newton_raphson_method.cpp File Reference

Solve the equation \(f(x)=0\) using Newton-Raphson method for both real and complex solutions. More...

#include <cmath>
#include <cstdint>
#include <ctime>
#include <iostream>
#include <limits>

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Functions
static double	eq (double i)
static double	eq_der (double i)
int	main ()

Variables
constexpr double	EPSILON = 1e-10
	system accuracy limit
constexpr int16_t	MAX_ITERATIONS = INT16_MAX
	Maximum number of iterations.

Detailed Description

Solve the equation \(f(x)=0\) using Newton-Raphson method for both real and complex solutions.

The \((i+1)^\text{th}\) approximation is given by:

\[ x_{i+1} = x_i - \frac{f(x_i)}{f'(x_i)} \]

Author

Krishna Vedala

See also

bisection_method.cpp, false_position.cpp

Function Documentation

eq()

static double eq ( double i )

static

define \(f(x)\) to find root for. Currently defined as:

\[ f(x) = x^3 - 4x - 9 \]

                           {
    return (std::pow(i, 3) - (4 * i) - 9);  // original equation
}

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eq_der()

static double eq_der ( double i )

static

define the derivative function \(f'(x)\) For the current problem, it is:

\[ f'(x) = 3x^2 - 4 \]

                               {
    return ((3 * std::pow(i, 2)) - 4);  // derivative of equation
}

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main()

int main

(

void

)

Main function

           {
    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;
}

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