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Numerical Algorithms

, Volume 71, Issue 2, pp 349–381 | Cite as

Bivariate Newton-Raphson method and toroidal attraction basins

  • Luis Javier Hernández Paricio
Original Paper

Abstract

When the numerical Newton-Raphson method is applied to find the intersections of two algebraic curves (that is, the roots of a pair of bivariate polynomials), some difficulties appear when the value of a denominator of the corresponding bivariate rational functions is zero. In this paper we give a solution to these problems by using adequate homogeneous coordinates and extending the domain of the iteration function. The iteration of a map given by a pair of bivariate rational maps is analyzed by taking a canonical extension, which is defined on the product of two copies of an (real or complex) augmented projective line. This method gives a global description of the basins of attraction of fixed points of an iteration. In particular, the attraction basins of fixed points associated to the intersection points of the two algebraic curves is obtained. As a consequence of our techniques, we are able to plot the attraction basins of a real root either in a torus (considered as a compactification of the real plane) or in an open square, which is homeomorphic to the global real plane containing the two algebraic curves.

Keywords

Roots of polynomial equations Bivariate Newthon-Raphson method Discrete semi-flow Basin of attraction Bivariate polynomials Intersection of algebraic curves Toroidal fractals Toroidal basins of attraction 

Mathematics Subject Classification (2010)

65H04 65S05 68W25 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of La RiojaLogroñoSpain

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