Computing all Pairs (λ,μ) Such That λ is a Double Eigenvalue of A+μB
Elias Jarlebring, Simen Kvaal, Wim Michiels · SIAM Journal on Matrix Analysis and Applications · 2011
Double eigenvalues are not generic for matrices without any particular structure. A matrix depending linearly on a scalar parameter, [Formula: see text], will, however, generically have double eigenvalues for some values of the parameter [Formula: see text]. In this paper, we consider the problem of finding those values. More precisely, we construct a method to accurately find all scalar pairs [Formula: see text] such that [Formula: see text] has a double eigenvalue [Formula: see text], where [Formula: see text] and [Formula: see text] are given arbitrary complex matrices. The general idea of the globally convergent method is that if [Formula: see text] is close to a solution, then [Formula: see text] has two eigenvalues which are 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 can be stated as 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 returns a very accurate solution, for a sufficiently good starting value. The approach is illustrated with one academic example and one application to a simple problem in computational quantum mechanics.