In this post I will show you how to calculate and plot the Chebyshev Polynomials of the second kind using a C Program and Gnuplot.

We will be using the following information:


and the recurrence relation:
where, n starts from 1.

Modifying the recurrence relation a little, so that n starts from 2, we get:

We will create a program that calculates the values of the Chebyshev Polynomials at various x values and for different n and store these values in a txt file. Then just plot it using Gnuplot.
We will create two functions called ‘U0’ and ‘U1’, that contain the definition of respectively.
Then we will create a function ‘Un’ that will use the first two functions and recursion to find the value of Chebyshev Polynomials for different x,n.

PROGRAM:

/******************************************
***********Chebyshev Polynomials***********
******************************************/
#include<stdio.h>
double U0(double x){
	return 1;
}
double U1(double x){
	return 2*x;
}
//General form of Chebyshev polynomial of second for a given value of n and x
double Un(double x, int n){
	if(n==0){
		return U0(x);
	}
	else if(n==1){
		return U1(x);
	}
	else{
		//using the recurrence relation
		return 2*x*Un(x,n-1)-Un(x,n-2);
	}
}
main(){
	double x;
	FILE *fp=NULL;
	fp=fopen("cheby.txt","w");
	//Write down the values to a file
	for(x=-1;x<=1;x=x+0.01){
		fprintf(fp,"%lf\t%lf\t%lf\t%lf\t%lf\t%lf\n",x,Un(x,0),Un(x,1),Un(x,2),Un(x,3),Un(x,4));
	}
}

When you run the above C, it will generate a file called ‘cheby.txt’ which would contain 6 columns of data-points.
The first column contains the ‘x’ values and the rest of them are for
These can be easily plotted using Gnuplot by using the following commands:

GnuPlot Command:

->set xlabel 'x'
->plot 'cheby.txt' u 1:2 w l t "U0(x)", '' u 1:3 w l t "U1(x)", '' u 1:4 w l t "U2(x)", '' u 1:5 w l t "U3(x)", '' u 1:6 w l t "U4(x)"


OUTPUT(Gnuplot):

References:

https://en.wikipedia.org/wiki/Chebyshev_polynomials
http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html

Manas Sharma

Ph.D. researcher at Friedrich-Schiller University Jena, Germany. I’m a physicist specializing in computational material science. 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/