# Legendre Polynomial – C PROGRAM

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

### Using Recurrence Relation

We will be using the following recurrence relation:
$(l+1)P_{l+1}(x)-(2l+1)x P_l (x) + lP_{l-1}(x)=0$
We would need two more relations, that is the relations for 0th and 1st order Legendre polynomials:
$P_0(x)=1$
$P_1(x)=x$
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 ‘P0’ and ‘P1’, that contain the definition of respectively.
Then we will create a function ‘Pn’ that will use the first two functions and recursion to find the value of Legendre polynomial for different x,l.
NOTE: I am using a slightly modified form of the recurrence relation. To get the form I am using, just replace l by l-1.
To get :
$P_{l}(x)=((2l-1)x P_{l-1} (x) - (l-1)P_{l-2}(x))/l$

#### CODE:

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

double P0(double x){
return 1;
}

double P1(double x){
return x;
}
//The following is a general functoin that returns the value of the Legendre Polynomial for any given x and n=0,1,2,3,...
double Pn(double x, int n){
if(n==0){
return P0(x);
}else if(n==1){
return P1(x);
}else{
return (double)((2*n-1)*x*Pn(x,n-1)-(n-1)*Pn(x,n-2))/n;
}
}
main(){
//We will create a data-file and store the values of first few Legendre polynomials for -1<x<1
FILE *fp=NULL;
//create data-file
fp=fopen("legendre1.txt","w");
double x;
//write the values of first 5 Legendre polynomials to data-file
for(x=-1;x<=1;x=x+0.1){
fprintf(fp,"%lf\t%lf\t%lf\t%lf\t%lf\t%lf\n",x,Pn(x,0),Pn(x,1),Pn(x,2),Pn(x,3),Pn(x,4));
}
}


#### OUTPUT:

The above program will create a data-file called legendre1.txt  and store the values of the first 5 Legendre polynomials for $-1\leq x \leq 1$. 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 './legendre1.txt' u 1:2 w l t 'P0(x)','' u 1:3 w l t 'P1(x)', '' u 1:4 w l t 'P2(x)', '' u 1:5 w l t 'P3(x)', '' u 1:6 w l t 'P4(x)'