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

using namespace std;

int main ( int argc, char *argv[] );
double arc_cosine ( double c );
char ch_cap ( char ch );
bool ch_eqi ( char ch1, char ch2 );
int ch_to_digit ( char ch );
int file_column_count ( string filename );
int file_row_count ( string filename );
int i4_max ( int i1, int i2 );
int i4_min ( int i1, int i2 );
int i4_modp ( int i, int j );
int i4_wrap ( int ival, int ilo, int ihi );
int i4col_compare ( int m, int n, int a[], int i, int j );
void i4col_sort_a ( int m, int n, int a[] );
void i4col_swap ( int m, int n, int a[], int icol1, int icol2 );
int *i4mat_data_read ( string input_filename, int m, int n );
void i4mat_header_read ( string input_filename, int *m, int *n );
void i4mat_transpose_print ( int m, int n, int a[], string title );
void i4mat_transpose_print_some ( int m, int n, int a[], int ilo, int jlo, 
  int ihi, int jhi, string title );
void mesh_base_zero ( int node_num, int element_order, int element_num, 
  int element_node[] );
double r8_huge ( );
double r8_min ( double x, double y );
double *r8mat_data_read ( string input_filename, int m, int n );
void r8mat_header_read ( string input_filename, int *m, int *n );
void r8mat_transpose_print ( int m, int n, double a[], string title );
void r8mat_transpose_print_some ( int m, int n, double a[], int ilo, int jlo, 
  int ihi, int jhi, string title );
double r8vec_max ( int n, double r8vec[] );
double r8vec_min ( int n, double r8vec[] );
int s_len_trim ( string s );
int s_to_i4 ( string s, int *last, bool *error );
bool s_to_i4vec ( string s, int n, int ivec[] );
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 sort_heap_external ( int n, int *indx, int *i, int *j, int isgn );
void timestamp ( );
double *triangle_angles_2d_new ( double t[2*3] );
double triangulation_delaunay_discrepancy_compute ( int node_num, 
  double node_xy[], int triangle_order, int triangle_num, int triangle_node[], 
  int triangle_neighbor[], double *angle_min, int *angle_min_triangle, 
  double *angle_max, int *angle_max_triangle );
int *triangulation_neighbor_triangles ( int triangle_order, int triangle_num,
  int triangle_node[] );

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

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

//****************************************************************************80
//
//  Purpose:
//
//    MAIN is the main program for TRIANGULATION_DELAUNAY_DISCREPANCY.
//
//  Discussion:
//
//    TRIANGULATION_DELAUNAY_DISCREPANCY measures the amount (possibly zero)
//    by which a triangulation fails the local Delaunay test.
//
//    The local Delaunay considers pairs of neighboring triangles.
//    The two triangles form a quadrilateral, and their common edge is one
//    diagonal of that quadrilateral.  The program considers the effect of
//    replacing that diagonal with the other one.  If the minimum angle
//    of the original configuration is smaller than in the new configuration,
//    then the pair of triangles has failed the local Delaunay test.
//    The amount by which the minimum angle would have increased is the
//    local discrepancy.
//
//    This program searches all pairs of triangles, and records the maximum
//    discrepancy found.  If this discrepancy is essentially zero, then the
//    triangulation is a Delaunay triangulation.  Otherwise, it is not a
//    Delaunay triangulation.  
//
//    The user supplies a node file and a triangle file, containing
//    the coordinates of the nodes, and the indices of the nodes that
//    make up each triangle.  Either 3-node or 6-node triangles may
//    be used.
//
//    The program reads the node and triangle data, computes the triangle
//    neighbor information, and writes it to a file.
//
//  Usage:
//
//    triangulation_delaunay_discrepancy prefix
//
//    where 'prefix' is the common filename prefix:
//
//    * prefix_nodes.txt contains the node coordinates,
//    * prefix_elements.txt contains the element definitions.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    03 October 2009
//
//  Author:
//
//    John Burkardt
//
{
  double angle_max;
  int angle_max_triangle;
  double angle_min;
  int angle_min_triangle;
  double discrepancy;
  int dim_num;
  int iarg;
  int iargc;
  string node_filename;
  int node_num;
  double *node_xy;
  string prefix;
  string element_filename;
  int *triangle_neighbor;
  int *triangle_node;
  int triangle_num;
  int triangle_order;

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

  cout << "\n";
  cout << "TRIANGULATION_DELAUNAY_DISCREPANCY:\n";
  cout << "  C++ version:\n";
  cout << "  Compiled on " << __DATE__ << " at " << __TIME__ << ".\n";
  cout << "\n";
  cout << "  Read a node dataset of NODE_NUM points in 2 dimensions.\n";
  cout << "  Read an associated triangulation dataset of \n";
  cout << "  TRIANGLE_NUM triangles using 3 or 6 nodes.\n";
  cout << "\n";
  cout << "  Determine the Delaunay discrepancy, that is, the amount\n";
  cout << "  by which the minimum angle in the triangulation could be\n";
  cout << "  changed by a single adjustment of a pair of triangles.\n";
  cout << "\n";
  cout << "  If this discrepancy is negative, \n";
  cout << "  then the triangulation is not a Delaunay triangulation.\n";
  cout << "\n";
  cout << "  If this discrepancy is 0 or essentially so, \n";
  cout << "  then the triangulation is a Delaunay triangulation.\n";
//
//  Get the filename prefix.
//
  if ( argc <= 1 ) 
  {
    cout << "\n";
    cout << "TRIANGULATION_DELAUNAY_DISCREPANCY:\n";
    cout << "  Please enter the filename prefix.\n";

    cin >> prefix;
  }
  else 
  {
    prefix = argv[1];
  }
//
//  Create the filenames.
//
  node_filename = prefix + "_nodes.txt";
  element_filename = prefix + "_elements.txt";
//
//  Read the node data.
//
  r8mat_header_read ( node_filename, &dim_num, &node_num );

  cout << "\n";
  cout << "  Read the header of \"" << node_filename << "\".\n";
  cout << "\n";
  cout << "  Spatial dimension DIM_NUM = " << dim_num << "\n";
  cout << "  Number of nodes NODE_NUM  = " << node_num << "\n";

  node_xy = r8mat_data_read ( node_filename, dim_num, node_num );

  cout << "\n";
  cout << "  Read the data in \"" << node_filename << "\".\n";

  r8mat_transpose_print_some ( dim_num, node_num, node_xy, 1, 1, dim_num, 5, 
    "  First 5 nodes:" );
//
//  Read the triangulation data.
//
  i4mat_header_read ( element_filename, &triangle_order, 
    &triangle_num );

  cout << "\n";
  cout << " Read the header of \"" << element_filename << "\".\n";
  cout << "\n";
  cout << "  Triangle order TRIANGLE_ORDER = " << triangle_order << "\n";
  cout << "  Number of triangles TRIANGLE_NUM  = " << triangle_num << "\n";

  triangle_node = i4mat_data_read ( element_filename, 
    triangle_order, triangle_num );

  cout << "\n";
  cout << "  Read the data in \"" << element_filename << "\".\n";

  i4mat_transpose_print_some ( triangle_order, triangle_num, triangle_node, 
    1, 1, triangle_order, 10, "  First 10 triangles:" );
//
//  Detect and correct 1-based node indexing.
//
  mesh_base_zero ( node_num, triangle_order, triangle_num, triangle_node );
//
//  Create the triangle neighbors.
//
  triangle_neighbor = triangulation_neighbor_triangles ( triangle_order, 
    triangle_num, triangle_node );

  i4mat_transpose_print_some ( 3, triangle_num, triangle_neighbor, 
    1, 1, 3, 10, "  First 10 triangle neighbors:" );
//
//  Now we are ready to check.
//
  discrepancy = triangulation_delaunay_discrepancy_compute ( node_num, node_xy, 
    triangle_order, triangle_num, triangle_node, triangle_neighbor, &angle_min, 
    &angle_min_triangle, &angle_max, &angle_max_triangle );

  cout << "\n";
  cout << "  Discrepancy (degrees) =   " << discrepancy << "\n";
  cout << "  Minimum angle (degrees) = " << angle_min << "\n";
  cout << "  occurred in triangle      " << angle_min_triangle << "\n";
  cout << "  Maximum angle (degrees) = " << angle_max << "\n";
  cout << "  occurred in triangle      " << angle_max_triangle << "\n";
//
//  Finish up.
//
  delete [] node_xy;
  delete [] triangle_neighbor;
  delete [] triangle_node;

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

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

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

double arc_cosine ( double c )

//****************************************************************************80
//
//  Purpose:
//
//    ARC_COSINE computes the arc cosine function, with argument truncation.
//
//  Discussion:
//
//    If you call your system ACOS routine with an input argument that is
//    outside the range [-1.0, 1.0 ], you may get an unpleasant surprise.
//    This routine truncates arguments outside the range.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    13 June 2002
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double C, the argument, the cosine of an angle.
//
//    Output, double ARC_COSINE, an angle whose cosine is C.
//
{
  double angle;
  double pi = 3.141592653589793;

  if ( c <= -1.0 )
  {
    angle = pi;
  } 
  else if ( 1.0 <= c )
  {
    angle = 0.0;
  }
  else
  {
    angle = acos ( c );
  }
  return angle;
}
//****************************************************************************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 September 2009
//
//  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.
//
{
  bool value;

  if ( 97 <= ch1 && ch1 <= 122 ) 
  {
    ch1 = ch1 - 32;
  } 
  if ( 97 <= ch2 && ch2 <= 122 ) 
  {
    ch2 = ch2 - 32;
  }     

  value = ( ch1 == ch2 );

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

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 ( ; ; )
  {
    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

int i4_max ( int i1, int i2 )

//****************************************************************************80
//
//  Purpose:
//
//    I4_MAX returns the maximum of two I4's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    13 October 1998
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int I1, I2, are two integers to be compared.
//
//    Output, int I4_MAX, the larger of I1 and I2.
//
{
  int value;

  if ( i2 < i1 )
  {
    value = i1;
  }
  else
  {
    value = i2;
  }
  return value;
}
//****************************************************************************80

int i4_min ( int i1, int i2 )

//****************************************************************************80
//
//  Purpose:
//
//    I4_MIN returns the minimum of two I4's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    13 October 1998
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int I1, I2, two integers to be compared.
//
//    Output, int I4_MIN, the smaller of I1 and I2.
//
{
  int value;

  if ( i1 < i2 )
  {
    value = i1;
  }
  else
  {
    value = i2;
  }
  return value;
}
//****************************************************************************80

int i4_modp ( int i, int j )

//****************************************************************************80
//
//  Purpose:
//
//    I4_MODP returns the nonnegative remainder of I4 division.
//
//  Discussion:
//
//    If 
//      NREM = I4_MODP ( I, J ) 
//      NMULT = ( I - NREM ) / J
//    then
//      I = J * NMULT + NREM
//    where NREM is always nonnegative.
//
//    The MOD function computes a result with the same sign as the
//    quantity being divided.  Thus, suppose you had an angle A,
//    and you wanted to ensure that it was between 0 and 360.
//    Then mod(A,360) would do, if A was positive, but if A
//    was negative, your result would be between -360 and 0.
//
//    On the other hand, I4_MODP(A,360) is between 0 and 360, always.
//
//        I         J     MOD  I4_MODP   I4_MODP Factorization
// 
//      107        50       7       7    107 =  2 *  50 + 7
//      107       -50       7       7    107 = -2 * -50 + 7
//     -107        50      -7      43   -107 = -3 *  50 + 43
//     -107       -50      -7      43   -107 =  3 * -50 + 43
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    26 May 1999
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int I, the number to be divided.
//
//    Input, int J, the number that divides I.
//
//    Output, int I4_MODP, the nonnegative remainder when I is 
//    divided by J.
//
{
  int value;

  if ( j == 0 )
  {
    cout << "\n";
    cout << "I4_MODP - Fatal error!\n";
    cout << "  I4_MODP ( I, J ) called with J = " << j << "\n";
    exit ( 1 );
  }

  value = i % j;

  if ( value < 0 )
  {
    value = value + abs ( j );
  }

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

int i4_wrap ( int ival, int ilo, int ihi )

//****************************************************************************80
//
//  Purpose:
//
//    I4_WRAP forces an I4 to lie between given limits by wrapping.
//
//  Example:
//
//    ILO = 4, IHI = 8
//
//    I   Value
//
//    -2     8
//    -1     4
//     0     5
//     1     6
//     2     7
//     3     8
//     4     4
//     5     5
//     6     6
//     7     7
//     8     8
//     9     4
//    10     5
//    11     6
//    12     7
//    13     8
//    14     4
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    19 August 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int IVAL, an integer value.
//
//    Input, int ILO, IHI, the desired bounds for the integer value.
//
//    Output, int I4_WRAP, a "wrapped" version of IVAL.
//
{
  int jhi;
  int jlo;
  int value;
  int wide;

  jlo = i4_min ( ilo, ihi );
  jhi = i4_max ( ilo, ihi );

  wide = jhi + 1 - jlo;

  if ( wide == 1 )
  {
    value = jlo;
  }
  else
  {
    value = jlo + i4_modp ( ival - jlo, wide );
  }

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

int i4col_compare ( int m, int n, int a[], int i, int j )

//****************************************************************************80
//
//  Purpose:
//
//    I4COL_COMPARE compares columns I and J of an I4COL.
//
//  Example:
//
//    Input:
//
//      M = 3, N = 4, I = 2, J = 4
//
//      A = (
//        1  2  3  4
//        5  6  7  8
//        9 10 11 12 )
//
//    Output:
//
//      I4COL_COMPARE = -1
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    12 June 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns.
//
//    Input, int A[M*N], an array of N columns of vectors of length M.
//
//    Input, int I, J, the columns to be compared.
//    I and J must be between 1 and N.
//
//    Output, int I4COL_COMPARE, the results of the comparison:
//    -1, column I < column J,
//     0, column I = column J,
//    +1, column J < column I.
//
{
  int k;
//
//  Check.
//
  if ( i < 1 )
  {
    cout << "\n";
    cout << "I4COL_COMPARE - Fatal error!\n";
    cout << "  Column index I = " << i << " is less than 1.\n";
    exit ( 1 );
  }

  if ( n < i )
  {
    cout << "\n";
    cout << "I4COL_COMPARE - Fatal error!\n";
    cout << "  N = " << n << " is less than column index I = " << i << ".\n";
    exit ( 1 );
  }

  if ( j < 1 )
  {
    cout << "\n";
    cout << "I4COL_COMPARE - Fatal error!\n";
    cout << "  Column index J = " << j << " is less than 1.\n";
    exit ( 1 );
  }

  if ( n < j )
  {
    cout << "\n";
    cout << "I4COL_COMPARE - Fatal error!\n";
    cout << "  N = " << n << " is less than column index J = " << j << ".\n";
    exit ( 1 );
  }

  if ( i == j )
  {
    return 0;
  }

  k = 1;

  while ( k <= m )
  {
    if ( a[k-1+(i-1)*m] < a[k-1+(j-1)*m] )
    {
      return (-1);
    }
    else if ( a[k-1+(j-1)*m] < a[k-1+(i-1)*m] )
    {
      return 1;
    }
    k = k + 1;
  }

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

void i4col_sort_a ( int m, int n, int a[] )

//****************************************************************************80
//
//  Purpose:
//
//    I4COL_SORT_A ascending sorts the columns of an I4COL.
//
//  Discussion:
//
//    In lexicographic order, the statement "X < Y", applied to two
//    vectors X and Y of length M, means that there is some index I, with
//    1 <= I <= M, with the property that
//
//      X(J) = Y(J) for J < I,
//    and
//      X(I) < Y(I).
//
//    In other words, X is less than Y if, at the first index where they
//    differ, the X value is less than the Y value.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    12 June 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, the number of rows of A.
//
//    Input, int N, the number of columns of A.
//
//    Input/output, int A[M*N].
//    On input, the array of N columns of M vectors;
//    On output, the columns of A have been sorted in ascending
//    lexicographic order.
//
{
  int i;
  int indx;
  int isgn;
  int j;
//
//  Initialize.
//
  i = 0;
  indx = 0;
  isgn = 0;
  j = 0;
//
//  Call the external heap sorter.
//
  for ( ; ; )
  {
    sort_heap_external ( n, &indx, &i, &j, isgn );
//
//  Interchange the I and J objects.
//
    if ( 0 < indx )
    {
      i4col_swap ( m, n, a, i, j );
    }
//
//  Compare the I and J objects.
//
    else if ( indx < 0 )
    {
      isgn = i4col_compare ( m, n, a, i, j );
    }
    else if ( indx == 0 )
    {
      break;
    }

  }

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

void i4col_swap ( int m, int n, int a[], int icol1, int icol2 )

//****************************************************************************80
//
//  Purpose:
//
//    I4COL_SWAP swaps two columns of an I4COL.
//
//  Discussion:
//
//    The two dimensional information is stored as a one dimensional
//    array, by columns.
//
//    The row indices are 1 based, NOT 0 based//  However, a preprocessor
//    variable, called OFFSET, can be reset from 1 to 0 if you wish to
//    use 0-based indices.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    03 April 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns.
//
//    Input/output, int A[M*N], an array of data.
//
//    Input, int ICOL1, ICOL2, the two columns to swap.
//    These indices should be between 1 and N.
//
{
# define OFFSET 1

  int i;
  int t;
//
//  Check.
//
  if ( icol1 - OFFSET < 0 || n-1 < icol1 - OFFSET )
  {
    cout << "\n";
    cout << "I4COL_SWAP - Fatal error!\n";
    cout << "  ICOL1 is out of range.\n";
    exit ( 1 );
  }

  if ( icol2 - OFFSET < 0 || n-1 < icol2 - OFFSET )
  {
    cout << "\n";
    cout << "I4COL_SWAP - Fatal error!\n";
    cout << "  ICOL2 is out of range.\n";
    exit ( 1 );
  }

  if ( icol1 == icol2 )
  {
    return;
  }
  for ( i = 0; i < m; i++ )
  {
    t                     = a[i+(icol1-OFFSET)*m];
    a[i+(icol1-OFFSET)*m] = a[i+(icol2-OFFSET)*m];
    a[i+(icol2-OFFSET)*m] = t;
  }

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

int *i4mat_data_read ( string input_filename, int m, int n )

//****************************************************************************80
//
//  Purpose:
//
//    I4MAT_DATA_READ reads data from an I4MAT file.
//
//  Discussion:
//
//    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, int I4MAT_DATA_READ[M*N], the table data.
//
{
  bool error;
  ifstream input;
  int i;
  int j;
  string line;
  int *table;
  int *x;

  input.open ( input_filename.c_str ( ) );

  if ( !input )
  {
    cerr << "\n";
    cerr << "I4MAT_DATA_READ - Fatal error!\n";
    cerr << "  Could not open the input file: \"" << input_filename << "\"\n";
    return NULL;
  }

  table = new int[m*n];

  x = new int[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_i4vec ( 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 i4mat_header_read ( string input_filename, int *m, int *n )
 
//****************************************************************************80
//
//  Purpose:
//
//    I4MAT_HEADER_READ reads the header from an I4MAT file.
//
//  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 << "I4MAT_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 << "I4MAT_HEADER_READ - Fatal error!\n";
    cerr << "  FILE_ROW_COUNT failed.\n";
    return;
  }
 
  return;
}
//****************************************************************************80

void i4mat_transpose_print ( int m, int n, int a[], string title )

//****************************************************************************80
//
//  Purpose:
//
//    I4MAT_TRANSPOSE_PRINT prints an I4MAT, transposed.
//
//  Discussion:
//
//    An I4MAT is an MxN array of I4's, stored by (I,J) -> [I+J*M].
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    31 January 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, the number of rows in A.
//
//    Input, int N, the number of columns in A.
//
//    Input, int A[M*N], the M by N matrix.
//
//    Input, string TITLE, a title.
//
{
  i4mat_transpose_print_some ( m, n, a, 1, 1, m, n, title );

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

void i4mat_transpose_print_some ( int m, int n, int a[], int ilo, int jlo, 
  int ihi, int jhi, string title )

//****************************************************************************80
//
//  Purpose:
//
//    I4MAT_TRANSPOSE_PRINT_SOME prints some of an I4MAT, transposed.
//
//  Discussion:
//
//    An I4MAT is an MxN array of I4's, stored by (I,J) -> [I+J*M].
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    14 June 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, the number of rows of the matrix.
//    M must be positive.
//
//    Input, int N, the number of columns of the matrix.
//    N must be positive.
//
//    Input, int A[M*N], the matrix.
//
//    Input, int ILO, JLO, IHI, JHI, designate the first row and
//    column, and the last row and column to be printed.
//
//    Input, string TITLE, a title.
//
{
# define INCX 10

  int i;
  int i2hi;
  int i2lo;
  int j;
  int j2hi;
  int j2lo;

  cout << "\n";
  cout << title << "\n";
//
//  Print the columns of the matrix, in strips of INCX.
//
  for ( i2lo = ilo; i2lo <= ihi; i2lo = i2lo + INCX )
  {
    i2hi = i2lo + INCX - 1;
    i2hi = i4_min ( i2hi, m );
    i2hi = i4_min ( i2hi, ihi );

    cout << "\n";
//
//  For each row I in the current range...
//
//  Write the header.
//
    cout << "  Row: ";
    for ( i = i2lo; i <= i2hi; i++ )
    {
      cout << setw(6) << i << "  ";
    }
    cout << "\n";
    cout << "  Col\n";
    cout << "\n";
//
//  Determine the range of the rows in this strip.
//
    j2lo = i4_max ( jlo, 1 );
    j2hi = i4_min ( jhi, n );

    for ( j = j2lo; j <= j2hi; j++ )
    {
//
//  Print out (up to INCX) entries in column J, that lie in the current strip.
//
      cout << setw(5) << j << "  ";
      for ( i = i2lo; i <= i2hi; i++ )
      {
        cout << setw(6) << a[i-1+(j-1)*m] << "  ";
      }
      cout << "\n";
    }
  }

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

void mesh_base_zero ( int node_num, int element_order, int element_num, 
  int element_node[] )

//****************************************************************************80
//
//  Purpose:
//
//    MESH_BASE_ZERO ensures that the element definition is zero-based.
//
//  Discussion:
//
//    The ELEMENT_NODE array contains nodes indices that form elements.
//    The convention for node indexing might start at 0 or at 1.
//    Since a C++ program will naturally assume a 0-based indexing, it is
//    necessary to check a given element definition and, if it is actually
//    1-based, to convert it.
//
//    This function attempts to detect 1-based node indexing and correct it.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 October 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int NODE_NUM, the number of nodes.
//
//    Input, int ELEMENT_ORDER, the order of the elements.
//
//    Input, int ELEMENT_NUM, the number of elements.
//
//    Input/output, int ELEMENT_NODE[ELEMENT_ORDER*ELEMENT_NUM], the element
//    definitions.
//
{
  int element;
  int node;
  int node_max;
  int node_min;
  int order;

  node_min = node_num + 1;
  node_max = -1;
  for ( element = 0; element < element_num; element++ )
  {
    for ( order = 0; order < element_order; order++ )
    {
      node = element_node[order+element*element_order];
      node_min = i4_min ( node_min, node );
      node_max = i4_max ( node_max, node );
    }
  }

  if ( node_min == 1 && node_max == node_num )
  {
    cout << "\n";
    cout << "MESH_BASE_ZERO:\n";
    cout << "  The element indexing appears to be 1-based!\n";
    cout << "  This will be converted to 0-based.\n";
    for ( element = 0; element < element_num; element++ )
    {
      for ( order = 0; order < element_order; order++ )
      {
        element_node[order+element*element_order] =
          element_node[order+element*element_order] - 1;
      }
    }
  }
  else if ( node_min == 0 && node_max == node_num - 1 )
  {
    cout << "\n";
    cout << "MESH_BASE_ZERO:\n";
    cout << "  The element indexing appears to be 0-based!\n";
    cout << "  No conversion is necessary.\n";
  }
  else
  {
    cout << "\n";
    cout << "MESH_BASE_ZERO - Warning!\n";
    cout << "  The element indexing is not of a recognized type.\n";
    cout << "  NODE_MIN = " << node_min << "\n";
    cout << "  NODE_MAX = " << node_max << "\n";
    cout << "  NODE_NUM = " << node_num << "\n";
  }
  return;
}
//****************************************************************************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_min ( double x, double y )

//****************************************************************************80
//
//  Purpose:
//
//    R8_MIN returns the minimum of two R8's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    31 August 2004
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double X, Y, the quantities to compare.
//
//    Output, double R8_MIN, the minimum of X and Y.
//
{
  double value;

  if ( y < x )
  {
    value = y;
  } 
  else
  {
    value = 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:
//
//    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.
//
//  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

void r8mat_transpose_print ( int m, int n, double a[], string title )

//****************************************************************************80
//
//  Purpose:
//
//    R8MAT_TRANSPOSE_PRINT prints an R8MAT, transposed.
//
//  Discussion:
//
//    An R8MAT is a doubly dimensioned array of R8 values,  stored as a vector 
//    in column-major order.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    10 September 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns.
//
//    Input, double A[M*N], an M by N matrix to be printed.
//
//    Input, string TITLE, a title.
//
{
  r8mat_transpose_print_some ( m, n, a, 1, 1, m, n, title );

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

void r8mat_transpose_print_some ( int m, int n, double a[], int ilo, int jlo, 
  int ihi, int jhi, string title )

//****************************************************************************80
//
//  Purpose:
//
//    R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed.
//
//  Discussion:
//
//    An R8MAT is a doubly dimensioned array of R8 values,  stored as a vector 
//    in column-major order.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    10 September 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns.
//
//    Input, double A[M*N], an M by N matrix to be printed.
//
//    Input, int ILO, JLO, the first row and column to print.
//
//    Input, int IHI, JHI, the last row and column to print.
//
//    Input, string TITLE, a title.
//
{
# define INCX 5

  int i;
  int i2;
  int i2hi;
  int i2lo;
  int inc;
  int j;
  int j2hi;
  int j2lo;

  cout << "\n";
  cout << title << "\n";

  for ( i2lo = i4_max ( ilo, 1 ); i2lo <= i4_min ( ihi, m ); i2lo = i2lo + INCX )
  {
    i2hi = i2lo + INCX - 1;
    i2hi = i4_min ( i2hi, m );
    i2hi = i4_min ( i2hi, ihi );

    inc = i2hi + 1 - i2lo;

    cout << "\n";
    cout << "  Row: ";
    for ( i = i2lo; i <= i2hi; i++ )
    {
      cout << setw(7) << i << "       ";
    }
    cout << "\n";
    cout << "  Col\n";
    cout << "\n";

    j2lo = i4_max ( jlo, 1 );
    j2hi = i4_min ( jhi, n );

    for ( j = j2lo; j <= j2hi; j++ )
    {
      cout << setw(5) << j << " ";
      for ( i2 = 1; i2 <= inc; i2++ )
      {
        i = i2lo - 1 + i2;
        cout << setw(14) << a[(i-1)+(j-1)*m];
      }
      cout << "\n";
    }
  }

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

double r8vec_max ( int n, double r8vec[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_MAX returns the value of the maximum element in an R8VEC.
//
//  Discussion:
//
//    An R8VEC is a vector of R8's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 July 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries in the array.
//
//    Input, double R8VEC[N], a pointer to the first entry of the array.
//
//    Output, double R8VEC_MAX, the value of the maximum element.  This
//    is set to 0.0 if N <= 0.
//
{
  int i;
  double value;

  value = - r8_huge ( );

  if ( n <= 0 ) 
  {
    return value;
  }

  for ( i = 0; i < n; i++ ) 
  {
    if ( value < r8vec[i] )
    {
      value = r8vec[i];
    }
  }
  return value;
}
//****************************************************************************80

double r8vec_min ( int n, double r8vec[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_MIN returns the value of the minimum element in an R8VEC.
//
//  Discussion:
//
//    An R8VEC is a vector of R8's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 July 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries in the array.
//
//    Input, double R8VEC[N], the array to be checked.
//
//    Output, double R8VEC_MIN, the value of the minimum element.
//
{
  int i;
  double value;

  value = r8_huge ( );

  if ( n <= 0 ) 
  {
    return value;
  }

  for ( i = 0; i < n; i++ ) 
  {
    if ( r8vec[i] < value )
    {
      value = r8vec[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

bool s_to_i4vec ( string s, int n, int ivec[] )

//****************************************************************************80
//
//  Purpose:
//
//    S_TO_I4VEC reads an I4VEC 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, int IVEC[N], the values read from the string.
//
//    Output, bool S_TO_I4VEC, 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++ )
  {
    ivec[i] = s_to_i4 ( s.substr(begin,length), &lchar, &error );

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

  return error;
}
//****************************************************************************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 sort_heap_external ( int n, int *indx, int *i, int *j, int isgn )

//****************************************************************************80
//
//  Purpose:
//
//    SORT_HEAP_EXTERNAL externally sorts a list of items into ascending order.
//
//  Discussion:
//
//    The actual list is not passed to the routine.  Hence it may
//    consist of integers, reals, numbers, names, etc.  The user,
//    after each return from the routine, will be asked to compare or
//    interchange two items.
//
//    The current version of this code mimics the FORTRAN version,
//    so the values of I and J, in particular, are FORTRAN indices.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    05 February 2004
//
//  Author:
//
//    Original FORTRAN77 version by Albert Nijenhuis, Herbert Wilf.
//    C++ version by John Burkardt
//
//  Reference:
//
//    Albert Nijenhuis, Herbert Wilf,
//    Combinatorial Algorithms,
//    Academic Press, 1978, second edition,
//    ISBN 0-12-519260-6.
//
//  Parameters:
//
//    Input, int N, the length of the input list.
//
//    Input/output, int *INDX.
//    The user must set INDX to 0 before the first call.
//    On return,
//      if INDX is greater than 0, the user must interchange
//      items I and J and recall the routine.
//      If INDX is less than 0, the user is to compare items I
//      and J and return in ISGN a negative value if I is to
//      precede J, and a positive value otherwise.
//      If INDX is 0, the sorting is done.
//
//    Output, int *I, *J.  On return with INDX positive,
//    elements I and J of the user's list should be
//    interchanged.  On return with INDX negative, elements I
//    and J are to be compared by the user.
//
//    Input, int ISGN. On return with INDX negative, the
//    user should compare elements I and J of the list.  If
//    item I is to precede item J, set ISGN negative,
//    otherwise set ISGN positive.
//
{
  static int i_save = 0;
  static int j_save = 0;
  static int k = 0;
  static int k1 = 0;
  static int n1 = 0;
//
//  INDX = 0: This is the first call.
//
  if ( *indx == 0 )
  {

    i_save = 0;
    j_save = 0;
    k = n / 2;
    k1 = k;
    n1 = n;
  }
//
//  INDX < 0: The user is returning the results of a comparison.
//
  else if ( *indx < 0 )
  {
    if ( *indx == -2 ) 
    {
      if ( isgn < 0 ) 
      {
        i_save = i_save + 1;
      }
      j_save = k1;
      k1 = i_save;
      *indx = -1;
      *i = i_save;
      *j = j_save;
      return;
    }

    if ( 0 < isgn ) 
    {
      *indx = 2;
      *i = i_save;
      *j = j_save;
      return;
    }

    if ( k <= 1 ) 
    {
      if ( n1 == 1 ) 
      {
        i_save = 0;
        j_save = 0;
        *indx = 0;
      }
      else 
      {
        i_save = n1;
        j_save = 1;
        n1 = n1 - 1;
        *indx = 1;
      }
      *i = i_save;
      *j = j_save;
      return;
    }
    k = k - 1;
    k1 = k;
  }
//
//  0 < INDX: the user was asked to make an interchange.
//
  else if ( *indx == 1 ) 
  {
    k1 = k;
  }

  for ( ; ; )
  {

    i_save = 2 * k1;

    if ( i_save == n1 ) 
    {
      j_save = k1;
      k1 = i_save;
      *indx = -1;
      *i = i_save;
      *j = j_save;
      return;
    }
    else if ( i_save <= n1 ) 
    {
      j_save = i_save + 1;
      *indx = -2;
      *i = i_save;
      *j = j_save;
      return;
    }

    if ( k <= 1 ) 
    {
      break;
    }

    k = k - 1;
    k1 = k;
  }

  if ( n1 == 1 ) 
  {
    i_save = 0;
    j_save = 0;
    *indx = 0;
    *i = i_save;
    *j = j_save;
  }
  else 
  {
    i_save = n1;
    j_save = 1;
    n1 = n1 - 1;
    *indx = 1;
    *i = i_save;
    *j = j_save;
  }

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

void timestamp ( )

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

  static char time_buffer[TIME_SIZE];
  const struct std::tm *tm_ptr;
  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
}
//****************************************************************************80

double *triangle_angles_2d_new ( double t[2*3] )

//****************************************************************************80
//
//  Purpose:
//
//    TRIANGLE_ANGLES_2D_NEW computes the angles of a triangle in 2D.
//
//  Discussion:
//
//    The law of cosines is used:
//
//      C * C = A * A + B * B - 2 * A * B * COS ( GAMMA )
//
//    where GAMMA is the angle opposite side C.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    11 September 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double T[2*3], the triangle vertices.
//
//    Output, double TRIANGLE_ANGLES_2D_NEW[3], the angles opposite
//    sides P1-P2, P2-P3 and P3-P1, in radians.
//
{
  double a;
  double *angle;
  double b;
  double c;
  double pi = 3.141592653589793;

  angle = new double[3];

  a = sqrt ( pow ( t[0+1*2] - t[0+0*2], 2 ) 
           + pow ( t[1+1*2] - t[1+0*2], 2 ) );

  b = sqrt ( pow ( t[0+2*2] - t[0+1*2], 2 ) 
           + pow ( t[1+2*2] - t[1+1*2], 2 ) );

  c = sqrt ( pow ( t[0+0*2] - t[0+2*2], 2 ) 
           + pow ( t[1+0*2] - t[1+2*2], 2 ) );
//
//  Take care of a ridiculous special case.
//
  if ( a == 0.0 && b == 0.0 && c == 0.0 )
  {
    angle[0] = 2.0 * pi / 3.0;
    angle[1] = 2.0 * pi / 3.0;
    angle[2] = 2.0 * pi / 3.0;
    return angle;
  }

  if ( c == 0.0 || a == 0.0 )
  {
    angle[0] = pi;
  }
  else
  {
    angle[0] = arc_cosine ( ( c * c + a * a - b * b ) / ( 2.0 * c * a ) );
  }

  if ( a == 0.0 || b == 0.0 )
  {
    angle[1] = pi;
  }
  else
  {
    angle[1] = arc_cosine ( ( a * a + b * b - c * c ) / ( 2.0 * a * b ) );
  }

  if ( b == 0.0 || c == 0.0 )
  {
    angle[2] = pi;
  }
  else
  {
    angle[2] = arc_cosine ( ( b * b + c * c - a * a ) / ( 2.0 * b * c ) );
  }

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

double triangulation_delaunay_discrepancy_compute ( int node_num, 
  double node_xy[], int triangle_order, int triangle_num, int triangle_node[], 
  int triangle_neighbor[], double *angle_min, int *angle_min_triangle, 
  double *angle_max, int *angle_max_triangle )

//****************************************************************************80
//
//  Purpose:
//
//    TRIANGULATION_DELAUNAY_DISCREPANCY_COMPUTE reports if a triangulation is Delaunay.
//
//  Discussion:
//
//    A (maximal) triangulation is Delaunay if and only if it is locally 
//    Delaunay.
//
//    A triangulation is Delaunay if the minimum angle over all triangles
//    in the triangulation is maximized.  That is, there is no other
//    triangulation of the points which has a larger minimum angle.
//
//    A triangulation is locally Delaunay if, for every pair of triangles
//    that share an edge, the minimum angle in the two triangles is larger
//    than the minimum angle in the two triangles formed by removing the
//    common edge and joining the two opposing vertices.
//
//    This function examines the question of whether a given triangulation
//    is locally Delaunay.  It does this by looking at every pair of
//    neighboring triangles and comparing the minimum angle attained
//    for the current triangle pair and the alternative triangle pair.
//
//    Let A(i,j) be the minimum angle formed by triangles T(i) and T(j),
//    which are two triangles in the triangulation which share a common edge.
//    Let B(I,J) be the minimum angle formed by triangles S(i) and S(j),
//    where S(i) and S(j) are formed by removing the common edge of T(i)
//    and T(j), and joining the opposing vertices.
//
//    Then the triangulation is Delaunay if B(i,j) <= A(i,j) for every
//    pair of neighbors T(i) and T(j).  
//
//    If A(i,j) < B(i,j) for at least one pair of neighbors, the triangulation
//    is not a Delaunay triangulation.
//
//    This program returns VALUE = min ( A(i,j) - B(i,j) ) over all
//    triangle neighbors.  VALUE is scaled to be in degrees, for
//    comprehensibility.  If VALUE is negative, then at least one pair
//    of triangles violates the Delaunay condition, and so the entire
//    triangulation is not a Delaunay triangulation.  If VALUE is nonnegative,
//    then the triangulation is a Delaunay triangulation.
//
//    It is useful to return VALUE, rather than a simple True/False value,
//    because there can be cases where the Delaunay condition is only
//    "slightly" violated.  A simple example is a triangulation formed
//    by starting with a mesh of squares and dividing each square into
//    two triangles by choosing one of the diagonals of the square.  
//    The Delaunay discrepancy for this mesh, if computed exactly, is 0,
//    but roundoff could easily result in discrepancies that were very
//    slightly negative.
//
//    If VALUE is positive, and not very small in magnitude, then every 
//    pair of triangles in the triangulation satisfies the local Delaunay 
//    condition, and so the triangulation is a Delaunay triangulation.
//
//    If VALUE is negative, and not very small in magnitude, then at least 
//    one pair of triangles violates the Delaunay condition, and to a 
//    significant degree.  The triangulation is not a Delaunay triangulation.
//
//    If the magnitude of VALUE is very close to zero, then the triangulation
//    is numerically ambiguous.  At least one pair of triangles violates
//    or almost violates the condition, but no triangle violates the
//    condition to a great extent.  The user must judge whether the
//    violation is significant or not.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    10 September 2009
//
//  Author:
//
//    John Burkardt.
//
//  Parameters:
//
//    Input, int NODE_NUM, the number of nodes.
//
//    Input, double NODE_XY[2*NODE_NUM], the coordinates of the nodes.
//
//    Input, int TRIANGLE_ORDER, the order of the triangles.
//
//    Input, int TRIANGLE_NUM, the number of triangles in 
//    the triangulation.
//
//    Input, int TRIANGLE_NODE[TRIANGLE_ORDER*TRIANGLE_NUM], 
//    the nodes that make up each triangle.
//
//    Input, int TRIANGLE_NEIGHBOR[3*TRIANGLE_NUM], the
//    triangle neighbor list.
//
//    Output, double *ANGLE_MIN, the minimum angle that occurred in 
//    the triangulation.
//
//    Output, int *ANGLE_MIN_TRIANGLE, the triangle in which
//    the minimum angle occurred.
//
//    Output, double *ANGLE_MAX, the maximum angle that occurred in 
//    the triangulation.
//
//    Output, int *ANGLE_MAX_TRIANGLE, the triangle in which
//    the maximum angle occurred.
//
//    Output, double TRIANGULATION_DELAUNAY_DISCREPANCY, 
//    the minimum value of ( A(i,j) - B(i,j) ).
//    POSITIVE indicates the triangulation is Delaunay.
//    VERY NEAR ZERO is a numerically ambiguous case.
//    NEGATIVE indicates the triangulation is not Delaunay.
//
{
  double angle_min1;
  double angle_min2;
  double *angles1;
  double *angles2;
  int i;
  int i1;
  int i2;
  int i3;
  int i4;
  int n1;
  int n2;
  int n3;
  int n4;
  int neighbor;
  double pi = 3.141592653589793;
  double t[2*3];
  int triangle_index;
  int triangle1;
  int triangle2;
  double value;

  *angle_max = 0.0;
  *angle_max_triangle = - 1;
  *angle_min = pi;
  *angle_min_triangle = -1;
  value = 0.0;
//
//  Consider triangle TRIANGLE1
//
  for ( triangle1 = 0; triangle1 < triangle_num; triangle1++ )
  {
//
//  Consider the side opposite vertex NEIGHBOR.
//
    for ( neighbor = 0; neighbor < 3; neighbor++ )
    {
      triangle2 = triangle_neighbor[neighbor+triangle1*3];
//
//  There might be no neighbor on side NEIGHBOR.
//
      if ( triangle2 < 0 )
      {
        continue;
      }
//
//  We only need to check a pair of triangles once.
//
      if ( triangle2 < triangle1 )
      {
        continue;
      }
//
//  List the vertices of the quadrilateral in such a way
//  that the nodes of triangle 1 come first.
//
//  We rely on a property of the TRIANGLE_NEIGHBOR array, namely, that
//  neighbor #1 is on the side opposite to vertex #1, and so on.
//
      i1 = i4_wrap ( neighbor + 2, 0, 2 );
      i2 = i4_wrap ( neighbor,     0, 2 );
      i3 = i4_wrap ( neighbor + 1, 0, 2 );

      n1 = triangle_node[i1+triangle1*triangle_order];
      n2 = triangle_node[i2+triangle1*triangle_order];
      n3 = triangle_node[i3+triangle1*triangle_order];
//
//  The "odd" or "opposing" node of the neighboring triangle
//  is the one which follows common node I3.
//
      n4 = -1;
      for ( i = 0; i < 3; i++ )
      {
        if ( triangle_node[i+triangle2*triangle_order] == n3 )
        {
          i4 = i + 1;
          i4 = i4_wrap ( i4, 0, 2 );
          n4 = triangle_node[i4+triangle2*triangle_order];
          break;
        }
      }

      if ( n4 == -1 ) 
      {
        cout << "\n";
        cout << "TRIANGULATION_DELAUNAY_DISCREPANCY_COMPUTE - Fatal error/!\n";
        cout << "  Could not identify the fourth node.\n";
        cout << "\n";
        cout << "  Triangle1 = " << triangle1 << "\n";
        cout << "  Nodes =     ";
        for ( i = 0; i < 3; i++ )
        {
          cout << "  " << triangle_node[i+triangle1*triangle_order];
        }
        cout << "\n";
        cout << "  Neighbors =     ";
        for ( i = 0; i < 3; i++ )
        {
          cout << "  " << triangle_neighbor[i+triangle1*3];
        }
        cout << "\n";
        cout << "\n";
        cout << "  Neighbor index = " << neighbor << "\n";
        cout << "\n";
        cout << "  Triangle2 = " << triangle2 << "\n";
        cout << "  Nodes =     ";
        for ( i = 0; i < 3; i++ )
        {
          cout << "  " << triangle_node[i+triangle2*triangle_order];
        }
        cout << "\n";
        cout << "  Neighbors =     ";
        for ( i = 0; i < 3; i++ )
        {
          cout << "  " << triangle_neighbor[i+triangle2*3];
        }
        cout << "\n";
        exit ( 1 );
      }
//
//  Compute the minimum angle for (I1,I2,I3) and (I1,I3,I4).
//
      t[0+0*2] = node_xy[0+n1*2];
      t[1+0*2] = node_xy[1+n1*2];
      t[0+1*2] = node_xy[0+n2*2];
      t[1+1*2] = node_xy[1+n2*2];
      t[0+2*2] = node_xy[0+n3*2];
      t[1+2*2] = node_xy[1+n3*2];
      angles1 = triangle_angles_2d_new ( t );

      t[0+0*2] = node_xy[0+n1*2];
      t[1+0*2] = node_xy[1+n1*2];
      t[0+1*2] = node_xy[0+n3*2];
      t[1+1*2] = node_xy[1+n3*2];
      t[0+2*2] = node_xy[0+n4*2];
      t[1+2*2] = node_xy[1+n4*2];
      angles2 = triangle_angles_2d_new ( t );

      angle_min1 = 
        r8_min ( r8vec_min ( 3, angles1 ), r8vec_min ( 3, angles2 ) );

      if ( *angle_max < r8vec_max ( 3, angles1 ) )
      {
        *angle_max = r8vec_max ( 3, angles1 );
        *angle_max_triangle = triangle1;
      }

      if ( *angle_max < r8vec_max ( 3, angles2 ) )
      {
        *angle_max = r8vec_max ( 3, angles2 );
        *angle_max_triangle = triangle2;
      }

      if ( r8vec_min ( 3, angles1 ) < *angle_min )
      {
        *angle_min = r8vec_min ( 3, angles1 );
        *angle_min_triangle = triangle1;
      }

      if ( r8vec_min ( 3, angles2 ) < *angle_min )
      {
        *angle_min = r8vec_min ( 3, angles2 );
        *angle_min_triangle = triangle2;
      }

     delete [] angles1;
     delete [] angles2;
//
//  Compute the minimum angle for (I1,I2,I4) and (I2,I3,I4).
//
      t[0+0*2] = node_xy[0+n1*2];
      t[1+0*2] = node_xy[1+n1*2];
      t[0+1*2] = node_xy[0+n2*2];
      t[1+1*2] = node_xy[1+n2*2];
      t[0+2*2] = node_xy[0+n4*2];
      t[1+2*2] = node_xy[1+n4*2];
      angles1 = triangle_angles_2d_new ( t );

      t[0+0*2] = node_xy[0+n3*2];
      t[1+0*2] = node_xy[1+n3*2];
      t[0+1*2] = node_xy[0+n3*2];
      t[1+1*2] = node_xy[1+n3*2];
      t[0+2*2] = node_xy[0+n4*2];
      t[1+2*2] = node_xy[1+n4*2];
      angles2 = triangle_angles_2d_new ( t );

      angle_min2 = 
        r8_min ( r8vec_min ( 3, angles1 ), r8vec_min ( 3, angles2 ) );

     delete [] angles1;
     delete [] angles2;
//
//  Compare this value to the current minimum.
//
      value = r8_min ( value, angle_min1 - angle_min2 );
    }
  }
//
//  Scale the results to degrees.
//
  value = value * 180.0 / pi;
  *angle_max = *angle_max * 180.0 / pi;
  *angle_min = *angle_min * 180.0 / pi;

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

int *triangulation_neighbor_triangles ( int triangle_order, int triangle_num,
  int triangle_node[] )

//****************************************************************************80
//
//  Purpose:
//
//    TRIANGULATION_ORDER3_NEIGHBOR_TRIANGLES determines triangle neighbors.
//
//  Discussion:
//
//    A triangulation of a set of nodes can be completely described by
//    the coordinates of the nodes, and the list of nodes that make up
//    each triangle.  However, in some cases, it is necessary to know
//    triangle adjacency information, that is, which triangle, if any,
//    is adjacent to a given triangle on a particular side.
//
//    This routine creates a data structure recording this information.
//
//    The primary amount of work occurs in sorting a list of 3 * TRIANGLE_NUM
//    data items.
//
//    This routine was modified to work with columns rather than rows.
//
//  Example:
//
//    The input information from TRIANGLE_NODE:
//
//    Triangle   Nodes
//    --------   ---------------
//     1         3      4      1
//     2         3      1      2
//     3         3      2      8
//     4         2      1      5
//     5         8      2     13
//     6         8     13      9
//     7         3      8      9
//     8        13      2      5
//     9         9     13      7
//    10         7     13      5
//    11         6      7      5
//    12         9      7      6
//    13        10      9      6
//    14         6      5     12
//    15        11      6     12
//    16        10      6     11
//
//    The output information in TRIANGLE_NEIGHBOR:
//
//    Triangle  Neighboring Triangles
//    --------  ---------------------
//
//     1        -1     -1      2
//     2         1      4      3
//     3         2      5      7
//     4         2     -1      8
//     5         3      8      6
//     6         5      9      7
//     7         3      6     -1
//     8         5      4     10
//     9         6     10     12
//    10         9      8     11
//    11        12     10     14
//    12         9     11     13
//    13        -1     12     16
//    14        11     -1     15
//    15        16     14     -1
//    16        13     15     -1
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    07 September 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int TRIANGLE_ORDER, the order of the triangles.
//
//    Input, int TRIANGLE_NUM, the number of triangles.
//
//    Input, int TRIANGLE_NODE[TRIANGLE_ORDER*TRIANGLE_NUM], the nodes that 
//    make up each triangle.
//
//    Output, int TRIANGLE_ORDER3_NEIGHBOR_TRIANGLES[3*TRIANGLE_NUM], 
//    the three triangles 
//    that are direct neighbors of a given triangle.  TRIANGLE_NEIGHBOR(1,I) 
//    is the index of the triangle which touches side 1, defined by nodes 2 
//    and 3, and so on.  TRIANGLE_NEIGHBOR(1,I) is negative if there is no 
//    neighbor on that side.  In this case, that side of the triangle lies 
//    on the boundary of the triangulation.
//
{
  int *col;
  int i;
  int icol;
  int j;
  int k;
  int side1;
  int side2;
  int tri;
  int tri1;
  int tri2;
  int *triangle_neighbor;

  triangle_neighbor = new int[3*triangle_num];
  col = new int[4*(3*triangle_num)];
//
//  Step 1.
//  From the list of nodes for triangle T, of the form: (I,J,K)
//  construct the three neighbor relations:
//
//    (I,J,3,T) or (J,I,3,T),
//    (J,K,1,T) or (K,J,1,T),
//    (K,I,2,T) or (I,K,2,T)
//
//  where we choose (I,J,3,T) if I < J, or else (J,I,3,T)
//
  for ( tri = 0; tri < triangle_num; tri++ )
  {
    i = triangle_node[0+tri*triangle_order];
    j = triangle_node[1+tri*triangle_order];
    k = triangle_node[2+tri*triangle_order];

    if ( i < j )
    {
      col[0+(3*tri+0)*4] = i;
      col[1+(3*tri+0)*4] = j;
      col[2+(3*tri+0)*4] = 3;
      col[3+(3*tri+0)*4] = tri + 1;
    }
    else
    {
      col[0+(3*tri+0)*4] = j;
      col[1+(3*tri+0)*4] = i;
      col[2+(3*tri+0)*4] = 3;
      col[3+(3*tri+0)*4] = tri + 1;
    }

    if ( j < k )
    {
      col[0+(3*tri+1)*4] = j;
      col[1+(3*tri+1)*4] = k;
      col[2+(3*tri+1)*4] = 1;
      col[3+(3*tri+1)*4] = tri + 1;
    }
    else
    {
      col[0+(3*tri+1)*4] = k;
      col[1+(3*tri+1)*4] = j;
      col[2+(3*tri+1)*4] = 1;
      col[3+(3*tri+1)*4] = tri + 1;
    }

    if ( k < i )
    {
      col[0+(3*tri+2)*4] = k;
      col[1+(3*tri+2)*4] = i;
      col[2+(3*tri+2)*4] = 2;
      col[3+(3*tri+2)*4] = tri + 1;
    }
    else
    {
      col[0+(3*tri+2)*4] = i;
      col[1+(3*tri+2)*4] = k;
      col[2+(3*tri+2)*4] = 2;
      col[3+(3*tri+2)*4] = tri + 1;
    }
  }
//
//  Step 2. Perform an ascending dictionary sort on the neighbor relations.
//  We only intend to sort on rows 1 and 2; the routine we call here
//  sorts on rows 1 through 4 but that won't hurt us.
//
//  What we need is to find cases where two triangles share an edge.
//  Say they share an edge defined by the nodes I and J.  Then there are
//  two columns of COL that start out ( I, J, ?, ? ).  By sorting COL,
//  we make sure that these two columns occur consecutively.  That will
//  make it easy to notice that the triangles are neighbors.
//
  i4col_sort_a ( 4, 3*triangle_num, col );
//
//  Step 3. Neighboring triangles show up as consecutive columns with
//  identical first two entries.  Whenever you spot this happening,
//  make the appropriate entries in TRIANGLE_NEIGHBOR.
//
  for ( j = 0; j < triangle_num; j++ )
  {
    for ( i = 0; i < 3; i++ )
    {
      triangle_neighbor[i+j*3] = -1;
    }
  }

  icol = 1;

  for ( ; ; )
  {
    if ( 3 * triangle_num <= icol )
    {
      break;
    }

    if ( col[0+(icol-1)*4] != col[0+icol*4] || 
         col[1+(icol-1)*4] != col[1+icol*4] )
    {
      icol = icol + 1;
      continue;
    }

    side1 = col[2+(icol-1)*4];
    tri1 =  col[3+(icol-1)*4];
    side2 = col[2+ icol   *4];
    tri2 =  col[3+ icol   *4];

    triangle_neighbor[side1-1+(tri1-1)*3] = tri2;
    triangle_neighbor[side2-1+(tri2-1)*3] = tri1;

    icol = icol + 2;
  }
 
  delete [] col;

  return triangle_neighbor;
}