/*********************** Numerically integrate a differential equation using Euler's method ************************/


//~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Symbols~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


//	h	::	Step size of integration

//	x_n	::	position at nth time

//	x_n1	::	position at (n+1)th time

//	c	::	integration constant

//~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~




//Include libraries

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



main()
{

	int i, n;

	double x_n, x_n1, h, c;

	FILE*fw = fopen( "euler.dat", "w" );

	FILE*fa = fopen( "analytical.dat", "w" );

	printf( "Enter the step size, h: \t" );

	scanf( "%lf", &h );

	printf( "Enter the number of steps: \t" );

	scanf( "%d", &n );

	printf( "Enter the initial value of x0 between 0 to 2 pi: \t" );

	scanf( "%lf", &x_n );

	fprintf( fw, "%d\t%lf\n", 0, x_n );

	c = log( tan( x_n / 2.0 ) );

//~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Euler's Method~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

	for( i = 1; i <= n; i++ )
	{

		x_n1 = x_n + h * sin( x_n );

		x_n = x_n1;

		fprintf( fw, "%0.6lf\t%0.6lf\n", i * h, x_n );

	}


//~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Analytical Results~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

	for( i = 1; i <= n; i++ )
	{

		x_n = 2 * atan( exp( i * h + c ) );

		fprintf( fa, "%0.6lf\t%0.6lf\n", i * h, x_n );

	}


	fclose(fw);

	fclose(fa);


}