Application of homotopy perturbation methods for solving systems of linear equations

Modified homotopy perturbation method for solving system of linear equations

“…To find the operator T, we may take either U 2 = 0 or U 3 = 0. We remark that, if U 2 = 0, then we obtain the same series of solution derived by Liu (2011). Here, we consider the case U 3 = 0.…”

“…(1.1). Liu (2011) and Karamati (2009) used homotopy perturbation method to suggest some iterative methods for solving a system of linear equations. We would like to mention that the Homotopy perturbation method was developed by He (1999) and has been used for solving a wide class of problems arising in various branches of pure and applied sciences.…”

“…(1.1). It has been shown that the modified homotopy perturbation method provides us the exact solution of the system of linear equations as compared to the series of solution obtained by using homotopy perturbation in Liu (2011). Yusufoglu (2009) has also considered the similar technique with different auxiliary parameter and has obtained the exact solution of the linear system of equations.…”

A New Approximation to the Linear Matrix Equation <i>AX</i> = <i>B</i> by Modification of He’s Homotopy Perturbation Method

“…However, this method does not converge for some systems when the spectrum radius is greater than one. To make the method available, the auxiliary parameter and the auxiliary matrix were added to the homotopy method by Liu [11]. He has adjusted the Richardson method, the Jacobi method, and the Gauss-Seidel method to choose the splitting matrix.…”

A new approach for computing the solution of Sylvester matrix equation

“…However, this method does not converge for some systems when the spectrum radius is greater than one. In order to solve this issue, Liu [23] was added the auxiliary parameter and the auxiliary matrix to the homotopy method. He has adjusted the Richardson method, the Jacobi method, and the Gauss-Seidel method to choose the splitting matrix.…”