Flexible BiCG and flexible Bi-CGSTAB for nonsymmetric linear systems

Flexible versions of the BiCG and Bi-CGSTAB algorithms are presented which allow for the use of a different preconditioner at each step of the algorithm. In particular, inexact solutions of the preconditioned equations are allowed, as well as the use of some (inner) iterative method as a preconditioner. Flexible BiCG is investigated theoretically, and a theorem is presented comparing the residual norms of Flexible BiCG and Flexible QMR that generalizes a well-known relation between the residual norms of BiCG and QMR. Numerical experiments are conducted for Flexible Bi-CGSTAB for several matrices including some indefinite and highly nonsymmetric matrices. These experiments illustrate the applicability of the flexible method.