Manchester Applied Mathematics and Numerical Analysis Seminars Winter 1998

October 7, 1998, 3.45 pm

Lecture Theatre 2.14, Maths Building, Oxford Road

Backward Error and Condition of Polynomial Eigenvalue Problems

Francoise Tisseur, Department of Mathematics, University of Manchester

Polynomial eigenvalue problems (PEP) appear often in industrial applications. An example is in the analysis of automobile brakes where vibrations between the friction pads on the caliper and the rotor generate sometimes uncomfortable noise and may have negative effect on the braking performance. When modeled with finite elements, this problem leads to a PEP containing three matrices. The standard way of dealing with the PEP is to reformulate it as a generalized eigenvalue problem (GEP). We aim to further our understanding concerning the sensitivity of these problems to perturbations and the stability of the numerical algorithms used to compute the solutions. We develop normwise backward errors and condition numbers for the (PEP). For the special case of the quadratic eigenvalue problem (QEP), we show that solving the QEP by applying the QZ algorithm to a corresponding GEP can be backward unstable. The QEP can be reformulated as a GEP in many ways. We investigate the sensitivity of a given eigenvalue to perturbations in each of the GEP formulations and identify which formulations are to be preferred for large and small eigenvalues, respectively.

For further info contact either Matthias Heil (mheil@ma.man.ac.uk), Mark Muldoon (M.Muldoon@umist.ac.uk)or the seminar secretary (Tel. 0161 275 5800).