Abstract
The problem treated in this work is the numerical solution of the complete eigenspectrum for Hermitian Toeplitz matrices. The core procedure utilized is Trench's algorithm which employs bisection on contiguous intervals and the Pegasus method to achieve estimates of distinct eigenvalues. Several modifications to Trench's algorithm are examined, the goals being an increase in the rate of convergence, even at some reduction in estimate accuracy, and an accommodation of eigenvalue multiplicity, or, practically speaking, eigenvalue clustering. A promising approach is found which contains three key ingredients: 1) a modification of Trench's procedure to employ noncontiguous intervals, 2) an addition of a procedure for multiplicity identification, and 3) a replacement of the Pegasus method by the modified Rayleigh quotient iteration (MRQI) of Hu and Kung. The result is the basis for a novel eigenspectrum solver possessing cubic convergence rate and good estimation accuracy. Simulation results are provided for high order Hermitian Toeplitz matrices.