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Comments on An Improvement to the Brentís Method
Steven A. Stage
Pages - 1 - 16     |    Revised - 15-01-2013     |    Published - 28-02-2013
Volume - 4   Issue - 1    |    Publication Date - January / February 2013  Table of Contents
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KEYWORDS
Brentís Method, Zhangís Method, Ridderís Method, Regula Falsi Method, Bisection Method, Root Finding, Simplification, Improvement.
ABSTRACT
Zhang (2011)[1] presented improvements to Brentís method for finding roots of a function of a single variable. Zhangís improvements make the algorithm simpler and much more understandable. He shows one test example and finds for that case that his method converges more rapidly than Brentís method. There are a few easily-correctible flaws in the algorithm as presented by Zhang which must be corrected in order to implement it. This paper shows these corrections.

We then proceed to compare the performance of several well-known root finding methods on a number of test functions. Methods tested are Zhangís method, Bisection, Regula Falsi with the Illinois algorithm, Ridderís method, and Brentís method. The results show that Brentís method and Regula Falsi generally give relatively slow initial convergence followed by very rapid final convergence and that Regula Falsi converges nearly as rapidly as Brentís method. Zhangís method and Ridderís method show similar convergence with both having faster initial convergence than Brent and Regula Falsi but slower final convergence. In many situations, the more rapid initial convergence of the Zhang method and Ridderís method leads to obtaining solutions with fewer total function evaluations than needed for Brent or Regula Falsi. Selection of the best method depends on the function being evaluated, the size of the initial interval, and the amount of accuracy required for the solution. Large initial intervals and low accuracy favor the Zhang and Ridder methods, while smaller intervals and high accuracy requirements favor Brent and Regula Falsi methods.

Guidance is presented to help the reader determine which root-finding method may be most efficient in a particular situation.
CITED BY (1)  
1 Gomes, A., & Morgado, J. (2013). A generalized regula falsi method for finding zeros and extrema of real functions. Mathematical Problems in Engineering, 2013.
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1 Z. Zhang, An Improvement to the Brentís Method, IJEA, vol. 2, pp. 21-26, May 31, 2011.
2 R.P. Brent. Algorithms for Minimization without Derivatives. Chapter 4. Prentice- Hall,Englewood Cliffs, NJ.
3 H.M. Antia. Numerical Methods for Scientists and Engineers, Birkhšuser, 2002, pp.362-365, 2 ed.
4 W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery. Numerical Recipes in C, The Art of Scientific Computing Second Edition, Cambridge University Press, November 27, 1992, pp.358Ė362.
5 G. Dahlguist and A. Bjorck. Numerical Methods. Dover Publications, 2003, p 232.
Dr. Steven A. Stage
IEM - United States of America
steve.stage@iem.com