Scaling, Sensitivity and Stability in the Numerical Solution of Quadratic Eigenvalue Problems

Higham, Nicholas J. and Mackey, D. Steven and Tisseur, Françoise and Garvey, Seamus D. (2007) Scaling, Sensitivity and Stability in the Numerical Solution of Quadratic Eigenvalue Problems. [MIMS Preprint]

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Abstract

The most common way of solving the quadratic eigenvalue problem (QEP) $(\l^2 M + \l D + K)x=0$ is to convert it into a linear problem $(\l X + Y)z=0$ of twice the dimension and solve the linear problem by the QZ algorithm or a Krylov method. In doing so, it is important to understand the influence of the linearization process on the accuracy and stability of the computed solution. We discuss these issues for three particular linearizations: the standard companion linearization and two linearizations that preserve symmetry in the problem. For illustration we employ a model QEP describing the motion of a beam simply supported at both ends and damped at the midpoint. We show that the above linearizations lead to poor numerical results for the beam problem, but that a two-parameter scaling proposed by Fan, Lin and Van Dooren cures the instabilities. We also show that half of the eigenvalues of the beam QEP are pure imaginary and are eigenvalues of the undamped problem. Our analysis makes use of recently developed theory explaining the sensitivity and stability of linearizations, the main conclusions of which are summarized. As well as arguing that scaling should routinely be used, we give guidance on how to choose a linearization and illustrate the practical value of condition numbers and backward errors.

Item Type:	MIMS Preprint
Additional Information:	To appear in International Journal for Numerical Methods in Engineering
Uncontrolled Keywords:	quadratic eigenvalue problem, sensitivity, condition number, backward error, stability, scaling, linearization, companion form, damped beam
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User:	Nick Higham
Date Deposited:	21 Mar 2007
Last Modified:	08 Nov 2017 18:18
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/711

Available Versions of this Item

• Scaling, Sensitivity and Stability in the Numerical Solution of Quadratic Eigenvalue Problems. (deposited 26 Nov 2006)
  • Scaling, Sensitivity and Stability in the Numerical Solution of Quadratic Eigenvalue Problems. (deposited 21 Mar 2007) [Currently Displayed]
    • Scaling, Sensitivity and Stability in the Numerical Solution of Quadratic Eigenvalue Problems. (deposited 14 Dec 2007)

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