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 two-parameter 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 eigenvalue solver method is used to solve the eliminated problem, leading to the conclusion that the error is benign in this situation.
Highlights
This paper concerns the two-parameter eigenvalue problem: Determine nontrivial quadruplets (\lambda, x, \mu, y) \in \BbbC \times \BbbC n \times \BbbC \times \BbbC m such that (1.1a) (1.1b)0 = A1x + \lambda A2x + \mu A3x, 0 = B1y + \lambda B2y + \mu B3y, where A1, A2, A3 \in \BbbC n\times n and B1, B2, B3 \in \BbbC m\times m
Some contributions are converse, i.e., we provide insight to nonlinear eigenvalue problem (NEP) based on the equivalence with two-parameter eigenvalue problems
The elimination of the B-equation (1.1b) in the two-parameter eigenvalue problem can be explicitly characterized as we describe
Summary
This paper concerns the two-parameter eigenvalue problem: Determine nontrivial quadruplets (\lambda , x, \mu , y) \in \BbbC \times \BbbC n \times \BbbC \times \BbbC m such that (1.1a) (1.1b). We denote the corresponding functions A(\lambda , \mu ) := A1 + \lambda A2 + \mu A3 and B(\lambda , \mu ) := B1 + \lambda B2 + \mu B3. This problem has been extensively studied in the literature; see, e.g., the fundamental work of Atkinson [2] and the summary of recent developments below. We view (1.1b) as a parameterized generalized linear eigenvalue problem, where \lambda is the parameter
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