# 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[] );
char ch_cap ( char ch );
bool ch_eqi ( char ch1, char ch2 );
int ch_to_digit ( char ch );
void dtable_close_write ( ofstream &output );
void dtable_data_write ( ofstream &output, int m, int n, double table[] );
void dtable_header_write ( char *output_filename, ofstream &output, int m, 
  int n );
void dtable_write ( char *output_filename, int m, int n, double table[], 
  bool header );
void gegenbauer_compute ( int order, double alpha, double xtab[], 
  double weight[] );
void gegenbauer_handle ( int order, double alpha, char *output );
void gegenbauer_recur ( double *p2, double *dp2, double *p1, double x, int order, 
  double alpha, double b[], double c[] );
void gegenbauer_root ( double *x, int order, double alpha, double *dp2, 
  double *p1, double c[] );
double r8_abs ( double x );
double r8_epsilon ( void );
double r8_gamma ( double x );
double r8_huge ( void );
bool s_eqi ( char *s1, char *s2 );
void timestamp ( void );
char *timestring ( void );

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

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

//****************************************************************************80
//
//  Purpose:
//
//    MAIN is the main program for GEGENBAUER_RULE.
//
//  Discussion:
//
//    This program computes a standard Gauss-Gegenbauer quadrature rule
//    and writes it to a file.
//
//    The user specifies:
//    * the ORDER (number of points) in the rule
//    * the ALPHA parameter,
//    * the root name of the output files.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    01 March 2008
//
//  Author:
//
//    John Burkardt
//
{
  double alpha;
  int order;
  char output[81];

  cout << "\n";
  timestamp ( );
  cout << "\n";
  cout << "GEGENBAUER_RULE\n";
  cout << "  C++ version\n";
  cout << "\n";
  cout << "  Compiled on " << __DATE__ << " at " << __TIME__ << ".\n";
  cout << "\n";
  cout << "  Compute a Gauss-Gegenbauer quadrature rule for approximating\n";
  cout << "\n";
  cout << "    Integral ( -1 <= x <= +1 ) (1-x^2)^ALPHA f(x) dx\n";
  cout << "\n";
  cout << "  of order ORDER.\n";
  cout << "\n";
  cout << "  The user specifies ORDER, ALPHA, and OUTPUT.\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;
  }

  cout << "\n";
  cout << "  The requested order of the rule is = " << order << "\n";
//
//  Get ALPHA.
//
  if ( 2 < argc )
  {
    alpha = atof ( argv[2] );
  }
  else
  {
    cout << "\n";
    cout << "  ALPHA is the exponent of (1-x^2) in the integral:\n";
    cout << "\n";
    cout << "      Integral ( -1 <= x <= +1 ) (1-x^2)^ALPHA f(x) dx\n";
    cout << "\n";
    cout << "  Note that -1.0 < ALPHA is required.\n";
    cout << "\n";
    cout << "  Enter the value of ALPHA:\n";
    cin >> alpha;
  }

  cout << "\n";
  cout << "  The requested value of ALPHA = " << alpha << "\n";
//
//  Get the output option or quadrature file root name:
//
  if ( 3 < argc )
  {
    strcpy ( 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 << "  OUTPUT option is \"" << output << "\".\n";
//
//  Construct the rule and output it.
//
  gegenbauer_handle ( order, alpha, output );

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

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

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

char ch_cap ( char ch )

//****************************************************************************80
//
//  Purpose:
//
//    CH_CAP capitalizes a single character.
//
//  Discussion:
//
//    This routine should be equivalent to the library "toupper" function.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    19 July 1998
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, char CH, the character to capitalize.
//
//    Output, char CH_CAP, the capitalized character.
//
{
  if ( 97 <= ch && ch <= 122 ) 
  {
    ch = ch - 32;
  }   

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

bool ch_eqi ( char ch1, char ch2 )

//****************************************************************************80
//
//  Purpose:
//
//    CH_EQI is true if two characters are equal, disregarding case.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    13 June 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, char CH1, CH2, the characters to compare.
//
//    Output, bool CH_EQI, is true if the two characters are equal,
//    disregarding case.
//
{
  if ( 97 <= ch1 && ch1 <= 122 ) 
  {
    ch1 = ch1 - 32;
  } 
  if ( 97 <= ch2 && ch2 <= 122 ) 
  {
    ch2 = ch2 - 32;
  }     

  return ( ch1 == ch2 );
}
//****************************************************************************80

int ch_to_digit ( char ch )

//****************************************************************************80
//
//  Purpose:
//
//    CH_TO_DIGIT returns the integer value of a base 10 digit.
//
//  Example:
//
//     CH  DIGIT
//    ---  -----
//    '0'    0
//    '1'    1
//    ...  ...
//    '9'    9
//    ' '    0
//    'X'   -1
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    13 June 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, char CH, the decimal digit, '0' through '9' or blank are legal.
//
//    Output, int CH_TO_DIGIT, the corresponding integer value.  If the 
//    character was 'illegal', then DIGIT is -1.
//
{
  int digit;

  if ( '0' <= ch && ch <= '9' )
  {
    digit = ch - '0';
  }
  else if ( ch == ' ' )
  {
    digit = 0;
  }
  else
  {
    digit = -1;
  }

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

void dtable_close_write ( ofstream &output )

//****************************************************************************80
//
//  Purpose:
//
//    DTABLE_CLOSE_WRITE closes a file to which a DTABLE was to be written.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    12 November 2007
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, ofstream output, the output file stream.
//
{
  output.close ( );

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

void dtable_data_write ( ofstream &output, int m, int n, double table[] )

//****************************************************************************80
//
//  Purpose:
//
//    DTABLE_DATA_WRITE writes data to a DTABLE file.
//
//  Discussion:
//
//    The file should already be open.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    12 November 2007
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, ofstream &OUTPUT, a pointer to the output stream.
//
//    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;
  char *s;

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      output << setprecision(16) << setw(24) << table[i+j*m] << "  ";
    }
    output << "\n";
  }

  output.close ( );

  return;
}
//****************************************************************************80
 
void dtable_header_write ( char *output_filename, ofstream &output, int m, 
  int n )
 
//****************************************************************************80
//
//  Purpose:
//
//    DTABLE_HEADER_WRITE writes the header of a DTABLE file.
//
//  Discussion:
//
//    The file should already be open.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    12 November 2007
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, char *OUTPUT_FILENAME, the output filename.
//
//    Input, ofstream &OUTPUT, the output stream.
//
//    Input, int M, the spatial dimension.
//
//    Input, int N, the number of points.
//
{
  char *s;

  s = timestring ( );

  output << "#  " << output_filename << "\n";
  output << "#  created by dtable_write in table_io.C" << "\n";
  output << "#  at " << s << "\n";
  output << "#\n";
  output << "#  Spatial dimension M = " << m << "\n";
  output << "#  Number of points N =  " << n << "\n";
  output << "#  EPSILON (unit roundoff) = " << r8_epsilon ( ) << "\n";
  output << "#\n";

  delete [] s;

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

void dtable_write ( char *output_filename, int m, int n, double table[], 
  bool header )

//****************************************************************************80
//
//  Purpose:
//
//    DTABLE_WRITE writes information to a DTABLE file.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    12 November 2007
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, char *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.
//
//    Input, bool HEADER, is TRUE if the header is to be included.
//
{
  ofstream output;

  output.open ( output_filename );

  if ( !output )
  if ( false )
  {
    cout << "\n";
    cout << "DTABLE_WRITE - Fatal error!\n";
    cout << "  Could not open the output file.\n";
    return;
  }

  if ( header )
  {
    dtable_header_write ( output_filename, output, m, n );
  }

  dtable_data_write ( output, m, n, table );

  dtable_close_write ( output );

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

void gegenbauer_compute ( int order, double alpha, double xtab[], 
  double weight[] )

//****************************************************************************80
//
//  Purpose:
//
//    GEGENBAUER_COMPUTE computes a Gauss-Gegenbauer quadrature rule.
//
//  Discussion:
//
//    The integration interval is [ -1, 1 ].
//
//    The integral to approximate:
//
//      Integral ( -1 <= X <= 1 ) (1-X^2)^ALPHA * 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:
//
//    26 February 2008
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Arthur Stroud, Don Secrest,
//    Gaussian Quadrature Formulas,
//    Prentice Hall, 1966,
//    LC: QA299.4G3S7.
//
//  Parameters:
//
//    Input, int ORDER, the order of the quadrature rule to be computed.
//
//    Input, double ALPHA, the exponent of (1-X^2).  -1.0 < ALPHA is required.
//
//    Output, double XTAB[ORDER], the abscissas.
//
//    Output, double WEIGHT[ORDER], the weights.
//
{
  double an;
  double *c;
  double cc;
  double delta;
  double dp2;
  int i;
  double p1;
  double prod;
  double r1;
  double r2;
  double r3;
  double temp;
  double x;

  c = new double[order];
//
//  Check ALPHA.
//
  if ( alpha <= -1.0 )
  {
    cout << "\n";
    cout << "GEGENBAUER_COMPUTE - Fatal error!\n";
    cout << "  -1.0 < ALPHA is required.\n";
    exit ( 1 );
  }
//
//  Set the recursion coefficients.
//
  for ( i = 1; i <= order; i++ )
  {
    if ( i == 1 )
    {
      c[i-1] = 0.0;
    }
    else
    {
      c[i-1] = ( double ) ( i - 1 ) 
            * ( alpha + alpha + ( double ) ( i - 1 ) ) / 
            ( ( alpha + alpha + ( double ) ( 2 * i - 1 ) ) 
            * ( alpha + alpha + ( double ) ( 2 * i - 3 ) ) );
    }
  }

  delta = r8_gamma ( alpha         + 1.0 ) 
        * r8_gamma (         alpha + 1.0 ) 
        / r8_gamma ( alpha + alpha + 2.0 );

  prod = 1.0;
  for ( i = 2; i <= order; i++ )
  {
    prod = prod * c[i-1];
  }
  cc = delta * pow ( 2.0, alpha + alpha + 1.0 ) * prod;

  for ( i = 1; i <= order; i++ )
  {
    if ( i == 1 )
    {
      an = alpha / ( double ) ( order );

      r1 = ( 1.0 + alpha ) 
        * ( 2.78 / ( 4.0 + ( double ) ( order * order ) ) 
        + 0.768 * an / ( double ) ( order ) );

      r2 = 1.0 + 2.44 * an + 1.282 * an * an;

      x = ( r2 - r1 ) / r2;
    }
    else if ( i == 2 )
    {
      r1 = ( 4.1 + alpha ) / 
        ( ( 1.0 + alpha ) * ( 1.0 + 0.156 * alpha ) );

      r2 = 1.0 + 0.06 * ( ( double ) ( order ) - 8.0 ) * 
        ( 1.0 + 0.12 * alpha ) / ( double ) ( order );

      r3 = 1.0 + 0.012 * alpha * 
        ( 1.0 + 0.25 * r8_abs ( alpha ) ) / ( double ) ( order );

      x = x - r1 * r2 * r3 * ( 1.0 - x );
    }
    else if ( i == 3 )
    {
      r1 = ( 1.67 + 0.28 * alpha ) / ( 1.0 + 0.37 * alpha );

      r2 = 1.0 + 0.22 * ( ( double ) ( order ) - 8.0 ) 
        / ( double ) ( order );

      r3 = 1.0 + 8.0 * alpha / 
        ( ( 6.28 + alpha ) * ( double ) ( order * order ) );

      x = x - r1 * r2 * r3 * ( xtab[0] - x );
    }
    else if ( i < order - 1 )
    {
      x = 3.0 * xtab[i-2] - 3.0 * xtab[i-3] + xtab[i-4];
    }
    else if ( i == order - 1 )
    {
      r1 = ( 1.0 + 0.235 * alpha ) / ( 0.766 + 0.119 * alpha );

      r2 = 1.0 / ( 1.0 + 0.639 
        * ( ( double ) ( order ) - 4.0 ) 
        / ( 1.0 + 0.71 * ( ( double ) ( order ) - 4.0 ) ) );

      r3 = 1.0 / ( 1.0 + 20.0 * alpha / ( ( 7.5 + alpha ) * 
        ( double ) ( order * order ) ) );

      x = x + r1 * r2 * r3 * ( x - xtab[i-3] );
    }
    else if ( i == order )
    {
      r1 = ( 1.0 + 0.37 * alpha ) / ( 1.67 + 0.28 * alpha );

      r2 = 1.0 / 
        ( 1.0 + 0.22 * ( ( double ) ( order ) - 8.0 ) 
        / ( double ) ( order ) );

      r3 = 1.0 / ( 1.0 + 8.0 * alpha / 
        ( ( 6.28 + alpha ) * ( double ) ( order * order ) ) );

      x = x + r1 * r2 * r3 * ( x - xtab[i-3] );
    }

    gegenbauer_root ( &x, order, alpha, &dp2, &p1, c );

    xtab[i-1] = x;
    weight[i-1] = cc / ( dp2 * p1 );
  }
//
//  Reverse the order of the values.
//
  for ( i = 1; i <= order/2; i++ )
  {
    temp          = xtab[i-1];
    xtab[i-1]     = xtab[order-i];
    xtab[order-i] = temp;
  }

  for ( i = 1; i <=order/2; i++ )
  {
    temp            = weight[i-1];
    weight[i-1]     = weight[order-i];
    weight[order-i] = temp;
  }

  delete [] c;

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

void gegenbauer_handle ( int order, double alpha, char *output )

//****************************************************************************80
//
//  Purpose:
//
//    GEGENBAUER_HANDLE handles the Gauss-Gegenbauer rule.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    01 March 2008
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int ORDER, the order of the rule.
//
//    Input, double ALPHA, the parameter.
//
//    Input, char *OUTPUT_FILE, 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.
// 
{
  bool header;
  int i;
  char output_r[81];
  char output_w[81];
  char output_x[81];
  double *r;
  double *w;
  double *x;

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

  r[0] = - 1.0;
  r[1] = + 1.0;

  gegenbauer_compute ( order, alpha, x, w );

  if ( s_eqi ( output, "C++" ) )
  {
    cout << "//\n";
    cout << "//  Weights W, abscissas X and range R\n";
    cout << "//  for a Gauss-Gegenbauer quadrature rule\n";
    cout << "//  ORDER = " << order << "\n";
    cout << "//  ALPHA = " << alpha << "\n";
    cout << "//\n";
    cout << "//  Standard rule:\n";
	cout << "//    Integral ( -1 <= x <= +1 ) (1-x^2)^ALPHA 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 ( s_eqi ( output, "F77" ) )
   {
    cout << "c\n";
    cout << "c  Weights W, abscissas X and range R\n";
    cout << "c  for a Gauss-Gegenbauer quadrature rule\n";
    cout << "c  ORDER = " << order << "\n";
    cout << "c  ALPHA = " << alpha << "\n";
    cout << "c\n";
    cout << "c  Standard rule:\n";
	cout << "c    Integral ( -1 <= x <= +1 ) (1-x^2)^ALPHA 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 ( s_eqi ( output, "F90" ) )
  {  
    cout << "!\n";
    cout << "!  Weights W, abscissas X and range R\n";
    cout << "!  for a Gauss-Gegenbauer quadrature rule\n";
    cout << "!  ORDER = " << order << "\n";
    cout << "!  ALPHA = " << alpha << "\n";
    cout << "!\n";
    cout << "!  Standard rule:\n";
	cout << "!    Integral ( -1 <= x <= +1 ) (1-x^2)^ALPHA 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 ( s_eqi ( output, "MAT" ) )
  {
    cout << "%\n";
    cout << "%  Weights W, abscissas X and range R\n";
    cout << "%  for a Gauss-Gegenbauer quadrature rule\n";
    cout << "%  ORDER = " << order << "\n";
    cout << "%  ALPHA = " << alpha << "\n";
    cout << "%\n";
    cout << "%  Standard rule:\n";
	cout << "%    Integral ( -1 <= x <= +1 ) (1-x^2)^ALPHA 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
  {
    sprintf ( output_w, "%s_w.txt", output );
    sprintf ( output_x, "%s_x.txt", output );
    sprintf ( output_r, "%s_r.txt", output );

    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";

    header = false;
            
    dtable_write ( output_w, 1, order, w, header );
    dtable_write ( output_x, 1, order, x, header );
    dtable_write ( output_r, 1, 2,     r, header );
  }
  
  delete [] r;
  delete [] w;
  delete [] x;

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

void gegenbauer_recur ( double *p2, double *dp2, double *p1, double x, int order, 
  double alpha, double c[] )

//****************************************************************************80
//
//  Purpose:
//
//    GEGENBAUER_RECUR finds the value and derivative of a Gegenbauer polynomial.
//
//  Modified:
//
//    26 February 2008
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Arthur Stroud, Don Secrest,
//    Gaussian Quadrature Formulas,
//    Prentice Hall, 1966,
//    LC: QA299.4G3S7.
//
//  Parameters:
//
//    Output, double *P2, the value of J(ORDER)(X).
//
//    Output, double *DP2, the value of J'(ORDER)(X).
//
//    Output, double *P1, the value of J(ORDER-1)(X).
//
//    Input, double X, the point at which polynomials are evaluated.
//
//    Input, int ORDER, the order of the polynomial to be computed.
//
//    Input, double ALPHA, the exponents of (1-X^2).
//
//    Input, double C[ORDER], the recursion coefficients.
//
{
  double dp0;
  double dp1;
  int i;
  double p0;

  *p1 = 1.0;
  dp1 = 0.0;

  *p2 = x;
  *dp2 = 1.0;

  for ( i = 2; i <= order; i++ )
  {
    p0 = *p1;
    dp0 = dp1;

    *p1 = *p2;
    dp1 = *dp2;

    *p2 = x *  ( *p1 ) - c[i-1] * p0;
    *dp2 = x * dp1 + ( *p1 ) - c[i-1] * dp0;
  }
  return;
}
//****************************************************************************80

void gegenbauer_root ( double *x, int order, double alpha,  double *dp2, 
  double *p1, double c[] )

//****************************************************************************80
//
//  Purpose:
//
//    GEGENBAUER_ROOT improves an approximate root of a Gegenbauer polynomial.
//
//  Modified:
//
//    26 February 2008
//
//  Author:
//
//    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 polynomial to be computed.
//
//    Input, double ALPHA, the exponents of (1-X^2).
//
//    Output, double *DP2, the value of J'(ORDER)(X).
//
//    Output, double *P1, the value of J(ORDER-1)(X).
//
//    Input, double C[ORDER], the recursion coefficients.
//
{
  double d;
  double eps;
  double p2;
  int step;
  int step_max = 10;

  eps = r8_epsilon ( );

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

    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 ( void )

//****************************************************************************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 function was originally named DGAMMA.
//
//    However, a number of Fortran compilers now include a library
//    function of this name.  To avoid conflicts, this function was
//    renamed R8_GAMMA.
//
//    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:
//
//    FORTRAN77 original 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 ( void )

//****************************************************************************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

bool s_eqi ( char *s1, char *s2 )

//****************************************************************************80
//
//  Purpose:
//
//    S_EQI reports whether two strings are equal, ignoring case.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    05 May 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, char *S1, char *S2, pointers to two strings.
//
//    Output, bool S_EQI, is true if the strings are equal. 
//
{
  int i;
  int nchar;
  int nchar1;
  int nchar2;

  nchar1 = strlen ( s1 );
  nchar2 = strlen ( s2 );
  if ( nchar1 < nchar2 )
  {
    nchar = nchar1;
  }
  else
  {
    nchar = nchar2;
  }
//
//  The strings are not equal if they differ over their common length.
//
  for ( i = 0; i < nchar; i++ ) 
  {

    if ( ch_cap ( s1[i] ) != ch_cap ( s2[i] ) ) 
    {
      return false;
    }
  }
//
//  The strings are not equal if the longer one includes nonblanks
//  in the tail.
//
  if ( nchar < nchar1 ) 
  {
    for ( i = nchar; i < nchar1; i++ ) 
    {
      if ( s1[i] != ' ' ) 
      {
        return false;
      }
    } 
  }
  else if ( nchar < nchar2 ) 
  {
    for ( i = nchar; i < nchar2; i++ )
    {
      if ( s2[i] != ' ' ) 
      {
        return false;
      }
    } 
  }

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

void timestamp ( void )

//****************************************************************************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:
//
//    24 September 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    None
//
{
# define TIME_SIZE 40

  static char time_buffer[TIME_SIZE];
  const struct tm *tm;
  size_t len;
  time_t now;

  now = time ( NULL );
  tm = localtime ( &now );

  len = strftime ( time_buffer, TIME_SIZE, "%d %B %Y %I:%M:%S %p", tm );

  cout << time_buffer << "\n";

  return;
# undef TIME_SIZE
}
//****************************************************************************80

char *timestring ( void )

//****************************************************************************80
//
//  Purpose:
//
//    TIMESTRING returns the current YMDHMS date as a string.
//
//  Example:
//
//    31 May 2001 09:45:54 AM
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    24 September 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Output, char *TIMESTRING, a string containing the current YMDHMS date.
//
{
# define TIME_SIZE 40

  const struct tm *tm;
  size_t len;
  time_t now;
  char *s;

  now = time ( NULL );
  tm = localtime ( &now );

  s = new char[TIME_SIZE];

  len = strftime ( s, TIME_SIZE, "%d %B %Y %I:%M:%S %p", tm );

  return s;
# undef TIME_SIZE
}