BiLQ: An Iterative Method for Nonsymmetric Linear Systems with a Quasi-Minimum Error Property

TL;DR

The paper introduces BiLQ, a reliable iterative method for solving general square linear systems, especially when the matrix is ill-conditioned or rank deficient, and explores its combination with other methods for enhanced performance.

Contribution

BiLQ is a new iterative method based on Lanczos biorthogonalization, providing reliable solutions for nonsymmetric systems where existing methods may fail, and it can be combined with QMR for simultaneous primal and adjoint system solutions.

Findings

BiLQ reliably solves ill-conditioned or rank-deficient systems.

BiLQR and TriLQR effectively estimate functionals involving primal and adjoint solutions.

BiLQR terminates earlier than TriLQR and MINRES-QLP in experiments.

Abstract

We introduce an iterative method named BiLQ for solving general square linear systems Ax = b based on the Lanczos biorthogonalization process defined by least-norm subproblems, and that is a natural companion to BiCG and QMR. Whereas the BiCG (Fletcher, 1976), CGS (Sonneveld, 1989) and BiCGSTAB (van der Vorst, 1992) iterates may not exist when the tridiagonal projection of A is singular, BiLQ is reliable on compatible systems even if A is ill-conditioned or rank deficient. As in the symmetric case, the BiCG residual is often smaller than the BiLQ residual and, when the BiCG iterate exists, an inexpensive transfer from the BiLQ iterate is possible. Although the Euclidean norm of the BiLQ error is usually not monotonic, it is monotonic in a different norm that depends on the Lanczos vectors. We establish a similar property for the QMR (Freund and Nachtigal, 1991) residual. BiLQ combines with QMR to take advantage of two initial vectors and solve a system and an adjoint system simultaneously at a cost similar to that of applying either method. We derive an analogous combination of USYMLQ and USYMQR based on the orthogonal tridiagonalization process (Saunders, Simon, and Yip, 1988). The resulting combinations, named BiLQR and TriLQR, may be used to estimate integral functionals involving the solution of a primal and an adjoint system. We compare BiLQR and TriLQR with Minres-qlp on a related augmented system, which performs a comparable amount of work and requires comparable storage. In our experiments, BiLQR terminates earlier than TriLQR and MINRES-QLP in terms of residual and error of the primal and adjoint systems.