Evaluate and plot the Cosine (Infinite) Series using C Programming and Gnuplot

In the last post I discussed and showed you how to write a program that finds the sum of the Sine series(Infinite so to speak).

Using the same concept, we will extend it a bit further in this post.

In this post we will evaluate the Cosine series, correct upto a certain decimal places, for a given range of x in radians. We would store the value of Cos(x) evaluated in a text file and then plot them using Gnuplot.

So let’s first start with writing a program that evaluates the Cosine series.
The series is given by:
$\cos (x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - ....$

Ratio of the successive terms(index the numbers from 0):
$R=\frac{t_i}{t_{i-1}}= -\frac{x^2}{2i(2i-1)}$

Since, we indexed the terms starting from 0, therefore, for the above relation to work, $i$ will go from 1 to $n$ .
Now, knowing the first( $t_0$ ) term, the successive terms can be calculated as :

$t_1=R\times t_0$

$t_2=R\times t_1$

and so on.

Therefore, the C program that calculates the sum of the cosine series upto a certain accuracy can be written as shown below.

PROGRAM:

/********************************
******INFINITE SERIES SUM**********
Series: cos(x) = 1 - (x^2/2!) + (x^4/4!) - ..... 
********************************/
#include<stdio.h>
#include<math.h>
main(){
	int i=1;
	double x,t0,t1,R,sum,eps;
	printf("Enter the value of x:\n");
	scanf("%lf",&x);
	printf("Enter the desired accuracy: ");
	scanf("%lf",&eps);
	//Initialize First Term
	t0=1;
	//Make sum equal to the first term
	sum=t0;
	do{
		//Find the ratio of the second term to the first term using already known relation
		R=-(x*x)/(2*i-1)/(2*i);
		//Calculate the second term
		t1=R*t0;
		//find the new sum
		sum=sum+t1;
		t0=t1;
		i++;
		//keep on summing terms until the required accuracy is reached
	}while(fabs(t1/sum)>eps);
	printf("\nThe sum [cos(%lf)] is: %lf with %d terms",x,sum,i);
}

OUTPUT:

The program asks the user to enter the value of x and the desired accuracy, and gives answer.

Now that we have a program for evaluating the cosine series, we can write a program that will evaluate the cosine series in a given range[0 to 4pi] and store the values in a file.

PROGRAM:

/********************************
******INFINITE SERIES SUM**********
Series: cos(x) = 1 - (x^2/2!) + (x^4/4!) - ..... 
********************************/
#include<stdio.h>
#include<math.h>
main(){
	FILE *fp=NULL;
	fp=fopen("cos(x).txt","w");
	double x,t0,t1,R,sum,eps;
	printf("Enter the desired accuracy: ");
	scanf("%lf",&eps);
	for(x=0.0000;x<=4*M_PI;x=x+0.001){
		int i=1;
		//Initialize First Term
		t0=1;
		//Make sum equal to the first term
		sum=t0;
		do{
			//Find the ratio of the second term to the first term using already known relation
			R=-(x*x)/(2*i-1)/(2*i);
			//Calculate the second term
			t1=R*t0;
			//find the new sum
			sum=sum+t1;
			t0=t1;
			i++;
			//keep on summing terms until the required accuracy is reached
		}while(fabs(t1/sum)>eps);	
		fprintf(fp,"%lf\t%lf\n",x,sum);
	}
}

OUPTUT:

When you run the above C program it will ask for the accuracy desired for the calculations.
When the execution is complete, it will create a txt file called ‘cos(x).txt’ that will contain the data to be plotted.

Gnuplot Command:

You can plot the data using Gnuplot, by giving the following command:
plot './cos(x).txt' w l

Gnuplot OUPTUT:

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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/