Graphical representations for the homogeneous bivariate Newton’s method

José M. García Calcines, José M. Gutiérrez, Luis J. Hernández Paricio and M. Teresa Rivas Rodríguez

Applied Mathematics and Computation, 2015, vol. 269, issue C, 988-1006

Abstract: 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.

Keywords: Roots of polynomial equations; Homogeneous bivariate Newton’s method; Discrete semi-flow; Intersection of algebraic curves; Fractals on the real projective plane; Basins of attraction on the Möbius band (search for similar items in EconPapers)
Date: 2015
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0096300315010243
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:269:y:2015:i:c:p:988-1006

DOI: 10.1016/j.amc.2015.07.102

Access Statistics for this article

Applied Mathematics and Computation is currently edited by Theodore Simos

More articles in Applied Mathematics and Computation from Elsevier
Bibliographic data for series maintained by Catherine Liu ().