Computing the Minimum Eigenvalue of a Symmetric Positive Definite Toeplitz Matrix by Newton-type Methods

A method for computing the smallest eigenvalue of a symmetric positive definite Toeplitz matrix by application of Newton's method to the characteristic polynomial has been recently introduced by Mastronardi and Boley [ SIAM J. Sci. Comput., 20 (1999), pp. 1921--1927]. Though considerably slower than methods developed by the authors [W. Mackens and H. Voss, SIAM J. Matrix. Anal. Appl., 18 (1997), pp. 521--534], [W. Mackens and H. Voss, Linear Algebra Appl., 275/276 (1998), pp. 401--415], [H. Voss, Linear Algebra Appl., 287 (1999), pp. 359--371] the new approach is conceptually much simpler. In this paper we improve the performance of the new method substantially while keeping its simplicity.