A Comparison Of The Implicit Determinant Method And Inverse Iteration

Abstract

It is well known that if the largest or smallest eigenvalue of a matrix has been computed by some numerical algorithms and one is interested in computing the corresponding eigenvector, one method that is known to give such good approximations to the eigenvector is inverse iteration with a shift. However, in a situation where the desired eigenvalue is defective, inverse iteration converges harmonically to the eigenvalue close to the shift. In this paper, we extend the implicit determinant method of Spence and Poulton [13] to compute a defective eigenvalue given a shift close to the eigenvalue of interest. For a defective eigenvalue, the proposed approach gives quadratic convergence and this is verified by some numerical experiments.