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In this paper, we present two different relaxed gradient based iterative (RGI) algorithms for solving the symmetric and skew symmetric solution of the Sylvester matrix equation

For the convenience of our statements, the following notations will be used throughout the paper:

Considering the symmetric (skew symmetric) solution of the Sylvester matrix equation

The Sylvester matrix equation (

As is well known, (

Due to these drawbacks, many other methods for the solution have appeared in the literature. The idea of transforming the coefficient matrix into a Schur or Hessenberg form to compute (

In this paper, inspired by [

The rest of the paper is organized as follows. In Section

In this section, we reviewed the ideas and principles of the gradient based iterative (GI) method, the relaxed gradient based iterative (RGI) method, and the modified gradient based iterative (MGI) method.

Let

The convergence of the gradient based iterative is stated as follows.

Assume matrix

In [

The authors [

Niu et al. [

Recently, in [

More recently, Xie and Ma [

In this section, we first study the necessary and sufficient conditions of the symmetric solution for (

The matrix equation (

If

Conversely, if the system of matrix equations (

According to Theorem

According to Theorem

In the following paragraph, we will investigate the convergence of Algorithm

Assume the matrix equations (

Further the sequence

Define the error matrix

From Theorem

Following the same line, the idea of Algorithm

The matrix equation (

The relaxed gradient based iterative algorithm for solving the skew symmetric solution of (

Similarly, we have the following theorem to ensure the convergence of the Algorithm

Assume the matrix equations (

Furthermore, the sequence

In this section, two numerical examples are used to show the efficiency of the RGI Method. All the computations were performed on Intel® Core™ i7-4500U CPU

In matrix equation (

It is easy to show that the matrix equation (

Taking

Convergence cure of symmetric solution (

In matrix equation (

it is easy to check that the matrix equation (

Also taking

Convergence cure of skew symmetric solution (

The authors declare that they have no conflicts of interest.

This project was supported by NSF China (no. 11471122), Anhui Provincial Natural Science Foundation (no. 1508085MA12), Key Projects of Anhui Provincial University Excellent Talent Support Program (no. gxyqZD2016188), and the University Natural Science Research Key Project of Anhui Province (no. KJ2015A161).