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  • Electronic Resource  (2)
  • 1
    Electronic Resource
    Electronic Resource
    Pseudozeros of polynomials and pseudospectra of companion matrices (1994)
    Springer
    Numerische Mathematik 68 (1994), S. 403-425 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991): 65F35
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. 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 ${\Bbb C}$ defined as the set of eigenvalues of matrices {$\hat{A} = A + E$} with $\Vert E\Vert \leq \epsilon$ ; 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 $Z_{\epsilon \Vert p\Vert }(p)$ and $\Lambda_{ \epsilon \Vert A\Vert }(A)$ 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 Jenkins-Traub (IMSL) and Madsen-Reid (Harwell) Fortran codes confirm that these three algorithms have roughly similar stability properties.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050069
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Primal-Dual Path-Following Algorithms for Determinant Maximization Problems With Linear Matrix Inequalities (1999)
    Springer
    Computational optimization and applications 14 (1999), S. 309-330 
    Details
    ISSN: 1573-2894
    Keywords: determinant optimization ; semidefinite programming ; predictor-corrector
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract Primal-dual path-following algorithms are considered for determinant maximization problem (maxdet-problem). These algorithms apply Newton's method to a primal-dual central path equation similar to that in semidefinite programming (SDP) to obtain a Newton system which is then symmetrized to avoid nonsymmetric search direction. Computational aspects of the algorithms are discussed, including Mehrotra-type predictor-corrector variants. Focusing on three different symmetrizations, which leads to what are known as the AHO, H..K..M and NT directions in SDP, numerical results for various classes of maxdet-problem are given. The computational results show that the proposed algorithms are efficient, robust and accurate.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1026400522929
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    Paper (German National Licenses)
    Fulltext
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