rungekutta.cpp

/**
 * @{
 * \file
 * \brief [Runge Kutta fourth
 * order](https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods) method
 * implementation
 *
 * \author [Rudra Prasad Das](http://github.com/rudra697)
 *
 * \details
 * It solves the unknown value of y
 * for a given value of x
 * only first order differential equations
 * can be solved
 * \example
 * it solves \frac{\mathrm{d} y}{\mathrm{d} x}= \frac{\left ( x-y \right )}{2}
 * given x for given initial
 * conditions
 * There can be many such equations
 */
#include <cassert>   /// asserting the test functions
#include <cstdint>
#include <iostream>  /// for io operations
#include <vector>    /// for using the vector container
/**
 * @brief The change() function is used
 * to return the updated iterative value corresponding
 * to the given function
 * @param x is the value corresponding to the x coordinate
 * @param y is the value corresponding to the y coordinate
 * @returns the computed function value at that call
 */
static double change(double x, double y) { return ((x - y) / 2.0); }

/**
 * @namespace numerical_methods
 * @brief Numerical Methods
 */
namespace numerical_methods {
/**
 * @namespace runge_kutta
 * @brief Functions for [Runge Kutta fourth
 * order](https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods) method
 */
namespace runge_kutta {
/**
 * @brief the Runge Kutta method finds the value of integration of a function in
 * the given limits. the lower limit of integration as the initial value and the
 * upper limit is the given x
 * @param init_x is the value of initial x and is updated after each call
 * @param init_y is the value of initial x and is updated after each call
 * @param x is current iteration at which the function needs to be evaluated
 * @param h is the step value
 * @returns the value of y at thr required value of x from the initial
 * conditions
 */
double rungeKutta(double init_x, const double &init_y, const double &x,
                  const double &h) {
    // Count number of iterations
    // using step size or
    // step height h

    // n calucates the number of iterations
    // k1, k2, k3, k4 are the Runge Kutta variables
    // used for calculation of y at each iteration

    auto n = static_cast<uint64_t>((x - init_x) / h);
    // used a vector container for the variables
    std::vector<double> k(4, 0.0);

    // Iterate for number of iterations

    double y = init_y;
    for (int i = 1; i <= n; ++i) {
        // Apply Runge Kutta Formulas
        // to find next value of y
        k[0] = h * change(init_x, y);
        k[1] = h * change(init_x + 0.5 * h, y + 0.5 * k[0]);
        k[2] = h * change(init_x + 0.5 * h, y + 0.5 * k[1]);
        k[3] = h * change(init_x + h, y + k[2]);

        // Update next value of y

        y += (1.0 / 6.0) * (k[0] + 2 * k[1] + 2 * k[2] + k[3]);

        // Update next value of x

        init_x += h;
    }

    return y;
}
}  // namespace runge_kutta
}  // namespace numerical_methods

/**
 * @brief Tests to check algorithm implementation.
 * @returns void
 */
static void test() {
    std::cout << "The Runge Kutta function will be tested on the basis of "
                 "precomputed values\n";

    std::cout << "Test 1...."
              << "\n";
    double valfirst = numerical_methods::runge_kutta::rungeKutta(
        2, 3, 4, 0.2);  // Tests the function with pre calculated values
    assert(valfirst == 3.10363932323749570);
    std::cout << "Passed Test 1\n";

    std::cout << "Test 2...."
              << "\n";
    double valsec = numerical_methods::runge_kutta::rungeKutta(
        1, 2, 5, 0.1);  // The value of step changed
    assert(valsec == 3.40600589380261409);
    std::cout << "Passed Test 2\n";

    std::cout << "Test 3...."
              << "\n";
    double valthird = numerical_methods::runge_kutta::rungeKutta(
        -1, 3, 4, 0.1);  // Tested with negative value
    assert(valthird == 2.49251005860244268);
    std::cout << "Passed Test 3\n";
}

/**
 * @brief Main function
 * @returns 0 on exit
 */
int main() {
    test();  // Execute the tests
    return 0;
}