We consider the following class of nonlinear eigenvalue problems: (Σ i=1 m Aipi(λ))v = 0, where A 1,⋯, Am are given n ×n matrices and the functions p1,⋯, pm are assumed to be entire. This does not only include polynomial eigenvalue problems but also eigenvalue problems arising from systems of delay differential equations. Our aim is to compute the ∈-pseudospectral abscissa, i.e., the supremum of the real parts of the points in the ∈-pseudospectrum, which is the complex set obtained by joining all solutions of the eigenvalue problem under perturbations {δA i} i=1 m, of norm at most ∈, of the matrices {A i} i=1 m. Under mild assumptions, guaranteeing the existence of a globally rightmost point of the ∈-pseudospectrum, we prove that it is sufficient to restrict the analysis to rank-one perturbations of the form δA i = β iuv *, where u ∈ ℂ n and v ∈ ℂ n with β i ∈ ℂ for all i. Using this main-and unexpected-result we present new iterative algorithms which require only the computation of the spectral abscissa of a sequence of problems obtained by adding rank one updates to the matrices Ai. These provide lower bounds to the pseudspectral abscissa and in most cases converge to it. A detailed analysis of the convergence of the algorithms is made. Their applicability and properties are illustrated by means of the delay and polynomial eigenvalue problem.

An iterative method for computing the pseudospectral abscissa for a class of nonlinear eigenvalue problems.

NICOLA
2012

Abstract

We consider the following class of nonlinear eigenvalue problems: (Σ i=1 m Aipi(λ))v = 0, where A 1,⋯, Am are given n ×n matrices and the functions p1,⋯, pm are assumed to be entire. This does not only include polynomial eigenvalue problems but also eigenvalue problems arising from systems of delay differential equations. Our aim is to compute the ∈-pseudospectral abscissa, i.e., the supremum of the real parts of the points in the ∈-pseudospectrum, which is the complex set obtained by joining all solutions of the eigenvalue problem under perturbations {δA i} i=1 m, of norm at most ∈, of the matrices {A i} i=1 m. Under mild assumptions, guaranteeing the existence of a globally rightmost point of the ∈-pseudospectrum, we prove that it is sufficient to restrict the analysis to rank-one perturbations of the form δA i = β iuv *, where u ∈ ℂ n and v ∈ ℂ n with β i ∈ ℂ for all i. Using this main-and unexpected-result we present new iterative algorithms which require only the computation of the spectral abscissa of a sequence of problems obtained by adding rank one updates to the matrices Ai. These provide lower bounds to the pseudspectral abscissa and in most cases converge to it. A detailed analysis of the convergence of the algorithms is made. Their applicability and properties are illustrated by means of the delay and polynomial eigenvalue problem.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.12571/2491
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