In this paper, we combine Graeffe matrices with the classical numerical method of Dandelin-Graeffe to estimate bounds for the moduli of the zeros of polynomials. Furthermore, we give some examples showing significant gain for the convergence towards the polynomials dominant zeros moduli.
References
[1]
Matthias, D. and Jurgen, K. (2007) On Bounds for the Zeros of Univariate Polynomials. Proceedings of World Academy of Science: Engineering & Technology, 20, 205.
[2]
Parodi, M. (1959) La Localisation des valeurs caractéristiques des Matrices et ses Applications. Gauthier-Villars, Paris.
[3]
Marden. M. (1949) The Geometry of the Zeros of a Polynomial in a Complex Variable. American Mathematical Society, New York. http://dx.doi.org/10.1090/surv/003
[4]
Dehmer, M. and Tsoy, Y.R. (2012) The Quality of Zero Bounds for Complex Polynomials. PLoS ONE, 7, e39537. http://dx.doi.org/10.1371/journal.pone.0039537
[5]
Linden, H. (1998) Bounds for the Zeros of Polynomials from Eigenvalues and Singular Values of Some Companion Matrices. Linear Algebra and Its Applications, 271, 41-82. http://dx.doi.org/10.1016/S0024-3795(97)00254-1
[6]
Mignotte, M. (1992) Mathematics for Computer Algebra. Springer Verlag, New York. http://dx.doi.org/10.1007/978-1-4613-9171-5
[7]
Mignotte, M. and Stefanescu, D. (2003) Linear Recurrent Sequences and Polynomial Roots. Journal of Symbolic Computation, 35, 637-649. http://dx.doi.org/10.1016/S0747-7171(03)00030-0
[8]
Kallol, P. and Santanu, B. (2012) On Numerical Radius of a Matrix and Estimation of Bounds for Zeros of a Polynomial. International Journal of Mathematics and Mathematical Sciences, 2012, Article ID: 129132.
[9]
Diouf, I. (2007) Méthode de Dandelin-Graeffe et Méthode de Baker. Thèses de Doctorat, Université Louis Pasteur de Strasbourg, Strasbourg.
[10]
Jacobson, N. (1985) Basic Algebra I. Fremann, New York.
