Simpson’s Rule for Approximating Definite Integrals in R

Simpson’s rule is another closed Newton-Cotes formula for approximating integrals over an interval with equally spaced nodes. Unlike the trapezoidal rule, which employs straight lines to approximate a definite integral, Simpson’s rule uses the third Lagrange polynomial, [latex]P_3(x)[/latex] to approximate the definite integral and as such can give exact results...

The Trapezoidal Rule of Numerical Integration in R

The Trapezoidal Rule is another of Closed Newton-Cotes formulas for approximating the definite integral of a function. The trapezoidal rule is so named due to the area approximated under the integral [latex]\int^a_b f(x) \space dx[/latex] representing a trapezoid. Although there exist much more accurate quadrature methods, the trapezoidal rule converges...

Numerical Differentiation with Finite Differences in R

Numerical differentiation is a method of approximating the derivative of a function [latex]f[/latex] at particular value [latex]x[/latex]. Often, particularly in physics and engineering, a function may be too complicated to merit the work necessary to find the exact derivative, or the function itself is unknown, and all that is available...

Divided Differences Method of Polynomial Interpolation

The divided differences method is a numerical procedure for interpolating a polynomial given a set of points. Unlike Neville’s method, which is used to approximate the value of an interpolating polynomial at a given point, the divided differences method constructs the interpolating polynomial in Newton form. Consider a table of values...

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...