On the singular two-parameter eigenvalue problem

Abstract

Abstract. In the 1960s, Atkinson introduced an abstract algebraic setting for multiparameter eigenvalue problems. He showed that a nonsingular multiparameter eigenvalue problem is equivalent to the associated system of generalized eigenvalue problems. Many theoretical results and numerical methods for nonsingular multiparameter eigenvalue problems are based on this relation. In this paper, the above relation to singular two-parameter eigenvalue problems is extended, and it is shown that the simple finite regular eigenvalues of a two-parameter eigenvalue problem and the associated system of generalized eigenvalue problems agree. This enables one to solve a singular two-parameter eigenvalue problem by computing the common regular eigenvalues of the associated system of two singular generalized eigenvalue problems.

Jacobi–Davidson methods for polynomial two-parameter eigenvalue problems

“…In this case, as shown in [14], if all eigenvalues of (1) are algebraically simple, which is the generic case, they agree with the finite regular eigenvalues of the pair of singular matrix pencils (3). A numerical algorithm from [15], based on the staircase algorithm by Van Dooren [20], may then be used to compute the common regular part of (3) and extract the finite regular eigenvalues.…”

“…To reveal the structure of the linearization for a general PMEP, we consider the cubic case. Details on a linearization for a general degree k can be found in [14,Appendix]. Problem 2.…”

“…We assume that the problem is regular singular, which means that neither of the polynomials p 1 (λ, µ) = det(W 1 (λ, µ)) and p 2 (λ, µ) = det(W 2 (λ, µ)) is identically zero and they do not share a common factor [14]. We obtain a problem with such properties when we linearize the PMEP (1) as a MEP [14]. Input: a regular singular MEP (A1 + λB1 + µC1) z1 = 0, (A2 + λB2 + µC2) z2 = 0 starting vectors u1, u2, a target (σ, τ ), and a tolerance ε.…”

“…The main difference to algorithms for nonsingular MEP is that the projected MEP might be singular, therefore we numerically solve it using the method for singular MEP from [14].…”

Functional Characterization of Intrinsic and Extrinsic Geometry

“…Lemma 3.4 from [Muhic and Plestenjak 2009] and the fact that B k are symmetric, we see that in that case for every choice of a k , there must exist a vector x such that x T B k x = 0 for every B k , and Bx = 0. Now, given the values of x on some triangle, this means that its values on the adjacent triangle are either uniquely determined by the corresponding two equations (one linear, one quadratic), or these equations cannot be satisfied.…”