Introduction

This program is an implementation of numerical integration methods sing the trapezoid method, the Simpson's method and Monte Carlo methods. It is part of Chapter 6 in the computational physics textbook by Landau Landau (2015)¹ on numerical integration. It is also a good refresher of the topics I learnt in MH3700 Numerical Analysis I that I took in NTU.

Numerical Integration

Integrating numerically involves finding the area under the curve of a function.

Trapezoid method

This method involves approximating the area under the curve of the function as a collection of $N$ trapezoids with area $\frac{1}{2}(a+b)h$. The algorithm for this method is then

$$\int_a^b f(x)dx= \frac{1}{2}\sum_{i=0}^{N-1}(f(a+ih)+f(a+(i+1)h))h$$

where $h=\frac{b-a}{N}$

Simpson's Method

This method invovles approximating the area under the curve of a function as a collection of $N$ parabolas. The algorithm for this method is

$$\int_a^b f(x)dx = \sum_{i=0}^{N-1}\left[\frac{4}{3}f(a+ih)+\frac{1}{3}f(a+\frac{3ih}{2})+\frac{4}{3}f((i+1)h)\right]$$

Monte Carlo Method

This method involves making use of the mean value theorem in calculus that states that

$$\int_a^bf(x)dx=(b-a)\langle f \rangle$$

Where $\langle f\rangle$ is the mean of the function in the interval. By the central limit theorem,

$$\langle f \rangle = \lim_{N\to \infty}\frac{1}{N}\sum_{i=0}^N f(x_i)$$

where $x_i$ is a uniformly distributed random number. This method however converges slower than the above methods. For high dimensional integration however, this method is preferred as it is less computationally intensive.

Numerical Integration

The function to be integrated is a function TheFunction(double x) that is defined in mainwindow.cpp. As it is a 1D integration it accepts a single variable x. An example that evaluates $\int_0^1 x^2+x+1dx$ is shown below. As can be seen, the first two methods converge to the correct answer while the Monte Carlo method does not converge but does get very close.

Footnotes

1. Landau, R. H. (2015). Computational physics problem solving with python. John Wiley & Sons, Incorporated. ↩