Bounds for the minimum eigenvalue of a symmetric Toeplitz matrix
Upper estimates for the smallest eigenvalue of a real symmetric Toeplitz matrix \(T\) are obtained by computing eigenvalues of reduced problems obtained by projection of the matrix eigenvalue problem onto subspaces of the Krylov space of \(T^{-1}\). It is proved that eigenvalue estimates obtained by \textit{A. Melman} [Bounds on the extreme eigenvalues of real symmetric Toeplitz matrices, Stanford Univ. Report SCCM 98-7, \url{http://etna.mcs.kent.edu/vol.8.1999/pp127-137.dir/pp127-137.html} and SIAM J. Matrix Anal. Appl. 21, No. 2, 362-378 (1999)] are obtained by using projections of dimension 2 and 3. Results of computations are presented employing an algorithm based on the new projection method for dimensions 2, 3, 4, 5 using an example considered by Melman. The higher-dimensional projections have smaller average relative error than the lower ones.
scientific article; zbMATH DE number 1336576