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

We present a new algorithm for computing eigenvectors of real symmetric tridiagonal matrices based on Godunov's two-sided Sturm sequence method and inverse iteration, which we call the Godunov-inverse iteration. We use eigenvector approximations computed recursively from two-sided Sturm sequences as starting vectors in inverse iteration, replacing any nonnumeric elements of these approximate eigenvectors with uniform random numbers. We use the left-hand bounds of the smallest machine presentable eigenvalue 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 or two iterations, producing accurate residuals.