Linearizable Eigenvector Nonlinearities

We present a method to linearize, without approximation, a specific class of eigenvalue problems with eigenvector nonlinearities (NEPv), where the nonlinearities are expressed by scalar functions that are defined by a quotient of linear functions of the eigenvector. The exact linearization relies on...

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Published in:SIAM journal on matrix analysis and applications Vol. 43; no. 2; pp. 764 - 786
Main Authors: Claes, Rob, Jarlebring, Elias, Meerbergen, Karl, Upadhyaya, Parikshit
Format: Journal Article
Language:English
Published: 01.01.2022
Subjects:
linearization
multiparameter eigenvalue problem
nonlinear eigenvalue problem
ISSN:0895-4798, 1095-7162, 1095-7162
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Abstract We present a method to linearize, without approximation, a specific class of eigenvalue problems with eigenvector nonlinearities (NEPv), where the nonlinearities are expressed by scalar functions that are defined by a quotient of linear functions of the eigenvector. The exact linearization relies on an equivalent multiparameter eigenvalue problem (MEP) that contains the exact solutions of the NEPv. Due to the characterization of MEPs in terms of a generalized eigenvalue problem this provides a direct way to compute all NEPv solutions for small problems, and it opens up the possibility to develop locally convergent iterative methods for larger problems. Moreover, the linear formulation allows us to easily determine the number of solutions of the NEPv. We propose two numerical schemes that exploit the structure of the linearization: inverse iteration and residual inverse iteration. We show how symmetry in the MEP can be used to improve reliability and reduce computational cost of both methods. Two numerical examples verify the theoretical results, and a third example shows the potential of a hybrid scheme that is based on a combination of the two proposed methods.
AbstractList We present a method to linearize, without approximation, a specific class of eigenvalue problems with eigenvector nonlinearities (NEPv), where the nonlinearities are expressed by scalar functions that are defined by a quotient of linear functions of the eigenvector. The exact linearization relies on an equivalent multiparameter eigenvalue problem (MEP) that contains the exact solutions of the NEPv. Due to the characterization of MEPs in terms of a generalized eigenvalue problem this provides a direct way to compute all NEPv solutions for small problems, and it opens up the possibility to develop locally convergent iterative methods for larger problems. Moreover, the linear formulation allows us to easily determine the number of solutions of the NEPv. We propose two numerical schemes that exploit the structure of the linearization: inverse iteration and residual inverse iteration. We show how symmetry in the MEP can be used to improve reliability and reduce computational cost of both methods. Two numerical examples verify the theoretical results, and a third example shows the potential of a hybrid scheme that is based on a combination of the two proposed methods.
Author Upadhyaya, Parikshit
Meerbergen, Karl
Claes, Rob
Jarlebring, Elias
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  givenname: Parikshit
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SubjectTerms linearization
multiparameter eigenvalue problem
nonlinear eigenvalue problem
Title Linearizable Eigenvector Nonlinearities
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