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 Σ i=1p (A i XB i + C i X 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. © 2014 Institute of Automation, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg.