Dynamical control of Newton's method for multiple roots of polynomials

Stef Graillat (1) , Fabienne Jézéquel (1, 2) , Moustadrani Saïd Ibrahim (1)

1 PEQUAN - Performance et Qualité des Algorithmes Numériques
2 UP2 - Université Panthéon-Assas

In this article, we show how to perform a dynamical control of Newton's method for the computation of multiple roots of polynomials. Using Discrete Stochastic Arithmetic, root approximations are computed until the difference between two successive approximations is numerical noise. With such a stopping criterion, the optimal number of iterations in Newton's method are performed. Moreover it is possible to estimate in the result obtained which digits are in common with the exact root. Two strategies to estimate the multiplicity of polynomials roots are compared: one requires root approximations computed at different precisions and the other three successive iterates of Newton's method. We show that using such a strategy and then the modified Newton's method, multiple roots can be computed with a requested accuracy.

Mots clés: Discrete Stochastic Arithmetic; floating-point arithmetic; numerical validation; rounding errors; polynomial; multiple roots; Newton's method

Domaines: Informatique [cs]; Mathématiques [math]

Stef Graillat, Fabienne Jézéquel, Moustadrani Saïd Ibrahim. Dynamical control of Newton's method for multiple roots of polynomials. Reliable Computing Journal, 2016, 21. ⟨hal-01363961⟩