Computation of generalized matrix functions with rational Krylov methods

Computation of Generalized Matrix Functions

We develop numerical algorithms for the efficient evaluation of quantities associated with generalized matrix functions [J. B. Hawkins and A. Ben-Israel, Linear and Multilinear Algebra, 1(2), 1973, pp. 163–171]. Our algorithms are based on Gaussian quadrature and Golub–Kahan bidiagonalization. Block variants are also investigated. Numerical experiments are performed to illustrate the effectiven...

Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection∗

Matrix functions are a central topic of linear algebra, and problems of their numerical approximation appear increasingly often in scientific computing. We review various rational Krylov methods for the computation of large-scale matrix functions. Emphasis is put on the rational Arnoldi method and variants thereof, namely, the extended Krylov subspace method and the shift-and-invert Arnoldi met...

Generalized Rational Krylov Decompositions with an Application to Rational Approximation

Generalized rational Krylov decompositions are matrix relations which, under certain conditions, are associated with rational Krylov spaces. We study the algebraic properties of such decompositions and present an implicit Q theorem for rational Krylov spaces. Transformations on rational Krylov decompositions allow for changing the poles of a rational Krylov space without recomputation, and two ...

Rational Krylov sequences and Orthogonal Rational Functions

In this paper we study the relationship between spectral decomposition, orthogonal rational functions and the rational Lanczos algorithm, based on a simple identity for rational Krylov sequences.

Krylov Projection Methods for Rational Interpolation