Updated numeric_constants.h. Added a first version of

a numerical integration method to /examples/adapt_simp.cpp

dlib/numeric_constants.h

@@ -12,13 +12,16 @@ namespace dlib
     // e  -- Euler's Constant
     const double e = 2.7182818284590452354;
 
-    // sqrt_2 -- the square root of 2
+    // sqrt_2 -- The square root of 2
     const double sqrt_2 = 1.4142135623730950488;
 
-    // sqrt_3 -- the square root of 3
+    // sqrt_3 -- The square root of 3
     const double sqrt_3 = 1.7320508075688772935;
 
-    // light_spd -- the speed of light in vacuum in meters per second
+    // log10_2 -- The logarithm base 10 of two
+    const double log10_2 = 0.30102999566398119521;
+
+    // light_spd -- The speed of light in vacuum in meters per second
     const double light_spd = 2.99792458e8;
 
     // newton_G  -- Newton's gravitational constant (in metric units of m^3/(kg*s^2))
@@ -27,6 +30,23 @@ namespace dlib
     // planck_cst -- Planck's constant (in units of Joules * seconds) 
     const double planck_cst = 6.62606957e-34;
 
+    // golden_ratio -- The Golden Ratio
+    const double golden_ratio = 1.6180339887498948482;
+
+    // euler_gamma -- The Euler Mascheroni Constant 
+    const double euler_gamma = 0.5772156649015328606065;
+
+    // catalan -- Catalan's Constant
+    const double catalan = 0.91596559417721901505; 
+
+    // glaisher -- Glaisher Kinkelin constant
+    const double glaisher = 1.2824271291006226369;
+
+    // khinchin -- Khinchin's constant
+    const double khinchin = 2.6854520010653064453;
+
+    // apery -- Apery's constant
+    const double apery = 1.2020569031595942854;
 }
 
 #endif //DLIB_NUMERIC_CONSTANTs_H_

examples/adapt_simp.cpp

+// 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
+
+// Test functions taken from Pedro Gonnet's dissertation at ETH: 
+// Adaptive Quadrature Re-Revisited
+// http://e-collection.library.ethz.ch/eserv/eth:65/eth-65-02.pdf
+
+#include <iostream>
+#include <iomanip>
+#include <stdint.h>
+#include <dlib/matrix.h>
+
+using namespace std;
+using namespace dlib;
+
+//***************************************************************//
+//*Begin definitions of test functions //
+
+//Initial Test Function
+double f(double x)
+{
+    return pow(x,0.5);
+}
+
+// The Lyness - Kaganove test functions from page 167 of Gonnet's thesis.
+
+// lambda in [0,1], alpha in [-0.5,0], x in [0,1]
+double LK1(double x, double lambda, double alpha)
+{
+    return pow(abs(x-lambda),alpha);
+}
+
+// lambda in [0,1], alpha in [0,1], x in [0,1]
+double LK2(double x, double lambda, double alpha)
+{
+    if(x > lambda)
+    {
+        return 0;
+    }
+    else
+    {
+        return pow(e, alpha*x);
+    }
+}
+
+// lambda in [0,1], alpha in [0,4], x in [0,1]
+double LK3(double x, double lambda, double alpha)
+{
+    return pow(e,-alpha*abs(x-lambda));
+}
+
+// lambda in [1,2], alpha in [-6,-3], x in [1,2]
+double LK4(double x, double lambda, double alpha)
+{
+    return pow(10,alpha)/((x-lambda)*(x-lambda)+pow(10,alpha));
+}
+
+// lambda_i in [1,2], alpha in [-5,-3], x in [1,2]
+double LK5(double x, double lambda1, double lambda2, double lambda3, double lambda4, double alpha)
+{
+    return   pow(10,alpha)/((x-lambda1)*(x-lambda1)+pow(10,alpha)) 
+           + pow(10,alpha)/((x-lambda2)*(x-lambda2)+pow(10,alpha))
+           + pow(10,alpha)/((x-lambda3)*(x-lambda3)+pow(10,alpha))
+           + pow(10,alpha)/((x-lambda4)*(x-lambda4)+pow(10,alpha));
+}
+
+// lambda in [0,1], alpha in [1.8,2], x in [0,1]
+double LK6(double x, double lambda, double alpha)
+{
+    double beta = pow(10,alpha)/max(lambda*lambda,(1-lambda)*(1-lambda)); 
+    return 2*beta*cos(beta*(x-lambda)*(x-lambda));
+}
+
+// Test Battery from reference [33] and p. 168 of Gonnet's thesis.
+
+// x in [0,1]
+double GG1(double x)
+{
+    return pow(e,x);
+}
+
+// x in [0,1]
+double GG2(double x)
+{
+    if(x > 0.3)
+    {
+        return 1.0;
+    }
+    else
+    {
+        return 0;
+    }
+}
+
+// x in [0,1]
+double GG3(double x)
+{
+    return pow(x,0.5);
+}
+
+// x in [0,1]
+double GG4(double x)
+{
+    return 22/25*cosh(x)-cos(x);
+}
+
+// x in [-1,1]
+double GG5(double x)
+{
+    return 1/(pow(x,4) + pow(x,2) + 0.9);
+}
+
+// x in [0,1]
+double GG6(double x)
+{
+    return pow(x,1.5);
+}
+
+// x in [0,1]
+double GG7(double x)
+{
+    return pow(x,-0.5);
+}
+
+// x in [0,1]
+double GG8(double x)
+{
+    return 1/(1 + pow(x,4));
+}
+
+// x in [0,1]
+double GG9(double x)
+{
+    return 2/(2 + sin(10*pi*x));
+}
+
+// x in [0,1]
+double GG10(double x)
+{
+    return 1/(1+x);
+}
+
+// x in [0,1]
+double GG11(double x)
+{
+    1/(1 + pow(e,x));
+}
+
+// x in [0,1]
+double GG12(double x)
+{
+    return x/(pow(e,x)-1);
+}
+
+// x in [0.1, 1]
+double GG13(double x)
+{
+    return sin(100.0*pi*x)/(pi*x);
+}
+
+// x in [0, 10]
+double GG14(double x)
+{
+    return sqrt(50)*pow(e,-50.0*pi*x*x);
+}
+
+// x in [0, 10]
+double GG15(double x)
+{
+    return 25.0*pow(e,-25.0*x);
+}
+
+// x in [0, 10]
+double GG16(double x)
+{
+    return 50.0/(pi*(2500.0*x*x+1));
+}
+
+// x in [0.01, 1]
+double GG17(double x)
+{
+    return 50.0*pow((sin(50.0*pi*x)/(50.0*pi*x)),2);
+}
+
+// x in [0, pi]
+double GG18(double x)
+{
+    return cos(cos(x)+3*sin(x)+2*cos(2*x)+3*cos(3*x));
+}
+
+// x in [0,1]
+double GG19(double x)
+{
+    return log10(x);
+}
+
+// x in [-1,1]
+double GG20(double x)
+{
+    return 1/(1.005+x*x);
+}
+
+// x in [0,1]
+double GG21(double x)
+{
+    return 1/cosh(20.0*(x-1/5)) + 1/cosh(400.0*(x-2/5)) + 1/cosh(8000.0*(x-3/5));
+}
+
+// x in [0,1]
+double GG22(double x)
+{
+    return 4*pi*pi*x*sin(20.0*pi*x)*cos(2*pi*x);
+}
+
+// x in [0,1]
+double GG23(double x)
+{
+    return 1/(1+(230*x-30)*(230*x-30));
+}
+
+// x in [0,3]
+double GG24(double x)
+{
+    return floor(pow(e,x));
+}
+
+// x in [0,5]
+double GG25(double x)
+{
+    if(x < 1)
+    {
+        return (x + 1);
+    }
+    else if(x >= 1 && x <= 3)
+    {
+        return 3 - x;
+    }
+    else
+    {
+        return 2;
+    }
+}
+
+// Returns double machine precision
+// Taken from Wikipedia en.wikipedia.org/wiki/Machine_epsilon
+template<typename float_t, typename int_t>
+float_t machine_eps()
+{
+    union
+    {
+        float_t f;
+        int_t   i;
+    }   one, one_plus, little, last_little;
+ 
+    one.f    = 1.0;
+    little.f = 1.0;
+    last_little.f = little.f;
+
+    while(true)
+    {
+        one_plus.f = one.f;
+        one_plus.f += little.f;
+ 
+        if( one.i != one_plus.i )
+        {
+            last_little.f = little.f;
+            little.f /= 2.0;
+        }
+        else
+        {
+            return last_little.f;
+        }
+    }
+}
+
+// Main Integration Function.
+// Supporting Integration Function
+template <typename T, typename funct>
+T AdaptSimpstp(const funct& f, T a, T b, T fa, T fm, T fb, T is)
+{
+    T m = (a + b)/2;
+    T h = (b - a)/4;
+    T fml = f(a + h);
+    T fmr = f(b - h);
+    T i1 = h/1.5*(fa+4*fm+fb);
+    T i2 = h/3.0*(fa+4*(fml+fmr)+2*fm+fb);
+    i1 = (16.0*i2 - i1)/15.0;
+    T Q = 0;
+
+    if((is+(i1-i2) == is) || (m <= a) || (b <= m))
+    {
+        if((m <= a) || (b <= m))
+        {
+            cout << "INT ERR" << endl;
+        }
+    
+        Q = i1;
+    }
+    else 
+    {
+       Q = AdaptSimpstp(f, a, m, fa, fml, fm, is) + AdaptSimpstp(f,m,b,fm,fmr,fb,is); 
+    }
+
+    return Q;
+}
+
+// Main integration function. 
+// f -- function to integrate, 
+// a -- left end point
+// b -- right end point
+// tol -- error tolerance 
+
+template <typename T, typename funct>
+T AdaptSimp(const funct& f, T a, T b, T tol)
+{
+    T eps = machine_eps<T, uint64_t>();
+
+    if(tol < eps)
+    {
+        tol = eps;
+    }
+
+    const T ba = b-a;
+    const T fa = f(a);
+    const T fb = f(b);
+    const T fm = f((a+b)/2);
+
+    T is =ba/8*(fa+fb+fm+ f(a + 0.9501*ba) + f(a + 0.2311*ba) + f(a + 0.6068*ba)
+                           + f(a + 0.4860*ba) + f(a + 0.8913*ba));
+
+    if(is == 0)
+    {
+        is = b-a;
+    }
+
+    is = is*tol/eps;
+    
+    T tstvl = AdaptSimpstp(f, a, b, fa, fm, fb, is);
+
+    return tstvl;
+
+}
+
+// Examples 
+int main()
+{
+
+typedef double T;
+
+T tol = 1e-10;
+T a = 0;
+T b = 5;
+
+T tstvl2 = AdaptSimp(&f, a, b, tol);
+
+cout << "Integral Value is: " << std::setprecision(18) << tstvl2 << endl;
+
+return 0;
+
+}
+