On the Danilewski method for computing the characteristic polynomial

Abstract

The Danilewski method, as commonly applied, tends to be a very unstable method for finding the characteristic polynomial of a matrix. In this paper we show that, by an appropriate computational procedure, the method seems, in general, to be relatively stable. For the examples tried, it has yielded the characteristic polynomial with high accuracy. It is a well known fact that the roots of a polynomial can vary greatly when small changes are made in the polynomial coefficients. Hence it has been argued that, to find matrix eigenvalues, one should not first obtain the characteristic polynomial. However, the Danilewski method is extremely fast and, as shown below, it seems possible to obtain eigenvalues of a given accuracy by the Danilewski method in less computing time than by other methods.