Two-Norm Normalization for the Matrix Pencil: Inverse Iteration with a Complex Shift

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. For complex eigenpairs, instead of using Ruhe’s normalization, we show that the natural two norm normalization for the matrix pencil, yields a quadratically convergent algorithm. Numerical experiment is given which confirms the theory.