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

We introduce an iterative method named BiLQ for solving general square linear systems $Ax=b$ that is based on the Lanczos biorthogonalization process defined by least-norm subproblems and that is a natural companion to BiCG and Qmr. Whereas the iterates of BiCG, Cgs, and BiCGStab may not exist when the tridiagonal projection of $A$ is singular, BiLQ is reliable on compatible systems even if $A$ is ill-conditioned or singular. 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 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 Usymlqr based on the orthogonal tridiagonalization process. 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.