Title :

On the rational approximation of Markov functions,with applications to the computation of Markovfunctions of Toeplitz matrices

Author(s) :

Beckermann, Bernhard [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Bisch, Joanna [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Luce, Robert [Auteur]
Gurobi Optimization

Journal title :

Numerical Algorithms

Pages :

109-144

Publisher :

Springer Verlag

Publication date :

2022

ISSN :

1017-1398

English keyword(s) :

positive Thiele continued fractions
rational interpolation
Markov function
Toeplitz matrices
matrix function

HAL domain(s) :

Mathématiques [math]/Analyse numérique [math.NA]

English abstract : [en]

We investigate the problem of approximating the matrix function $f(A)$ by $r(A)$, with $f$ a Markov function, $r$ a rational interpolant of $f$, and $A$ a symmetric Toeplitz matrix. In a first step, we obtain a new upper ...
Show more >We investigate the problem of approximating the matrix function $f(A)$ by $r(A)$, with $f$ a Markov function, $r$ a rational interpolant of $f$, and $A$ a symmetric Toeplitz matrix. In a first step, we obtain a new upper bound for the relative interpolation error $1-r/f$ on the spectral interval of $A$. By minimizing this upper bound over all interpolation points, we obtain a new, simple and sharp a priori bound for the relative interpolation error. We then consider three different approaches of representing and computing the rational interpolant $r$. Theoretical and numerical evidence is given that any of these methods for a scalar argument allows to achieve high precision, even in the presence of finite precision arithmetic. We finally investigate the problem of efficiently evaluating $r(A)$, where it turns out that the relative error for a matrix argument is only small if we use a partial fraction decomposition for $r$ following Antoulas and Mayo. An important role is played by a new stopping criterion which ensures to automatically find the degree of $r$ leading to a small error, even in presence of finite precision arithmetic.Show less >

Language :

Anglais

Popular science :

Non

ANR Project :

Centre Européen pour les Mathématiques, la Physique et leurs Interactions

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL