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