# Real and complex pseudozero sets for polynomials with applications

Stef Graillat; Philippe Langlois

RAIRO - Theoretical Informatics and Applications (2007)

- Volume: 41, Issue: 1, page 45-56
- ISSN: 0988-3754

## Access Full Article

top## Abstract

top## How to cite

topGraillat, Stef, and Langlois, Philippe. "Real and complex pseudozero sets for polynomials with applications." RAIRO - Theoretical Informatics and Applications 41.1 (2007): 45-56. <http://eudml.org/doc/250033>.

@article{Graillat2007,

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.
},

author = {Graillat, Stef, Langlois, Philippe},

journal = {RAIRO - Theoretical Informatics and Applications},

keywords = {Polynomial root; pseudozero set; uncertainty; perturbation; stability.; pseudozero of of polynomial; root perturbation; computational stability; control theory},

language = {eng},

month = {4},

number = {1},

pages = {45-56},

publisher = {EDP Sciences},

title = {Real and complex pseudozero sets for polynomials with applications},

url = {http://eudml.org/doc/250033},

volume = {41},

year = {2007},

}

TY - JOUR

AU - Graillat, Stef

AU - Langlois, Philippe

TI - Real and complex pseudozero sets for polynomials with applications

JO - RAIRO - Theoretical Informatics and Applications

DA - 2007/4//

PB - EDP Sciences

VL - 41

IS - 1

SP - 45

EP - 56

AB -
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.

LA - eng

KW - Polynomial root; pseudozero set; uncertainty; perturbation; stability.; pseudozero of of polynomial; root perturbation; computational stability; control theory

UR - http://eudml.org/doc/250033

ER -

## References

top- 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 , (in French). URIhttp://www-anp.lip6.fr/cadna/
- W. Gautschi, On the condition of algebraic equations. Numer. Math.21 (1973) 405–424. Zbl0278.65044
- 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).
- D. Hinrichsen and B. Kelb, Spectral value sets: a graphical tool for robustness analysis. Systems Control Lett.21 (1993) 127–136. Zbl0785.93030
- D. Hinrichsen and A.J. Pritchard, Robustness measures for linear systems with application to stability radii of Hurwitz and Schur polynomials. Internat. J. Control55 (1992) 809–844. Zbl0747.93017
- WWW resources about Interval Arithmetic. . URIhttp://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). Zbl1023.65037
- D.G. Luenberger, Optimization by vector space methods. John Wiley & Sons Inc., New York (1969). Zbl0176.12701
- R.G. Mosier, Root neighborhoods of a polynomial. Math. Comp.47 (1986) 265–273. Zbl0598.65023
- A.M. Ostrowski, Solution of equations and systems of equations. Second edition. Academic Press, New York. Pure Appl. Math.9 (1966). Zbl0222.65070
- H.J. Stetter, Polynomials with coefficients of limited accuracy, in Computer algebra in scientific computing – CASC'99 (Munich), Springer, Berlin (1999) 409–430. Zbl1072.65509
- H.J. Stetter, Numerical Polynomial Algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2004). Zbl1058.65054
- K.-C. Toh and L.N. Trefethen, Pseudozeros of polynomials and pseudospectra of companion matrices. Numer. Math.68 (1994) 403–425. Zbl0808.65053
- J. Vignes, A stochastic arithmetic for reliable scientific computation. Math. Comp. Sim.35 (1993) 233–261.
- J.H. Wilkinson, Rounding errors in algebraic processes. Dover Publications Inc., New York (1994). Zbl0868.65027

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.