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Real and complex pseudozero sets for polynomials with applications

Published online by Cambridge University Press:  24 April 2007

Stef Graillat  and
Philippe Langlois
Stef Graillat
Affiliation:
Laboratoire LIP6, département Calcul scientifique, université Pierre et Marie Curie, 4 place Jussieu, F-75252, Paris Cedex 05, France; [email protected]
Philippe Langlois
Affiliation:
DALI Project, Laboratory LP2A, université de Perpignan, 52, avenue Paul Alduy, 66860 Perpignan Cedex, France; [email protected]
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Abstract

Pseudozeros are useful to describe how perturbations of polynomial coefficients affect its zeros. We compare two types of pseudozero sets: the complex and the real pseudozero sets. These sets differ with respect to the type of perturbations. The first set – complex perturbations of a complex polynomial – has been intensively studied while the second one – real perturbations of a real polynomial – seems to have received little attention. We present a computable formula for the real pseudozero set and a comparison between these two pseudozero sets. We conclude that the complex pseudozero sets have to be preferred except when the perturbed real polynomials admit non-real zeros. We also give some applications of pseudozero set in control theory.

Keywords

Polynomial rootpseudozero setuncertaintyperturbationstability.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 41 , Issue 1: Real Numbers , January 2007 , pp. 45 - 56
Copyright
© EDP Sciences, 2007

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References

J.-M. Chesneaux, S. Guilain and J. Vignes, La bibliothèque CADNA : présentation et utilisation. Manual, Laboratoire d'Informatique de Paris 6, Université P. et M. Curie, Paris, France, November 1996. Available at http://www-anp.lip6.fr/cadna/, (in French).
Gautschi, W., On the condition of algebraic equations. Numer. Math. 21 (1973) 405–424. CrossRef
S. Graillat and P. Langlois, Testing polynomial primality with pseudozeros, in RNC-5, Real Numbers and Computer Conference, Lyon, France, edited by M. Daumas (September 2003) 121–137.
S. Graillat and P. Langlois, Pseudozero set decides on polynomial stability, in Proceedings of the Symposium on Mathematical Theory of Networks and Systems, Leuven, Belgium, edited by B. de Moor, B. Motmans, J. Willems, P. Van Dooren and V. Blondel (July 2004) (CD-ROM, papers/537.pdf).
Hinrichsen, D. and Kelb, B., Spectral value sets: a graphical tool for robustness analysis. Systems Control Lett. 21 (1993) 127–136. CrossRef
Hinrichsen, D. and Pritchard, A.J., Robustness measures for linear systems with application to stability radii of Hurwitz and Schur polynomials. Internat. J. Control 55 (1992) 809–844. CrossRef
WWW resources about Interval Arithmetic. http://www.cs.utep.edu/interval-comp/main.html.
L. Jaulin, M. Kieffer, O. Didrit and É. Walter, Applied interval analysis. Springer-Verlag London Ltd., London (2001).
D.G. Luenberger, Optimization by vector space methods. John Wiley & Sons Inc., New York (1969).
Mosier, R.G., Root neighborhoods of a polynomial. Math. Comp. 47 (1986) 265–273. CrossRef
A.M. Ostrowski, Solution of equations and systems of equations. Second edition. Academic Press, New York. Pure Appl. Math. 9 (1966).
H.J. Stetter, Polynomials with coefficients of limited accuracy, in Computer algebra in scientific computing – CASC'99 (Munich), Springer, Berlin (1999) 409–430.
H.J. Stetter, Numerical Polynomial Algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2004).
Toh, K.-C. and Trefethen, L.N., Pseudozeros of polynomials and pseudospectra of companion matrices. Numer. Math. 68 (1994) 403–425. CrossRef
Vignes, J., A stochastic arithmetic for reliable scientific computation. Math. Comp. Sim. 35 (1993) 233–261. CrossRef
J.H. Wilkinson, Rounding errors in algebraic processes. Dover Publications Inc., New York (1994).