Numerical Algorithms

, Volume 47, Issue 1, pp 95–107

Simple geometric constructions of quadratically and cubically convergent iterative functions to solve nonlinear equations

Original Paper

DOI: 10.1007/s11075-007-9149-4

Cite this article as:
Kanwar, V., Singh, S. & Bakshi, S. Numer Algor (2008) 47: 95. doi:10.1007/s11075-007-9149-4

Abstract

In this paper, we derive one-parameter families of Newton, Halley, Chebyshev, Chebyshev-Halley type methods, super-Halley, C-methods, osculating circle and ellipse methods respectively for finding simple zeros of nonlinear equations, permitting f ′ (x) = 0 at some points in the vicinity of the required root. Halley, Chebyshev, super-Halley methods and, as an exceptional case, Newton method are seen as the special cases of the family. All the methods of the family and various others are cubically convergent to simple roots except Newton’s or a family of Newton’s method.

Keywords

Nonlinear equationsIterative methodsOne-parameter familyNewton’s methodHalley’s methodChebyshev’s methodsuper-Halley method

AMS subject classifications

65H05

Copyright information

© Springer Science+Business Media, LLC. 2007

Authors and Affiliations

  1. 1.University Institute of Engineering and TechnologyPanjab UniversityChandigarhIndia
  2. 2.Department of MathematicsSant Longowal Institute of Engineering and TechnologyLongowalIndia
  3. 3.Department of Applied SciencesIndo Global College of EngineeringMohaliIndia