Computational Methods for Matrix Eigenproblems

Accurate polynomial root-finding methods for symmetric tridiagonal matrix eigenproblems

In this paper we consider the application of polynomial root-finding methods to the solution of the tridiagonal matrix eigenproblem. All considered solvers are based on evaluating the Newton correction. We show that the use of scaled three-term recurrence relations complemented with error free transformations yields some compensated schemes which significantly improve the accuracy of computed r...

Square & Stretch Multigrid for Stochastic Matrix Eigenproblems∗∗

A novel multigrid algorithm for computing the principal eigenvector of column stochastic matrices is developed. The method is based on the Exact Interpolation Scheme multigrid approach of Brandt and Ron, whereby the prolongation is adapted to yield a better and better coarse representation of the sought eigenvector. A special feature of the present approach is the squaring of the stochastic mat...

Computational Methods for Linear Matrix Equations

Given the matrices A,B,C,D and E of conforming dimensions, we consider the linear matrix equation AXE + DXB = C in the unknown matrix X. Our aim is to provide an overview of the major algorithmic developments that have taken place in the past few decades in the numerical solution of this and of related problems, which are becoming a reliable tool in the numerical formulation of advanced applica...

Jacobi-Davidson Methods for Symmetric Eigenproblems

1 Why and how The Lanczos method is quite eeective if the desired eigenvalue is either max or min and if this eigenvalue is relatively well separated from the remaining spectrum, or when the method is applied with (A ? I) ?1 , for some reasonable guess for an eigenvalue. If none of these conditions is fulllled, for instance the computation of a vector (A ? I) ?1 y for given y may be computation...

Matrix methods for radial Schrödinger eigenproblems defined on a semi-infinite domain