Block minimum perturbation algorithm based on block Arnoldi process for nonsymmetric linear systems with multiple right-hand sides

In many applications in scientific computing and engineering one has to solve huge sparse linear systems of equations with several right-hand sides. Customarily, one uses the residual error as a stopping condition, but small residue does not imply accurate approximate solutions. Therefore, minimizing quantities other than the residual norm may be more adequate in this context. Based on the above consideration a block minimum perturbation Krylov subspace method (BMinPet) for the solution, Xm satisfying (A−ΔA)Xm=(B+ΔB) at each step, of nonsymmetric linear systems AX=B is shown such that the normwise backward error meets some optimality condition with minimizing the block joint backward perturbation norm ‖(ΔA, ΔB)‖F of the matrix (A, B), and a stopping criterion is given for consistent system with multiple right-hand sides to get a more robust way to stop the iterations. This process is a generalization of minimum perturbation algorithm (MinPert) by Kasenally and Simoncini (1997) (Analysis of a minimum perturbation algorithm for nonsymmetric linear systems. SIAM J. Numer. Anal., 34, 48–66) to multiple right-hand sides. For the benefit of the amount of calculation, we give lower and upper bounds about min ‖(ΔA, ΔB)‖F that are easier to compute. As a by-product, we come up with a conception of Function ψΦ(F, G) which is a generalization of Function ψΦ that is a symmetric gauge (SG) function. The algorithm is used to analyze the performance of BGMRES method by evaluating the proximity of their solutions to block joint backward perturbation optimality. The relationships between BMinPet method and relative methods are investigated. Numerical examples show us that BMinPert shows its superiority compared with BFGMRES-S(m, pf), GsGMRES, Bl-BiCG-rQ, BGMRES and BArnoldi in solving the large sparse ill-conditioned problems.