The Relation between Galerkin-Type and 1-Norm Quasi-Minimal Residual Iterative Methods

The main ingredients of any Krylov subspace method for the solution of systems of linear equations with nonsingular, in general non-Hermitian coefficient matrix are the generation of a suitable basis and the definition of the actual iterates. Two different strategies for defining the iterates are the Galerkin-type approach and the 1-norm quasi-minimal residual approach. Given any process to form a basis, it is shown that applying the 1-norm quasi-minimal residual approach corresponds to trivial residual smoothing of Galerkin-type iterative methods. An example involving the non-Hermitian Lanczos algorithm without look-ahead as the underlying technique for the generation of a basis is used to illustrate this relationship