# Solving polynomial eigenvalue problems via linearizations is backward stable

## Document Type

Presentation Abstract

## Presentation Date

3-17-2016

## Abstract

Polynomial eigenvalue problems arise directly from applications in mechanics and control theory, from finite element discretizations of continuous models, or as approximations of nonlinear eigenvalue problems, and are still a challenge for modern eigenvalue methods. The most widely used approach for solving the polynomial eigenvalue problem associated with a matrix polynomial *P*(λ)= λ* ^{d}A_{d}* +…+ λ

*A*

_{1}+ λ

*A*

_{0}is to linearize to produce a larger order pencil (i.e. a linear matrix polynomial)

*L*(λ)=λ

*X*+

*Y*, whose eigenstructure is then found by any method for generalized eigenproblems (such as the QR algorithm). Though this may not be the best way to address the polynomial eigenvalue problem, from the point of view of efficiency and storage, it has been used extensively, because of the advantages of the QR algorithm (robustness and stability).

To determine whether this method to solve polynomial eigenvalue problems is backward stable or not has been an open problem until very recently.

The goal of this talk will be to explain our most recent results about the backward stability of this approach to solve polynomial eigenvalue problems. In the first part of the talk I will introduce polynomial eigenvalue problems and the concept of a matrix polynomial. In the second part of the talk I will introduce the concept of a linearization of a matrix polynomials. In the third part of the talk I will make a gentle introduction to Numerical Linear Algebra, and, in particular, I will introduce the ideas of a backward error analysis and a backward stable numerical algorithm. Finally, I will explain some of the ideas behind our backward error analysis of the polynomial eigenvalue problem solved via linearizations.

## Recommended Citation

Pérez Álvaro, Javier, "Solving polynomial eigenvalue problems via linearizations is backward stable" (2016). *Colloquia of the Department of Mathematical Sciences*. 499.

https://scholarworks.umt.edu/mathcolloquia/499

## Additional Details

Applied Math/Statistics Seminar

Thursday, March 17 at 3:10 p.m. in Math 108