Homotopy method for the numerical solution of the eigenvalue problem of self-adjoint partial differential operators

Given A₁, the discrete approximation of a linear self-adjoint partial differential operator, the smallest few eigenvalues and eigenvectors of A₁ are computed by the homotopy (continuation) method. The idea of the method is very simple. From some initial operator A₀ with known eigenvalues and eigenvectors, define the homotopy H(t)=(1-t)A₀+tA₁, 0≤t≤1. If the eigenvectors of H(t₀) are known, then they are used to determine the eigenpairs of H(t₀+dt) via the Rayleigh quotient iteration, for some value of dt. This is repeated until t becomes 1, when the solution to the original problem is found. A fundamental problem is the selection of the step size dt. A simple criterion to select dt is given. It is shown that the iterative solver used to find the eigenvector at each step can be stabilized by applying a low-rank perturbation to the relevant matrix. By carrying out a small part of the calculation in higher precision, it is demonstrated that eigenvectors corresponding to clustered eigenvalues can be computed to high accuracy. Some numerical results for the Schrödinger eigenvalue problem are given. This algorithm will also be used to compute the bifurcation point of a parametrized partial differential equation. © 1995 J.C. Baltzer AG, Science Publishers