Stabilizing the Richardson Algorithm by Controlling Chaos

By viewing the operations of the Richardson purification algorithm as a discrete time dynamical process, we propose a method to overcome the instability of the algorithm by controlling chaos. We present theoretical analysis and numerical results on the behavior and performance of the stabilized algorithm.Comment: Send email to [email protected] or [email protected] for uuencoded tarred gzipped postscript files for the five figure

Numerical Computation of the Complex Eigenvalues of a Matrix by solving a Square System of Equations

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 an underdetermined system of equation and by adding an extra equation, the augmented system becomes square which can be solved by LU factorization at a cheaper rate and quadratic convergence is guaranteed. While the underdetermined system of equations can be solved using QR factorization as shown in an earlier work by the same authors, converting it to a square system of equations has the added advantage that besides using LU factorization, it can be solved by several approaches including iterative methods. We show both theoretically and numerically that both algorithms are equivalent in the absence of roundoff errors