On efficient numerical approximation of the bilinear form c*A(-1)b

• ID: 2639, RIV: 10104026
• ISSN: 1064-8275, ISBN: not specified
• source: SIAM Journal of Scientific Computing
• keywords: bilinear forms; scattering amplitude; method of moments; Krylov subspace methods; Lanczos algorithm; Arnoldi algorithm; Gauss-Christoffel quadrature; model reduction
• authors: Zdeněk Strakoš, Petr Tichý
• authors from KNM: Strakoš Zdeněk

Abstract

Let A be a nonsingular complex matrix and b and c be complex vectors. We investigate 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.