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On Efficient Numerical Approximation of the Bilinear Form c* A(-1)b
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SYSNO ASEP 0358802 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Článek ve WOS Title On Efficient Numerical Approximation of the Bilinear Form c* A(-1)b Author(s) Strakoš, Z. (CZ)
Tichý, Petr (UIVT-O) SAI, RID, ORCIDSource Title SIAM Journal on Scientific Computing. - : SIAM Society for Industrial and Applied Mathematics - ISSN 1064-8275
Roč. 33, č. 2 (2011), s. 565-587Number of pages 23 s. Language eng - English Country US - United States Keywords bilinear forms ; scattering amplitude ; method of moments ; Krylov subspace methods ; conjugate gradient method ; biconjugate gradient method ; Lanczos algorithm ; Arnoldi algorithm ; Gauss-Christoffel quadrature ; model reduction Subject RIV BA - General Mathematics R&D Projects IAA100300802 GA AV ČR - Academy of Sciences of the Czech Republic (AV ČR) CEZ AV0Z10300504 - UIVT-O (2005-2011) UT WOS 000289973500005 EID SCOPUS 79957583077 DOI 10.1137/090753723 Annotation Let $A$ be a nonsingular complex matrix and $b$ and $c$ be complex vectors. We investigates approaches for efficient approximations of the bilinear form $c^*A^{-1}b$. Equivalently, we wish to approximate the scalar value $c^*x$, where $x$ solves the linear system $Ax = b$. Here the matrix $A$ can be very large or its elements can be too costly to compute so that $A$ is not explicitly available and it is used only in the form of the matrix-vector product. Therefore a direct method is not an option. For $A$ Hermitian positive definite, $b^*A^{-1}b$ can be efficiently approximated as a by-product of the conjugate-gradient iterations, which is mathematically equivalent to the matching moment approximations computed via the Gauss–Christoffel quadrature. We propose a new method using the biconjugate gradient iterations which is applicable to the general complex case. The proposed approach is compared with existing ones using analytic arguments and numerical experiments. Workplace Institute of Computer Science Contact Tereza Šírová, sirova@cs.cas.cz, Tel.: 266 053 800 Year of Publishing 2012
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