/*T Math/CS 466/666 Midterm Solutions
\medskip\noindent
Problem 1(iv).  Consider the approximation
$$
    f''(x)\approx {f(x+h)-2f(x)+f(x-h)\over h^2}.
$$
Let $f(x)=\sin(x^2)$.  Write or modify a computer program to create
a table showing the approximation and the errors in the approximation
when $x=2$ and $h=2^{-n}$ for $n=0,1,\ldots,30.$ */

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

double f(double x){
    return sin(x*x);
}
double ddf(double x){
/*T Always set this to $f''(x)$ where $f(x)$ is defined above.
Since $f'(x)=2x\cos(x^2)$ then the exact second derivative is
$f''(x)=2\cos(x^2)-4x^2\sin(x^2)$. */
    double xx=x*x;
    return 2*cos(xx)-4*xx*sin(xx);
}
double addf(double x,double h){
    return (f(x+h)-2*f(x)+f(x-h))/(h*h);
}

int main(){
    printf("Math/CS 466/666 Midterm\nProblem 1(iv)\n\n");
    double h=1,x=2;
    printf("%3s %22s %22s %22s\n","n","h","approximation","error");
    for(int n=0;n<=30;n++,h/=2){
        printf("%3d %22.14e %22.14e %22.14e\n",
            n,h,addf(x,h),addf(x,h)-ddf(x));
    }
    return 0;
}