source: CIVL/examples/accuracy/secondDerivativeBad.cvl@ 6df6487

/* Commandline execution:
 *              civl verify -inputnum_elements=5 secondDerivativeBad.cvl
 *
 * Note: based on Quarteroni, Sacco, Saleri. "Numerical Mathematics" 2nd ed. sec 10.10.1
 *
 * This version erroneously assumes that the method is 3rd order accurate in terms of h.
 * */
#include<civlc.cvh>

$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*h*h*h));
}

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