Norm bounds for rational matrix functions
type: Item
instance of: scholarly article
title: Norm bounds for rational matrix functions
zbMATH Open document ID: 0541.65022
DOI: 10.1007/BF01389581
published in: Numerische Mathematik
publication date: 1983-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/132884
review text: In the solution of a linear differential equation \(y'=Ay,\) it is appropriate to consider bounds of the forms \(\| f(A)\| \leq 1\) and \(\| f(hA)\| \leq 1+ch\) for all \(0<h\leq h_ 0\), where f is an approximation to the exponential function characteristic of a particular numerical method. The constants c and \(h_ 0\) should not be dependent on the norm of the matrix A, which corresponds to an arbitrarily stiff problem. Let \(G_ A\) denote the numerical range, that is the set of possible values of the Rayleigh quotient, of A. The main result of the paper deals with rational functions of the form \(f(z)=\prod^{s}_{j=1}(1-z/w_ j)/\prod^{s}_{k=1}(1-z/z_ k)\) in which the poles and zeros can be so ordered that \(| w_ k| \geq | z_ k|\) and \(Re(1/w_ k)<Re(1/z_ k)\) for \(k=1,2,...,s\). In this case it is shown how to construct a stability region S such that \(\| f(A)\| \leq 1\) whenever \(G_ A\subset S\). Furthermore, it is shown how to obtain a stability angle \(\alpha\) and constants c, \(\zeta_ 0>0\) such that for \(G_ A\subset Z_{\alpha}+\beta\), with \(Z_{\alpha}=\{z\in {\mathbb{C}}:| \arg(-z)| \leq \alpha \}\) and \(\beta \in {\mathbb{R}}\), \(\| f(hA)\| \leq 1+h\beta c\quad for\quad h\beta \leq \zeta_ 0.\) The results are applied to Padé approximations and to implicit Runge-Kutta methods based on equidistant knots.
Mathematics Subject Classification ID: 65F35
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 41A21
zbMATH DE Number: 3860289
zbMATH Keywords: norm bounds
zbMATH Keywords: rational matrix functions
zbMATH Keywords: spectral norm
zbMATH Keywords: stiff problem
zbMATH Keywords: Rayleigh quotient
zbMATH Keywords: stability region
zbMATH Keywords: implicit Runge-Kutta methods
author: B. A. Schmitt
MaRDI profile type: Publication
cites work: Numerical range of a matrix: Some effective criteria
cites work: An implementation of singly-implicit Runge-Kutta methods
cites work: Order and stepsize control in extrapolation methods
cites work: Iterated defect correction for the efficient solution of stiff systems of ordinary differential equations
cites work: On the stability of semi-implicit methods for ordinary differential equations
cites work: High-Accuracy Stable Difference Schemes for Well-Posed Initial-Value Problems
cites work: Semidiscretization in Time for Parabolic Problems
cites work: Q3673995
cites work: Q4133641
description: scientific article; zbMATH DE number 3860289