/* Commandline execution:
 *		civl verify -inputnum_elements=5 secondDerivative.cvl
 *
 * Note: based on Quarteroni, Sacco, Saleri. "Numerical Mathematics" 2nd ed. sec 10.10.1
 *
 * */
#include<civlc.h>

$input double h;
$input int num_elements = 5;
$input double initial[num_elements];
double working[num_elements];
$abstract $contin(4) $real rho($real x);
$assume h > 0;

void secondDerivative(double h, int n, double y[], double result[]){
	int i;
	
	$assume $forall {m=0 .. n-1} y[m] == rho(m*h);
	for(i = 1; i < n-1; i++)
	{
		result[i] = (y[i+1]-2*y[i]+y[i-1])/(h*h);
	}
	result[0] = (y[2]-2*y[1]+y[0])/h; 
	result[n-1] = (y[n-3] - 2*y[n-2]-y[n-1])/h;
	$assert($uniform {k=1 .. n-2} result[k]-$D[rho,{x,2}](k*h) == $O(h*h));
}

void main() {
	secondDerivative(h, num_elements, initial, working);
}