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 byA, it is necessary to decide whether to construct the Lanczos vectorsv n +1 andw n +1 as regular or inner vectors. For a regular step it is necessary thatD k =W T k V k is nonsingular. Therefore, in the floating-point arithmetic, the smallest singular value of matrix Dk,σ min (D k ), is computed and an inner step is performed ifσ min (D k )<∈, 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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Faezeh Toutounian received her B. Sc in Mathematics from Ferdowsi University of Mashhad, Iran, two degree of M. Sc in Mathematical statistics and applied computer and her Ph. D in Mathematics from Paris VI University, France. She spent two sabbatical years in 1985 and 1996 at Paris VI University. She is currently a professor of Mathematics at Ferdowsi University of mashhad. Her research interests are mainly numerical linear algebra, iterative methods and error analysis.
Davod Khojasteh Salkuyeh received his B. Sc from Sharif University of Technology, Tehran, Iran and his M. Sc from Ferdowsi University of Mashhad, Mashhad, Iran. He received his Ph. D degree under supervision of professor Faezeh Toutounian at Ferdowsi University of Mashhad in 2003. He is currently an assistant professor of Mathematics at Mohaghegh Ardabili University of Ardabil, Iran. His research interests are mainly iterative methods for sparse linear systems and finite element method.
Bahram Asadi received his M. Sc degree under supervision of professor Faezeh Toutounian at Ferdowsi University of Mashhad, Iran. He is currently a lecturer in Islamic Azad Univrsity of Hamadan, Iran. His research interest is mainly iterative methods for sparse linear systems.
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Toutounian, F., Salkuyeh, D.K. & Asadi, B. Numerical implementation of the QMR algorithm by using discrete stochastic arithmetic. JAMC 17, 457–473 (2005). https://doi.org/10.1007/BF02936068
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DOI: https://doi.org/10.1007/BF02936068