A 1-norm quasi-minimal residual variant of the Bi-CGSTAB algorithm for nonsymmetric linear systems

Abstract

Due to the sheer size of sparse systems of linear equations arising from real-world applications in science and engineering, parallel computing as well as iterative methods are almost mandatory. For the iterative solution of large sparse nonsymmetric linear systems, a 1-norm quasi-minimal residual variant of the biconjugate gradient stabilized method (Bi-CGSTAB) is proposed. The algorithm is inspired by a recent transpose-free 1-norm quasi-minimal residual method (TFQMR/sub 1/) in that it applies the 1-norm quasi-minimal residual approach to Bi-CGSTAB in the same way as TFQMR/sub 1/ is derived from the conjugate gradient squared method (CGS). There is also an intimate connection to a method called QMRCGSTAB that is based on applying the (Euclidean norm) quasi-minimal residual approach to Bi-CGSTAB. Numerical examples are used to compare the convergence behavior of Bi-CGSTAB and its 1-norm quasi-minimal residual variant.