Laguerre Polynomial – C PROGRAM

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

Using Recurrence Relation

We will be using the following recurrence relation:
(n+1)L_{n+1}(x)=(2n+1-x) L_n (x) - nL_{n-1}(x)
We would need two more relations, that is the relations for 0th and 1st order Laguerre polynomials:
We will create a program that calculates the values of the Laguerre 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 ‘l0’ and ‘l1’, that contain the definition of respectively.
Then we will create a function ‘ln’ 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.



double l0(double x){
	return 1;

double l1(double x){
	return -x+1;
//The following is a general functoin that returns the value of the Laguerre Polynomial for any given x and n=0,1,2,3,...
double ln(double x, int n){
		return l0(x);
	else if(n==1){
		return l1(x);
		return ((2*(n-1)+1-x)*ln(x,n-1)-(n-1)*ln(x,n-2))/n;
	//We will create a data-file and store the values of first few Legendre polynomials for -1<x<5
	//create data-file
	double x;
	//write the values of first 5 Lagurre  polynomials to data-file


The above program will create a data-file called laguerre1.txt and store the values of the first 5 Legendre polynomials for -1\leq x \leq 5 . Now, you can just open the file and select the data and plot it using Excel, GnuPlot, Origin, etc.
For GnuPlot, the command is:
plot './laguerre1.txt' u 1:2 w l t 'L0(x)','' u 1:3 w l t 'L1(x)', '' u 1:4 w l t 'L2(x)', '' u 1:5 w l t 'L3(x)', '' u 1:6 w l t 'L4(x)'

First few Laguerre polynomials using recurrence relation

Using Series Representation

Using Numerical Integration


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One thought on “Laguerre Polynomial – C PROGRAM

  1. Hi! Thank you for the code 🙂 Why do all of your polynomial passes from x=1? In the Wolfram page in the reference, they don’t, and I don’t know why. Also, I am using your code to find the zeroes of the function, which are different from what is tabulated on the internet (I need to write a code to integrate functions with the Gaussian-Laguerre method).

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