**Previous:** BiConjugate Gradient (BiCG)

**Up:** Nonstationary Iterative Methods

**Next:** Conjugate Gradient Squared Method (CGS)

**Previous Page:** Implementation

**Next Page:** Convergence

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 [100], [101] 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.