midpoint_integral_method.cpp

#include <iostream>
#include <cmath>
#include <cassert>
#include <cstdlib>
#include <functional>
#include <map>

/*!
 * @title Calculate definite integrals with midpoint method
 * @see https://en.wikipedia.org/wiki/Midpoint_method
 * @brief A numerical method for easy approximation of integrals
 * @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)}
 *
 * In this program there are 4 sample test functions f, g, k, l that are evaluated in the same interval [1, 3.
 *
 * 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.
 *
 * In the end of the main() and if and only if N, a, b are on their default values,
 * i compare the program's result with the one from mathematical software with 2 decimal points margin.
 *
 * Add your own sample function by replacing one of the f, g, k, l and the corresponding assert
 *
 * @author ggkogkou
*/

/**
 * @namespace midpoint_rule
 * @brief Contains the function of the midpoint method implementation
*/
namespace midpoint_rule{
    /*!
     * @fn double midpoint(const int N, const double h, const double a, const std::function<double (double)>& func)
     * @brief Implement midpoint method
     * @param N number of intervals
     * @param h step
     * @param a x0
     * @param func The function that will be evaluated
     * @returns the result of the integration
    */
    double midpoint(const int 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;
        for(int i=0; i<N; i++){
            temp = func(xi + h/2); /// find f(xi+h/2)
            data_table.insert(std::pair<int ,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(int 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;
    }

} // midpoint_rule ends here

/**
 * @fn double f(double x)
 * @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)
*/
double f(double x);
/**
 * @brief Another test function
*/
double g(double x);
/**
 * @brief Another test function
*/
double k(double x);
/**
 * @brief Another test function
*/
double l(double x);

int main(int argc, char** argv){
    int 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; /// 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 tst cases

    // Get user input (by the command line parameters or the console after displaying messages)
    if(argc == 4){
        N = std::atoi(argv[1]);
        a = (double) std::atof(argv[2]);
        b = (double) 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;

    // Call midpoint() for each of the test functions f, g, k, l
    // Assert with two decimal point precision
    double result_f = midpoint_rule::midpoint(N, h, a, 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 = midpoint_rule::midpoint(N, h, a, 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 = midpoint_rule::midpoint(N, h, a, 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 = midpoint_rule::midpoint(N, h, a, 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;

    return 0;
}

double f(double x){
    return std::sqrt(x) + std::log(x);
}

double g(double x){
    return std::exp(-x) * (4 - std::pow(x, 2));
}

double k(double x){
    return std::sqrt(2*std::pow(x, 3)+3);
}

double l(double x){
    return x + std::log(2*x+1);
}