# include <cstdlib>
# include <cstdio>
# include <cmath>
# include <iostream>
# include <fstream>
# include <iomanip>
# include <ctime>
# include <cstring>

using namespace std;

int main ( int argc, char *argv[] );
void hermite_compute ( int order, double xtab[], double weight[] );
void hermite_handle ( int order, double alpha, int option, string output );
void hermite_recur ( double *p2, double *dp2, double *p1, double x, int order );
void hermite_root ( double *x, int order, double *dp2, double *p1 );
double r8_abs ( double x );
double r8_epsilon ( );
double r8_gamma ( double x );
double r8_huge ( );
void r8mat_write ( string output_filename, int m, int n, double table[] );
void timestamp ( );

//****************************************************************************80

int main ( int argc, char *argv[] )

//****************************************************************************80
//
//  Purpose:
//
//    MAIN is the main program for HERMITE_RULE.
//
//  Discussion:
//
//    This program computes a Gauss-Hermite quadrature rule 
//    and writes it to a file.
//
//    The user specifies:
//    * the ORDER (number of points) in the rule
//    * the OPTION (unweighted/physicist weight/probabilist weight)
//    * the root name of the OUTPUT files.
//
//    The parameter ALPHA is fixed to zero in this program.  If nonzero, a
//    generalized Gauss-Hermite quadrature rule would be generated, in which
//    the weight function would be multiplied by |x|^(2*ALPHA).
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    19 May 2009
//
//  Author:
//
//    John Burkardt
//
{
  double alpha = 0.0;
  int option;
  int order;
  string output;

  cout << "\n";
  timestamp ( );
  cout << "\n";
  cout << "HERMITE_RULE\n";
  cout << "  C++ version\n";
  cout << "\n";
  cout << "  Compiled on " << __DATE__ << " at " << __TIME__ << ".\n";
  cout << "\n";
  cout << "  Compute a Gauss-Hermite quadrature rule for approximating\n";
  cout << "\n";
  cout << "    Integral ( -oo < x < +oo ) w(x) f(x) dx\n";
  cout << "\n";
  cout << "  of order ORDER.\n";
  cout << "\n";
  cout << "  The user specifies ORDER, OPTION, and OUTPUT.\n";
  cout << "\n";
  cout << "  OPTION specifies the weight function w(x):\n";
  cout << "\n";
  cout << "    0, the unweighted rule for:\n";
  cout << "      Integral ( -oo < x < +oo )             f(x) dx\n";
  cout << "\n";
  cout << "    1, the physicist weighted rule for:\n";
  cout << "      Integral ( -oo < x < +oo ) exp(-x*x)   f(x) dx\n";
  cout << "\n";
  cout << "    2, the probabilist weighted rule for:\n";
  cout << "      Integral ( -oo < x < +oo ) exp(-x*x/2) f(x) dx\n";
  cout << "\n";
  cout << "  OUTPUT is:\n";
  cout << "\n";
  cout << "  \"C++\" for printed C++ output;\n";
  cout << "  \"F77\" for printed Fortran77 output;\n";
  cout << "  \"F90\" for printed Fortran90 output;\n";
  cout << "  \"MAT\" for printed MATLAB output;\n";
  cout << "\n";
  cout << "  or:\n";
  cout << "\n";
  cout << "  \"filename\" to generate 3 files:\n";
  cout << "\n";
  cout << "    filename_w.txt - the weight file\n";
  cout << "    filename_x.txt - the abscissa file.\n";
  cout << "    filename_r.txt - the region file.\n";
//
//  Get the order.
//
  if ( 1 < argc )
  {
    order = atoi ( argv[1] );
  }
  else
  {
    cout << "\n";
    cout << "  Enter the value of ORDER (1 or greater)\n";
    cin >> order;
  }
//
//  Get the option.
//
  if ( 2 < argc )
  {
    option = atoi ( argv[2] );
  }
  else
  {
    cout << "\n";
    cout << "  OPTION = 0 to get the unweighted rule for:\n";
    cout << "      Integral ( -oo < x < +oo )             f(x) dx\n";
    cout << "\n";
    cout << "  OPTION = 1 to get the physicist weighted rule for:\n";
    cout << "      Integral ( -oo < x < +oo ) exp(-x*x)   f(x) dx\n";
    cout << "\n";
    cout << "  OPTION = 2 to get the probabilist weighted rule for:\n";
    cout << "      Integral ( -oo < x < +oo ) exp(-x*x/2) f(x) dx\n";
    cout << "\n";
    cout << "  Enter the value of OPTION:\n";
    cin >> option;
  }

  if ( option < 0 || 2 < option )
  {
    cerr << "\n";
    cerr << "HERMITE_RULE - Fatal error!\n";
    cerr << "  0 <= OPTION <= 2 is required.\n";
    return 1;
  }
//
//  Get the output option or quadrature file root name:
//
  if ( 3 < argc )
  {
    output = argv[3];
  }
  else
  {
    cout << "\n";
    cout << "  Enter OUTPUT (one of C++, F77, F90, MAT\n";
    cout << "  or else the \"root name\" of the quadrature files).\n";
    cin >> output;
  }

  cout << "\n";
  cout << "  ORDER =  " << order << "\n";
  cout << "  OPTION = " << option << "\n";
  cout << "  OUTPUT = \"" << output << "\".\n";
//
//  Construct the rule and output it.
//
  hermite_handle ( order, alpha, option, output );

  cout << "\n";
  cout << "HERMITE_RULE:\n";
  cout << "  Normal end of execution.\n";

  cout << "\n";
  timestamp ( );

  return 0;
}
//****************************************************************************80

void hermite_compute ( int order, double xtab[], double weight[] )

//****************************************************************************80
//
//  Purpose:
//
//    HERMITE_COMPUTE computes a Gauss-Hermite quadrature rule.
//
//  Discussion:
//
//    The abscissas are the zeros of the N-th order Hermite polynomial.
//
//    The integration interval is ( -oo, +oo ).
//
//    The weight function is w(x) = exp ( - x*x ).
//
//    The integral to approximate:
//
//      Integral ( -oo < X < +oo ) exp ( - X*X ) * F(X) dX
//
//    The quadrature rule:
//
//      Sum ( 1 <= I <= ORDER ) WEIGHT(I) * F ( XTAB(I) )
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    18 February 2008
//
//  Author:
//
//    Original FORTRAN77 version by Arthur Stroud, Don Secrest.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Arthur Stroud, Don Secrest,
//    Gaussian Quadrature Formulas,
//    Prentice Hall, 1966,
//    LC: QA299.4G3S7.
//
//  Parameters:
//
//    Input, int ORDER, the order of the formula to be computed.
//
//    Output, double XTAB[ORDER], the Gauss-Hermite abscissas.
//
//    Output, double WEIGHT[ORDER], the Gauss-Hermite weights.
//
{
  double cc;
  double dp2;
  int i;
  double p1;
  double s;
  double temp;
  double x;

  cc = 1.7724538509 * r8_gamma ( ( double ) ( order ) ) 
    / pow ( 2.0, order - 1 );

  s = pow ( 2.0 * ( double ) ( order ) + 1.0, 1.0 / 6.0 );

  for ( i = 0; i < ( order + 1 ) / 2; i++ )
  {
    if ( i == 0 )
    {
      x = s * s * s - 1.85575 / s;
    }
    else if ( i == 1 )
    {
      x = x - 1.14 * pow ( ( double ) ( order ), 0.426 ) / x;
    }
    else if ( i == 2 )
    {
      x = 1.86 * x - 0.86 * xtab[0];
    }
    else if ( i == 3 )
    {
      x = 1.91 * x - 0.91 * xtab[1];
    }
    else
    {
      x = 2.0 * x - xtab[i-2];
    }

    hermite_root ( &x, order, &dp2, &p1 );

    xtab[i] = x;
    weight[i] = ( cc / dp2 ) / p1;

    xtab[order-i-1] = -x;
    weight[order-i-1] = weight[i];
  }
//
//  Reverse the order of the abscissas.
//
  for ( i = 1; i <= order/2; i++ )
  {
    temp          = xtab[i-1];
    xtab[i-1]     = xtab[order-i];
    xtab[order-i] = temp;
  }

  return;
}
//****************************************************************************80

void hermite_handle ( int order, double alpha, int option, string output )

//****************************************************************************80
//
//  Purpose:
//
//    HERMITE_HANDLE computes the requested Gauss-Hermite rule and outputs it.
//
//  Discussion:
//
//    This routine assumes that the input parameter ALPHA is zero, so that
//    a basic Gauss-Hermite rule is requested, not a generalized rule.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    19 May 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int ORDER, the order of the rule.
//
//    Input, double ALPHA, the parameter.
//
//    Input, int OPTION.
//    0, the integral has the form Integral ( -oo < x < +oo )             f(x) dx.
//    1, the integral has the form Integral ( -oo < x < +oo ) exp(-x*x)   f(x) dx
//    2, the integral has the form Integral ( -oo < x < +oo ) exp(-x*x/2) f(x) dx
//
//    Input, string OUTPUT, specifies the output.
//    * "C++'", print as C++ code.
//    * "F77", print as FORTRAN77 code.
//    * "F90", print as FORTRAN90 code.
//    * "MAT", print as MATLAB code.
//    * file,  write files "file_w.txt", "file_x.txt", "file_r.txt" 
//      defining weights, abscissas, and region.
// 
{
  int i;
  string output_r;
  string output_w;
  string output_x;
  double *r;
  double *w;
  double *x;

  r = new double[2];
  w = new double[order];
  x = new double[order];

  r[0] = - r8_huge ( );
  r[1] =   r8_huge ( );

  hermite_compute ( order, x, w );
//
//  Modify weights if requested.
//
  if ( option == 0 )
  {
    for ( i = 0; i < order; i++ )
    {
      w[i] = exp ( x[i] * x[i] ) * w[i];
    }
  }
  else if ( option == 2 )
  {
    for ( i = 0; i < order; i++ )
    {
      w[i] = exp ( x[i] * x[i]  / 2.0 ) * w[i];
    }
  }

  if ( output == "C++" || output == "c++" )
  {
    cout << "//\n";
    cout << "//  Weights W, abscissas X and range R\n";
    cout << "//  for a Gauss-Hermite quadrature rule\n";
    cout << "//  ORDER = " << order << "\n";
    cout << "//\n";

    if ( option == 0 )
    {
      cout << "//  OPTION = 0, unweighted rule:\n";
      cout << "//    Integral ( -oo < x < +oo ) f(x) dx\n";
      cout << "//    is to be approximated by\n";
      cout << "//    sum ( 1 <= I <= ORDER ) w(i) * f(x(i)).\n";

    }
    else if ( option == 1 )
    {
      cout << "//  OPTION = 1, physicist weighted rule:\n";
	  cout << "//    Integral ( -oo < x < +oo ) exp(-x*x) f(x) dx\n";
	  cout << "//    is to be approximated by\n";
	  cout << "//    sum ( 1 <= I <= ORDER ) w(i) * f(x(i)).\n";
    }
    else if ( option == 2 )
    {
      cout << "//  OPTION = 2, probabilist weighted rule:\n";
	  cout << "//    Integral ( -oo < x < +oo ) exp(-x*x/2) f(x) dx\n";
	  cout << "//    is to be approximated by\n";
	  cout << "//    sum ( 1 <= I <= ORDER ) w(i) * f(x(i)).\n";
    }
    cout << "//\n";

    for ( i = 0; i < order; i++ )
    {
      cout << "  w[" << i << "] = " 
           << setprecision(16) << w[i] << ";\n";
    }
    cout << "\n";
    for ( i = 0; i < order; i++ )
    {
      cout << "  x[" << i << "] = " 
           << setprecision(16) << x[i] << ";\n";
    }
    cout << "\n";
    for ( i = 0; i < 2; i++ )
    {
      cout << "  r[" << i << "] = " << r[i] << ";\n";
    }
   }
   else if ( output == "F77" || output == "f77" )
   {
    cout << "c\n";
    cout << "c  Weights W, abscissas X and range R\n";
    cout << "c  for a Gauss-Hermite quadrature rule\n";
    cout << "c  ORDER = " << order << "\n";
    cout << "c\n";
    if ( option == 0 )
    {
      cout << "c  OPTION = 0, unweighted rule:\n";
      cout << "c    Integral ( -oo < x < +oo ) f(x) dx\n";
      cout << "c    is to be approximated by\n";
      cout << "c    sum ( 1 <= I <= ORDER ) w(i) * f(x(i)).\n";
    }
    else if ( option == 1 )
    {
      cout << "c  OPTION = 1, physicist weighted rule:\n";
	  cout << "c    Integral ( -oo < x < +oo ) exp(-x*x) f(x) dx\n";
	  cout << "c    is to be approximated by\n";
	  cout << "c    sum ( 1 <= I <= ORDER ) w(i) * f(x(i)).\n";
    }
    else if ( option == 2 )
    {
      cout << "c  OPTION = 2, probabilist weighted rule:\n";
	  cout << "c    Integral ( -oo < x < +oo ) exp(-x*x/2) f(x) dx\n";
	  cout << "c    is to be approximated by\n";
	  cout << "c    sum ( 1 <= I <= ORDER ) w(i) * f(x(i)).\n";
    }
    cout << "c\n";

    for ( i = 0; i < order; i++ )
    {
      cout << "      w(" << i + 1 << ") = " 
           << setprecision(16) << w[i] << "\n";
    }
    cout << "\n";
    for ( i = 0; i < order; i++ )
    {
      cout << "      x(" << i + 1 << ") = " 
           << setprecision(16) << x[i] << "\n";
    }
    cout << "\n";
    for ( i = 0; i < 2; i++ )
    {
      cout << "      r(" << i + 1 << ") = " << r[i] << "\n";
    }
  }
  else if ( output == "F90" || output == "f90" )
  {  
    cout << "!\n";
    cout << "!  Weights W, abscissas X and range R\n";
    cout << "!  for a Gauss-Hermite quadrature rule\n";
    cout << "!  ORDER = " << order << "\n";
    cout << "!\n";
    if ( option == 0 )
    {
      cout << "!  OPTION = 0, unweighted rule rule:\n";
      cout << "!    Integral ( -oo < x < +oo ) f(x) dx\n";
      cout << "!    is to be approximated by\n";
      cout << "!    sum ( 1 <= I <= ORDER ) w(i) * f(x(i)).\n";
    }
    else if ( option == 1 )
    {
      cout << "!  OPTION = 1, physicist weighted rule:\n";
	  cout << "!    Integral ( -oo < x < +oo ) exp(-x*x) f(x) dx\n";
	  cout << "!    is to be approximated by\n";
	  cout << "!    sum ( 1 <= I <= ORDER ) w(i) * f(x(i)).\n";
    }
    else if ( option == 2 )
    {
      cout << "!  OPTION = 2, probabilist weighted rule:\n";
	  cout << "!    Integral ( -oo < x < +oo ) exp(-x*x/2) f(x) dx\n";
	  cout << "!    is to be approximated by\n";
	  cout << "!    sum ( 1 <= I <= ORDER ) w(i) * f(x(i)).\n";
    }
    cout << "!\n";

    for ( i = 0; i < order; i++ )
    {
      cout << "  w(" << i + 1 << ") = " 
           << setprecision(16) << w[i] << "\n";
    }
    cout << "\n";
    for ( i = 0; i < order; i++ )
    {
      cout << "  x(" << i + 1 << ") = " 
           << setprecision(16) << x[i] << "\n";
    }
    cout << "\n";
    for ( i = 0; i < 2; i++ )
    {
      cout << "  r(" << i + 1 << ") = " << r[i] << "\n";
    }
  }
  else if ( output == "MAT" || output == "mat" )
  {
    cout << "%\n";
    cout << "%  Weights W, abscissas X and range R\n";
    cout << "%  for a Gauss-Hermite quadrature rule\n";
    cout << "%  ORDER = " << order << "\n";
    cout << "%\n";
    if ( option == 0 )
    {
      cout << "%  OPTION = 0, unweighted rule:\n";
      cout << "%    Integral ( -oo < x < +oo ) f(x) dx\n";
      cout << "%    is to be approximated by\n";
      cout << "%    sum ( 1 <= I <= ORDER ) w(i) * f(x(i)).\n";
    }
    else if ( option == 1 )
    {
      cout << "%  OPTION = 1, physicist weighted rule:\n";
	  cout << "%    Integral ( -oo < x < +oo ) exp(-x*x) f(x) dx\n";
	  cout << "%    is to be approximated by\n";
	  cout << "%    sum ( 1 <= I <= ORDER ) w(i) * f(x(i)).\n";
    }
    else if ( option == 2 )
    {
      cout << "%  OPTION = 2, probabilist weighted rule:\n";
	  cout << "%    Integral ( -oo < x < +oo ) exp(-x*x) f(x) dx\n";
	  cout << "%    is to be approximated by\n";
	  cout << "%    sum ( 1 <= I <= ORDER ) w(i) * f(x(i)).\n";
    }
    cout << "%\n";

    for ( i = 0; i < order; i++ )
    {
      cout << "  w(" << i + 1 << ") = " 
           << setprecision(16) << w[i] << ";\n";
    }
    cout << "\n";
    for ( i = 0; i < order; i++ )
    {
      cout << "  x(" << i + 1 << ") = " 
           << setprecision(16) << x[i] << ";\n";
    }
    cout << "\n";
    for ( i = 0; i < 2; i++ )
    {
      cout << "  r(" << i + 1 << ") = " << r[i] << ";\n";
    }
  } 
  else
  {
    if ( option == 0 )
    {
      output_w = output + "_unweighted_w.txt";
      output_x = output + "_unweighted_x.txt";
      output_r = output + "_unweighted_r.txt";
    }
    else if ( option == 1 )
    {
      output_w = output + "_physicist_w.txt";
      output_x = output + "_physicist_x.txt";
      output_r = output + "_physicist_r.txt";
    }
    else if ( option == 2 )
    {
      output_w = output + "_probabilist_w.txt";
      output_x = output + "_probabilist_x.txt";
      output_r = output + "_probabilist_r.txt";
    }
    cout << "\n";
    cout << "  Creating quadrature files.\n";
    cout << "\n";
    cout << "  Root file name is     \"" << output   << "\".\n";
    cout << "\n";
    cout << "  Weight file will be   \"" << output_w << "\".\n";
    cout << "  Abscissa file will be \"" << output_x << "\".\n";
    cout << "  Region file will be   \"" << output_r << "\".\n";
            
    r8mat_write ( output_w, 1, order, w );
    r8mat_write ( output_x, 1, order, x );
    r8mat_write ( output_r, 1, 2,     r );
  }
  
  delete [] r;
  delete [] w;
  delete [] x;

  return;
}
//****************************************************************************80

void hermite_recur ( double *p2, double *dp2, double *p1, double x, int order )

//****************************************************************************80
//
//  Purpose:
//
//    HERMITE_RECUR finds the value and derivative of a Hermite polynomial.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    20 February 2008
//
//  Author:
//
//    Original FORTRAN77 version by Arthur Stroud, Don Secrest.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Arthur Stroud, Don Secrest,
//    Gaussian Quadrature Formulas,
//    Prentice Hall, 1966,
//    LC: QA299.4G3S7.
//
//  Parameters:
//
//    Output, double *P2, the value of H(ORDER)(X).
//
//    Output, double *DP2, the value of H'(ORDER)(X).
//
//    Output, double *P1, the value of H(ORDER-1)(X).
//
//    Input, double X, the point at which polynomials are evaluated.
//
//    Input, int ORDER, the order of the polynomial to be computed.
//
{
  int i;
  double dq0;
  double dq1;
  double dq2;
  double q0;
  double q1;
  double q2;

  q1 = 1.0;
  dq1 = 0.0;

  q2 = x;
  dq2 = 1.0;

  for ( i = 2; i <= order; i++ )
  {
    q0 = q1;
    dq0 = dq1;

    q1 = q2;
    dq1 = dq2;

    q2  = x * q1 - 0.5 * ( ( double ) ( i ) - 1.0 ) * q0;
    dq2 = x * dq1 + q1 - 0.5 * ( ( double ) ( i ) - 1.0 ) * dq0;
  }

  *p2 = q2;
  *dp2 = dq2;
  *p1 = q1;

  return;
}
//****************************************************************************80

void hermite_root ( double *x, int order, double *dp2, double *p1 )

//****************************************************************************80
//
//  Purpose:
//
//    HERMITE_ROOT improves an approximate root of a Hermite polynomial.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    19 February 2008
//
//  Author:
//
//    Original FORTRAN77 version by Arthur Stroud, Don Secrest.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Arthur Stroud, Don Secrest,
//    Gaussian Quadrature Formulas,
//    Prentice Hall, 1966,
//    LC: QA299.4G3S7.
//
//  Parameters:
//
//    Input/output, double *X, the approximate root, which
//    should be improved on output.
//
//    Input, int ORDER, the order of the Hermite polynomial.
//
//    Output, double *DP2, the value of H'(ORDER)(X).
//
//    Output, double *P1, the value of H(ORDER-1)(X).
//
{
  double d;
  double eps;
  double p2;
  int step;
  int step_max = 10;

  eps = r8_epsilon ( );

  for ( step = 1; step <= step_max; step++ )
  {
    hermite_recur ( &p2, dp2, p1, *x, order );

    d = p2 / ( *dp2 );
    *x = *x - d;

    if ( r8_abs ( d ) <= eps * ( r8_abs ( *x ) + 1.0 ) )
    {
      return;
    }
  }

  return;
}
//****************************************************************************80

double r8_abs ( double x )

//****************************************************************************80
//
//  Purpose:
//
//    R8_ABS returns the absolute value of an R8.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    14 November 2006
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double X, the quantity whose absolute value is desired.
//
//    Output, double R8_ABS, the absolute value of X.
//
{
  double value;

  if ( 0.0 <= x )
  {
    value = x;
  } 
  else
  {
    value = -x;
  }
  return value;
}
//****************************************************************************80

double r8_epsilon ( )

//****************************************************************************80
//
//  Purpose:
//
//    R8_EPSILON returns the R8 roundoff unit.
//
//  Discussion:
//
//    The roundoff unit is a number R which is a power of 2 with the 
//    property that, to the precision of the computer's arithmetic,
//      1 < 1 + R
//    but 
//      1 = ( 1 + R / 2 )
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    01 July 2004
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Output, double R8_EPSILON, the R8 round-off unit.
//
{
  double value;

  value = 1.0;

  while ( 1.0 < ( double ) ( 1.0 + value )  )
  {
    value = value / 2.0;
  }

  value = 2.0 * value;

  return value;
}
//****************************************************************************80

double r8_gamma ( double x )

//****************************************************************************80
//
//  Purpose:
//
//    R8_GAMMA evaluates Gamma(X) for a real argument.
//
//  Discussion:
//
//    This routine calculates the gamma function for a real argument X.
//
//    Computation is based on an algorithm outlined in reference 1.
//    The program uses rational functions that approximate the gamma
//    function to at least 20 significant decimal digits.  Coefficients
//    for the approximation over the interval (1,2) are unpublished.
//    Those for the approximation for 12 <= X are from reference 2.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    18 January 2008
//
//  Author:
//
//    Original FORTRAN77 version by William Cody, Laura Stoltz.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    William Cody,
//    An Overview of Software Development for Special Functions,
//    in Numerical Analysis Dundee, 1975,
//    edited by GA Watson,
//    Lecture Notes in Mathematics 506,
//    Springer, 1976.
//
//    John Hart, Ward Cheney, Charles Lawson, Hans Maehly,
//    Charles Mesztenyi, John Rice, Henry Thatcher,
//    Christoph Witzgall,
//    Computer Approximations,
//    Wiley, 1968,
//    LC: QA297.C64.
//
//  Parameters:
//
//    Input, double X, the argument of the function.
//
//    Output, double R8_GAMMA, the value of the function.
//
{
//
//  Coefficients for minimax approximation over (12, INF).
//
  double c[7] = {
   -1.910444077728E-03, 
    8.4171387781295E-04, 
   -5.952379913043012E-04, 
    7.93650793500350248E-04, 
   -2.777777777777681622553E-03, 
    8.333333333333333331554247E-02, 
    5.7083835261E-03 };
  double eps = 2.22E-16;
  double fact;
  double half = 0.5;
  int i;
  int n;
  double one = 1.0;
  double p[8] = {
  -1.71618513886549492533811E+00,
   2.47656508055759199108314E+01, 
  -3.79804256470945635097577E+02,
   6.29331155312818442661052E+02, 
   8.66966202790413211295064E+02,
  -3.14512729688483675254357E+04, 
  -3.61444134186911729807069E+04,
   6.64561438202405440627855E+04 };
  bool parity;
  double pi = 3.1415926535897932384626434;
  double q[8] = {
  -3.08402300119738975254353E+01,
   3.15350626979604161529144E+02, 
  -1.01515636749021914166146E+03,
  -3.10777167157231109440444E+03, 
   2.25381184209801510330112E+04,
   4.75584627752788110767815E+03, 
  -1.34659959864969306392456E+05,
  -1.15132259675553483497211E+05 };
  double res;
  double sqrtpi = 0.9189385332046727417803297;
  double sum;
  double twelve = 12.0;
  double two = 2.0;
  double value;
  double xbig = 171.624;
  double xden;
  double xinf = 1.79E+308;
  double xminin = 2.23E-308;
  double xnum;
  double y;
  double y1;
  double ysq;
  double z;
  double zero = 0.0;;

  parity = false;
  fact = one;
  n = 0;
  y = x;
//
//  Argument is negative.
//
  if ( y <= zero )
  {
    y = - x;
    y1 = ( double ) ( int ) ( y );
    res = y - y1;

    if ( res != zero )
    {
      if ( y1 != ( double ) ( int ) ( y1 * half ) * two )
      {
        parity = true;
      }

      fact = - pi / sin ( pi * res );
      y = y + one;
    }
    else
    {
      res = xinf;
      value = res;
      return value;
    }
  }
//
//  Argument is positive.
//
  if ( y < eps )
  {
//
//  Argument < EPS.
//
    if ( xminin <= y )
    {
      res = one / y;
    }
    else
    {
      res = xinf;
      value = res;
      return value;
    }
  }
  else if ( y < twelve )
  {
    y1 = y;
//
//  0.0 < argument < 1.0.
//
    if ( y < one )
    {
      z = y;
      y = y + one;
    }
//
//  1.0 < argument < 12.0.
//  Reduce argument if necessary.
//
    else
    {
      n = ( int ) ( y ) - 1;
      y = y - ( double ) ( n );
      z = y - one;
    }
//
//  Evaluate approximation for 1.0 < argument < 2.0.
//
    xnum = zero;
    xden = one;
    for ( i = 0; i < 8; i++ )
    {
      xnum = ( xnum + p[i] ) * z;
      xden = xden * z + q[i];
    }
    res = xnum / xden + one;
//
//  Adjust result for case  0.0 < argument < 1.0.
//
    if ( y1 < y )
    {
      res = res / y1;
    }
//
//  Adjust result for case 2.0 < argument < 12.0.
//
    else if ( y < y1 )
    {
      for ( i = 1; i <= n; i++ )
      {
        res = res * y;
        y = y + one;
      }
    }
  }
  else
  {
//
//  Evaluate for 12.0 <= argument.
//
    if ( y <= xbig )
    {
      ysq = y * y;
      sum = c[6];
      for ( i = 0; i < 6; i++ )
      {
        sum = sum / ysq + c[i];
      }
      sum = sum / y - y + sqrtpi;
      sum = sum + ( y - half ) * log ( y );
      res = exp ( sum );
    }
    else
    {
      res = xinf;
      value = res;
      return value;
    }
  }
//
//  Final adjustments and return.
//
  if ( parity )
  {
    res = - res;
  }

  if ( fact != one )
  {
    res = fact / res;
  }

  value = res;

  return value;
}
//****************************************************************************80

double r8_huge ( )

//****************************************************************************80
//
//  Purpose:
//
//    R8_HUGE returns a "huge" R8.
//
//  Discussion:
//
//    The value returned by this function is NOT required to be the
//    maximum representable R8.  This value varies from machine to machine,
//    from compiler to compiler, and may cause problems when being printed.
//    We simply want a "very large" but non-infinite number.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    06 October 2007
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Output, double R8_HUGE, a "huge" R8 value.
//
{
  double value;

  value = 1.0E+30;

  return value;
}
//****************************************************************************80

void r8mat_write ( string output_filename, int m, int n, double table[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8MAT_WRITE writes an R8MAT file with no header.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    29 June 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string OUTPUT_FILENAME, the output filename.
//
//    Input, int M, the spatial dimension.
//
//    Input, int N, the number of points.
//
//    Input, double TABLE[M*N], the table data.
//
{
  int i;
  int j;
  ofstream output;
//
//  Open the file.
//
  output.open ( output_filename.c_str ( ) );

  if ( !output )
  {
    cerr << "\n";
    cerr << "R8MAT_WRITE - Fatal error!\n";
    cerr << "  Could not open the output file.\n";
    return;
  }
//
//  Write the data.
//
  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      output << "  " << setw(24) << setprecision(16) << table[i+j*m];
    }
    output << "\n";
  }
//
//  Close the file.
//
  output.close ( );

  return;
}
//****************************************************************************80

void timestamp ( )

//****************************************************************************80
//
//  Purpose:
//
//    TIMESTAMP prints the current YMDHMS date as a time stamp.
//
//  Example:
//
//    31 May 2001 09:45:54 AM
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    08 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    None
//
{
# define TIME_SIZE 40

  static char time_buffer[TIME_SIZE];
  const struct std::tm *tm_ptr;
  std::time_t now;

  now = std::time ( NULL );
  tm_ptr = std::localtime ( &now );

  std::strftime ( time_buffer, TIME_SIZE, "%d %B %Y %I:%M:%S %p", tm_ptr );

  std::cout << time_buffer << "\n";

  return;
# undef TIME_SIZE
}