An Example Showing Severe Difficulties in the Evaluation of Eigenvalues of Simple Tridiagonal Non-Symmetric Matrices

Abstract

Non-normal matrices do not possess a complete set of orthogonal eigenvectors. Consequently no guaranty exists against the collapse of at least some of the eigenvectors in the basis. For such cases the eigenvectors basis may become strongly non-orthogonal, without control. Thus, the results of
evaluating eigenpairs may be extremely sensitive to small perturbations originated, for example, by numerical errors. The aim of this short note is to provide a simple and detailed example of tridiagonal matrices with constant entries along each diagonal, for which the spectra calculated by ARPACK package (Arnoldi's approach) or by IMSL's routine EIGZF are completely irrealistic.
For such kind of problems, no calculation is possible without a very high machine precision. This is our main conclusion. In the paticular example discussed here, the codes would need about 50 digits i.e., more than three times the usual double precision.