CGS, a fast Lanczos-type solver for nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
SIAM Journal on Scientific and Statistical Computing
Variants of BICGSTAB for matrices with complex spectrum
SIAM Journal on Scientific Computing
Matrix computations (3rd ed.)
Estimating the Attainable Accuracy of Recursively Computed Residual Methods
SIAM Journal on Matrix Analysis and Applications
LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares
ACM Transactions on Mathematical Software (TOMS)
Algorithm 583: LSQR: Sparse Linear Equations and Least Squares Problems
ACM Transactions on Mathematical Software (TOMS)
Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems
Applied Numerical Mathematics
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Quadrature Rules Based on the Arnoldi Process
SIAM Journal on Matrix Analysis and Applications
Modified Gram-Schmidt (MGS), Least Squares, and Backward Stability of MGS-GMRES
SIAM Journal on Matrix Analysis and Applications
Matrices, Moments and Quadrature with Applications
Matrices, Moments and Quadrature with Applications
A Posteriori Error Estimates Including Algebraic Error and Stopping Criteria for Iterative Solvers
SIAM Journal on Scientific Computing
Block conjugate gradient type methods for the approximation of bilinear form CHA-1B
Computers & Mathematics with Applications
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Let $A\in\mathbb{C}^{N\times N}$ be a nonsingular complex matrix and $b$ and $c$ be complex vectors of length $N$. The goal of this paper is to 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. In this paper we propose a new method using the biconjugate gradient iterations which is applicable to the general complex case. The proposed approach will be compared with existing ones using analytic arguments and numerical experiments.