Many numerical one-step methods for solving initial value problems of ordinary differential equations and for treating initial boundary value problems of partial differential equations are of the form \(u_ n=\phi (hA)u_{n-1},\) \((n=1,2,3,...)\), where \(u_ n\in R^ s\), A is a square matrix of order s, h a stepsize and \(\phi\) is usually a rational function with \(\phi (0)=\phi '(0)=1.\)
In the paper the stability of such numerical processes is analyzed and especially the error propagation is investigated if the process is started with a slightly perturbed initial vector \(\tilde u_ 0\) instead of \(u_ 0\). Bounds of the error \(v_ n=\tilde u_ n-u_ n\) are considered of the type \(| v_ n| \leq \gamma s^ pn^ q| v_ 0|,\) for \(s\geq 1\), \(n\geq 1\), where \(\gamma\), p, q denote nonnegative constants independent of s, n, \(v_ 0\), and the estimates are valid for general norms \(| \cdot |.\)
A natural way to study the stability leads to the usual stability region of the function \(\phi\) that depends on the spectrum of the matrix hA. Due to the fact that the normal conditions, obtained in such a way, may be unreliable, finer stability estimates are derived using the concept of the M-numerical range \(\tau\) [A]. A series of applications illustrates the main result of the paper.