# include <cstdlib>
# include <iostream>
# include <iomanip>
# include <cmath>
# include <ctime>
# include <fstream>
# include <string>

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 comp_next ( int n, int k, int a[], bool *more, int *h, int *t );
int file_column_count ( string input_filename );
int file_row_count ( string input_filename );
double monomial_integral_generalized_hermite ( int expon, double alpha );
double monomial_integral_generalized_laguerre ( int expon, double alpha );
double monomial_integral_hermite ( int expon );
double monomial_integral_jacobi ( int expon, double alpha, double beta );
double monomial_integral_laguerre ( int expon );
double monomial_integral_legendre ( int expon );
double monomial_integral_mixed ( int dim_num, int rule[], double alpha[],
  double beta[], int expon[] );
double monomial_quadrature ( int dim_num, int point_num, int rule[], 
  double alpha[], double beta[], int expon[], double weight[], double x[] );
double *monomial_value ( int dim_num, int point_num, int expon[], double x[] );
double r8_abs ( double x );
double r8_factorial ( int n );
double r8_factorial2 ( int n );
double r8_gamma ( double x );
double r8_huge ( );
double r8_hyper_2f1 ( double a, double b, double c, double x );
double r8_psi ( double xx );
double *r8mat_data_read ( string input_filename, int m, int n );
void r8mat_header_read ( string input_filename, int *m, int *n );
double r8vec_dot ( int n, double a1[], double a2[] );
int s_len_trim (string s );
int s_to_i4 ( string s, int *last, bool *error );
double s_to_r8 ( string s, int *lchar, bool *error );
bool s_to_r8vec ( string s, int n, double rvec[] );
int s_word_count ( string s );
void timestamp ( );

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

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

//****************************************************************************80
//
//  Purpose:
//
//    MAIN is the main program for NINT_EXACTNESS_MIXED.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    25 January 2010
//
//  Author:
//
//    John Burkardt
//
{
  double *alpha;
  double *beta;
  int degree;
  int degree_max;
  int dim;
  int dim_num;
  int dim_num2;
  bool error;
  int *expon;
  int h;
  int last;
  bool more;
  int point;
  int point_num;
  int point_num2;
  double quad_error;
  string quad_exact_filename;
  string quad_filename;
  string quad_a_filename;
  string quad_b_filename;
  string quad_r_filename;
  string quad_w_filename;
  string quad_x_filename;
  int *rule;
  int t;
  double *weight;
  double *x;
  double *x_range;

  timestamp ( );
  cout << "\n";
  cout << "NINT_EXACTNESS_MIXED\n";
  cout << "  C++ version\n";
  cout << "\n";
  cout << "  Compiled on " << __DATE__ << " at " << __TIME__ << ".\n";
  cout << "\n";
  cout << "  Investigate the polynomial exactness of\n";
  cout << "  a multidimensional quadrature rule\n";
  cout << "  for a region R = R1 x R2 x ... x RM.\n";
  cout << "\n";
  cout << "  Individual rules may be for:\n";
  cout << "\n";
  cout << "  Legendre:\n";
  cout << "  region: [-1,+1]\n";
  cout << "  weight: w(x)=1\n";
  cout << "  rules: Gauss-Legendre, Clenshaw-Curtis, Fejer2, Gauss-Patterson\n";
  cout << "\n";
  cout << "  Jacobi:\n";
  cout << "  region: [-1,+1]\n";
  cout << "  weight: w(x)=(1-x)^alpha (1+x)^beta\n";
  cout << "  rules: Gauss-Jacobi\n";
  cout << "\n";
  cout << "  Laguerre:\n";
  cout << "  region: [0,+oo)\n";
  cout << "  weight: w(x)=exp(-x)\n";
  cout << "  rules: Gauss-Laguerre\n";
  cout << "\n";
  cout << "  Generalized Laguerre:\n";
  cout << "  region: [0,+oo)\n";
  cout << "  weight: w(x)=x^alpha exp(-x)\n";
  cout << "  rules: Generalized Gauss-Laguerre\n";
  cout << "\n";
  cout << "  Hermite:\n";
  cout << "  region: (-oo,+o)\n";
  cout << "  weight: w(x)=exp(-x*x)\n";
  cout << "  rules: Gauss-Hermite\n";
  cout << "\n";
  cout << "  Generalized Hermite:\n";
  cout << "  region: (-oo,+oo)\n";
  cout << "  weight: w(x)=|x|^alpha exp(-x*x)\n";
  cout << "  rules: generalized Gauss-Hermite\n";
//
//  Get the quadrature file root name:
//
  if ( 1 < argc ) 
  {
    quad_filename = argv[1];
  }
  else
  {
    cout << "\n";
    cout << "NINT_EXACTNESS_MIXED:\n";
    cout << "  Enter the \"root\" name of the quadrature files.\n";

    cin >> quad_filename;
  }
//
//  Create the names of:
//    the quadrature X file;
//    the quadrature W file;
//    the quadrature R file;
//    the output "exactness" file.
//
  quad_a_filename = quad_filename + "_a.txt";
  quad_b_filename = quad_filename + "_b.txt";
  quad_r_filename = quad_filename + "_r.txt";
  quad_w_filename = quad_filename + "_w.txt";
  quad_x_filename = quad_filename + "_x.txt";
  quad_exact_filename = quad_filename + "_exact.txt";
//
//  The second command line argument is the maximum degree.
//
  if ( 2 < argc )
  {
    degree_max = s_to_i4 ( argv[2], &last, &error );
  }
  else
  {
    cout << "\n";
    cout << "NINT_EXACTNESS_MIXED:\n";
    cout << "  Please enter the maximum total degree to check.\n";

    cin >> degree_max;
  }
//
//  Summarize the input.
//
  cout << "\n";
  cout << "NINT_EXACTNESS: User input:\n";
  cout << "  Quadrature rule A file = \"" << quad_a_filename << "\".\n";
  cout << "  Quadrature rule B file = \"" << quad_b_filename << "\".\n";
  cout << "  Quadrature rule R file = \"" << quad_r_filename << "\".\n";
  cout << "  Quadrature rule W file = \"" << quad_w_filename << "\".\n";
  cout << "  Quadrature rule X file = \"" << quad_x_filename << "\".\n";
  cout << "  Maximum total degree to check = " << degree_max << "\n";
//
//  Read the X file.
//
  r8mat_header_read ( quad_x_filename, &dim_num, &point_num );

  cout << "\n";
  cout << "  Spatial dimension = " << dim_num   << "\n";
  cout << "  Number of points  = " << point_num << "\n";

  x = r8mat_data_read ( quad_x_filename, dim_num, point_num );
//
//  Read the W file.
//
  r8mat_header_read ( quad_w_filename, &dim_num2, &point_num2 );

  if ( dim_num2 != 1 ) 
  {
    cout << "\n";
    cout << "NINT_EXACTNESS_MIXED - Fatal error!\n";
    cout << "  The quadrature weight file should have exactly\n";
    cout << "  one value on each line.\n";
    exit ( 1 );
  }

  if ( point_num2 != point_num )
  {
    cout << "\n";
    cout << "NINT_EXACTNESS_MIXED - Fatal error!\n";
    cout << "  The quadrature weight file should have exactly\n";
    cout << "  the same number of lines as the abscissa file.\n";
    exit ( 1 );
  }

  weight = r8mat_data_read ( quad_w_filename, 1, point_num );
//
//  Read the R file.
//
  r8mat_header_read ( quad_r_filename, &dim_num2, &point_num2 );

  if ( dim_num2 != dim_num )
  {
    cout << "\n";
    cout << "NINT_EXACTNESS_MIXED - Fatal error!\n";
    cout << "  The quadrature region file should have the same\n";
    cout << "  number of values on each line as the abscissa file.\n";
    exit ( 1 );
  }

  if ( point_num2 != 2 )
  {
    cout << "\n";
    cout << "NINT_EXACTNESS_MIXED - Fatal error!\n";
    cout << "  The quadrature region file should have two lines.\n";
    exit ( 1 );
  }

  x_range = r8mat_data_read ( quad_r_filename, dim_num, 2 );
//
//  Read the A file.
//
  r8mat_header_read ( quad_a_filename, &dim_num2, &point_num2 );

  if ( dim_num2 != dim_num )
  {
    cout << "\n";
    cout << "NINT_EXACTNESS_MIXED - Fatal error!\n";
    cout << "  The quadrature A file should have the same\n";
    cout << "  number of values on each line as the abscissa file.\n";
    exit ( 1 );
  }

  if ( point_num2 != 1 )
  {
    cout << "\n";
    cout << "NINT_EXACTNESS_MIXED - Fatal error!\n";
    cout << "  The quadrature A file should have 1 line.\n";
    exit ( 1 );
  }

  alpha = r8mat_data_read ( quad_a_filename, dim_num, 1 );
//
//  Read the B file.
//
  r8mat_header_read ( quad_b_filename, &dim_num2, &point_num2 );

  if ( dim_num2 != dim_num )
  {
    cout << "\n";
    cout << "NINT_EXACTNESS_MIXED - Fatal error!\n";
    cout << "  The quadrature B file should have the same\n";
    cout << "  number of values on each line as the abscissa file,\n";
    exit ( 1 );
  }

  if ( point_num2 != 1 )
  {
    cout << "\n";
    cout << "NINT_EXACTNESS_MIXED - Fatal error!\n";
    cout << "  The quadrature B file should have 1 line.\n";
    exit ( 1 );
  }

  beta = r8mat_data_read ( quad_b_filename, dim_num, 1 );
//
//  Try to determine the rule types.
//
  rule = new int[dim_num];

  cout << "\n";
  cout << "  Analysis of integration region:\n";
  cout << "\n";

  for ( dim = 0; dim < dim_num; dim++ )
  {
    if ( x_range[dim+0*dim_num] == -1.0 &&
         x_range[dim+1*dim_num] == +1.0 )
    {
      if ( alpha[dim] == 0.0 && beta[dim] == 0.0 )
      {
        rule[dim] = 1;
        cout << "  " << setw(4) << dim << "  Gauss Legendre.\n";
      }
      else
      {
        rule[dim] = 2;
        cout << "  " << setw(4) << dim << "  Gauss Jacobi" 
             << ", ALPHA = " << alpha[dim]
             << ", BETA = " << beta[dim] << "\n";
      }
    }
    else if ( x_range[dim+0*dim_num] == 0.0 &&
              x_range[dim+1*dim_num] == r8_huge ( ) )
    {
      if ( alpha[dim] == 0.0 )
      {
        rule[dim] = 3;
        cout << "  " << setw(4) << dim << "  Gauss Laguerre.\n";
      }
      else
      {
        rule[dim] = 4;
        cout << "  " << setw(4) << dim << "  Generalized Gauss Laguerre" 
             << ", ALPHA = " << alpha[dim] << "\n";
      }
    }
    else if ( x_range[dim+0*dim_num] == - r8_huge ( ) &&
              x_range[dim+1*dim_num] == + r8_huge ( ) )
    {
      if ( alpha[dim] == 0.0 )
      {
        rule[dim] = 5;
        cout << "  " << setw(4) << dim << "  Gauss Hermite dimension.\n";
      }
      else
      {
        rule[dim] = 6;
        cout << "  " << setw(4) << dim << "  Generalized Gauss Hermite" 
             << ", ALPHA = " << alpha[dim] << "\n";
      }
    }
    else
    {
      cout << "\n";
      cout << "NINT_EXACTNESS_MIXED - Fatal error!\n";
      cout << "  Did not recognize region component.\n";
      cout << "  Dimension DIM = " << dim << "\n";
      cout << "  A = " << x_range[dim+0*dim_num] << "\n";
      cout << "  B = " << x_range[dim+1*dim_num] << "\n";
      exit ( 1 );
    }
  }
//
//  Explore the monomials.
//
  expon = new int[dim_num];

  cout << "\n";
  cout << "      Error    Degree  Exponents\n";
  cout << "\n";

  for ( degree = 0; degree <= degree_max; degree++ )
  {
    more = false;
    h = 0;
    t = 0;

    for ( ; ; )
    {
      comp_next ( degree, dim_num, expon, &more, &h, &t );

      quad_error = monomial_quadrature ( dim_num, point_num, rule, alpha,
        beta, expon, weight, x );

      cout << "  " << setw(12) << quad_error
           << "     " << setw(2)  << degree
           << "  ";

      for ( dim = 0; dim < dim_num; dim++ )
      {
        cout << setw(3) << expon[dim];
      }
      cout << "\n";

      if ( !more )
      {
        break;
      }
    }
    cout << "\n";
  }
//
//  Terminate.
//
  delete [] alpha;
  delete [] beta;
  delete [] expon;
  delete [] rule;
  delete [] weight;
  delete [] x;
  delete [] x_range;

  cout << "\n";
  cout << "NINT_EXACTNESS_MIXED:\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 comp_next ( int n, int k, int a[], bool *more, int *h, int *t )

//****************************************************************************80
//
//  Purpose:
//
//    COMP_NEXT computes the compositions of the integer N into K parts.
//
//  Discussion:
//
//    A composition of the integer N into K parts is an ordered sequence
//    of K nonnegative integers which sum to N.  The compositions (1,2,1)
//    and (1,1,2) are considered to be distinct.
//
//    The routine computes one composition on each call until there are no more.
//    For instance, one composition of 6 into 3 parts is
//    3+2+1, another would be 6+0+0.
//
//    On the first call to this routine, set MORE = FALSE.  The routine
//    will compute the first element in the sequence of compositions, and
//    return it, as well as setting MORE = TRUE.  If more compositions
//    are desired, call again, and again.  Each time, the routine will
//    return with a new composition.
//
//    However, when the LAST composition in the sequence is computed 
//    and returned, the routine will reset MORE to FALSE, signaling that
//    the end of the sequence has been reached.
//
//    This routine originally used a SAVE statement to maintain the
//    variables H and T.  I have decided that it is safer
//    to pass these variables as arguments, even though the user should
//    never alter them.  This allows this routine to safely shuffle
//    between several ongoing calculations.
//
//
//    There are 28 compositions of 6 into three parts.  This routine will
//    produce those compositions in the following order:
//
//     I         A
//     -     ---------
//     1     6   0   0
//     2     5   1   0
//     3     4   2   0
//     4     3   3   0
//     5     2   4   0
//     6     1   5   0
//     7     0   6   0
//     8     5   0   1
//     9     4   1   1
//    10     3   2   1
//    11     2   3   1
//    12     1   4   1
//    13     0   5   1
//    14     4   0   2
//    15     3   1   2
//    16     2   2   2
//    17     1   3   2
//    18     0   4   2
//    19     3   0   3
//    20     2   1   3
//    21     1   2   3
//    22     0   3   3
//    23     2   0   4
//    24     1   1   4
//    25     0   2   4
//    26     1   0   5
//    27     0   1   5
//    28     0   0   6
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 July 2008
//
//  Author:
//
//    Original FORTRAN77 version by Albert Nijenhuis, Herbert Wilf.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Albert Nijenhuis, Herbert Wilf,
//    Combinatorial Algorithms for Computers and Calculators,
//    Second Edition,
//    Academic Press, 1978,
//    ISBN: 0-12-519260-6,
//    LC: QA164.N54.
//
//  Parameters:
//
//    Input, int N, the integer whose compositions are desired.
//
//    Input, int K, the number of parts in the composition.
//
//    Input/output, int A[K], the parts of the composition.
//
//    Input/output, bool *MORE.
//    Set MORE = FALSE on first call.  It will be reset to TRUE on return
//    with a new composition.  Each new call returns another composition until
//    MORE is set to FALSE when the last composition has been computed
//    and returned.
//
//    Input/output, int *H, *T, two internal parameters needed for the
//    computation.  The user should allocate space for these in the calling
//    program, include them in the calling sequence, but never alter them!
//
{
  int i;

  if ( !( *more ) )
  {
    *t = n;
    *h = 0;
    a[0] = n;
    for ( i = 1; i < k; i++ )
    {
       a[i] = 0;
    }
  }
  else
  {
    if ( 1 < *t )
    {
      *h = 0;
    }
    *h = *h + 1;
    *t = a[*h-1];
    a[*h-1] = 0;
    a[0] = *t - 1;
    a[*h] = a[*h] + 1;
  }

  *more = ( a[k-1] != n );

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

int file_column_count ( string filename )

//****************************************************************************80
//
//  Purpose:
//
//    FILE_COLUMN_COUNT counts the columns in the first line of a file.
//
//  Discussion:
//
//    The file is assumed to be a simple text file.
//
//    Most lines of the file are presumed to consist of COLUMN_NUM words, 
//    separated by spaces.  There may also be some blank lines, and some 
//    comment lines, which have a "#" in column 1.
//
//    The routine tries to find the first non-comment non-blank line and
//    counts the number of words in that line.
//
//    If all lines are blanks or comments, it goes back and tries to analyze
//    a comment line.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    05 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string FILENAME, the name of the file.
//
//    Output, int FILE_COLUMN_COUNT, the number of columns assumed 
//    to be in the file.
//
{
  int column_num;
  ifstream input;
  bool got_one;
  string text;
//
//  Open the file.
//
  input.open ( filename.c_str ( ) );

  if ( !input )
  {
    column_num = -1;
    cerr << "\n";
    cerr << "FILE_COLUMN_COUNT - Fatal error!\n";
    cerr << "  Could not open the file:\n";
    cerr << "  \"" << filename << "\"\n";
    return column_num;
  }
//
//  Read one line, but skip blank lines and comment lines.
//
  got_one = false;

  for ( ; ; )
  {
    getline ( input, text );

    if ( input.eof ( ) )
    {
      break;
    }

    if ( s_len_trim ( text ) <= 0 )
    {
      continue;
    }

    if ( text[0] == '#' )
    {
      continue;
    }
    got_one = true;
    break;
  }

  if ( !got_one )
  {
    input.close ( );

    input.open ( filename.c_str ( ) );

    for ( ; ; )
    {
      input >> text;

      if ( input.eof ( ) )
      {
        break;
      }

      if ( s_len_trim ( text ) == 0 )
      {
        continue;
      }
      got_one = true;
      break;
    }
  }

  input.close ( );

  if ( !got_one )
  {
    cerr << "\n";
    cerr << "FILE_COLUMN_COUNT - Warning!\n";
    cerr << "  The file does not seem to contain any data.\n";
    return -1;
  }

  column_num = s_word_count ( text );

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

int file_row_count ( string filename )

//****************************************************************************80
//
//  Purpose:
//
//    FILE_ROW_COUNT counts the number of row records in a file.
//
//  Discussion:
//
//    It does not count lines that are blank, or that begin with a
//    comment symbol '#'.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    30 January 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string FILENAME, the name of the input file.
//
//    Output, int FILE_ROW_COUNT, the number of rows found.
//
{
  int bad_num;
  int comment_num;
  ifstream input;
  int record_num;
  int row_num;
  string text;

  row_num = 0;
  comment_num = 0;
  record_num = 0;
  bad_num = 0;

  input.open ( filename.c_str ( ) );

  if ( !input )
  {
    cerr << "\n";
    cerr << "FILE_ROW_COUNT - Fatal error!\n";
    cerr << "  Could not open the file: \"" << filename << "\"\n";
    exit ( 1 );
  }

  for ( ; ; )
  {
    getline ( input, text );

    if ( input.eof ( ) )
    {
      break;
    }

    record_num = record_num + 1;

    if ( text[0] == '#' )
    {
      comment_num = comment_num + 1;
      continue;
    }

    if ( s_len_trim ( text ) == 0 )
    {
      comment_num = comment_num + 1;
      continue;
    }

    row_num = row_num + 1;
  }

  input.close ( );

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

double monomial_integral_generalized_hermite ( int expon, double alpha )

//****************************************************************************80
//
//  Purpose:
//
//    MONOMIAL_INTEGRAL_GENERALIZED_HERMITE evaluates a 1D monomial generalized Hermite integral.
//
//  Discussion:
//
//    The integral being computed is
//
//      integral ( -oo < x < +oo ) x^n |x|^alpha exp(-x*x) dx 
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    17 October 2008
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int EXPON, the exponent of the monomial.
//    0 <= EXPON.
//
//    Input, double ALPHA, the exponent of |x| in the weight.
//    -1.0 < ALPHA.
//
//    Output, double MONOMIAL_INTEGRAL_GENERALIZED_HERMITE, 
//    the value of the integral.
//
{
  double arg;
  double value;

  if ( ( expon % 2 ) == 1 )
  {
    value = 0.0;
  }
  else
  {
    arg = alpha + ( double ) ( expon );

    if ( arg <= - 1.0 )
    {
      value = - r8_huge ( );
    }
    else
    {
      arg = ( arg + 1.0 ) / 2.0;
      value = r8_gamma ( arg );
    }
  }
  return value;
}
//****************************************************************************80

double monomial_integral_generalized_laguerre ( int expon, double alpha )

//****************************************************************************80
//
//  Purpose:
//
//    MONOMIAL_INTEGRAL_GENERALIZED_LAGUERRE evaluates a 1D monomial generalized Laguerre integral.
//
//  Discussion:
//
//    The integral being computed is
//
//      integral ( 0 <= x < +oo ) x^n x^alpha exp(-x) dx 
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    10 October 2008
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int EXPON, the exponent of the monomial.
//    0 <= EXPON.
//
//    Input, double ALPHA, the exponent of x in the weight.
//    -1.0 < ALPHA.
//
//    Output, double MONOMIAL_INTEGRAL_GENERALIZED_LAGUERRE, 
//    the value of the integral.
//
{
  double arg;
  double value;

  arg = alpha + ( double ) ( expon + 1 );

  value = r8_gamma ( arg );

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

double monomial_integral_hermite ( int expon )

//****************************************************************************80
//
//  Purpose:
//
//    MONOMIAL_INTEGRAL_HERMITE evaluates a 1D monomial Hermite integral.
//
//  Discussion:
//
//    H(n) = Integral ( -oo < x < +oo ) x^n exp(-x*x) dx
//
//    H(n) is 0 for n odd.
//
//    H(n) = (n-1)!! * sqrt(pi) / 2^(n/2) for n even.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    31 August 2008
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int EXPON, the exponent.  
//    0 <= EXPON.
//
//    Output, double MONOMIAL_INTEGRAL_HERMITE, 
//    the value of the integral.
//
{
  double pi = 3.141592653589793;
  double value;

  if ( expon < 0 )
  {
    value = - r8_huge ( );
  }
  else if ( ( expon % 2 ) == 1 )
  {
    value = 0.0;
  }
  else
  {
    value = r8_factorial2 ( expon - 1 ) * sqrt ( pi ) / pow ( 2.0, expon / 2 );
  }
  return value;
}
//****************************************************************************80

double monomial_integral_jacobi ( int expon, double alpha, double beta )

//****************************************************************************80
//
//  Purpose:
//
//    MONOMIAL_INTEGRAL_JACOBI evaluates the integral of a monomial with Jacobi weight.
//
//  Discussion:
//
//    VALUE = Integral ( -1 <= X <= +1 ) x^EXPON (1-x)^ALPHA (1+x)^BETA dx
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    17 October 2008
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int EXPON, the exponent.
//
//    Input, double ALPHA, the exponent of (1-X) in the weight factor.
//
//    Input, double BETA, the exponent of (1+X) in the weight factor.
//
//    Output, double MONOMIAL_INTEGRAL_JACOBI, the value of the integral.
//
{
  double arg1;
  double arg2;
  double arg3;
  double arg4;
  double c;
  double s;
  double value;
  double value1;
  double value2;

  c = ( double ) ( expon );

  if ( ( expon % 2 ) == 0 )
  {
    s = +1.0;
  }
  else
  {
    s = -1.0;
  }

  arg1 = - alpha;
  arg2 =   1.0 + c;
  arg3 =   2.0 + beta + c;
  arg4 = - 1.0;

  value1 = r8_hyper_2f1 ( arg1, arg2, arg3, arg4 );

  arg1 = - beta;
  arg2 =   1.0 + c;
  arg3 =   2.0 + alpha + c;
  arg4 = - 1.0;

  value2 = r8_hyper_2f1 ( arg1, arg2, arg3, arg4 );

  value = r8_gamma ( 1.0 + c ) * ( s * r8_gamma ( 1.0 + beta  ) * value1 
    / r8_gamma ( 2.0 + beta  + c ) 
    +     r8_gamma ( 1.0 + alpha ) * value2 
    / r8_gamma ( 2.0 + alpha + c ) );

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

double monomial_integral_laguerre ( int expon )

//****************************************************************************80
//
//  Purpose:
//
//    MONOMIAL_INTEGRAL_LAGUERRE evaluates a 1D monomial Laguerre integral.
//
//  Discussion:
//
//    The integral being computed is
//
//      integral ( 0 <= x < +oo ) x^n * exp ( -x ) dx
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    31 August 2008
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int EXPON, the exponent.
//    0 <= EXPON.
//
//    Output, double MONOMIAL_INTEGRAL_LAGUERRE, 
//    the value of the integral.
//
{
  double value;

  value = r8_factorial ( expon );

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

double monomial_integral_legendre ( int expon )

//****************************************************************************80
//
//  Purpose:
//
//    MONOMIAL_INTEGRAL_LEGENDRE evaluates a 1D monomial Legendre integral.
//
//  Discussion:
//
//    The integral being computed is
//
//      integral ( -1 <= x < +1 ) x^n dx 
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    31 August 2008
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int EXPON, the exponent.
//    0 <= EXPON.
//
//    Output, double MONOMIAL_INTEGRAL_LEGENDRE, 
//    the value of the integral.
//
{
  double value;

  if ( ( expon % 2 ) == 1 ) 
  {
    value = 0.0;
  }
  else if ( ( expon % 2 ) == 0 )
  {
    value = 2.0 / ( double ) ( expon + 1 );
  }

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

double monomial_integral_mixed ( int dim_num, int rule[], double alpha[],
  double beta[], int expon[] )

//****************************************************************************80
//
//  Purpose:
//
//    MONOMIAL_INTEGRAL_MIXED evaluates a multi-D monomial mixed integral.
//
//  Discussion:
//
//    This routine evaluates a monomial of the form
//
//      product ( 1 <= dim <= dim_num ) x(dim)^expon(dim)
//
//    where the exponents are nonnegative integers.  Note that
//    if the combination 0^0 is encountered, it should be treated
//    as 1.
//
//    The integration is carried out in a region that is a direct product
//    of 1D factors that may be of Legendre, Laguerre or Hermite type,
//    and the integration includes the weight functions associated with
//    the 1D factors.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    31 August 2008
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int DIM_NUM, the spatial dimension.
//
//    Input, int RULE[DIM_NUM], the component rules.
//    1, Gauss-Legendre rule on [-1,+1];
//    2, Gauss-Jacobi rule on [-1,+1];
//    3, Gauss-Laguerre rule on [0,+oo);
//    4, Generalized Gauss-Laguerre rule on [0,+oo);
//    5, Gauss-Hermite rule on (-oo,+oo);
//    6, Generalized Gauss-Hermite rule on (-oo,+oo).
//
//    Input, double ALPHA[DIM_NUM], BETA[DIM_NUM], parameters that
//    may be needed for Jacobi, Generalized-Laguerre, or Generalized Hermite rules.
//
//    Input, int EXPON[DIM_NUM], the exponents.
//
//    Output, double MONOMIAL_INTEGRAL_MIXED, 
//    the value of the integral.
//
{
  int dim;
  double value;

  value = 1.0;

  for ( dim = 0; dim < dim_num; dim++ )
  {
    if ( rule[dim] == 1 )
    {
      value = value * monomial_integral_legendre ( expon[dim] );
    }
    else if ( rule[dim] == 2 )
    {
      value = value 
        * monomial_integral_jacobi ( expon[dim], alpha[dim], beta[dim] );
    }
    else if ( rule[dim] == 3 )
    {
      value = value * monomial_integral_laguerre ( expon[dim] );
    }
    else if ( rule[dim] == 4 )
    {
      value = value 
        * monomial_integral_generalized_laguerre ( expon[dim], alpha[dim] );
    }
    else if ( rule[dim] == 5 )
    {
      value = value * monomial_integral_hermite ( expon[dim] );
    }
    else if ( rule[dim] == 6 )
    {
      value = value 
        * monomial_integral_generalized_hermite ( expon[dim], alpha[dim] );
    }
  }
  return value;
}
//****************************************************************************80

double monomial_quadrature ( int dim_num, int point_num, int rule[], 
  double alpha[], double beta[], int expon[], double weight[], double x[] )

//****************************************************************************80
//
//  Purpose:
//
//    MONOMIAL_QUADRATURE applies a quadrature rule to a monomial.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    17 October 2008
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int DIM_NUM, the spatial dimension.
//
//    Input, int POINT_NUM, the number of points in the rule.
//
//    Input, int RULE[DIM_NUM], the component rules.
//    1, Gauss-Legendre rule on [-1,+1];
//    2, Gauss-Jacobi rule on [-1,+1];
//    3, Gauss-Laguerre rule on [0,+oo);
//    4, Generalized Gauss-Laguerre rule on [0,+oo);
//    5, Gauss-Hermite rule on (-oo,+oo);
//    6, Generalized Gauss-Hermite rule on (-oo,+oo).
//
//    Input, double ALPHA[DIM_NUM], BETA[DIM_NUM], parameters that
//    may be needed for Jacobi, Generalized-Laguerre, or Generalized Hermite rules.
//
//    Input, int EXPON[DIM_NUM], the exponents.
//
//    Input, double WEIGHT[POINT_NUM], the quadrature weights.
//
//    Input, double X[DIM_NUM*POINT_NUM], the quadrature points.
//
//    Output, double MONOMIAL_QUADRATURE, the quadrature error.
//
{
  double exact;
  double quad;
  double quad_error;
  double *value;
//
//  Get the exact value of the integral of the unscaled monomial.
//
  exact = monomial_integral_mixed ( dim_num, rule, alpha, beta, expon );
//
//  Evaluate the monomial at the quadrature points.
//
  value = monomial_value ( dim_num, point_num, expon, x );
//
//  Compute the weighted sum and divide by the exact value.
//
  quad = r8vec_dot ( point_num, weight, value );
//
//  Error:
//
  if ( exact == 0.0 )
  {
    quad_error = r8_abs ( quad - exact );
  }
  else
  {
    quad_error = r8_abs ( quad - exact ) / r8_abs ( exact );
  }

  delete [] value;

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

double *monomial_value ( int dim_num, int point_num, int expon[], double x[] )

//****************************************************************************80
//
//  Purpose:
//
//    MONOMIAL_VALUE evaluates a monomial.
//
//  Discussion:
//
//    This routine evaluates a multidimensional monomial which is a product 
//    of 1D factors of the form x(dim)^expon(dim).
//
//    The exponents are nonnegative integers.  
//
//    Note that if the combination 0^0 is encountered, it should be treated
//    as 1.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    17 October 2008
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int DIM_NUM, the spatial dimension.
//
//    Input, int POINT_NUM, the number of points at which the
//    monomial is to be evaluated.
//
//    Input, int EXPON[DIM_NUM], the exponents.
//
//    Input, double X[DIM_NUM*POINT_NUM], the point coordinates.
//
//    Output, double MONOMIAL_VALUE[POINT_NUM], the value of the monomial.
//
{
  int dim;
  int point;
  double *value;

  value = new double[point_num];

  for ( point = 0; point < point_num; point++ )
  {
    value[point] = 1.0;
  }

  for ( dim = 0; dim < dim_num; dim++ )
  {
    if ( 0 != expon[dim] )
    {
      for ( point = 0; point < point_num; point++ )
      {
        value[point] = value[point] * pow ( x[dim+point*dim_num], expon[dim] );
      }
    }
  }
  return value;
}
//****************************************************************************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_factorial ( int n )

//****************************************************************************80
//
//  Purpose:
//
//    R8_FACTORIAL computes the factorial of N.
//
//  Discussion:
//
//    factorial ( N ) = product ( 1 <= I <= N ) I
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    16 January 1999
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the argument of the factorial function.
//    If N is less than 1, the function value is returned as 1.
//
//    Output, double R8_FACTORIAL, the factorial of N.
//
{
  int i;
  double value;

  value = 1.0;

  for ( i = 1; i <= n; i++ )
  {
    value = value * ( double ) ( i );
  }

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

double r8_factorial2 ( int n )

//****************************************************************************80
//
//  Purpose:
//
//    R8_FACTORIAL2 computes the double factorial function.
//
//  Discussion:
//
//    FACTORIAL2( N ) = Product ( N * (N-2) * (N-4) * ... * 2 )  (N even)
//                    = Product ( N * (N-2) * (N-4) * ... * 1 )  (N odd)
//
//  Example:
//
//     N    Factorial2(N)
//
//     0     1
//     1     1
//     2     2
//     3     3
//     4     8
//     5    15
//     6    48
//     7   105
//     8   384
//     9   945
//    10  3840
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    22 January 2008
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the argument of the double factorial 
//    function.  If N is less than 1, R8_FACTORIAL2 is returned as 1.0.
//
//    Output, double R8_FACTORIAL2, the value of Factorial2(N).
//
{
  int n_copy;
  double value;

  value = 1.0;

  if ( n < 1 )
  {
    return value;
  }

  n_copy = n;

  while ( 1 < n_copy )
  {
    value = value * ( double ) n_copy;
    n_copy = n_copy - 2;
  }

  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

double r8_hyper_2f1 ( double a, double b, double c, double x )

//****************************************************************************80
//
//  Purpose:
//
//    R8_HYPER_2F1 evaluates the hypergeometric function F(A,B,C,X).
//
//  Discussion:
//
//    A minor bug was corrected.  The HW variable, used in several places as
//    the "old" value of a quantity being iteratively improved, was not
//    being initialized.  JVB, 11 February 2008.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    05 July 2009
//
//  Author:
//
//    Original FORTRAN77 version by Shanjie Zhang, Jianming Jin.
//    C++ version by John Burkardt.
//
//    The original FORTRAN77 version of this routine is copyrighted by
//    Shanjie Zhang and Jianming Jin.  However, they give permission to
//    incorporate this routine into a user program provided that the copyright
//    is acknowledged.
//
//  Reference:
//
//    Shanjie Zhang, Jianming Jin,
//    Computation of Special Functions,
//    Wiley, 1996,
//    ISBN: 0-471-11963-6,
//    LC: QA351.C45
//
//  Parameters:
//
//    Input, double A, B, C, X, the arguments of the function.
//    C must not be equal to a nonpositive integer.
//    X < 1.
//
//    Output, double R8_HYPER_2F1, the value of the function.
//
{
  double a0;
  double aa;
  double bb;
  double c0;
  double c1;
  double el = 0.5772156649015329;
  double eps;
  double f0;
  double f1;
  double g0;
  double g1;
  double g2;
  double g3;
  double ga;
  double gabc;
  double gam;
  double gb;
  double gbm;
  double gc;
  double gca;
  double gcab;
  double gcb;
  double gm;
  double hf;
  double hw;
  int j;
  int k;
  bool l0;
  bool l1;
  bool l2;
  bool l3;
  bool l4;
  bool l5;
  int m;
  int nm;
  double pa;
  double pb;
  double pi = 3.141592653589793;
  double r;
  double r0;
  double r1;
  double rm;
  double rp;
  double sm;
  double sp;
  double sp0;
  double x1;

  l0 = ( c == ( int ) ( c ) ) && ( c < 0.0 );
  l1 = ( 1.0 - x < 1.0E-15 ) && ( c - a - b <= 0.0 );
  l2 = ( a == ( int ) ( a ) ) && ( a < 0.0 );
  l3 = ( b == ( int ) ( b ) ) && ( b < 0.0 );
  l4 = ( c - a == ( int ) ( c - a ) ) && ( c - a <= 0.0 );
  l5 = ( c - b == ( int ) ( c - b ) ) && ( c - b <= 0.0 );

  if ( l0 || l1 )
  {
    cout << "\n";
    cout << "R8_HYPER_2F1 - Fatal error!\n";
    cout << "  The hypergeometric series is divergent.\n";
    hf = 0.0;
    return hf;
  }

  if ( 0.95 < x )
  {
    eps = 1.0E-08;
  }
  else
  {
    eps = 1.0E-15;
  }

  if ( x == 0.0 || a == 0.0 || b == 0.0 )
  {
    hf = 1.0;
    return hf;
  }
  else if ( 1.0 - x == eps && 0.0 < c - a - b )
  {
    gc = r8_gamma ( c );
    gcab = r8_gamma ( c - a - b );
    gca = r8_gamma ( c - a );
    gcb = r8_gamma ( c - b );
    hf = gc * gcab / ( gca * gcb );
    return hf;
  }
  else if ( 1.0 + x <= eps && r8_abs ( c - a + b - 1.0 ) <= eps )
  {
    g0 = sqrt ( pi ) * pow ( 2.0, - a );
    g1 = r8_gamma ( c );
    g2 = r8_gamma ( 1.0 + a / 2.0 - b );
    g3 = r8_gamma ( 0.5 + 0.5 * a );
    hf = g0 * g1 / ( g2 * g3 );
    return hf;
  }
  else if ( l2 || l3 )
  {
    if ( l2 )
    {
      nm = ( int ) ( r8_abs ( a ) );
    }

    if ( l3 )
    {
      nm = ( int ) ( r8_abs ( b ) );
    }

    hf = 1.0;
    r = 1.0;

    for ( k = 1; k <= nm; k++ )
    {
      r = r * ( a + k - 1.0 ) * ( b + k - 1.0 ) 
        / ( k * ( c + k - 1.0 ) ) * x;
      hf = hf + r;
    }

    return hf;
  }
  else if ( l4 || l5 )
  {
    if ( l4 )
    {
      nm = ( int ) ( r8_abs ( c - a ) );
    }

    if ( l5 )
    {
      nm = ( int ) ( r8_abs ( c - b ) );
    }

    hf = 1.0;
    r  = 1.0;
    for ( k = 1; k <= nm; k++ )
    {
      r = r * ( c - a + k - 1.0 ) * ( c - b + k - 1.0 ) 
        / ( k * ( c + k - 1.0 ) ) * x;
      hf = hf + r;
    }
    hf = pow ( 1.0 - x, c - a - b ) * hf;
    return hf;
  }

  aa = a;
  bb = b;
  x1 = x;

  if ( x < 0.0 )
  {
    x = x / ( x - 1.0 );
    if ( a < c && b < a && 0.0 < b )
    {
      a = bb;
      b = aa;
    }
    b = c - b;
  }

  if ( 0.75 <= x )
  {
    gm = 0.0;

    if ( r8_abs ( c - a - b - ( int ) ( c - a - b ) ) < 1.0E-15 )
    {
      m = int ( c - a - b );
      ga = r8_gamma ( a );
      gb = r8_gamma ( b );
      gc = r8_gamma ( c );
      gam = r8_gamma ( a + m );
      gbm = r8_gamma ( b + m );

      pa = r8_psi ( a );
      pb = r8_psi ( b );

      if ( m != 0 )
      {
        gm = 1.0;
      }

      for ( j = 1; j <= abs ( m ) - 1; j++ )
      {
        gm = gm * j;
      }

      rm = 1.0;
      for ( j = 1; j <= abs ( m ); j++ )
      {
        rm = rm * j;
      }

      f0 = 1.0;
      r0 = 1.0;;
      r1 = 1.0;
      sp0 = 0.0;;
      sp = 0.0;

      if ( 0 <= m )
      {
        c0 = gm * gc / ( gam * gbm );
        c1 = - gc * pow ( x - 1.0, m ) / ( ga * gb * rm );

        for ( k = 1; k <= m - 1; k++ )
        {
          r0 = r0 * ( a + k - 1.0 ) * ( b + k - 1.0 ) 
            / ( k * ( k - m ) ) * ( 1.0 - x );
          f0 = f0 + r0;
        }

        for ( k = 1; k <= m; k++ )
        {
          sp0 = sp0 + 1.0 / ( a + k - 1.0 ) + 1.0 / ( b + k - 1.0 ) 
          - 1.0 / ( double ) ( k );
        }

        f1 = pa + pb + sp0 + 2.0 * el + log ( 1.0 - x );
        hw = f1;

        for ( k = 1; k <= 250; k++ )
        {
          sp = sp + ( 1.0 - a ) / ( k * ( a + k - 1.0 ) ) 
            + ( 1.0 - b ) / ( k * ( b + k - 1.0 ) );

          sm = 0.0;
          for ( j = 1; j <= m; j++ )
          {
            sm = sm + ( 1.0 - a ) 
              / ( ( j + k ) * ( a + j + k - 1.0 ) ) 
              + 1.0 / ( b + j + k - 1.0 );
          }

          rp = pa + pb + 2.0 * el + sp + sm + log ( 1.0 - x );

          r1 = r1 * ( a + m + k - 1.0 ) * ( b + m + k - 1.0 ) 
            / ( k * ( m + k ) ) * ( 1.0 - x );

          f1 = f1 + r1 * rp;

          if ( r8_abs ( f1 - hw ) < r8_abs ( f1 ) * eps )
          {
            break;
          }
          hw = f1;
        }
        hf = f0 * c0 + f1 * c1;
      }
      else if ( m < 0 )
      {
        m = - m;
        c0 = gm * gc / ( ga * gb * pow ( 1.0 - x, m ) );
        c1 = - pow ( - 1.0, m ) * gc / ( gam * gbm * rm );

        for ( k = 1; k <= m - 1; k++ )
        {
          r0 = r0 * ( a - m + k - 1.0 ) * ( b - m + k - 1.0 ) 
            / ( k * ( k - m ) ) * ( 1.0 - x );
          f0 = f0 + r0;
        }

        for ( k = 1; k <= m; k++ )
        {
          sp0 = sp0 + 1.0 / ( double ) ( k );
        }

        f1 = pa + pb - sp0 + 2.0 * el + log ( 1.0 - x );
        hw = f1;

        for ( k = 1; k <= 250; k++ )
        {
          sp = sp + ( 1.0 - a ) 
            / ( k * ( a + k - 1.0 ) ) 
            + ( 1.0 - b ) / ( k * ( b + k - 1.0 ) );

          sm = 0.0;
          for ( j = 1; j <= m; j++ )
          {
            sm = sm + 1.0 / ( double ) ( j + k );
          }

          rp = pa + pb + 2.0 * el + sp - sm + log ( 1.0 - x );

          r1 = r1 * ( a + k - 1.0 ) * ( b + k - 1.0 ) 
            / ( k * ( m + k ) ) * ( 1.0 - x );

          f1 = f1 + r1 * rp;

          if ( r8_abs ( f1 - hw ) < r8_abs ( f1 ) * eps )
          {
            break;
          }

          hw = f1;
        }

        hf = f0 * c0 + f1 * c1;
      }
    }
    else
    {
      ga = r8_gamma ( a );
      gb = r8_gamma ( b );
      gc = r8_gamma ( c );
      gca = r8_gamma ( c - a );
      gcb = r8_gamma ( c - b );
      gcab = r8_gamma ( c - a - b );
      gabc = r8_gamma ( a + b - c );
      c0 = gc * gcab / ( gca * gcb );
      c1 = gc * gabc / ( ga * gb ) * pow ( 1.0 - x, c - a - b );
      hf = 0.0;
      hw = hf;
      r0 = c0;
      r1 = c1;

      for ( k = 1; k <= 250; k++ )
      {
        r0 = r0 * ( a + k - 1.0 ) * ( b + k - 1.0 ) 
          / ( k * ( a + b - c + k ) ) * ( 1.0 - x );

        r1 = r1 * ( c - a + k - 1.0 ) * ( c - b + k - 1.0 ) 
          / ( k * ( c - a - b + k ) ) * ( 1.0 - x );

        hf = hf + r0 + r1;

        if ( r8_abs ( hf - hw ) < r8_abs ( hf ) * eps )
        {
          break;
        }
        hw = hf;
      }
      hf = hf + c0 + c1;
    }
  }
  else
  {
    a0 = 1.0;

    if ( a < c && c < 2.0 * a && b < c && c < 2.0 * b )
    {
      a0 = pow ( 1.0 - x, c - a - b );
      a = c - a;
      b = c - b;
    }

    hf = 1.0;
    hw = hf;
    r = 1.0;

    for ( k = 1; k <= 250; k++ )
    {
      r = r * ( a + k - 1.0 ) * ( b + k - 1.0 ) 
        / ( k * ( c + k - 1.0 ) ) * x;

      hf = hf + r;

      if ( r8_abs ( hf - hw ) <= r8_abs ( hf ) * eps )
      {
        break;
      }

      hw = hf;
    }
    hf = a0 * hf;
  }

  if ( x1 < 0.0 )
  {
    x = x1;
    c0 = 1.0 / pow ( 1.0 - x, aa );
    hf = c0 * hf;
  }

  a = aa;
  b = bb;

  if ( 120 < k )
  {
    cout << "\n";
    cout << "R8_HYPER_2F1 - Warning!\n";
    cout << "  A large number of iterations were needed.\n";
    cout << "  The accuracy of the results should be checked.\n";
  }

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

double r8_psi ( double xx )

//****************************************************************************80
//
//  Purpose:
//
//    R8_PSI evaluates the function Psi(X).
//
//  Discussion:
//
//    This routine evaluates the logarithmic derivative of the
//    Gamma function,
//
//      PSI(X) = d/dX ( GAMMA(X) ) / GAMMA(X)
//             = d/dX LN ( GAMMA(X) )
//
//    for real X, where either
//
//      - XMAX1 < X < - XMIN, and X is not a negative integer,
//
//    or
//
//      XMIN < X.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    09 February 2008
//
//  Author:
//
//    Original FORTRAN77 version by William Cody
//    C++ version by John Burkardt
//
//  Reference:
//
//    William Cody, Anthony Strecok, Henry Thacher,
//    Chebyshev Approximations for the Psi Function,
//    Mathematics of Computation,
//    Volume 27, Number 121, January 1973, pages 123-127.
//
//  Parameters:
//
//    Input, double XX, the argument of the function.
//
//    Output, double, the value of the function.
//
{
  double aug;
  double den;
  double four = 4.0;
  double fourth = 0.25;
  double half = 0.5;
  int i;
  int n;
  int nq;
  double one = 1.0;
  double p1[9] = { 
   4.5104681245762934160E-03, 
   5.4932855833000385356, 
   3.7646693175929276856E+02, 
   7.9525490849151998065E+03, 
   7.1451595818951933210E+04, 
   3.0655976301987365674E+05, 
   6.3606997788964458797E+05, 
   5.8041312783537569993E+05, 
   1.6585695029761022321E+05 };
  double p2[7] = { 
  -2.7103228277757834192, 
  -1.5166271776896121383E+01, 
  -1.9784554148719218667E+01, 
  -8.8100958828312219821, 
  -1.4479614616899842986, 
  -7.3689600332394549911E-02, 
  -6.5135387732718171306E-21 };
  double piov4 = 0.78539816339744830962;
  double q1[8] = { 
   9.6141654774222358525E+01, 
   2.6287715790581193330E+03, 
   2.9862497022250277920E+04, 
   1.6206566091533671639E+05, 
   4.3487880712768329037E+05, 
   5.4256384537269993733E+05, 
   2.4242185002017985252E+05, 
   6.4155223783576225996E-08 };
  double q2[6] = { 
   4.4992760373789365846E+01, 
   2.0240955312679931159E+02, 
   2.4736979003315290057E+02, 
   1.0742543875702278326E+02, 
   1.7463965060678569906E+01, 
   8.8427520398873480342E-01 };
  double sgn;
  double three = 3.0;
  double upper;
  double value;
  double w;
  double x;
  double x01 = 187.0;
  double x01d = 128.0;
  double x02 = 6.9464496836234126266E-04;
  double xinf = 1.70E+38;
  double xlarge = 2.04E+15;
  double xmax1 = 3.60E+16;
  double xmin1 = 5.89E-39;
  double xsmall = 2.05E-09;
  double z;
  double zero = 0.0;

  x = xx;
  w = r8_abs ( x );
  aug = zero;
//
//  Check for valid arguments, then branch to appropriate algorithm.
//
  if ( xmax1 <= - x || w < xmin1 )
  {
    if ( zero < x )
    {
      value = - xinf;
    }
    else
    {
      value = xinf;
    }
    return value;
  }

  if ( x < half )
  {
//
//  X < 0.5, use reflection formula: psi(1-x) = psi(x) + pi * cot(pi*x)
//  Use 1/X for PI*COTAN(PI*X)  when  XMIN1 < |X| <= XSMALL.
//
    if ( w <= xsmall )
    {
      aug = - one / x;
    }
//
//  Argument reduction for cotangent.
//
    else
    {
      if ( x < zero )
      {
        sgn = piov4;
      }
      else
      {
        sgn = - piov4;
      }

      w = w - ( double ) ( ( int ) ( w ) );
      nq = int ( w * four );
      w = four * ( w - ( double ) ( nq ) * fourth );
//
//  W is now related to the fractional part of 4.0 * X.
//  Adjust argument to correspond to values in the first
//  quadrant and determine the sign.
//
      n = nq / 2;

      if ( n + n != nq )
      {
        w = one - w;
      }

      z = piov4 * w;

      if ( ( n % 2 ) != 0 )
      {
        sgn = - sgn;
      }
//
//  Determine the final value for  -pi * cotan(pi*x).
//
      n = ( nq + 1 ) / 2;
      if ( ( n % 2 ) == 0 )
      {
//
//  Check for singularity.
//
        if ( z == zero )
        {
          if ( zero < x )
          {
            value = -xinf;
          }
          else
          {
            value = xinf;
          }
          return value;
        }
        aug = sgn * ( four / tan ( z ) );
      }
      else
      {
        aug = sgn * ( four * tan ( z ) );
      }
    }
    x = one - x;
  }
//
//  0.5 <= X <= 3.0.
//
  if ( x <= three )
  {
    den = x;
    upper = p1[0] * x;
    for ( i = 1; i <= 7; i++ )
    {
      den = ( den + q1[i-1] ) * x;
      upper = ( upper + p1[i]) * x;
    }
    den = ( upper + p1[8] ) / ( den + q1[7] );
    x = ( x - x01 / x01d ) - x02;
    value = den * x + aug;
    return value;
  }
//
//  3.0 < X.
//
  if ( x < xlarge )
  {
    w = one / ( x * x );
    den = w;
    upper = p2[0] * w;
    for ( i = 1; i <= 5; i++ )
    {
      den = ( den + q2[i-1] ) * w;
      upper = ( upper + p2[i] ) * w;
    }
    aug = ( upper + p2[6] ) / ( den + q2[5] ) - half / x + aug;
  }

  value = aug + log ( x );

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

double *r8mat_data_read ( string input_filename, int m, int n )

//****************************************************************************80
//
//  Purpose:
//
//    R8MAT_DATA_READ reads the data from an R8MAT file.
//
//  Discussion:
//
//    An R8MAT is an array of R8's.
//
//    The file is assumed to contain one record per line.
//
//    Records beginning with '#' are comments, and are ignored.
//    Blank lines are also ignored.
//
//    Each line that is not ignored is assumed to contain exactly (or at least)
//    M real numbers, representing the coordinates of a point.
//
//    There are assumed to be exactly (or at least) N such records.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    23 February 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string INPUT_FILENAME, the name of the input file.
//
//    Input, int M, the number of spatial dimensions.
//
//    Input, int N, the number of points.  The program
//    will stop reading data once N values have been read.
//
//    Output, double R8MAT_DATA_READ[M*N], the table data.
//
{
  bool error;
  ifstream input;
  int i;
  int j;
  string line;
  double *table;
  double *x;

  input.open ( input_filename.c_str ( ) );

  if ( !input )
  {
    cerr << "\n";
    cerr << "R8MAT_DATA_READ - Fatal error!\n";
    cerr << "  Could not open the input file: \"" << input_filename << "\"\n";
    return NULL;
  }

  table = new double[m*n];

  x = new double[m];

  j = 0;

  while ( j < n )
  {
    getline ( input, line );

    if ( input.eof ( ) )
    {
      break;
    }

    if ( line[0] == '#' || s_len_trim ( line ) == 0 )
    {
      continue;
    }

    error = s_to_r8vec ( line, m, x );

    if ( error )
    {
      continue;
    }

    for ( i = 0; i < m; i++ )
    {
      table[i+j*m] = x[i];
    }
    j = j + 1;

  }

  input.close ( );

  delete [] x;

  return table;
}
//****************************************************************************80
 
void r8mat_header_read ( string input_filename, int *m, int *n )
 
//****************************************************************************80
//
//  Purpose:
//
//    R8MAT_HEADER_READ reads the header from an R8MAT file.
//
//  Discussion:
//
//    An R8MAT is an array of R8's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    23 February 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string INPUT_FILENAME, the name of the input file.
//
//    Output, int *M, the number of spatial dimensions.
//
//    Output, int *N, the number of points.
//
{
  *m = file_column_count ( input_filename );

  if ( *m <= 0 )
  {
    cerr << "\n";
    cerr << "R8MAT_HEADER_READ - Fatal error!\n";
    cerr << "  FILE_COLUMN_COUNT failed.\n";
    *n = -1;
    return;
  }

  *n = file_row_count ( input_filename );

  if ( *n <= 0 )
  {
    cerr << "\n";
    cerr << "R8MAT_HEADER_READ - Fatal error!\n";
    cerr << "  FILE_ROW_COUNT failed.\n";
    return;
  }

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

double r8vec_dot ( int n, double a1[], double a2[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_DOT computes the dot product of a pair of R8VEC's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    03 July 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries in the vectors.
//
//    Input, double A1[N], A2[N], the two vectors to be considered.
//
//    Output, double R8VEC_DOT, the dot product of the vectors.
//
{
  int i;
  double value;

  value = 0.0;
  for ( i = 0; i < n; i++ )
  {
    value = value + a1[i] * a2[i];
  }
  return value;
}
//****************************************************************************80

int s_len_trim ( string s )

//****************************************************************************80
//
//  Purpose:
//
//    S_LEN_TRIM returns the length of a string to the last nonblank.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    05 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string S, a string.
//
//    Output, int S_LEN_TRIM, the length of the string to the last nonblank.
//    If S_LEN_TRIM is 0, then the string is entirely blank.
//
{
  int n;

  n = s.length ( );

  while ( 0 < n ) 
  {
    if ( s[n-1] != ' ' )
    {
      return n;
    }
    n = n - 1;
  }

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

int s_to_i4 ( string s, int *last, bool *error )

//****************************************************************************80
//
//  Purpose:
//
//    S_TO_I4 reads an I4 from a string.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    05 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string S, a string to be examined.
//
//    Output, int *LAST, the last character of S used to make IVAL.
//
//    Output, bool *ERROR is TRUE if an error occurred.
//
//    Output, int *S_TO_I4, the integer value read from the string.
//    If the string is blank, then IVAL will be returned 0.
//
{
  char c;
  int i;
  int isgn;
  int istate;
  int ival;

  *error = false;
  istate = 0;
  isgn = 1;
  i = 0;
  ival = 0;

  for ( ; ; ) 
  {
    c = s[i];
    i = i + 1;
//
//  Haven't read anything.
//
    if ( istate == 0 )
    {
      if ( c == ' ' )
      {
      }
      else if ( c == '-' )
      {
        istate = 1;
        isgn = -1;
      }
      else if ( c == '+' )
      {
        istate = 1;
        isgn = + 1;
      }
      else if ( '0' <= c && c <= '9' )
      {
        istate = 2;
        ival = c - '0';
      }
      else
      {
        *error = true;
        return ival;
      }
    }
//
//  Have read the sign, expecting digits.
//
    else if ( istate == 1 )
    {
      if ( c == ' ' )
      {
      }
      else if ( '0' <= c && c <= '9' )
      {
        istate = 2;
        ival = c - '0';
      }
      else
      {
        *error = true;
        return ival;
      }
    }
//
//  Have read at least one digit, expecting more.
//
    else if ( istate == 2 )
    {
      if ( '0' <= c && c <= '9' )
      {
        ival = 10 * (ival) + c - '0';
      }
      else
      {
        ival = isgn * ival;
        *last = i - 1;
        return ival;
      }

    }
  }
//
//  If we read all the characters in the string, see if we're OK.
//
  if ( istate == 2 )
  {
    ival = isgn * ival;
    *last = s_len_trim ( s );
  }
  else
  {
    *error = true;
    *last = 0;
  }

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

double s_to_r8 ( string s, int *lchar, bool *error )

//****************************************************************************80
//
//  Purpose:
//
//    S_TO_R8 reads an R8 from a string.
//
//  Discussion:
//
//    This routine will read as many characters as possible until it reaches
//    the end of the string, or encounters a character which cannot be
//    part of the real number.
//
//    Legal input is:
//
//       1 blanks,
//       2 '+' or '-' sign,
//       2.5 spaces
//       3 integer part,
//       4 decimal point,
//       5 fraction part,
//       6 'E' or 'e' or 'D' or 'd', exponent marker,
//       7 exponent sign,
//       8 exponent integer part,
//       9 exponent decimal point,
//      10 exponent fraction part,
//      11 blanks,
//      12 final comma or semicolon.
//
//    with most quantities optional.
//
//  Example:
//
//    S                 R
//
//    '1'               1.0
//    '     1   '       1.0
//    '1A'              1.0
//    '12,34,56'        12.0
//    '  34 7'          34.0
//    '-1E2ABCD'        -100.0
//    '-1X2ABCD'        -1.0
//    ' 2E-1'           0.2
//    '23.45'           23.45
//    '-4.2E+2'         -420.0
//    '17d2'            1700.0
//    '-14e-2'         -0.14
//    'e2'              100.0
//    '-12.73e-9.23'   -12.73 * 10.0**(-9.23)
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    05 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string S, the string containing the
//    data to be read.  Reading will begin at position 1 and
//    terminate at the end of the string, or when no more
//    characters can be read to form a legal real.  Blanks,
//    commas, or other nonnumeric data will, in particular,
//    cause the conversion to halt.
//
//    Output, int *LCHAR, the number of characters read from
//    the string to form the number, including any terminating
//    characters such as a trailing comma or blanks.
//
//    Output, bool *ERROR, is true if an error occurred.
//
//    Output, double S_TO_R8, the real value that was read from the string.
//
{
  char c;
  int ihave;
  int isgn;
  int iterm;
  int jbot;
  int jsgn;
  int jtop;
  int nchar;
  int ndig;
  double r;
  double rbot;
  double rexp;
  double rtop;
  char TAB = 9;

  nchar = s_len_trim ( s );
  *error = false;
  r = 0.0;
  *lchar = -1;
  isgn = 1;
  rtop = 0.0;
  rbot = 1.0;
  jsgn = 1;
  jtop = 0;
  jbot = 1;
  ihave = 1;
  iterm = 0;

  for ( ; ; )
  {
    c = s[*lchar+1];
    *lchar = *lchar + 1;
//
//  Blank or TAB character.
//
    if ( c == ' ' || c == TAB )
    {
      if ( ihave == 2 )
      {
      }
      else if ( ihave == 6 || ihave == 7 )
      {
        iterm = 1;
      }
      else if ( 1 < ihave )
      {
        ihave = 11;
      }
    }
//
//  Comma.
//
    else if ( c == ',' || c == ';' )
    {
      if ( ihave != 1 )
      {
        iterm = 1;
        ihave = 12;
        *lchar = *lchar + 1;
      }
    }
//
//  Minus sign.
//
    else if ( c == '-' )
    {
      if ( ihave == 1 )
      {
        ihave = 2;
        isgn = -1;
      }
      else if ( ihave == 6 )
      {
        ihave = 7;
        jsgn = -1;
      }
      else
      {
        iterm = 1;
      }
    }
//
//  Plus sign.
//
    else if ( c == '+' )
    {
      if ( ihave == 1 )
      {
        ihave = 2;
      }
      else if ( ihave == 6 )
      {
        ihave = 7;
      }
      else
      {
        iterm = 1;
      }
    }
//
//  Decimal point.
//
    else if ( c == '.' )
    {
      if ( ihave < 4 )
      {
        ihave = 4;
      }
      else if ( 6 <= ihave && ihave <= 8 )
      {
        ihave = 9;
      }
      else
      {
        iterm = 1;
      }
    }
//
//  Exponent marker.
//
    else if ( ch_eqi ( c, 'E' ) || ch_eqi ( c, 'D' ) )
    {
      if ( ihave < 6 )
      {
        ihave = 6;
      }
      else
      {
        iterm = 1;
      }
    }
//
//  Digit.
//
    else if ( ihave < 11 && '0' <= c && c <= '9' )
    {
      if ( ihave <= 2 )
      {
        ihave = 3;
      }
      else if ( ihave == 4 )
      {
        ihave = 5;
      }
      else if ( ihave == 6 || ihave == 7 )
      {
        ihave = 8;
      }
      else if ( ihave == 9 )
      {
        ihave = 10;
      }

      ndig = ch_to_digit ( c );

      if ( ihave == 3 )
      {
        rtop = 10.0 * rtop + ( double ) ndig;
      }
      else if ( ihave == 5 )
      {
        rtop = 10.0 * rtop + ( double ) ndig;
        rbot = 10.0 * rbot;
      }
      else if ( ihave == 8 )
      {
        jtop = 10 * jtop + ndig;
      }
      else if ( ihave == 10 )
      {
        jtop = 10 * jtop + ndig;
        jbot = 10 * jbot;
      }

    }
//
//  Anything else is regarded as a terminator.
//
    else
    {
      iterm = 1;
    }
//
//  If we haven't seen a terminator, and we haven't examined the
//  entire string, go get the next character.
//
    if ( iterm == 1 || nchar <= *lchar + 1 )
    {
      break;
    }

  }
//
//  If we haven't seen a terminator, and we have examined the
//  entire string, then we're done, and LCHAR is equal to NCHAR.
//
  if ( iterm != 1 && (*lchar) + 1 == nchar )
  {
    *lchar = nchar;
  }
//
//  Number seems to have terminated.  Have we got a legal number?
//  Not if we terminated in states 1, 2, 6 or 7!
//
  if ( ihave == 1 || ihave == 2 || ihave == 6 || ihave == 7 )
  {
    *error = true;
    return r;
  }
//
//  Number seems OK.  Form it.
//
  if ( jtop == 0 )
  {
    rexp = 1.0;
  }
  else
  {
    if ( jbot == 1 )
    {
      rexp = pow ( 10.0, jsgn * jtop );
    }
    else
    {
      rexp = jsgn * jtop;
      rexp = rexp / jbot;
      rexp = pow ( 10.0, rexp );
    }

  }

  r = isgn * rexp * rtop / rbot;

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

bool s_to_r8vec ( string s, int n, double rvec[] )

//****************************************************************************80
//
//  Purpose:
//
//    S_TO_R8VEC reads an R8VEC from a string.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    05 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string S, the string to be read.
//
//    Input, int N, the number of values expected.
//
//    Output, double RVEC[N], the values read from the string.
//
//    Output, bool S_TO_R8VEC, is true if an error occurred.
//
{
  int begin;
  bool error;
  int i;
  int lchar;
  int length;

  begin = 0;
  length = s.length ( );
  error = 0;

  for ( i = 0; i < n; i++ )
  {
    rvec[i] = s_to_r8 ( s.substr(begin,length), &lchar, &error );

    if ( error )
    {
      return error;
    }
    begin = begin + lchar;
    length = length - lchar;
  }

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

int s_word_count ( string s )

//****************************************************************************80
//
//  Purpose:
//
//    S_WORD_COUNT counts the number of "words" in a string.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    05 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string S, the string to be examined.
//
//    Output, int S_WORD_COUNT, the number of "words" in the string.
//    Words are presumed to be separated by one or more blanks.
//
{
  bool blank;
  int char_count;
  int i;
  int word_count;

  word_count = 0;
  blank = true;

  char_count = s.length ( );

  for ( i = 0; i < char_count; i++ )
  {
    if ( isspace ( s[i] ) )
    {
      blank = true;
    }
    else if ( blank )
    {
      word_count = word_count + 1;
      blank = false;
    }
  }

  return word_count;
}
//****************************************************************************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;
  size_t len;
  std::time_t now;

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

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

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

  return;
# undef TIME_SIZE
}