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

Abstract

top
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.

How to cite

top

Graillat, 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
  1. 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/
  2. W. Gautschi, On the condition of algebraic equations. Numer. Math.21 (1973) 405–424.  Zbl0278.65044
  3. 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.  
  4. 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).  
  5. D. Hinrichsen and B. Kelb, Spectral value sets: a graphical tool for robustness analysis. Systems Control Lett.21 (1993) 127–136.  Zbl0785.93030
  6. 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
  7. WWW resources about Interval Arithmetic. .  URIhttp://www.cs.utep.edu/interval-comp/main.html
  8. L. Jaulin, M. Kieffer, O. Didrit and É. Walter, Applied interval analysis. Springer-Verlag London Ltd., London (2001).  Zbl1023.65037
  9. D.G. Luenberger, Optimization by vector space methods. John Wiley & Sons Inc., New York (1969).  Zbl0176.12701
  10. R.G. Mosier, Root neighborhoods of a polynomial. Math. Comp.47 (1986) 265–273.  Zbl0598.65023
  11. A.M. Ostrowski, Solution of equations and systems of equations. Second edition. Academic Press, New York. Pure Appl. Math.9 (1966).  Zbl0222.65070
  12. 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
  13. H.J. Stetter, Numerical Polynomial Algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2004).  Zbl1058.65054
  14. K.-C. Toh and L.N. Trefethen, Pseudozeros of polynomials and pseudospectra of companion matrices. Numer. Math.68 (1994) 403–425.  Zbl0808.65053
  15. J. Vignes, A stochastic arithmetic for reliable scientific computation. Math. Comp. Sim.35 (1993) 233–261.  
  16. J.H. Wilkinson, Rounding errors in algebraic processes. Dover Publications Inc., New York (1994).  Zbl0868.65027

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.