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 correspond- ing 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 eigen-value 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.