Developing Bi-CG and Bi-CR Methods to Solve Generalized Sylvester-transpose Matrix Equations

Abstract

The bi-conjugate gradients (Bi-CG) and bi-conjugate residual (Bi-CR) methods are powerful tools for solving nonsymmetric linear systems Ax = b. By using Kronecker product and vectorization operator, this paper develops the Bi-CG and Bi-CR methods for the solution of the generalized Sylvester-transpose matrix equation \(\sum _{i = 1}^p({A_i}X{B_i} + {C_i}{X^{\rm{T}}}{D_i}) = E\) (including Lyapunov, Sylvester and Sylvester-transpose matrix equations as special cases). Numerical results validate that the proposed algorithms are much more efficient than some existing algorithms.