Related bounds for the extreme eigenvalues

We show how to construct an increasing (decreasing) sequence of lower (uppe r) bounds for the smallest (largest) eigenvalue of a matrix with real spect rum by using the knowledge of the largest (smallest) eigenvalue. We apply t hese results to obtain a decreasing sequence of upper bounds for the larges t singular value of a complex matrix. Finally, we show that the new bounds improve some previous results. Examples in which the bounds could be useful are given. (C) 1999 Elsevier Science Ltd. All rights reserved.