The Godunov-inverse iteration: A fast and accurate solution to the symmetric tridiagonal eigenvalue problem

Abstract

We present a new hybrid algorithm based on Godunov's method for computing eigenvectors of symmetric tridiagonal matrices and Inverse Iteration, which we call the Godunov-Inverse Iteration. We use eigenvectors computed according to Godunov's method as starting vectors in the Inverse Iteration, replacing any nonnumeric elements of Godunov's eigenvectors with random uniform numbers. We use the right-hand bounds of the Ritz intervals found by the bisection method as Inverse Iteration shifts, while staying within guaranteed error bounds. In most test cases convergence is reached after only one step of the iteration, producing error estimates that are as good as or superior to those produced by standard Inverse Iteration implementations.

Bidiagonal SVD Computation via an Associated Tridiagonal Eigenproblem

“…• Matrices used in Dhillon [1997] and Dhillon et al [1997] (the matrices in the latter reference inspired the development of the MRRR algorithm). • Matrices used in Matsekh [2005] to test Godunov's two-sided Sturm sequence.…”

Algorithm 880

“…(1) to experiment with and select algorithmic variants as in Marques et al [2006], (2) to experiment with parameter and threshold choices such as those to be set in LAPACK's ilaenv, (3) to carry out large scale performance comparisons such as Marques et al [2006]; Demmel et al [2006], and (4) to validate and benchmark new algorithms such as Matsekh [2005].…”