Numerical implementation of the QMR algorithm by using discrete stochastic arithmetic - Toutounian, Faezeh; Salkuyeh, Davod Khojasteh; Asadi, Bahram (Journal of applied mathematics & computing, 2005)
In each step of the quasi-minimal residual (QMR) method which uses a look-ahead variant of the nonsymmetric Lanczos process to generate basis vectors for the Krylov subspaces induced byA, it is necessary to decide whether to construct the Lanczos vectorsvn+1 andwn+1 as regular or inner vectors. For a regular step it is necessary thatDk=WkTVk is nonsingular. Therefore, in the floating-point arithmetic, the smallest singular value of matrix Dk,σmin(Dk), is computed and an inner step is performed ifσmin(Dk)<∈, where ∈ is a suitably chosen tolerance. In practice it is absolutely impossible to choose correctly the value of the tolerance ∈. The subject of this paper is to show how discrete stochastic arithmetic remedies the problem of this tolerance, as well as the problem of the other tolerances which are needed in the other checks of the QMR method with the estimation of the accuracy of some intermediate results. Numerical examples are used to show the good numerical properties.