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Title:  Pseudozeros of Polynomials and Pseudospectra of Companion Matrices 
Authors:  Toh, KimChuan Trefethen, Lloyd N. 
Keywords:  computer science technical report 
Issue Date:  Jun1993 
Publisher:  Cornell University 
Citation:  http://techreports.library.cornell.edu:8081/Dienst/UI/1.0/Display/cul.cs/TR931360 
Abstract:  It is well known that the zeros of a polynomial $p$ are equal to the eigenvalues of the associated companion matrix $A$. In this paper, we take a geometric view of the conditioning of these two problems and of the stability of algorithms for polynomial zerofinding. The $\epsilon$pseudozero set $Z_{\epsilon}(p)$ is the set of zeros of all polynomials $\hat{p}$ obtained by coefficientwise perturbations of $p$ of size $\leq \epsilon$; this is a subset of the complex plane considered earlier by Mosier, and is bounded by a certain generalized lemniscate. The $\epsilon$pseudospectrum $\Lambda_{\epsilon}(A)$ is another subset of C defined as the set of eigenvalues of matrices $\hat{A}=A+E$ with $E \leq\epsilon$; it is bounded by a level curve of the resolvent of $A$. We find that if $A$ is first balanced in the usual EISPACK sense, then $Z_{\epsilonp}(p)$ and $\Lambda_{\epsilonA}(A)$ are usually quite close to one another. It follows that the Matlab ROOTS algorithm of balancing the companion matrix, then computing its eigenvalues, is a stable algorithm for polynomial zerofinding. Experimental comparisons with the JenkinsTraub (IMSL) and MadsenReid (Harwell) Fortran codes confirm that these three algorithms have roughly similar stability properties. Key words: polynomial zeros, comapnion matrix, pseudospectrum. 
URI:  http://hdl.handle.net/1813/6130 
Appears in Collections:  Computer Science Technical Reports

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