Real and complex pseudozero sets for polynomials with applications
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 41 (2007) no. 1, pp. 45-56.

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.

Classification : 65F35,  68W30
Mots clés : polynomial root, pseudozero set, uncertainty, perturbation, stability
     author = {Graillat, Stef and Langlois, Philippe},
     title = {Real and complex pseudozero sets for polynomials with applications},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {45--56},
     publisher = {EDP-Sciences},
     volume = {41},
     number = {1},
     year = {2007},
     doi = {10.1051/ita:2007006},
     mrnumber = {2330042},
     language = {en},
     url = {}
Graillat, Stef; Langlois, Philippe. Real and complex pseudozero sets for polynomials with applications. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 41 (2007) no. 1, pp. 45-56. doi : 10.1051/ita:2007006.

[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).

[2] W. Gautschi, On the condition of algebraic equations. Numer. Math. 21 (1973) 405-424. | Zbl 0278.65044

[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. | Zbl 0785.93030

[6] D. Hinrichsen and A.J. Pritchard, Robustness measures for linear systems with application to stability radii of Hurwitz and Schur polynomials. Internat. J. Control 55 (1992) 809-844. | Zbl 0747.93017

[7] WWW resources about Interval Arithmetic.

[8] L. Jaulin, M. Kieffer, O. Didrit and É. Walter, Applied interval analysis. Springer-Verlag London Ltd., London (2001). | MR 1989308 | Zbl 1023.65037

[9] D.G. Luenberger, Optimization by vector space methods. John Wiley & Sons Inc., New York (1969). | MR 238472 | Zbl 0176.12701

[10] R.G. Mosier, Root neighborhoods of a polynomial. Math. Comp. 47 (1986) 265-273. | Zbl 0598.65023

[11] A.M. Ostrowski, Solution of equations and systems of equations. Second edition. Academic Press, New York. Pure Appl. Math. 9 (1966). | MR 216746 | Zbl 0222.65070

[12] H.J. Stetter, Polynomials with coefficients of limited accuracy, in Computer algebra in scientific computing - CASC'99 (Munich), Springer, Berlin (1999) 409-430. | Zbl 1072.65509

[13] H.J. Stetter, Numerical Polynomial Algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2004). | MR 2048781 | Zbl 1058.65054

[14] K.-C. Toh and L.N. Trefethen, Pseudozeros of polynomials and pseudospectra of companion matrices. Numer. Math. 68 (1994) 403-425. | Zbl 0808.65053

[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). | MR 1280465 | Zbl 0868.65027