Numerical Algorithms

, Volume 47, Issue 1, pp 95–107 | Cite as

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

Original Paper

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 equations Iterative methods One-parameter family Newton’s method Halley’s method Chebyshev’s method super-Halley method 

AMS subject classifications

65H05 

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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

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