Nonlinearizing Two-parameter Eigenvalue Problems

Abstract

We investigate a technique to transform a linear two-parameter eigenvalue problem into a nonlinear eigenvalue problem (NEP). The transformation stems from an elimination of one of the equations in the two-parameter eigenvalue problem, by considering it as a (standard) generalized eigenvalue problem. We characterize the equivalence between the original and the nonlinearized problem theoretically and show how to use the transformation computationally. Special cases of the transformation can be interpreted as a reversed companion linearization for polynomial eigenvalue problems, as well as a reversed (less known) linearization technique for certain algebraic eigenvalue problems with square-root terms. Moreover, by exploiting the structure of the NEP we present algorithm specializations for NEP methods, although the technique also allows general solution methods for NEPs to be directly applied. The nonlinearization is illustrated in examples and simulations, with focus on problems where the eliminated equation is of much smaller size than the other twoparameter eigenvalue equation. This situation arises naturally in domain decomposition techniques. A general error analysis is also carried out under the assumption that a backward stable eigensolver is used to solve the eliminated problem, leading to the conclusion that the error is benign in this situation.

On Nonlinearization of 3-parameter Eigenvalue Problems

“…Thus, nonlinearization technique works if a solution to 2PEP corresponds to a simple eigenvalue a 2 of the GEP represented by the equation (4.1). Again, in [33] it is also showed that the Jacobian J(a 1 , a 2 ) is nonsingualar if we take the simple eigenvalue a 1 of the GEP such that s is not orthogonal to the corresponding eigenvector, which confirms the existence of nonlinerization. Therefore, the nonlinearization exists simple eigenvalues of GEP given in equation (4.1).…”

“…But it is singular in non generic situation only. In [33], Ringh and Jarlebring showed that the Jacobian J(a 1 , a 2 ) is singular if and only if the GEP (4.1) has at least Jordan chain of length two or more. Thus, nonlinearization technique works if a solution to 2PEP corresponds to a simple eigenvalue a 2 of the GEP represented by the equation (4.1).…”

“…Condition number plays important role in the sensitivity analysis of eigenvalue problem. Normwise condition number of NEP, based on L 1i -matrices, i = 0, 1, 2 are generally used to measure the sensitivity of a 1 and is defined as [33].…”

“…Denote u and z 1 as the left and right eigenvectors of the NEP respectively, then it follows [33] that…”

“…Formula defined in equation (5.3) is further improved in [33] by considering the perturbations with respect to M 3i -matrices, i = 0, 1, 2 as follows:…”