Numerical Solutions of Quantum Mechanical Eigenvalue Problems

The Numerical Treatment of Large Eigenvalue Problems

This paper surveys techniques for calculating eigenvalues and eigenvectors of very large matrices.

Numerical Solution of Linear Eigenvalue Problems

We review numerical methods for computing eigenvalues of matrices. We start by considering the computation of the dominant eigenpair of a general dense matrix using the power method, and then generalize to orthogonal iterations and the QR iteration with shifts. We also consider divide-and-conquer algorithms for tridiagonal matrices. The second part of this survey involves the computation of eig...

Numerical Analysis of Nonlinear Eigenvalue Problems

We provide a priori error estimates for variational approximations of the ground state eigenvalue and eigenvector of nonlinear elliptic eigenvalue problems of the form −div(A∇u) + V u + f(u)u = λu, ‖u‖L2 = 1. We focus in particular on the Fourier spectral approximation (for periodic problems) and on the P1 and P2 finite-element discretizations. Denoting by (uδ, λδ) a variational approximation o...

Numerical Solution of Large Nonsymmetric Eigenvalue Problems

Numerical methods for large eigenvalue problems

Over the past decade considerable progress has been made towards the numerical solution of large-scale eigenvalue problems, particularly for nonsymmetric matrices. Krylov methods and variants of subspace iteration have been improved to the point that problems of the order of several million variables can be solved. The methods and software that have led to these advances are surveyed.