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diff --git a/ml/dlib/examples/integrate_function_adapt_simp_ex.cpp b/ml/dlib/examples/integrate_function_adapt_simp_ex.cpp
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+// 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_simp().  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 <dlib/matrix.h>
+#include <dlib/numeric_constants.h>
+#include <dlib/numerical_integration.h>
+
+using namespace std;
+using namespace dlib;
+
+// Here we the set of functions that we wish to integrate and comment in the domain of
+// integration.
+
+// x in [0,1]
+double gg1(double x)
+{
+    return pow(e,x);
+}   
+
+// x in [0,1]
+double gg2(double x)
+{
+    return x*x;
+}
+
+// x in [0, pi]
+double gg3(double x)
+{
+    return 1/(x*x + cos(x)*cos(x));
+}
+
+// x in [-pi, pi]
+double gg4(double x)
+{
+    return sin(x);
+}
+
+// x in [0,2]
+double gg5(double x)
+{
+    return 1/(1 + x*x);
+}
+
+int main()
+{
+    // We first define a tolerance parameter.  Roughly speaking, a lower tolerance will
+    // 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 compute the integrals of the five functions defined above using the same 
+    // tolerance level for each.
+
+    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);
+
+    // We finally print out the values of each of the approximated integrals to ten
+    // significant digits.
+
+    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;
+    cout << endl;
+
+    return 0;
+}
+