Generalization of the Secant Method for Nonlinear Equations (extended version)

TL;DR

This paper generalizes the secant method for solving nonlinear equations by using higher-order polynomial interpolation, achieving faster convergence rates that approach quadratic as the interpolation order increases.

Contribution

It introduces a new family of root-finding methods based on $(k+1)$-point interpolation, with proven higher convergence order and efficiency compared to the classical secant method.

Findings

Convergence order increases with interpolation degree $k$.

Order of convergence approaches 2 as $k$ increases.

The method maintains computational efficiency by evaluating $f(x)$ only once per iteration.

Abstract

The secant method is a very effective numerical procedure used for solving nonlinear equations of the form $f(x)=0$. It is derived via a linear interpolation procedure and employs only values of $f(x)$ at the approximations to the root of $f(x)=0$, hence it computes $f(x)$ only once per iteration. In this note, we generalize it by replacing the relevant linear interpolant by a suitable $(k+1)$-point polynomial of interpolation, where $k$ is an integer at least 2. Just as the secant method, this generalization too enjoys the property that it computes $f(x)$ only once per iteration. We provide its error in closed form and analyze its order of convergence $s_k$. We show that this order of convergence is greater than that of the secant method, and it increases towards $2$ as $k\to \infty$. (Indeed, $s_7=1.9960\cdots$, for example.) This is true for the efficiency index of the method too. We also confirm the theory via an illustrative example.