## 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 "
// 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;
}
``````