//numerical derivatives, calculated using difference tables
//this takes experimental data for x and f(x)
//the data is used to calculate the values for 
//the function in areas where we have no data


//this example uses hard coded data for the square root of x


public class derivatives
{
    
    public static void main ( String args[] )
    {
	int arraysize = 6;
	double x[] = { 2.50, 2.55, 2.60, 2.65, 2.70, 2.75 };
	double f_of_x[] = { 1.58114, 1.59687, 1.61245, 1.62788, 1.64317, 1.65831 };
	
	double x1[] = new double[arraysize];
	double x2[] = new double[arraysize];
	
	
	
	//initialize stuff
	for ( int i=0; i<arraysize; i++){
	    x1[i] = 0.0;
	    x2[i] = 0.0;
	}
	
	
	//calculate differences
	for ( int i=0; i<(arraysize-1); i++ ){
	    x1[i] = (f_of_x[i+1] - f_of_x[i]);
	}
	
	for ( int i=0; i<(arraysize-2); i++ ){
	    x2[i] = (x1[i+1] - x1[i]);
	}
	
	
	
	
	
	
	for ( int i=0; i<arraysize; i++){
	    System.out.println ( x[i] + "\t" + f_of_x[i]);
	    if ( x1[i] != 0 ) System.out.println ( "\t\t" + x1[i] );
	    if ( x2[i] != 0 ) System.out.println ( "\t\t\t" + x2[i] );
	}
	
	
	
	
	System.out.println();
	System.out.println ( "First derivative @ x = 2.5    1/" + (x[1]-x[0]) + " * ( " + x1[0] +" - " + "(1/2)" + " * " + (x2[0])  + ")"   );
	System.out.println ( "First derivative of sqrt of x, x=2.5 = " + 1/(x[1]-x[0] ) * ( x1[0] - (0.50)*(x2[0]) ) );
	
	System.out.println ( "Second derivative @ x = 2.5     1/ " + "(" + (x[1]-x[0]) + " * " + (x[1]-x[0]) + ") * " + ( x2[0] )  );
	System.out.println ( "Second derivative of sqrt of x, x=2.5 = " + 1/((x[1]-x[0])*(x[1]-x[0])) * ( x2[0] ));
	
	System.out.println();
    
    }
    
    
}