dlib C++ Library - optimization_test_functions.cpp

// Copyright (C) 2010 Davis E. King (davis@dlib.net)
// License: Boost Software License See LICENSE.txt for the full license.
#include "optimization_test_functions.h"
/*
 Most of the code in this file is converted from the set of Fortran 90 routines 
 created by John Burkardt.
 The original Fortran can be found here: http://orion.math.iastate.edu/burkardt/f_src/testopt/testopt.html
*/
namespace dlib
{
 namespace test_functions
 {
 // ----------------------------------------------------------------------------------------
 matrix<double,0,1> chebyquad_residuals(const matrix<double,0,1>& x)
 {
 matrix<double,0,1> fvec(x.size());
 const int n = x.size();
 int i;
 int j;
 double t;
 double t1;
 double t2;
 double th;
 fvec = 0;
 for (j = 1; j <= n; ++j)
 {
 t1 = 1.0E+00;
 t2 = 2.0E+00 * x(j-1) - 1.0E+00;
 t = 2.0E+00 * t2;
 for (i = 1; i <= n; ++i)
 {
 fvec(i-1) = fvec(i-1) + t2;
 th = t * t2 - t1;
 t1 = t2;
 t2 = th;
 }
 }
 for (i = 1; i <= n; ++i)
 {
 fvec(i-1) = fvec(i-1) / (double) ( n );
 if ( ( i%2 ) == 0 ) 
 fvec(i-1) = fvec(i-1) + 1.0E+00 / ( (double)i*i - 1.0E+00 );
 }
 return fvec;
 }
 // ----------------------------------------------------------------------------------------
 double chebyquad_residual(int i, const matrix<double,0,1>& x)
 {
 return chebyquad_residuals(x)(i);
 }
 // ----------------------------------------------------------------------------------------
 int& chebyquad_calls() 
 {
 static int count = 0;
 return count;
 }
 double chebyquad(const matrix<double,0,1>& x )
 {
 chebyquad_calls()++;
 return sum(squared(chebyquad_residuals(x)));
 }
 // ----------------------------------------------------------------------------------------
 matrix<double,0,1> chebyquad_derivative (const matrix<double,0,1>& x)
 {
 const int n = x.size();
 matrix<double,0,1> fvec = chebyquad_residuals(x);
 matrix<double,0,1> g(n);
 int i;
 int j;
 double s1;
 double s2;
 double t;
 double t1;
 double t2;
 double th;
 for (j = 1; j <= n; ++j)
 {
 g(j-1) = 0.0E+00;
 t1 = 1.0E+00;
 t2 = 2.0E+00 * x(j-1) - 1.0E+00;
 t = 2.0E+00 * t2;
 s1 = 0.0E+00;
 s2 = 2.0E+00;
 for (i = 1; i <= n; ++i)
 {
 g(j-1) = g(j-1) + fvec(i-1) * s2;
 th = 4.0E+00 * t2 + t * s2 - s1;
 s1 = s2;
 s2 = th;
 th = t * t2 - t1;
 t1 = t2;
 t2 = th;
 }
 }
 g = 2.0E+00 * g / (double) ( n );
 return g;
 }
 // ----------------------------------------------------------------------------------------
 matrix<double,0,1> chebyquad_start (int n)
 {
 int i;
 matrix<double,0,1> x(n);
 for (i = 1; i <= n; ++i)
 x(i-1) = double ( i ) / double ( n + 1 );
 return x;
 }
 // ----------------------------------------------------------------------------------------
 matrix<double,0,1> chebyquad_solution (int n)
 {
 matrix<double,0,1> x(n);
 x = 0;
 switch (n)
 {
 case 2:
 x = 0.2113249E+00, 0.7886751E+00;
 break;
 case 4:
 x = 0.1026728E+00, 0.4062037E+00, 0.5937963E+00, 0.8973272E+00;
 break;
 case 6:
 x = 0.066877E+00, 0.288741E+00, 0.366682E+00, 0.633318E+00, 0.711259E+00, 0.933123E+00;
 break;
 case 8:
 x = 0.043153E+00, 0.193091E+00, 0.266329E+00, 0.500000E+00, 0.500000E+00, 0.733671E+00, 0.806910E+00, 0.956847E+00;
 break;
 default:
 std::ostringstream sout;
 sout << "don't know chebyquad solution for n = " << n;
 throw dlib::error(sout.str());
 break;
 }
 return x;
 }
 // ----------------------------------------------------------------------------------------
 matrix<double> chebyquad_hessian(const matrix<double,0,1>& x)
 {
 const int lda = x.size();
 const int n = x.size();
 double d1;
 double d2;
 matrix<double,0,1> fvec = chebyquad_residuals(x);
 matrix<double,0,1> gvec(n);
 matrix<double> h(lda,n);
 int i;
 int j;
 int k;
 double p1;
 double p2;
 double s1;
 double s2;
 double ss1;
 double ss2;
 double t;
 double t1;
 double t2;
 double th;
 double tt;
 double tth;
 double tt1;
 double tt2;
 h = 0;
 d1 = 1.0E+00 / double ( n );
 d2 = 2.0E+00 * d1;
 for (j = 1; j <= n; ++j)
 {
 h(j-1,j-1) = 4.0E+00 * d1;
 t1 = 1.0E+00;
 t2 = 2.0E+00 * x(j-1) - 1.0E+00;
 t = 2.0E+00 * t2;
 s1 = 0.0E+00;
 s2 = 2.0E+00;
 p1 = 0.0E+00;
 p2 = 0.0E+00;
 gvec(0) = s2;
 for (i = 2; i <= n; ++i)
 {
 th = 4.0E+00 * t2 + t * s2 - s1;
 s1 = s2;
 s2 = th;
 th = t * t2 - t1;
 t1 = t2;
 t2 = th;
 th = 8.0E+00 * s1 + t * p2 - p1;
 p1 = p2;
 p2 = th;
 gvec(i-1) = s2;
 h(j-1,j-1) = h(j-1,j-1) + fvec(i-1) * th + d1 * s2*s2;
 }
 h(j-1,j-1) = d2 * h(j-1,j-1);
 for (k = 1; k <= j-1; ++k)
 {
 h(j-1,k-1) = 0.0;
 tt1 = 1.0E+00;
 tt2 = 2.0E+00 * x(k-1) - 1.0E+00;
 tt = 2.0E+00 * tt2;
 ss1 = 0.0E+00;
 ss2 = 2.0E+00;
 for (i = 1; i <= n; ++i)
 {
 h(j-1,k-1) = h(j-1,k-1) + ss2 * gvec(i-1);
 tth = 4.0E+00 * tt2 + tt * ss2 - ss1;
 ss1 = ss2;
 ss2 = tth;
 tth = tt * tt2 - tt1;
 tt1 = tt2;
 tt2 = tth;
 }
 h(j-1,k-1) = d2 * d1 * h(j-1,k-1);
 }
 }
 h = make_symmetric(h);
 return h;
 }
 // ----------------------------------------------------------------------------------------
 // ----------------------------------------------------------------------------------------
 // ----------------------------------------------------------------------------------------
 // ----------------------------------------------------------------------------------------
 double brown_residual (int i, const matrix<double,4,1>& x)
 /*!
 requires
 - 1 <= i <= 20
 ensures
 - returns the ith brown residual
 !*/
 {
 double c;
 double f;
 double f1;
 double f2;
 f = 0.0E+00;
 c = double ( i ) / 5.0E+00;
 f1 = x(0) + c * x(1) - std::exp ( c );
 f2 = x(2) + std::sin ( c ) * x(3) - std::cos ( c );
 f = f1*f1 + f2*f2; 
 return f;
 }
 // ----------------------------------------------------------------------------------------
 double brown ( const matrix<double,4,1>& x)
 {
 double f;
 int i;
 f = 0;
 for (i = 1; i <= 20; ++i)
 {
 f += std::pow(brown_residual(i, x), 2);
 }
 return f;
 }
 // ----------------------------------------------------------------------------------------
 matrix<double,4,1> brown_derivative ( const matrix<double,4,1>& x)
 {
 double c;
 double df1dx1;
 double df1dx2;
 double df2dx3;
 double df2dx4;
 double f1;
 double f2;
 matrix<double,4,1> g;
 int i;
 g = 0;
 for (i = 1; i <= 20; ++i)
 {
 c = double ( i ) / 5.0E+00;
 f1 = x(0) + c * x(1) - std::exp ( c );
 f2 = x(2) + std::sin ( c ) * x(3) - std::cos ( c );
 df1dx1 = 1.0E+00;
 df1dx2 = c;
 df2dx3 = 1.0E+00;
 df2dx4 = std::sin ( c );
 using std::pow;
 g(0) = g(0) + 4.0E+00 * ( pow(f1,3) * df1dx1 + f1 * pow(f2,2) * df1dx1 );
 g(1) = g(1) + 4.0E+00 * ( pow(f1,3) * df1dx2 + f1 * pow(f2,2) * df1dx2 );
 g(2) = g(2) + 4.0E+00 * ( pow(f1,2) * f2 * df2dx3 + pow(f2,3) * df2dx3 );
 g(3) = g(3) + 4.0E+00 * ( pow(f1,2) * f2 * df2dx4 + pow(f2,3) * df2dx4 );
 }
 return g;
 }
 // ----------------------------------------------------------------------------------------
 matrix<double,4,4> brown_hessian ( const matrix<double,4,1>& x)
 {
 double c;
 double df1dx1;
 double df1dx2;
 double df2dx3;
 double df2dx4;
 double f1;
 double f2;
 matrix<double,4,4> h;
 int i;
 h = 0;
 for (i = 1; i <= 20; ++i)
 {
 c = double ( i ) / 5.0E+00;
 f1 = x(0) + c * x(1) - std::exp ( c );
 f2 = x(2) + std::sin ( c ) * x(3) - std::cos ( c );
 df1dx1 = 1.0E+00;
 df1dx2 = c;
 df2dx3 = 1.0E+00;
 df2dx4 = std::sin ( c );
 using std::pow;
 h(0,0) = h(0,0) + 12.0E+00 * pow(f1,2) * df1dx1 * df1dx1 + 4.0E+00 * pow(f2,2) * df1dx1 * df1dx1;
 h(0,1) = h(0,1) + 12.0E+00 * pow(f1,2) * df1dx1 * df1dx2 + 4.0E+00 * pow(f2,2) * df1dx1 * df1dx2;
 h(0,2) = h(0,2) + 8.0E+00 * f1 * f2 * df1dx1 * df2dx3;
 h(0,3) = h(0,3) + 8.0E+00 * f1 * f2 * df1dx1 * df2dx4;
 h(1,0) = h(1,0) + 12.0E+00 * pow(f1,2) * df1dx2 * df1dx1 + 4.0E+00 * pow(f2,2) * df1dx2 * df1dx1;
 h(1,1) = h(1,1) + 12.0E+00 * pow(f1,2) * df1dx2 * df1dx2 + 4.0E+00 * pow(f2,2) * df1dx2 * df1dx2;
 h(1,2) = h(1,2) + 8.0E+00 * f1 * f2 * df1dx2 * df2dx3;
 h(1,3) = h(1,3) + 8.0E+00 * f1 * f2 * df1dx2 * df2dx4;
 h(2,0) = h(2,0) + 8.0E+00 * f1 * f2 * df2dx3 * df1dx1;
 h(2,1) = h(2,1) + 8.0E+00 * f1 * f2 * df2dx3 * df1dx2;
 h(2,2) = h(2,2) + 4.0E+00 * pow(f1,2) * df2dx3 * df2dx3 + 12.0E+00 * pow(f2,2) * df2dx3 * df2dx3;
 h(2,3) = h(2,3) + 4.0E+00 * pow(f1,2) * df2dx4 * df2dx3 + 12.0E+00 * pow(f2,2) * df2dx3 * df2dx4;
 h(3,0) = h(3,0) + 8.0E+00 * f1 * f2 * df2dx4 * df1dx1;
 h(3,1) = h(3,1) + 8.0E+00 * f1 * f2 * df2dx4 * df1dx2;
 h(3,2) = h(3,2) + 4.0E+00 * pow(f1,2) * df2dx3 * df2dx4 + 12.0E+00 * pow(f2,2) * df2dx4 * df2dx3;
 h(3,3) = h(3,3) + 4.0E+00 * pow(f1,2) * df2dx4 * df2dx4 + 12.0E+00 * pow(f2,2) * df2dx4 * df2dx4;
 }
 return make_symmetric(h);
 }
 // ----------------------------------------------------------------------------------------
 matrix<double,4,1> brown_start ()
 {
 matrix<double,4,1> x;
 x = 25.0E+00, 5.0E+00, -5.0E+00, -1.0E+00;
 return x;
 }
 // ----------------------------------------------------------------------------------------
 matrix<double,4,1> brown_solution ()
 {
 matrix<double,4,1> x;
 // solution from original documentation.
 //x = -11.5844E+00, 13.1999E+00, -0.406200E+00, 0.240998E+00;
 x = -11.594439905669450042, 13.203630051593080452, -0.40343948856573402795, 0.23677877338218666914;
 return x;
 }
 // ----------------------------------------------------------------------------------------
 }
}