midpoint_integral_method.cpp

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

namespace numerical_methods {
namespace midpoint_rule {
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;
}

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); }
 
}  // namespace midpoint_rule
}  // namespace numerical_methods

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;
}

int main(int argc, char** argv) {
    std::int32_t N =
        16;  
    double a = 1, b = 3;  
    double h = NAN;  
 
    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;
}