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Constructive Analysis of Eigenvalue Problems in Control under Numerical Uncertainty

Pavel Osinenko*, Grigory Devadze, and Stefan Streif
International Journal of Control, Automation, and Systems, vol. 18, no. 9, pp.2177-2185, 2020

Abstract : The eigenvalue problem plays a central role in linear algebra and its applications in control and optimization methods. In particular, many matrix decompositions rely upon computation of eigenvalue-eigenvector pairs, such as diagonal or Jordan normal forms. Perturbation theory and various regularization techniques help address some numerical difficulties of computation eigenvectors, but often rely on per se uncomputable quantities, such as a minimal gap between eigenvalues. In this note, the eigenvalue problem is revisited within constructive analysis allowing to explicitly consider numerical uncertainty. Exact eigenvectors are substituted by approximate ones in a suitable format. Examples showing influence of computation precision are provided.

Keyword : Approximate solutions, constructive analysis, eigenvalues, eigenvectors, fundamental theorem of algebra.

 
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