The BiConjugate Gradient method often displays rather irregular convergence behavior. Moreover, the implicit decomposition of the reduced tridiagonal system may not exist, resulting in breakdown of the algorithm. A related algorithm, the Quasi-Minimal Residual method of Freund and Nachtigal [102], [103] attempts to overcome these problems. The main idea behind this algorithm is to solve the reduced tridiagonal system in a least squares sense, similar to the approach followed in GMRES. Since the constructed basis for the Krylov subspace is bi-orthogonal , rather than orthogonal as in GMRES, the obtained solution is viewed as a quasi-minimal residual solution, which explains the name. Additionally, QMR uses look-ahead techniques to avoid breakdowns in the underlying Lanczos process, which makes it more robust than BiCG.

**Figure:** The Preconditioned Quasi Minimal Residual Method
without Look-ahead

Mon Nov 20 08:52:54 EST 1995