Norm bounds for rational matrix functions
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.
scientific article; zbMATH DE number 3860289