In this post I’m gonna show you how to calculate Hermite polynomials using three different techniques: using recurrence relations, series representations, and numerical integration.
The programs will calculate and plot the first few Hermite polynomials.

Using Recurrence Relation

We will be using the following recurrence relation:

We would need two more relations, that is the relations for 0th and 1st order Legendre polynomials:


We will create a program that calculates the values of the Legendre polynomial at various x values and for different l and store these values in a txt file. Then just plot it using Gnuplot.
We will create two functions called ‘h0’ and ‘h1’, that contain the definition of respectively.
Then we will create a function ‘hn’ that will use the first two functions and recursion to find the value of Legendre polynomial for different x,n.
NOTE: I am using a slightly modified form of the recurrence relation. To get the form I am using, just replace n by n-1.

CODE:

#include<stdio.h>
#include<math.h>

double h0(double x){
	return 1;
}

double h1(double x){
	return 2*x;
}
//The following is a general function that returns the value of the Hermite Polynomial for any given x and n=0,1,2,3,...
double hn(double x,int n){
	if(n==0){
		return h0(x);
	}
	else if(n==1){
		return h1(x);
	}
	else{
		return 2*x*hn(x,n-1)-2*(n-1)*hn(x,n-2); 	
	} 
}
main(){
	//We will create a data-file and store the values of first few Hermite polynomials for -1<x<5
	FILE *fp=NULL;
	//create data-file
	fp=fopen("hermite1.txt","w");
	double x;
	//write the values of first 5 Hermite  polynomials to data-file
	for(x=-2;x<=2;x=x+0.1){
		fprintf(fp,"%lf\t%lf\t%lf\t%lf\t%lf\t%lf\n",x,hn(x,0),hn(x,1),hn(x,2),hn(x,3),hn(x,4));
	}
}

OUTPUT:

The above program will create a data-file called legendre1.txt and store the values of the first 5 Hermite polynomials for . Now, you can just open the file and select the data and plot it using Excel, GnuPlot, Origin, etc.
For GnuPlot, the command is:

Using Series Representation

Using Numerical Integration

References:

http://mathworld.wolfram.com/HermitePolynomial.html

Manas Sharma

I’m a physicist specializing in computational material science with a PhD in Physics from Friedrich-Schiller University Jena, Germany. I write efficient codes for simulating light-matter interactions at atomic scales. I like to develop Physics, DFT, and Machine Learning related apps and software from time to time. Can code in most of the popular languages. I like to share my knowledge in Physics and applications using this Blog and a YouTube channel.

manas.bragitoff.com/