"2"^^ . "33" . "[B5714B90A55B]" . . "On Efficient Numerical Approximation of the Bilinear Form c* A(-1)b"@en . . "Strako\u0161, Z." . "RIV/67985807:_____/11:00358802" . . . "SIAM Journal on Scientific Computing" . . . "P(GA201/09/0917), P(IAA100300802), Z(AV0Z10300504), Z(MSM0021620839)" . . . "On Efficient Numerical Approximation of the Bilinear Form c* A(-1)b" . "23"^^ . . . . "2" . . . "10.1137/090753723" . . "On Efficient Numerical Approximation of the Bilinear Form c* A(-1)b" . . "US - Spojen\u00E9 st\u00E1ty americk\u00E9" . . "218006" . . "Tich\u00FD, Petr" . "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\u2013Christoffel 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."@en . . . . "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\u2013Christoffel 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." . "On Efficient Numerical Approximation of the Bilinear Form c* A(-1)b"@en . . . "RIV/67985807:_____/11:00358802!RIV12-AV0-67985807" . . "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"@en . "1064-8275" . . . . . "000289973500005" . "1"^^ .