Double eigenvalues are not generic for matrices without any particular structure. A matrix depending linearly on a scalar parameter, A+μ B, will however generically have double eigenvalues for some values of the parameter μ. In this paper we consider the problem of finding those values. More precisely, we construct a method to accurately find all scalar pairs (λ,μ) such that A+μ B has a double eigenvalue λ, where A and B are given arbitrary complex matrices.
Before presenting the numerical scheme, we prove some properties necessary for a problem to be solvable numerically in a reliable way. In particular, we show that the problem is (under mild assumptions) well conditioned.
The general idea of the globally convergent method is that if μ is close to a solution then A+μ B has two eigenvalues close to each other. We fix the relative distance between these two eigenvalues and construct a method to solve and study it by observing that the resulting problem is a two-parameter eigenvalue problem, which is already studied in the literature. The method, which we call the method of fixed relative distance (MFRD), involves solving a two-parameter eigenvalue problem which returns approximations of all solutions. It is unfortunately not possible to get full accuracy with MFRD. In order to compute solutions with full accuracy, we present an iterative method which, when given a sufficiently good starting value, returns a very accurate solution. The method returns accurate solutions for non-semisimple as well as semisimple eigenvalues.
The approach is illustrated with one academic example and one application to a simple problem in computational quantum mechanics.