The right–left preconditioning technique for the solution of the large matrix equation AXB = C

Abstract

In this paper, the linear matrix equation $AXB=C$ is considered, where $A\in {\mathbb{R}}^{n\times m}$, $B\in {\mathbb{R}}^{q\times s}$ are given large matrices and $C\in {\mathbb{R}}^{n\times s}$, $X\in {\mathbb{R}}^{m\times q}$ are right-hand side and unknown matrices, respectively. The global least squares algorithm is applied to approximate the solution of this group of matrix equations. The right–left preconditioned global least squares algorithm is presented for obtaining the approximate solution of the mentioned matrix equation. This preconditioner is based on the C-orthogonalization process, where C is a symmetric positive definite matrix. Also, the iterative method and its preconditioned algorithm are proposed to find the approximate generalized inverses of nearly singular matrices and rectangular matrices. Finally, some numerical experiments are given to illustrate the efficiency of the new preconditioners.

Keywords:

• preconditioner
• matrix equation
• GL-LSQR algorithm
• iterative method
• symmetric positive matrix
• pseudo-inverse

AMS Subject Classifications:

• 65F08
• 65F10

Acknowledgments

I would like to thank anonymous referees for their comments and suggestions, which helped me to considerably improve the manuscript. Also, I thank Prof. D. Khojasteh Salkuyeh for his helps to improve the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.