Graphical representations for the homogeneous bivariate Newton's method

  1. García Calcines, J.M. 1
  2. Gutiérrez, J.M. 2
  3. Hernández Paricio, L.J. 2
  4. Rivas Rodríguez, M.T. 2
  1. 1 Universidad de La Laguna
    info

    Universidad de La Laguna

    San Cristobal de La Laguna, España

    ROR https://ror.org/01r9z8p25

  2. 2 Universidad de La Rioja
    info

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

Revista:
Applied Mathematics and Computation

ISSN: 0096-3003

Año de publicación: 2015

Volumen: 269

Páginas: 988-1006

Tipo: Artículo

DOI: 10.1016/J.AMC.2015.07.102 SCOPUS: 2-s2.0-84940703024 WoS: WOS:000361771500084 GOOGLE SCHOLAR

Otras publicaciones en: Applied Mathematics and Computation

Proyectos relacionados

Ecuaciones no lineales y métodos iterativos. Aplicaciones a problemas de optimización y ecuaciones matriciales. Nonlinear equations and iterative methods. Applications to optimization problems and matrix equations.

2015/00069/001

Resumen

In this paper we propose a new and effective strategy to apply Newton's method to the problem of finding the intersections of two real algebraic curves, that is, the roots of a pair of real bivariate polynomials. The use of adequate homogeneous coordinates and the extension of the domain where the iteration function is defined allow us to avoid some numerical difficulties, such as divisions by values close to zero. In fact, we consider an iteration map defined on a real augmented projective plane. So, we obtain a global description of the basins of attraction of the fixed points associated to the intersection of the curves. As an application of our techniques, we can plot the basins of attraction of the roots in the following geometric models: hemisphere, hemicube, Möbius band, square and disk. We can also give local graphical representations on any rectangle of the plane. © 2015 Elsevier Inc.