#### Newton Raphson Method

/**
* \file
* \brief Solve the equation \f$f(x)=0\f$ using [Newton-Raphson
* method](https://en.wikipedia.org/wiki/Newton%27s_method) for both real and
* complex solutions
*
* The \f$(i+1)^\text{th}\f$ approximation is given by:
* \f[
* x_{i+1} = x_i - \frac{f(x_i)}{f'(x_i)}
* \f]
*
* \author [Krishna Vedala](https://github.com/kvedala)
* \see bisection_method.cpp, false_position.cpp
*/
#include <cmath>
#include <ctime>
#include <iostream>
#include <limits>

constexpr double EPSILON = 1e-10;              ///< system accuracy limit
constexpr int16_t MAX_ITERATIONS = INT16_MAX;  ///< Maximum number of iterations

/** define \f$f(x)\f$ to find root for.
* Currently defined as:
* \f[
* f(x) = x^3 - 4x - 9
* \f]
*/
static double eq(double i) {
return (std::pow(i, 3) - (4 * i) - 9);  // original equation
}

/** define the derivative function \f$f'(x)\f$
* For the current problem, it is:
* \f[
* f'(x) = 3x^2 - 4
* \f]
*/
static double eq_der(double i) {
return ((3 * std::pow(i, 2)) - 4);  // derivative of equation
}

/** Main function */
int main() {
std::srand(std::time(nullptr));  // initialize randomizer

double z = NAN, c = std::rand() % 100, m = NAN, n = NAN;
int i = 0;

std::cout << "\nInitial approximation: " << c;

// start iterations
for (i = 0; i < MAX_ITERATIONS; i++) {
m = eq(c);
n = eq_der(c);

z = c - (m / n);
c = z;

if (std::abs(m) < EPSILON) {  // stoping criteria
break;
}
}

std::cout << "\n\nRoot: " << z << "\t\tSteps: " << i << std::endl;
return 0;
}