Numerical implementation of the QMR algorithm by using discrete stochastic arithmetic

다국어 초록 (Multilingual Abstract)

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 by A, it is necessary
to decide whether to construct the Lanczos vectors vn+1 and wn+1 as regular
or inner vectors. For a regular step it is necessary that Dk = WT
k Vk is nonsingular. Therefore, in the floating-point arithmetic, the smallest
singular value of matrix Dk, min(Dk), is computed and an inner step is
performed if min(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.

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