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# On Efficient Numerical Approximation of the Bilinear Form c* A(-1)b

1. 1.
0358802 - UIVT-O 2012 RIV US eng J - Článek v odborném periodiku
Strakoš, Z. - Tichý, Petr
On Efficient Numerical Approximation of the Bilinear Form c* A(-1)b.
SIAM Journal on Scientific Computing. Roč. 33, č. 2 (2011), s. 565-587 ISSN 1064-8275
Grant CEP: GA AV ČR IAA100300802
Grant ostatní:GA ČR(CZ) GA201/09/0917; GA AV ČR(CZ) M100300901
Program:GA
Výzkumný záměr: CEZ:AV0Z10300504
Klíčová slova: 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
Kód oboru RIV: BA - Obecná matematika
Impakt faktor: 1.569, rok: 2011

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.