[Paper Review] A three point formula for finding roots of equations by the method of least squares

Ababu Teklemariam Tiruneh, N. Ndlela+1 more|arXiv (Cornell University)|Feb 8, 2013

Iterative Methods for Nonlinear Equations12 references3 citations

TL;DR

This paper proposes a derivative-free iterative root-finding method that uses three equidistant points to fit a power function P(x) = a(x−b)^N via least squares, iteratively refining the root estimate. It achieves quadratic convergence comparable to Newton’s method without requiring derivative evaluations, with demonstrated robustness and faster convergence in some cases compared to secant and Newton methods.

ABSTRACT

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.

Motivation & Objective

• To develop a derivative-free iterative method for solving nonlinear equations.
• To improve convergence speed and stability in root-finding by adapting the power N of the fitted polynomial at each iteration.
• To avoid divergence or oscillation issues common in traditional methods like Newton-Raphson.
• To provide a robust alternative to Newton and secant methods that does not require derivative computation.
• To demonstrate quadratic convergence behavior through numerical examples and comparisons.

Proposed method

• The method selects three equidistant points on the function f(x) to fit a power function P(x) = a(x−b)^N using the least squares method.
• The fitted polynomial is used to estimate the root by solving P(x) = 0, yielding a new iterate.
• The exponent N is adaptively adjusted at each iteration based on the function's local behavior to enhance convergence.
• The process is repeated iteratively until the root is located within a specified tolerance.
• The method avoids derivative evaluation by relying solely on function values at three points per iteration.
• Convergence is monitored by checking the difference between successive iterates and the residual.

Experimental results

Research questions

• RQ1 Can a three-point least squares fitting of a power function achieve quadratic convergence without derivative evaluation?
• RQ2 How does the adaptive selection of the power N affect convergence speed and stability?
• RQ3 Does the method outperform or match the convergence rate of Newton’s and secant methods in practical cases?
• RQ4 What are the conditions under which the method avoids divergence or oscillation?
• RQ5 How does the method perform on functions with varying curvature or nonlinearity?

Key findings

• The method achieves quadratic convergence, matching the rate of Newton’s method, as confirmed by numerical experiments.
• The convergence rate is maintained even when the function is highly nonlinear, provided the power N is adaptively adjusted.
• The method avoids divergence and oscillation in cases where Newton’s method fails due to poor initial guesses or zero derivatives.
• In tested examples, the method converges faster than the secant method and comparably to Newton’s method, without requiring derivatives.
• The least squares fitting of the power function provides a stable and accurate approximation of the root-seeking behavior near the solution.
• The method is robust across a range of test functions, including those with complex curvature or multiple roots.