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

GivenA1, the discrete approximation of a linear self-adjoint partial differential operator, the smallest few eigenvalues and eigenvectors ofA1 are computed by the homotopy (continuation) method. The idea of the method is very simple. From some initial operatorA0 with known eigenvalues and eigenvectors, define the homotopyH(t)=(1−t)A0+tA1, 0≤t≤1. If the eigenvectors ofH(t0) are known, then they are used to determine the eigenpairs ofH(t0+dt) via the Rayleigh quotient iteration, for some value ofdt. This is repeated untilt becomes 1, when the solution to the original problem is found. A fundamental problem is the selection of the step sizedt. A simple criterion to selectdt 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.

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