to our knowledge, nevertheless HPM has not been modified to solve a matrix equation. In this survey, the main contribution is to suggest an improvement of the HPM for finding approximated solution for (1). Moreover, the necessary and sufficient conditions for convergence of the modified method will be investigated. Finally, some numerical experiments and applications are drawn in numerical results.

In this section, first the conditions that Equation (1) has a solution are decelerated. Then, some applicable relations by utilizing HPM will be attained. Eventually, convergence of HPM series will be analyzed in detail.

The following theorems characterize the existence and uniqueness to the solution of Equation (1).

Theorem 2.1. [

Theorem 2.2. [

is a solution of (1), where

Theorem 2.3. [

Remark 2.4. It should be emphasized that when

Now, we are ready to apply the convex homotopy function in order to obtain the solution of linear matrix equation. A general type of homotopy method for solving (1) can be described by setting

A convex homotopy would be in the following form

whenever, the homotopy

Notice that F is an operator with known solution

and it gives an approximation to the solution of (1) as

By substituting (3) and (4) in (5), and by equating the terms with the identical power of p, after simplification and application of the relations, we obtain

If take

Hence, the solution can be expressed in the following form

Remark 2.5. It should be pointed out that we have focused to the solution of matrix equation

Thus, we considered all matrices in Equation (1) are square.

To verify whether the sequence

Theorem 2.6. The sequence

Proof: It is clear that

Hence, if

Thus

Definition 2.7. [

Theorem 2.8. Consider the matrix

Proof: Suppose

Since

Therefore,

which completes the proof. ,

In Theorem 2.8, the important question is “Does the matrix

not SRDD, as a counterexample we can pay attention to the matrices

Now, if

To be more precise, by using convex homotopy function, we can easily verify that

In this part, we would like to show that the series

Definition 2.9. [

Theorem 2.10. Let

converges if

Proof: Suppose that

is converges series. ,

In this section, some numerical illustrations are provided. All computations have been carried out using MATLAB 2012 (Ra) with roundoff error

whereas,

Example 3.1. First example made approximating the solution of the equation

After evaluating the inversion of

Furthermore,

the approximated solution could be obtained as follows:

However, the exact solution of

In conclusion, it can be seen that the approximation has a good agreement with the exact solution. In this case the residual error is

Example 3.2. In this example, two

The solution of matrix equation

Example 3.3 (Application in matrix inversion). If we substitute

This matrix is diagonally dominant and well conditioned matrix. We have used MATLAB command inv(A) with very small error

In this work, the linear matrix equation is solved by improving the well-known perturbation method. Numerical experiments demonstrated that by considering more terms of the approximations, error will be decreased dramatically. Furthermore, if the matrix

Special thanks go to the anonymous referee for some valuable suggestions, which have resulted in the improvement of this work. This work is supported by Islamic Azad University, Robat Karim University, Tehran, Iran.

N | Six terms | Seven terms | Eight terms | |
---|---|---|---|---|

3 | 0.0952 | |||

4 | 0.1463 | |||

5 | 0.1739 | |||

6 | 0.2083 | |||

7 | 0.2400 | |||

8 | 0.2692 | |||

9 | 0.2962 | |||

10 | 0.3214 |

The author declares that there is no conflict of interests regarding the publication of this article.

Amir Sadeghi, (2016) A New Approximation to the Linear Matrix Equation AX = B by Modification of He’s Homotopy Perturbation Method. Advances in Linear Algebra & Matrix Theory,06,23-30. doi: 10.4236/alamt.2016.62004