2013-05-20 08:52:19 +08:00
|
|
|
|
// The contents of this file are in the public domain. See
|
|
|
|
|
// LICENSE_FOR_EXAMPLE_PROGRAMS.txt
|
|
|
|
|
|
|
|
|
|
/*
|
|
|
|
|
|
|
|
|
|
This example demonstrates the usage of the numerical quadrature function
|
|
|
|
|
integrate_function_adapt_simpson. This function takes as input a single variable
|
|
|
|
|
function, the endpoints of a domain over which the function will be integrated, and a
|
|
|
|
|
tolerance parameter. It outputs an approximation of the integral of this function
|
|
|
|
|
over the specified domain. The algorithm is based on the adaptive Simpson method outlined in:
|
|
|
|
|
|
|
|
|
|
Numerical Integration method based on the adaptive Simpson method in
|
|
|
|
|
Gander, W. and W. Gautschi, "Adaptive Quadrature – Revisited,"
|
|
|
|
|
BIT, Vol. 40, 2000, pp. 84-101
|
|
|
|
|
|
|
|
|
|
*/
|
|
|
|
|
|
|
|
|
|
#include <iostream>
|
|
|
|
|
#include <stdint.h>
|
|
|
|
|
#include <dlib/matrix.h>
|
|
|
|
|
#include <dlib/numeric_constants.h>
|
|
|
|
|
#include <dlib/integrate_function_adapt_simpson.h>
|
|
|
|
|
|
|
|
|
|
using namespace std;
|
|
|
|
|
using namespace dlib;
|
|
|
|
|
|
|
|
|
|
// Here we define a class that consists of the set of functions that we
|
|
|
|
|
// wish to integrate and comment in the domain of integration.
|
|
|
|
|
|
2013-05-23 10:36:33 +08:00
|
|
|
|
// x in [0,1]
|
|
|
|
|
static double gg1(double x)
|
|
|
|
|
{
|
|
|
|
|
return pow(e,x);
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
// x in [0,1]
|
|
|
|
|
static double gg2(double x)
|
|
|
|
|
{
|
|
|
|
|
return x*x;
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
// x in [0, pi]
|
|
|
|
|
static double gg3(double x)
|
|
|
|
|
{
|
|
|
|
|
return 1/(x*x + cos(x)*cos(x));
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
// x in [-pi, pi]
|
|
|
|
|
static double gg4(double x)
|
|
|
|
|
{
|
|
|
|
|
return sin(x);
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
// x in [0,2]
|
|
|
|
|
static double gg5(double x)
|
|
|
|
|
{
|
|
|
|
|
return 1/(1 + x*x);
|
|
|
|
|
}
|
2013-05-20 08:52:19 +08:00
|
|
|
|
|
|
|
|
|
// Examples
|
|
|
|
|
int main()
|
|
|
|
|
{
|
2013-05-23 10:36:33 +08:00
|
|
|
|
// We first define a tolerance parameter. Roughly speaking, a lower tolerance will
|
2013-05-20 08:52:19 +08:00
|
|
|
|
// result in a more accurate approximation of the true integral. However, there are
|
|
|
|
|
// instances where too small of a tolerance may yield a less accurate approximation
|
|
|
|
|
// than a larger tolerance. We recommend taking the tolerance to be in the
|
|
|
|
|
// [1e-10, 1e-8] region.
|
|
|
|
|
|
|
|
|
|
double tol = 1e-10;
|
|
|
|
|
|
|
|
|
|
// Here we instantiate a class which contains the numerical quadrature method.
|
2013-05-23 10:36:33 +08:00
|
|
|
|
// adapt_simp ad;
|
2013-05-20 08:52:19 +08:00
|
|
|
|
|
|
|
|
|
// Here we compute the integrals of the five functions defined above using the same
|
|
|
|
|
// tolerance level for each.
|
|
|
|
|
|
2013-05-23 10:36:33 +08:00
|
|
|
|
double m1 = integrate_function_adapt_simp(&gg1, 0.0, 1.0, tol);
|
|
|
|
|
double m2 = integrate_function_adapt_simp(&gg2, 0.0, 1.0, tol);
|
|
|
|
|
double m3 = integrate_function_adapt_simp(&gg3, 0.0, pi, tol);
|
|
|
|
|
double m4 = integrate_function_adapt_simp(&gg4, -pi, pi, tol);
|
|
|
|
|
double m5 = integrate_function_adapt_simp(&gg5, 0.0, 2.0, tol);
|
2013-05-20 08:52:19 +08:00
|
|
|
|
|
|
|
|
|
// We finally print out the values of each of the approximated integrals to ten
|
|
|
|
|
// significant digits.
|
|
|
|
|
|
2013-05-23 10:36:33 +08:00
|
|
|
|
cout << "\nThe integral of exp(x) for x in [0,1] is " << std::setprecision(10) << m1 << endl;
|
|
|
|
|
cout << "The integral of x^2 for in [0,1] is " << std::setprecision(10) << m2 << endl;
|
|
|
|
|
cout << "The integral of 1/(x^2 + cos(x)^2) for in [0,pi] is " << std::setprecision(10) << m3 << endl;
|
|
|
|
|
cout << "The integral of sin(x) for in [-pi,pi] is " << std::setprecision(10) << m4 << endl;
|
|
|
|
|
cout << "The integral of 1/(1+x^2) for in [0,2] is " << std::setprecision(10) << m5 << endl;
|
2013-05-20 08:52:19 +08:00
|
|
|
|
cout << "" << endl;
|
|
|
|
|
|
|
|
|
|
return 0;
|
|
|
|
|
}
|
|
|
|
|
|