Lanczos version of BCR algorithm for solving the generalised second‐order Sylvester matrix equation

Abstract

It is well known that the biconjugate residual (BCR) algorithm and its variants are powerful procedures to find the solution of large sparse non-symmetric systems equation A x = b . In this study, the authors develop the Lanczos version of BCR algorithm for computing the solution pair ( V , W ) of the generalised second-order Sylvester matrix equation E V F + G V H + B V C = D W E + M , which includes the second-order Sylvester, Lyapunov and Stein matrix equations as special cases. The convergence results show that the algorithm with any initial matrices converges to the solutions within a finite number of iterations in the absence of round-off errors. Finally, two numerical examples are provided to support the theoretical findings and to testify the effectiveness and usefulness of the algorithm.