Loss of biorthogonality and linear system solvers

This paper is devoted to the analysis of the behaviour, in nite precision arithmetic, of the quasi-minimal residual method without look ahead for the solution of linear systems Ax = b where A is a real or complex matrix of order n. In exact arithmetic, the behaviour of the method as applied to symmetric deenite matrices is well understood; but results about convergence usually suppose that the basis generated during the iterative process satisses the property required in exact arithmetic. This is not the case when we run the computation using nite precision arithmetic, when the orthogonality or biorthogonality condition may no longer be satissed. We show how some distributions of eigenvalues may slow down the convergence of the iterative method, and we consider also a technique that may improve the convergence.