Efficient solution of parameter‐dependent quasiseparable systems and computation of meromorphic matrix functions

Abstract

Summary In this paper, we focus on the solution of shifted quasiseparable systems and of more general parameter‐dependent matrix equations with quasiseparable representations. We propose an efficient algorithm exploiting the invariance of the quasiseparable structure under diagonal shifting and inversion. This algorithm is applied to compute various functions of matrices. Numerical experiments show that this approach is fast and numerically robust.

Computing the Reciprocal of a $ϕ$-function by Rational Approximation

“…The matrix function ψ 1 (A) can thus be manipulated efficiently using its resulting data-sparse format combined with the rank-structured matrix technology [8]. Furthermore, when A is rank-structured the action of the matrix ψ 1 (A) on a vector can be computed efficiently using the direct fast solver for shifted linear systems proposed in [4].…”

7th Workshop on Matrix Equations and Tensor Techniques

“…Boito et al consider shifted linear systems with a quasiseparable structure or, equivalently, a Sylvester matrix equation with a quasiseparable and a diagonal matrix coefficient. Such equations arise, for example, from the discretization of nonlocal boundary value problems and, more generally, the approximation of meromorphic functions.…”