Variants of BICGSTAB for Matrices with Complex Spectrum

Recently Van der Vorst [SIAM J. Sci. Statist. Comput.,13 (1992), pp. 631–644] proposed for solving nonsymmetric linear systems $Az = b$ a biconjugate gradient (BICG)-based Krylov space method called BICGSTAB that, like the biconjugate gradient squared (BICGS) method of Sonneveld, does not require matrix–vector multiplications with the transposed matrix AT, and that has typically a much smoother convergence behavior than BICG and BICGS. Its nth residual polynomial is the product of the one of BICG (i.e., the nth Lanczos polynomial) with a polynomial of the same degree with real zeros. Therefore, nonreal eigenvalues of A are not approximated well by the second polynomial factor. Here, the author presents for real nonsymmetric matrices a method BICGSTAB2 in which the second factor may have complex conjugate zeros. Moreover, versions suitable for complex matrices are given for both methods.