C++, , midpoint_integral_method.cpp

/**
 * @file
 * @brief A numerical method for easy [approximation of
 * integrals](https://en.wikipedia.org/wiki/Midpoint_method)
 * @details The idea is to split the interval into N of intervals and use as
 * interpolation points the xi for which it applies that xi = x0 + i*h, where h
 * is a step defined as h = (b-a)/N where a and b are the first and last points
 * of the interval of the integration [a, b].
 *
 * We create a table of the xi and their corresponding f(xi) values and we
 * evaluate the integral by the formula: I = h * {f(x0+h/2) + f(x1+h/2) + ... +
 * f(xN-1+h/2)}
 *
 * Arguments can be passed as parameters from the command line argv[1] = N,
 * argv[2] = a, argv[3] = b. In this case if the default values N=16, a=1, b=3
 * are changed then the tests/assert are disabled.
 *
 *
 * @author [ggkogkou](https://github.com/ggkogkou)
 */
#include <cassert>     /// for assert
#include <cmath>       /// for math functions
#include <cstdint>     /// for integer allocation
#include <cstdlib>     /// for std::atof
#include <functional>  /// for std::function
#include <iostream>    /// for IO operations
#include <map>         /// for std::map container

/**
 * @namespace numerical_methods
 * @brief Numerical algorithms/methods
 */
namespace numerical_methods {
/**
 * @namespace midpoint_rule
 * @brief Functions for the [Midpoint
 * Integral](https://en.wikipedia.org/wiki/Midpoint_method) method
 * implementation
 */
namespace midpoint_rule {
/**
 * @fn double midpoint(const std::int32_t N, const double h, const double a,
 * const std::function<double (double)>& func)
 * @brief Main function for implementing the Midpoint Integral Method
 * implementation
 * @param N is the number of intervals
 * @param h is the step
 * @param a is x0
 * @param func is the function that will be integrated
 * @returns the result of the integration
 */
double midpoint(const std::int32_t N, const double h, const double a,
                const std::function<double(double)>& func) {
    std::map<int, double>
        data_table;  // Contains the data points, key: i, value: f(xi)
    double xi = a;   // Initialize xi to the starting point x0 = a

    // Create the data table
    // Loop from x0 to xN-1
    double temp = NAN;
    for (std::int32_t i = 0; i < N; i++) {
        temp = func(xi + h / 2);  // find f(xi+h/2)
        data_table.insert(
            std::pair<std::int32_t, double>(i, temp));  // add i and f(xi)
        xi += h;  // Get the next point xi for the next iteration
    }

    // Evaluate the integral.
    // Remember: {f(x0+h/2) + f(x1+h/2) + ... + f(xN-1+h/2)}
    double evaluate_integral = 0;
    for (std::int32_t i = 0; i < N; i++) evaluate_integral += data_table.at(i);

    // Multiply by the coefficient h
    evaluate_integral *= h;

    // If the result calculated is nan, then the user has given wrong input
    // interval.
    assert(!std::isnan(evaluate_integral) &&
           "The definite integral can't be evaluated. Check the validity of "
           "your input.\n");
    // Else return
    return evaluate_integral;
}

/**
 * @brief A function f(x) that will be used to test the method
 * @param x The independent variable xi
 * @returns the value of the dependent variable yi = f(xi) = sqrt(xi) + ln(xi)
 */
double f(double x) { return std::sqrt(x) + std::log(x); }
/**
 * @brief A function g(x) that will be used to test the method
 * @param x The independent variable xi
 * @returns the value of the dependent variable yi = g(xi) = e^(-xi) * (4 -
 * xi^2)
 */
double g(double x) { return std::exp(-x) * (4 - std::pow(x, 2)); }
/**
 * @brief A function k(x) that will be used to test the method
 * @param x The independent variable xi
 * @returns the value of the dependent variable yi = k(xi) = sqrt(2*xi^3 + 3)
 */
double k(double x) { return std::sqrt(2 * std::pow(x, 3) + 3); }
/**
 * @brief A function l(x) that will be used to test the method
 * @param x The independent variable xi
 * @returns the value of the dependent variable yi = l(xi) = xi + ln(2*xi + 1)
 */
double l(double x) { return x + std::log(2 * x + 1); }

}  // namespace midpoint_rule
}  // namespace numerical_methods

/**
 * @brief Self-test implementations
 * @param N is the number of intervals
 * @param h is the step
 * @param a is x0
 * @param b is the end of the interval
 * @param used_argv_parameters is 'true' if argv parameters are given and
 * 'false' if not
 */
static void test(std::int32_t N, double h, double a, double b,
                 bool used_argv_parameters) {
    // Call midpoint() for each of the test functions f, g, k, l
    // Assert with two decimal point precision
    double result_f = numerical_methods::midpoint_rule::midpoint(
        N, h, a, numerical_methods::midpoint_rule::f);
    assert((used_argv_parameters || (result_f >= 4.09 && result_f <= 4.10)) &&
           "The result of f(x) is wrong");
    std::cout << "The result of integral f(x) on interval [" << a << ", " << b
              << "] is equal to: " << result_f << std::endl;

    double result_g = numerical_methods::midpoint_rule::midpoint(
        N, h, a, numerical_methods::midpoint_rule::g);
    assert((used_argv_parameters || (result_g >= 0.27 && result_g <= 0.28)) &&
           "The result of g(x) is wrong");
    std::cout << "The result of integral g(x) on interval [" << a << ", " << b
              << "] is equal to: " << result_g << std::endl;

    double result_k = numerical_methods::midpoint_rule::midpoint(
        N, h, a, numerical_methods::midpoint_rule::k);
    assert((used_argv_parameters || (result_k >= 9.06 && result_k <= 9.07)) &&
           "The result of k(x) is wrong");
    std::cout << "The result of integral k(x) on interval [" << a << ", " << b
              << "] is equal to: " << result_k << std::endl;

    double result_l = numerical_methods::midpoint_rule::midpoint(
        N, h, a, numerical_methods::midpoint_rule::l);
    assert((used_argv_parameters || (result_l >= 7.16 && result_l <= 7.17)) &&
           "The result of l(x) is wrong");
    std::cout << "The result of integral l(x) on interval [" << a << ", " << b
              << "] is equal to: " << result_l << std::endl;
}

/**
 * @brief Main function
 * @param argc commandline argument count (ignored)
 * @param argv commandline array of arguments (ignored)
 * @returns 0 on exit
 */
int main(int argc, char** argv) {
    std::int32_t N =
        16;  /// Number of intervals to divide the integration interval.
    /// MUST BE EVEN
    double a = 1, b = 3;  /// Starting and ending point of the integration in
    /// the real axis
    double h = NAN;  /// Step, calculated by a, b and N

    bool used_argv_parameters =
        false;  // If argv parameters are used then the assert must be omitted
    // for the test cases

    // Get user input (by the command line parameters or the console after
    // displaying messages)
    if (argc == 4) {
        N = std::atoi(argv[1]);
        a = std::atof(argv[2]);
        b = std::atof(argv[3]);
        // Check if a<b else abort
        assert(a < b && "a has to be less than b");
        assert(N > 0 && "N has to be > 0");
        if (N < 4 || a != 1 || b != 3) {
            used_argv_parameters = true;
        }
        std::cout << "You selected N=" << N << ", a=" << a << ", b=" << b
                  << std::endl;
    } else {
        std::cout << "Default N=" << N << ", a=" << a << ", b=" << b
                  << std::endl;
    }

    // Find the step
    h = (b - a) / N;

    test(N, h, a, b, used_argv_parameters);  // run self-test implementations

    return 0;
}