Rigorous machine bounds for the eigensystem of a general complex matrix

Introduction. We are concerned here with giving rigorous error bounds for the eigensystem of a general complex n X n matrix A, given an approximate eigensystem such as is furnished by [2]. In Section 1, we outline the technique in general terms and show that the bounds can be found in terms of computed quantities if ll-Elloo = ||/ — XFU«, < 1, where X is the matrix of approximate eigenvectors and Y is an approximate inverse for X. Then in Sections 2, 3, and 4 we give the specific roundoff error bounds for these general error terms, which include all the rounding errors made during the computation. An Algol program using the method is given in the microfiche section, and the results for the matrix example given in [2] are presented in Section 5, using the results of [2] as the initial approximation.