/*T Math/CS 466/666 Midterm Solutions
\medskip\noindent
Problem 1(i).  Write or modify a computer program to implement Newton's
method and use it to approximate the solution to $x^3=\cos x$ starting
with an initial guess of $x_0=1.$  Print the first 5 iterations of the
method. */

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

double f(double x){
    return x*x*x-cos(x);
}
double df(double x){
    //T Always set this to $f'(x)$ where $f(x)$ is defined above.
    return 3*x*x+sin(x);
}

int main(){
    printf(
        "Math/CS 466/666 Midterm\nProblem 1(i).\n\n"
        "%3s %24s\n","n","xn");
    double x=1;
    for(int i=0;;i++){
        printf("%3d %24.14e\n",i,x);
        if(i>=5) break;
        x=x-f(x)/df(x);             //T $x_{i+1}=x_{i}-f(x_i)/f'(x_i)$
    }
}