A mixed product Krylov subspace method for solving nonsymmetric linear systems (Q1382024)

The paper presents the so-called mixed BiCGSTAB-CGS method. Since difficulties encountered in the standard CGS and BiCGSTAB methods are of a different type, the authors propose a combination of both methods with a possibility to switch between them at iterations and to avoid using the one for which difficulties arise. The authors derive the recurrence relations needed for the building of the general mixed algorithm and its preconditioned form. The main purpose of the introduced mixed method is to improve the stability of the CGS method. In particular, a CGS based implementation is presented which takes a BiCGSTAB step only if it is necessary. The cost of this variant of the mixed method is more effective. To minimize the extra cost associated with the number of the switchings, it is necessary to choice a good switching criterion. The mixed method can be implemented to control local increase in the residual norm. Numerical examples demonstrate the stabilizing effect of the mixed algorithm.