Neville’s Method of Polynomial Interpolation

Neville’s method evaluates a polynomial that passes through a given set of [latex]x[/latex] and [latex]y[/latex] points for a particular [latex]x[/latex] value using the Newton polynomial form. Neville’s method is similar to a now-defunct procedure named Aitken’s algorithm and is based on the divided differences recursion relation (“Neville’s Algorithm”, n.d). It was...

Lagrangian Polynomial Interpolation with R

Polynomial interpolation is the method of determining a polynomial that fits a set of given points. There are several approaches to polynomial interpolation, of which one of the most well known is the Lagrangian method. This post will introduce the Lagrangian method of interpolating polynomials and how to perform the...

The Bisection Method of Root-Finding with R

The bisection method is another approach to finding the root of a continuous function [latex]f(x)[/latex] on an interval [latex][a, b][/latex]. The method takes advantage of a corollary of the intermediate value theorem called Bolzano’s theorem which states that if the values of [latex]f(a)[/latex] and [latex]f(b)[/latex] have opposite signs, the interval...

The Secant Method Root-Finding Algorithm in R

The secant method for finding roots of nonlinear equations is a common and popular variation of the Newton-Raphson method that has been used for several millennia before the invention of Newton-Raphson (Papakonstantinou, J. as cited in Wikipedia). The secant method is an iterative method that takes two initial guesses of...

The Newton-Raphson Root-Finding Algorithm in R

The Newton-Raphson method is an approach for finding the roots of nonlinear equations and is one of the most common root-finding algorithms due to its relative simplicity and speed. The root of a function is the point at which [latex]f(x) = 0[/latex]. Many equations have more than one root. Every...