# include <cmath>
# include <cstdlib>
# include <ctime>
# include <fstream>
# include <iostream>
# include <iomanip>
# 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 filename );
int file_row_count ( string filename );
double *monomial_value ( int dim_num, int point_num, int expon[], double x[] );
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[] );
double r8vec_sum ( int n, double a[] );
bool s_eqi ( string s1, string s2 );
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 );
double sphere01_monomial_integral ( int e[] );
double sphere01_monomial_quadrature ( int expon[], int point_num, double xyz[], 
  double w[] );
void timestamp ( );

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

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

//****************************************************************************80
//
//  Purpose:
//
//    MAIN is the main program for SPHERE_EXACTNESS.
//
//  Discussion:
//
//    This program investigates the polynomial exactness of a quadrature
//    rule for the unit sphere
//
//  Usage:
//
//    sphere_exactness files prefix degree_max
//
//    where
//
//    * files explains how the quadrature rule is stored:
//      'XYZW'  for file 'prefix.xyzw' containing (X,Y,Z,Weight);
//      'RTPW'  for file 'prefix.rtpw' containing  (Theta, Phi, Weight) (radians);
//      'DTPW'  for file 'prefix.dtpw' containing  (Theta, Phi, Weight) (degrees);
//      'XYZ+W' for file 'prefix.xyz' containing (X,Y,Z)
//              and file 'prefix.w' containing Weight;
//      'RTP+W' for file 'prefix.rtp' containing (Theta, Phi ) in radians,
//              and file 'prefix.w' containing Weight;
//      'DTP+W' for file 'prefix.dtp' containing (Theta, Phi ) in degrees,
//              and file 'prefix.w' containing Weight;
//      'XYZ1'  for file 'prefix.xyz' containing (X,Y,Z), 
//              and equal weights, which do not need to be read in.
//      'RTP1'  for file 'prefix.rtp' containing (Theta, Phi ) in radians,
//              and equal weights, which do not need to be read in.
//      'DTP1'  for file 'prefix.dtp' containing (Theta, Phi ) in degrees,'
//              and equal weights, which do not need to be read in.
//    * prefix is the common file prefix;
//    * degree_max is the maximum monomial degree to check.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    10 September 2010
//
//  Author:
//
//    John Burkardt
//
{
  int degree;
  int degree_max;
  int dim;
  int dim_num;
  double *dtp;
  double *dtpw;
  bool error;
  int *expon;
  string filename;
  string files;
  int h;
  int i;
  int j;
  int last;
  bool more;
  double pi = 3.141592653589793;
  int point_num;
  string prefix;
  double quad_error;
  double *rtp;
  double *rtpw;
  int t;
  double *w;
  double w_sum;
  double *xyz;
  double *xyzw;

  timestamp ( );
  cout << "\n";
  cout << "SPHERE_EXACTNESS\n";
  cout << "  C++ version\n";
  cout << "\n";
  cout << "  Compiled on " << __DATE__ << " at " << __TIME__ << ".\n";
  cout << "\n";
  cout << "  Investigate the polynomial exactness of a quadrature\n";
  cout << "  rule for the unit sphere by integrating all monomials\n";
  cout << "  of a given degree.\n";
//
//  Get the file structure;
//
  if ( 1 < argc ) 
  {
    files = argv[1];
  }
  else
  {
    cout << "\n";
    cout << "SPHERE_EXACTNESS:\n";
    cout << "  Describe the files to be read:\n";
    cout << "\n";
    cout << "  For coordinates and weights in one file:\n";
    cout << "    XYZW     (X,Y,Z,Weight)\n";
    cout << "    RTPW     (Theta, Phi, Weight) (radians)\n";
    cout << "    DTPW     (Theta, Phi, Weight) (degrees)\n";
    cout << "  For coordinates in one file and weights in another:\n";
    cout << "    XYZ+W    (X,Y,Z)       + Weight\n";
    cout << "    RTP+W    (Theta, Phi ) + Weight\n";
    cout << "    DTP+W    (Theta, Phi ) + Weight\n";
    cout << "  For coordinates in one file, and equal weights:\\n";
    cout << "    XYZ1     (X,Y,Z)\n";
    cout << "    RTP1     (Theta, Phi ) (radians)\n";
    cout << "    DTP1     (Theta, Phi ) (degrees)\n";

    cin >> files;
  }
//
//  Get the file prefix.
//
  if ( 2 < argc ) 
  {
    prefix = argv[2];
  }
  else
  {
    cout << "\n";
    cout << "SPHERE_EXACTNESS:\n";
    cout << "  Enter the filename prefix.\n";

    cin >> prefix;
  }
//
//  Get the maximum degree.
//
  if ( 3 < argc )
  {
    degree_max = s_to_i4 ( argv[3], &last, &error );
  }
  else
  {
    cout << "\n";
    cout << "SPHERE_EXACTNESS:\n";
    cout << "  Please enter the maximum total degree to check.\n";

    cin >> degree_max;
  }
//
//  Summarize the input.
//
  cout << "\n";
  cout << "SPHERE_EXACTNESS: User input:\n";
  cout << "  File structure = \"" << files << "\".\n";
  cout << "  Filename prefix = \"" << prefix << "\".\n";
  cout << "  Maximum total degree to check = " << degree_max << "\n";
//
//  Read data needed to create XYZ and W arrays.
//
  if ( s_eqi ( files, "xyzw" ) )
  {
    filename = prefix + ".xyzw";

    r8mat_header_read ( filename, &dim_num, &point_num );

    xyzw = r8mat_data_read ( filename, 4, point_num );

    xyz = new double[3*point_num];
    w = new double[point_num];

    for ( j = 0; j < point_num; j++ )
    {
      for ( i = 0; i < 3; i++ )
      {
         xyz[i+j*3] = xyzw[i+j*4];
      }
      w[j] = xyzw[3+j*4];
    }
    delete [] xyzw;
  }
  else if ( s_eqi ( files, "xyz+w" ) )
  { 
    filename = prefix + ".xyz";

    r8mat_header_read ( filename, &dim_num, &point_num );

    xyz = r8mat_data_read ( filename, 3, point_num );

    filename = prefix + ".w";

    w = r8mat_data_read ( filename, 1, point_num );
  }
  else if ( s_eqi ( files, "xyz1" ) )
  { 
    filename = prefix + ".xyz";

    r8mat_header_read ( filename, &dim_num, &point_num );

    xyz = r8mat_data_read ( filename, 3, point_num );

    w = new double[point_num];
    for ( i = 0; i < point_num; i++ )
    {
      w[i] = 4.0 * pi / point_num;
    }
  }
  else if ( s_eqi ( files, "rtpw" ) )
  {
    filename = prefix + ".rtpw";

    r8mat_header_read ( filename, &dim_num, &point_num );

    rtpw = r8mat_data_read ( filename, 3, point_num );
 
    xyz = new double[3*point_num];
    w = new double[point_num];

    for ( j = 0; j < point_num; j++ )
    {
      xyz[0+j*3] = cos ( rtpw[0+j*3] ) * sin ( rtpw[1+j*3] );
      xyz[1+j*3] = sin ( rtpw[0+j*3] ) * sin ( rtpw[1+j*3] );
      xyz[2+j*3] =                       cos ( rtpw[1+j*3] );
      w[j] = rtpw[2+j*3];
    }
    delete [] rtpw;
  }
  else if ( s_eqi ( files, "rtp+w" ) )
  {
    filename = prefix + ".rtp";

    r8mat_header_read ( filename, &dim_num, &point_num );

    rtp = r8mat_data_read ( filename, 2, point_num );

    filename = prefix + ".w";

    w = r8mat_data_read ( filename, 1, point_num );

    xyz = new double[3*point_num];

    for ( j = 0; j < point_num; j++ )
    {
      xyz[0+j*3] = cos ( rtp[0+j*2] ) * sin ( rtp[1+j*2] );
      xyz[1+j*3] = sin ( rtp[0+j*2] ) * sin ( rtp[1+j*2] );
      xyz[2+j*3] =                      cos ( rtp[1+j*2] );
    }
    delete [] rtp;
  }
  else if ( s_eqi ( files, "rtp1" ) )
  {
    filename = prefix + ".rtp";

    r8mat_header_read ( filename, &dim_num, &point_num );

    rtp = r8mat_data_read ( filename, 2, point_num );

    xyz = new double[3*point_num];

    for ( j = 0; j < point_num; j++ )
    {
      xyz[0+j*3] = cos ( rtp[0+j*2] ) * sin ( rtp[1+j*2] );
      xyz[1+j*3] = sin ( rtp[0+j*2] ) * sin ( rtp[1+j*2] );
      xyz[2+j*3] =                      cos ( rtp[1+j*2] );
    }

    delete [] rtp;

    w = new double[point_num];
    for ( i = 0; i < point_num; i++ )
    {
      w[i] = 4.0 * pi / point_num;
    }
  }
  else if ( s_eqi ( files, "dtpw" ) )
  { 
    filename = prefix + ".dtpw";

    r8mat_header_read ( filename, &dim_num, &point_num );

    dtpw = r8mat_data_read ( filename, 3, point_num );

    xyz = new double[3*point_num];
    w = new double[point_num];

    for ( j = 0; j < point_num; j++ )
    {
      dtpw[0+j*3] = dtpw[0+j*3] * pi / 180.0;
      dtpw[1+j*3] = dtpw[1+j*3] * pi / 180.0;
    }
    for ( j = 0; j < point_num; j++ )
    {
      xyz[0+j*3] = cos ( dtpw[0+j*3] ) * sin ( dtpw[1+j*3] );
      xyz[1+j*3] = sin ( dtpw[0+j*3] ) * sin ( dtpw[1+j*3] );
      xyz[2+j*3] =                       cos ( dtpw[1+j*3] );
      w[j] = dtpw[2+j*3];
    }

    delete [] dtpw;
  }
  else if ( s_eqi ( files, "dtp+w" ) )
  {
    filename = prefix + ".dtp";

    r8mat_header_read ( filename, &dim_num, &point_num );

    dtp = r8mat_data_read ( filename, 2, point_num );

    filename = prefix + ".w";

    w = r8mat_data_read ( filename, 1, point_num );

    xyz = new double[3*point_num];

    for ( j = 0; j < point_num; j++ )
    {
      dtp[0+j*2] = dtp[0+j*2] * pi / 180.0;
      dtp[1+j*2] = dtp[1+j*2] * pi / 180.0;
    }

    for ( j = 0; j < point_num; j++ )
    {
      xyz[0+j*3] = cos ( dtp[0+j*2] ) * sin ( dtp[1+j*2] );
      xyz[1+j*3] = sin ( dtp[0+j*2] ) * sin ( dtp[1+j*2] );
      xyz[2+j*3] =                      cos ( dtp[1+j*2] );
    }
    delete [] dtp;
  }
  else if ( s_eqi ( files, "dtp1" ) )
  { 
    filename = prefix + ".dtp";

    r8mat_header_read ( filename, &dim_num, &point_num );

    dtp = r8mat_data_read ( filename, 2, point_num );

    xyz = new double[3*point_num];

    for ( j = 0; j < point_num; j++ )
    {
      dtp[0+j*2] = dtp[0+j*2] * pi / 180.0;
      dtp[1+j*2] = dtp[1+j*2] * pi / 180.0;
    }

    for ( j = 0; j < point_num; j++ )
    {
      xyz[0+j*3] = cos ( dtp[0+j*2] ) * sin ( dtp[1+j*2] );
      xyz[1+j*3] = sin ( dtp[0+j*2] ) * sin ( dtp[1+j*2] );
      xyz[2+j*3] =                      cos ( dtp[1+j*2] );
    }
    delete [] dtp;

    w = new double[point_num];
    for ( i = 0; i < point_num; i++ )
    {
      w[i] = 4.0 * pi / point_num;
    }
  }
  else
  {
    cerr << "\n";
    cerr << "SPHERE_EXACTNESS - Fatal error!\n";
    cerr << "  Unrecognized file structure choice\n";
    exit ( 1 );
  }

  cout << "\n";
  cout << "  Number of points  = " << point_num << "\n";
//
//  The W's should sum to 4 * PI.
//
  w_sum = r8vec_sum ( point_num, w );

  for ( i = 0; i < point_num; i++ )
  {
    w[i] = 4.0 * pi * w[i] / w_sum;
  }
//
//  Explore the monomials.
//
  expon = new int[dim_num];

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

  for ( degree = 0; degree <= degree_max; degree++ )
  {
    cout << "\n";
    more = false;
    h = 0;
    t = 0;

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

      quad_error = sphere01_monomial_quadrature ( expon, point_num, xyz, w ); 

      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;
      }
    }

  }

  delete [] expon;
  delete [] w;
  delete [] xyz;
//
//  Terminate.
//
  cout << "\n";
  cout << "SPHERE_EXACTNESS:\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 MIT 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 MIT 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 MIT 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 MIT 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 MIT 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 input_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 MIT license. 
//
//  Modified:
//
//    23 February 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string INPUT_FILENAME, the name of the input file.
//
//    Output, int FILE_ROW_COUNT, the number of rows found.
//
{
  int comment_num;
  ifstream input;
  string line;
  int record_num;
  int row_num;

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

  input.open ( input_filename.c_str ( ) );

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

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

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

    record_num = record_num + 1;

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

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

    row_num = row_num + 1;

  }

  input.close ( );

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

double *monomial_value ( int m, int n, int e[], double x[] )

//****************************************************************************80
//
//  Purpose:
//
//    MONOMIAL_VALUE evaluates a monomial.
//
//  Discussion:
//
//    This routine evaluates a monomial of the form
//
//      product ( 1 <= i <= m ) x(i)^e(i)
//
//    The combination 0.0^0 is encountered is treated as 1.0.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    17 August 2014
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, the spatial dimension.
//
//    Input, int N, the number of evaluation points.
//
//    Input, int E[M], the exponents.
//
//    Input, double X[M*N], the point coordinates.
//
//    Output, double MONOMIAL_VALUE[N], the monomial values.
//
{
  int i;
  int j;
  double *v;

  v = new double[n];
  for ( j = 0; j < n; j++)
  {
    v[j] = 1.0;
  }
//v = r8vec_ones_new ( n );

  for ( i = 0; i < m; i++ )
  {
    if ( 0 != e[i] )
    {
      for ( j = 0; j < n; j++ )
      {
        v[j] = v[j] * pow ( x[i+j*m], e[i] );
      }
    }
  }

  return v;
}
//****************************************************************************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 MIT 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 MIT 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 MIT 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

double r8vec_sum ( int n, double a[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_SUM returns the sum of an R8VEC.
//
//  Discussion:
//
//    An R8VEC is a vector of R8's.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    15 October 2004
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries in the vector.
//
//    Input, double A[N], the vector.
//
//    Output, double R8VEC_SUM, the sum of the vector.
//
{
  int i;
  double value;

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

bool s_eqi ( string s1, string s2 )

//****************************************************************************80
//
//  Purpose:
//
//    S_EQI reports whether two strings are equal, ignoring case.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    05 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string S1, S2, two strings.
//
//    Output, bool S_EQI, is true if the strings are equal. 
//
{
  int i;
  int nchar;
  int s1_length;
  int s2_length;

  s1_length = s1.length ( );
  s2_length = s2.length ( );

  if ( s1_length < s2_length )
  {
    nchar = s1_length;
  }
  else
  {
    nchar = s2_length;
  }
//
//  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 < s1_length ) 
  {
    for ( i = nchar; i < s1_length; i++ ) 
    {
      if ( s1[i] != ' ' ) 
      {
        return false;
      }
    } 
  }
  else if ( nchar < s2_length ) 
  {
    for ( i = nchar; i < s2_length; i++ )
    {
      if ( s2[i] != ' ' ) 
      {
        return false;
      }
    } 
  }

  return true;
}
//****************************************************************************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 MIT 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 MIT 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 MIT 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 MIT 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 MIT 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

double sphere01_monomial_integral ( int e[] )

//****************************************************************************80
//
//  Purpose:
//
//    SPHERE01_MONOMIAL_INT returns monomial integrals on the unit sphere.
//
//  Discussion:
//
//    The integration region is 
//
//      X^2 + Y^2 + Z^2 = 1.
//
//    The monomial is F(X,Y,Z) = X^E(1) * Y^E(2) * Z^E(3).
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    10 September 2010
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Philip Davis, Philip Rabinowitz,
//    Methods of Numerical Integration,
//    Second Edition,
//    Academic Press, 1984, page 263.
//
//  Parameters:
//
//    Input, int E[3], the exponents of X, Y and Z in the 
//    monomial.  Each exponent must be nonnegative.
//
//    Output, double SPHERE01_MONOMIAL_INTEGRAL, the integral.
//
{
  int i;
  double integral;
  const double r8_pi = 3.141592653589793;

  if ( e[0] < 0 || e[1] < 0 || e[2] < 0 )
  {
    cerr << "\n";
    cerr << "SPHERE01_MONOMIAL_INTEGRAL - Fatal error!\n";
    cerr << "  All exponents must be nonnegative.\n";
    exit ( 1 );
  }

  if ( e[0] == 0 && e[1] == 0 && e[2] == 0 )
  {
    integral = 2.0 * sqrt ( r8_pi * r8_pi * r8_pi ) / tgamma ( 1.5 );
  }
  else if ( ( e[0] % 2 == 1 ) || ( e[1] % 2 == 1 ) || ( e[2] % 2 == 1 ) )
  {
    integral = 0.0;
  }
  else
  {
    integral = 2.0;
    for ( i = 0; i < 3; i++ )
    {
      integral = integral * tgamma ( 0.5 * ( double ) ( e[i] + 1 ) );
    }
    integral = integral / tgamma ( 0.5 * ( double ) ( e[0] + e[1] + e[2] + 3 ) );
  }
  return integral;
}
//****************************************************************************80

double sphere01_monomial_quadrature ( int expon[], int point_num, double xyz[], 
  double w[] )

//****************************************************************************80
//
//  Purpose:
//
//    SPHERE01_MONOMIAL_QUADRATURE applies quadrature to a monomial in a sphere.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    05 July 2007
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int DIM_NUM, the spatial dimension.
//
//    Input, int EXPON[DIM_NUM], the exponents.
//
//    Input, int POINT_NUM, the number of points in the rule.
//
//    Input, double XYZ[DIM_NUM*POINT_NUM], the quadrature points.
//
//    Input, double W[POINT_NUM], the quadrature weights.
//
//    Output, double SPHERE01_MONOMIAL_QUADRATURE, the quadrature error.
//
{
  double exact;
  double quad;
  double quad_error;
  double *value;
//
//  Get the exact value of the integral.
//
  exact = sphere01_monomial_integral ( expon );
//
//  Evaluate the monomial at the quadrature points.
//
  value = monomial_value ( 3, point_num, expon, xyz );
//
//  Compute the weighted sum.
//
  quad = r8vec_dot ( point_num, w, value );
//
//  Error:
//
  quad_error = fabs ( quad - exact );

  delete [] value;

  return quad_error;
}
//****************************************************************************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 MIT license. 
//
//  Modified:
//
//    08 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    None
//
{
# define TIME_SIZE 40

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

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

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

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

  return;
# undef TIME_SIZE
}