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.