Recursive and iterative algorithms for computing eigenvalues of Hermitian Toeplitz matrices

The numerical solution of the complete eigenspectrum for Hermitian Toeplitz matrices is presented. Trench's algorithm (1989), which employs bisection on contiguous intervals, and the Pegasus method are used to achieve estimates of distinct eigenvalues. Several modifications of Trench's algorithm are examined; the goals are an increase in the rate of convergence, even at some reduction in estimate accuracy, and an accommodation of eigenvalue multiplicity or clustering. A promising approach that contains three key ingredients is found. They are: a modification of Trench's procedure to employ noncontiguous intervals, a procedure for multiplicity identification, and a replacement of the Pegasus method by the modified Rayleigh quotient iteration. The result is the basis for a novel eigenspectrum solver with a cubic convergence rate and good estimation accuracy. Simulation results for high-order Hermitian Toeplitz matrices are provided.<
>