A three point formula for finding roots of equations by the method of least squares

A new method of root finding is formulated that uses a numerical iterative process involving three points. A given function y = f(x) whose roots are desired is fitted and approximated by a polynomial function of the form P(x)= a(x-b)^N that passes between three equi-spaced points using the method of...

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Bibliographic Details
Published in:arXiv.org 2013-02
Main Authors: Ababu Teklemariam Tiruneh, Ndlela, William N, Nkambule, Stanley J
Format: Article
Language:English
Subjects:
Convergence
Curve fitting
Divergence
Economic models
Estimating techniques
Iterative methods
Least squares method
Mathematical analysis
Newton methods
Polynomials
Roots
Roots of equations
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Summary:A new method of root finding is formulated that uses a numerical iterative process involving three points. A given function y = f(x) whose roots are desired is fitted and approximated by a polynomial function of the form P(x)= a(x-b)^N that passes between three equi-spaced points using the method of least squares. Successive iterations using the same procedure of curve fitting is used to locate the root within a given level of tolerance. The power N of the curve suitable for a given function form can be appropriately varied at each step of the iteration to give a faster rate of convergence and avoid cases where oscillation, divergence or off shooting to an invalid domain may be encountered. An estimate of the rate of convergence is provided. It is shown that the method has a quadratic convergence similar to that of Newton's method. Examples are provided showing the procedure as well as comparison of the rate of convergence with the secant and Newton methods. The method does not require evaluation of function derivatives.
ISSN:2331-8422