function triangulation_l2q ( prefix )

%*****************************************************************************80
%
%% MAIN is the main program for TRIANGULATION_L2Q.
%
%  Discussion:
%
%    TRIANGULATION_L2Q makes a quadratic triangulation from a linear one.
%
%    Thanks to Zhu Wang for pointing out a problem caused by a change
%    in the ordering of elements in the triangle neighbor array, 25 August 2010.
%
%  Usage:
%
%    triangulation_l2q ( 'prefix' )
%
%    where 'prefix' is the common filename prefix:
%
%    * 'prefix'_nodes.txt contains the node coordinates,
%    * 'prefix'_elements.txt contains the element definitions.
%    * 'prefix'_l2q_nodes.txt will contain the quadratic node coordinates,
%    * 'prefix'_l2q_elements.txt will contain the quadratic element definitions.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    25 August 2010
%
%  Author:
%
%    John Burkardt
%
  timestamp ( );

  fprintf ( 1, '\n' );
  fprintf ( 1, 'TRIANGULATION_L2Q\n' );
  fprintf ( 1, '  MATLAB version\n' );
  fprintf ( 1, '  Read a "linear" triangulation and\n' );
  fprintf ( 1, '  write out a "quadratic" triangulation.\n' );
  fprintf ( 1, '\n' );
  fprintf ( 1, '  Read a dataset of NODE_NUM1 points in 2 dimensions.\n' );
  fprintf ( 1, '  Read an associated triangulation dataset of TRIANGLE_NUM \n' );
  fprintf ( 1, '  triangles which uses 3 nodes per triangle.\n' );
  fprintf ( 1, '\n' );
  fprintf ( 1, '  Create new nodes which are triangle midpoints,\n' );
  fprintf ( 1, '  generate new node and triangulation data for\n' );
  fprintf ( 1, '  quadratic 6-node triangles, and write them out.\n' );
%
%  The command line argument is the common filename prefix.
%
  if ( nargin < 1 )

    fprintf ( 1, '\n' );
    fprintf ( 1, 'TRIANGULATION_L2Q:\n' );

    prefix = input ( ...
      'Please enter the filename prefix:' );

  end
%
%  Create the filenames.
%
  node_filename = strcat ( prefix, '_nodes.txt' );
  element_filename = strcat ( prefix, '_elements.txt' );
  node_l2q_filename = strcat ( prefix, '_l2q_nodes.txt' );
  element_l2q_filename = strcat ( prefix, '_l2q_elements.txt' );
%
%  Read the data.
%
  [ dim_num, node_num1 ] = r8mat_header_read (  node_filename );

  fprintf ( 1, '\n' );
  fprintf ( 1, '  Read the header of "%s".\n', node_filename );
  fprintf ( 1, '\n' );
  fprintf ( 1, '  Spatial dimension DIM_NUM = %d\n', dim_num );
  fprintf ( 1, '  Number of points NODE_NUM1  = %d\n', node_num1 );

  node_xy1(1:dim_num,1:node_num1) = r8mat_data_read ( node_filename, ...
    dim_num, node_num1 );

  fprintf ( 1, '\n' );
  fprintf ( 1, '  Read the data in "%s".\n', node_filename );

  r8mat_transpose_print_some ( dim_num, node_num1, node_xy1, 1, 1, 5, 5, ...
    '  5 by 5 portion of data read from file:' );
%
%  Read the triangulation data.
%
  [ triangle_order1, triangle_num ] = i4mat_header_read ( element_filename );

  if ( triangle_order1 ~= 3 )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'TRIANGULATION_L2Q - Fatal error!\n' );
    fprintf ( 1, '  Data is not for a 3-node triangulation.\n' );
    error ( 'TRIANGULATION_L2Q - Fatal error!' );
  end

  fprintf ( 1, '\n' );
  fprintf ( 1, '  Read the header of ""%s".\n', element_filename );
  fprintf ( 1, '\n' );
  fprintf ( 1, '  Triangle order = %d\n', triangle_order1 );
  fprintf ( 1, '  Number of triangles TRIANGLE_NUM  = %d\n', triangle_num );

  triangle_node1(1:triangle_order1,1:triangle_num) = i4mat_data_read ( ...
    element_filename, triangle_order1, triangle_num );

  fprintf ( 1, '\n' );
  fprintf ( 1, '  Read the data in ""%s".\n', element_filename );

  i4mat_transpose_print_some ( triangle_order1, triangle_num, triangle_node1, ...
    1, 1, triangle_order1, 10, '  3 by 10 portion TRIANGLE_NODE1:' );
%
%  Detect and correct 0-based indexing.
%
  triangle_node1 = mesh_base_one ( node_num1, triangle_order1, triangle_num, ...
    triangle_node1 );
%
%  Determine the number of midside nodes that will be added.
%
  boundary_num = triangulation_order3_boundary_edge_count ( triangle_num, ...
    triangle_node1 );

  interior_num = ( 3 * triangle_num - boundary_num ) / 2;
  edge_num = interior_num + boundary_num;
  fprintf ( 1, '\n' );
  fprintf ( 1, '  Number of midside nodes to add = %d\n', edge_num );
%
%  Build the triangle neighbor array.
%
  triangle_neighbor = triangulation_neighbor_triangles ( triangle_order1, ...
    triangle_num, triangle_node1 );

  i4mat_transpose_print ( 3, triangle_num, triangle_neighbor, ...
    '  Triangle_neighbor' );
%
%  Create the midside nodes.
%
  triangle_order2 = 6;
  triangle_node2(1:3,1:triangle_num) = triangle_node1(1:3,1:triangle_num);
  triangle_node2(4:6,1:triangle_num) = -1;
  node_xy2(1:2,1:node_num1) = node_xy1(1:2,1:node_num1);

  node_num2 = node_num1;

  fprintf ( 1, '\n' );
  fprintf ( 1, '  Generate midside nodes\n' );
  fprintf ( 1, '\n' );

  for triangle = 1 : triangle_num

    for i = 1 : 3
%
%  CORRECTION #1 because element neighbor definition changed.
%
      iii = i4_wrap ( i + 2, 1, 3 );
      triangle2 = triangle_neighbor(iii,triangle);

      if ( 0 < triangle2 && triangle2 < triangle )
        continue
      end

      ip1 = i4_wrap ( i + 1, 1, 3 );

      k1 = triangle_node2(i,triangle);
      k2 = triangle_node2(ip1,triangle);

      node_num2 = node_num2 + 1;
      node_xy2(1:dim_num,node_num2) = 0.5 ...
        * ( node_xy1(1:dim_num,k1) + node_xy1(1:dim_num,k2) );

      fprintf ( 1, '%8d  %14f  %14f\n', node_num2, node_xy2(1:dim_num,node_num2) );

      triangle_node2(3+i,triangle) = node_num2;

      if ( 0 < triangle2 )
        for ii = 1 : 3
%
%  CORRECTION #2 because element neighbor definition changed.
%
          iii = i4_wrap ( ii + 2, 1, 3 );
          if ( triangle_neighbor(iii,triangle2) == triangle )
            triangle_node2(ii+3,triangle2) = node_num2;
          end
        end
      end

    end
  end

  i4mat_transpose_print ( triangle_order2, triangle_num, triangle_node2, ...
    '  TRIANGLE_NODE2' );
%
%  Write out the node and triangle data for the quadratic mesh.
%
  r8mat_transpose_print ( dim_num, node_num2, node_xy2, '  NODE_XY2:' );

  r8mat_write ( node_l2q_filename, dim_num, node_num2, node_xy2 );

  i4mat_write ( element_l2q_filename, triangle_order2, triangle_num, ...
    triangle_node2 );
%
%  Terminate.
%
  fprintf ( 1, '\n' );
  fprintf ( 1, 'TRIANGULATION_L2Q\n' );
  fprintf ( 1, '  Normal end of execution.\n' );

  fprintf ( 1, '\n' );
  timestamp ( );

  return
end
function column_num = file_column_count ( input_file_name )

%*****************************************************************************80
%
%% 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:
%
%    21 February 2004
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, string INPUT_FILE_NAME, the name of the file.
%
%    Output, integer COLUMN_NUM, the number of columns in the file.
%
  FALSE = 0;
  TRUE = 1;
%
%  Open the file.
%
  input_unit = fopen ( input_file_name );

  if ( input_unit < 0 ) 
    fprintf ( 1, '\n' );
    fprintf ( 1, 'FILE_COLUMN_COUNT - Error!\n' );
    fprintf ( 1, '  Could not open the file "%s".\n', input_file_name );
    error ( 'FILE_COLUMN_COUNT - Error!' );
  end
%
%  Read one line, but skip blank lines and comment lines.
%  Use FGETL so we drop the newline character!
%
  got_one = FALSE;

  while ( 1 )

    line = fgetl ( input_unit );

    if ( line == -1 )
      break;
    end

    if ( s_len_trim ( line ) == 0 )

    elseif ( line(1) == '#' )

    else
      got_one = TRUE;
      break;
    end

  end

  fclose ( input_unit );

  if ( got_one == FALSE ) 
    fprintf ( 1, '\n' );
    fprintf ( 1, 'FILE_COLUMN_COUNT - Warning!\n' );
    fprintf ( 1, '  The file does not seem to contain any data.\n' );
    column_num = -1;
    return;
  end

  column_num = s_word_count ( line );

  return
end
function row_num = file_row_count ( input_file_name )

%*****************************************************************************80
%
%% FILE_ROW_COUNT counts the number of row records in a file.
%
%  Discussion:
%
%    Each input line is a "RECORD".
%
%    The records are divided into three groups:
%    
%    * BLANK LINES (nothing but blanks)
%    * COMMENT LINES (begin with a '#')
%    * DATA RECORDS (anything else)
%
%    The value returned by the function is the number of data records.
%
%    By the way, if the MATLAB routine FGETS is used, instead of
%    FGETL, then the variable LINE will include line termination 
%    characters, which means that a blank line would not actually
%    have zero characters.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license. 
%
%  Modified:
%
%    31 December 2006
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, string INPUT_FILE_NAME, the name of the input file.
%
%    Output, integer ROW_NUM, the number of rows found. 
%
  input_unit = fopen ( input_file_name );

  if ( input_unit < 0 ) 
    fprintf ( 1, '\n' );
    fprintf ( 1, 'FILE_ROW_COUNT - Error!\n' );
    fprintf ( 1, '  Could not open the file "%s".\n', input_file_name );
    error ( 'FILE_ROW_COUNT - Error!' );
  end

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

  while ( 1 )

    line = fgetl ( input_unit );

    if ( line == -1 )
      break;
    end

    record_num = record_num + 1;
    record_length = s_len_trim ( line );
    
    if ( record_length <= 0 )
      blank_num = blank_num + 1;
    elseif ( line(1) == '#' )
      comment_num = comment_num + 1;
    else
      row_num = row_num + 1;
    end

  end

  fclose ( input_unit );

  return
end
function value = i4_modp ( i, j )

%*****************************************************************************80
%
%% 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.
%
%  Example:
%
%        I     J     MOD  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:
%
%    02 March 1999
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, integer I, the number to be divided.
%
%    Input, integer J, the number that divides I.
%
%    Output, integer VALUE, the nonnegative remainder when I is
%    divided by J.
%
  if ( j == 0 )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'I4_MODP - Fatal error!\n' );
    fprintf ( 1, '  Illegal divisor J = %d\n', j );
    error ( 'I4_MODP - Fatal error!' );
  end

  value = mod ( i, j );

  if ( value < 0 )
    value = value + abs ( j );
  end

  return
end
function value = i4_wrap ( ival, ilo, ihi )

%*****************************************************************************80
%
%% I4_WRAP forces an integer 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:
%
%    02 October 2006
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, integer IVAL, an integer value.
%
%    Input, integer ILO, IHI, the desired bounds for the integer value.
%
%    Output, integer I4_WRAP, a "wrapped" version of IVAL.
%
  jlo = min ( ilo, ihi );
  jhi = max ( ilo, ihi );

  wide = jhi - jlo + 1;

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

  return
end
function isgn = i4col_compare ( m, n, a, i, j )

%*****************************************************************************80
%
%% I4COL_COMPARE compares columns I and J of a integer array.
%
%  Example:
%
%    Input:
%
%      M = 3, N = 4, I = 2, J = 4
%
%      A = (
%        1  2  3  4
%        5  6  7  8
%        9 10 11 12 )
%
%    Output:
%
%      ISGN = -1
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    12 June 2005
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, integer M, N, the number of rows and columns.
%
%    Input, integer A(M,N), an array of N columns of vectors of length M.
%
%    Input, integer I, J, the columns to be compared.
%    I and J must be between 1 and N.
%
%    Output, integer ISGN, the results of the comparison:
%    -1, column I < column J,
%     0, column I = column J,
%    +1, column J < column I.
%

%
%  Check.
%
  if ( i < 1)
    fprintf ( 1, '\n' );
    fprintf ( 1, 'I4COL_COMPARE - Fatal error!\n' );
    fprintf ( 1, '  Column index I = %d < 1.\n', i );
    error ( 'I4COL_COMPARE - Fatal error!' );
  end

  if ( n < i )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'I4COL_COMPARE - Fatal error!\n' );
    fprintf ( 1, '  N = %d < column index I = %d.\n', n, i );
    error ( 'I4COL_COMPARE - Fatal error!' );
  end

  if ( j < 1 )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'I4COL_COMPARE - Fatal error!\n' );
    fprintf ( 1, '  Column index J = %d < 1.\n', j );
    error ( 'I4COL_COMPARE - Fatal error!' );
  end

  if ( n < j )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'I4COL_COMPARE - Fatal error!\n' );
    fprintf ( 1, '  N = %d < column index J = %d.\n', n, j );
    error ( 'I4COL_COMPARE - Fatal error!' );
  end

  isgn = 0;

  if ( i == j )
    return
  end

  k = 1;

  while ( k <= m )

    if ( a(k,i) < a(k,j) )
      isgn = -1;
      return
    elseif ( a(k,j) < a(k,i) )
      isgn = +1;
      return
    end

    k = k + 1;

  end

  return
end
function a = i4col_sort_a ( m, n, a )

%*****************************************************************************80
%
%% I4COL_SORT_A ascending sorts an I4COL.
%
%  Discussion:
%
%    In lexicographic order, the statement "X < Y", applied to two real
%    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, the first time they differ, X is smaller.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    20 February 2005
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, integer M, the number of rows of A, and the length of
%    a vector of data.
%
%    Input, integer N, the number of columns of A.
%
%    Input, integer A(M,N), the array of N columns of M-vectors.
%
%    Output, integer A(M,N), the columns of A have been sorted in ascending
%    lexicographic order.
%
  if ( m <= 0 )
    return
  end

  if ( n <= 1 )
    return
  end
%
%  Initialize.
%
  indx = 0;
  isgn = 0;
%
%  Call the external heap sorter.
%
  while ( 1 )

    [ indx, i, j ] = sort_heap_external ( n, indx, isgn );
%
%  Interchange the I and J objects.
%
    if ( 0 < indx )

      a = i4col_swap ( m, n, a, i, j );
%
%  Compare the I and J objects.
%
    elseif ( indx < 0 )

      isgn = i4col_compare ( m, n, a, i, j );

    elseif ( indx == 0 )

      break

    end

  end

  return
end
function unique_num = i4col_sorted_unique_count ( m, n, a )

%*****************************************************************************80
%
%% I4COL_SORTED_UNIQUE_COUNT counts unique elements in an I4COL.
%
%  Discussion:
%
%    The columns of the array may be ascending or descending sorted.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    14 June 2005
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, integer M, N, the number of rows and columns.
%
%    Input, integer A(M,N), a sorted array, containing
%    N columns of data.
%
%    Output, integer UNIQUE_NUM, the number of unique columns.
%
  if ( n <= 0 )
    unique_num = 0;
    return
  end

  unique_num = 1;
  j1 = 1;

  for j2 = 2 : n

    if ( any ( a(1:m,j1) ~= a(1:m,j2) ) )
      unique_num = unique_num + 1;
      j1 = j2;
    end

  end

  return
end
function a = i4col_swap ( m, n, a, i, j )

%*****************************************************************************80
%
%% I4COL_SWAP swaps columns I and J of a integer array of column data.
%
%  Example:
%
%    Input:
%
%      M = 3, N = 4, I = 2, J = 4
%
%      A = (
%        1  2  3  4
%        5  6  7  8
%        9 10 11 12 )
%
%    Output:
%
%      A = (
%        1  4  3  2
%        5  8  7  6
%        9 12 11 10 )
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    19 February 2005
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, integer M, N, the number of rows and columns in the array.
%
%    Input, integer A(M,N), an array of N columns of length M.
%
%    Input, integer I, J, the columns to be swapped.
%
%    Output, integer A(M,N), the array, with columns I and J swapped.
%
  if ( i < 1 || n < i || j < 1 || n < j )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'I4COL_SWAP - Fatal error!\n' );
    fprintf ( 1, '  I or J is out of bounds.\n' );
    fprintf ( 1, '  I =    %d\n', i );
    fprintf ( 1, '  J =    %d\n', j );
    fprintf ( 1, '  N =    %d\n', n );
    error ( 'I4COL_SWAP - Fatal error!' );
  end

  if ( i == j )
    return
  end

  col(1:m) = a(1:m,i)';
  a(1:m,i) = a(1:m,j);
  a(1:m,j) = col(1:m)';

  return
end
function table = i4mat_data_read ( input_filename, m, n )

%*****************************************************************************80
%
%% I4MAT_DATA_READ reads data from an I4MAT file.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    27 January 2006
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, string INPUT_FILENAME, the name of the input file.
%
%    Input, integer M, N, the number of rows and columns in the data.
%
%    Output, integer TABLE(M,N), the point coordinates.
%
  table = zeros ( m, n );
%
%  Build up the format string for reading M real numbers.
%
  string = ' ';

  for i = 0 : m
    string = strcat ( string, ' %d' );
  end

  input_unit = fopen ( input_filename );

  if ( input_unit < 0 )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'I4MAT_DATA_READ - Error!\n' );
    fprintf ( 1, '  Could not open the input file.\n' );
    error ( 'I4MAT_DATA_READ - Error!' );
  end

  i = 0;

  while ( i < n )

    line = fgets ( input_unit );

    if ( line == -1 )
      fprintf ( 1, '\n' );
      fprintf ( 1, 'I4MAT_DATA_READ - Error!\n' );
      fprintf ( 1, '  End of input while reading data.\n' );
      error ( 'I4MAT_DATA_READ - Error!' );
    end

    if ( line(1) == '#' )

    elseif ( s_len_trim ( line ) == 0 )
      
    else

      [ x, count ] = sscanf ( line, string );

      if ( count == m )
        i = i + 1;
        table(1:m,i) = x(1:m);
      end

    end

  end

  fclose ( input_unit );

  return
end
function [ m, n ] = i4mat_header_read ( input_filename )

%*****************************************************************************80
%
%% I4MAT_HEADER_READ reads the header from an I4MAT file.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    22 October 2004
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, string INPUT_FILENAME, the name of the input file.
%
%    Output, integer M, the spatial dimension.
%
%    Output, integer N, the number of points.
%
  m = file_column_count ( input_filename );

  if ( m <= 0 )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'I4MAT_HEADER_READ - Fatal error!\n' );
    fprintf ( 1, '  There was some kind of I/O problem while trying\n' );
    fprintf ( 1, '  to count the number of data columns in\n' );
    fprintf ( 1, '  the file %s.\n', input_filename );
  end

  n = file_row_count ( input_filename );

  if ( n <= 0 )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'I4MAT_HEADER_READ - Fatal error!\n' );
    fprintf ( 1, '  There was some kind of I/O problem while trying\n' );
    fprintf ( 1, '  to count the number of data rows in\n' );
    fprintf ( 1, '  the file %s\n', input_filename );
  end

  return
end
function i4mat_transpose_print ( m, n, a, title )

%*****************************************************************************80
%
%% I4MAT_TRANSPOSE_PRINT prints an I4MAT, transposed.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    31 January 2005
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, integer M, N, the number of rows and columns.
%
%    Input, integer A(M,N), an M by N matrix to be printed.
%
%    Input, string TITLE, a title.
%
  i4mat_transpose_print_some ( m, n, a, 1, 1, m, n, title );

  return
end
function i4mat_transpose_print_some ( m, n, a, ilo, jlo, ihi, jhi, title )

%*****************************************************************************80
%
%% I4MAT_TRANSPOSE_PRINT_SOME prints some of an I4MAT, transposed.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    21 June 2005
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, integer M, N, the number of rows and columns.
%
%    Input, integer A(M,N), an M by N matrix to be printed.
%
%    Input, integer ILO, JLO, the first row and column to print.
%
%    Input, integer IHI, JHI, the last row and column to print.
%
%    Input, string TITLE, a title.
%
  incx = 10;

  fprintf ( 1, '\n' );
  fprintf ( 1, '%s\n', title );

  for i2lo = max ( ilo, 1 ) : incx : min ( ihi, m )

    i2hi = i2lo + incx - 1;
    i2hi = min ( i2hi, m );
    i2hi = min ( i2hi, ihi );

    inc = i2hi + 1 - i2lo;

    fprintf ( 1, '\n' );
    fprintf ( 1, '  Row: ' );
    for i = i2lo : i2hi
      fprintf ( 1, '%7d  ', i );
    end
    fprintf ( 1, '\n' );
    fprintf ( 1, '  Col\n' );
    fprintf ( 1, '\n' );

    j2lo = max ( jlo, 1 );
    j2hi = min ( jhi, n );

    for j = j2lo : j2hi

      fprintf ( 1, '%5d: ', j );
      for i2 = 1 : inc
        i = i2lo - 1 + i2;
        fprintf ( 1, '%7d  ', a(i,j) );
      end
      fprintf ( 1, '\n' );

    end

  end

  return
end
function i4mat_write ( output_filename, m, n, table )

%*****************************************************************************80
%
%% I4MAT_WRITE writes an I4MAT file.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    20 July 2005
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, string OUTPUT_FILENAME, the output filename.
%
%    Input, integer M, the spatial dimension.
%
%    Input, integer N, the number of points.
%
%    Input, integer TABLE(M,N), the points.
%
%    Input, logical HEADER, is TRUE if the header is to be included.
%

%
%  Open the file.
%
  output_unit = fopen ( output_filename, 'wt' );

  if ( output_unit < 0 ) 
    fprintf ( 1, '\n' );
    fprintf ( 1, 'I4MAT_WRITE - Error!\n' );
    fprintf ( 1, '  Could not open the output file.\n' );
    error ( 'I4MAT_WRITE - Error!' );
  end
%
%  Write the data.
%
  for j = 1 : n
    for i = 1 : m
      fprintf ( output_unit, '  %12d', round ( table(i,j) ) );
    end
    fprintf ( output_unit, '\n' );
  end
%
%  Close the file.
%
  fclose ( output_unit );

  return
end
function element_node = mesh_base_one ( node_num, element_order, ...
  element_num, element_node )

%*****************************************************************************80
%
%% MESH_BASE_ONE ensures that the element definition is one-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 MATLAB program will naturally assume a 1-based indexing, it is
%    necessary to check a given element definition and, if it is actually
%    0-based, to convert it.
%
%    This function attempts to detect 0-based node indexing and correct it.
%
%    Thanks to Feifei Xu for pointing out that I was subtracting 1 when I
%    should have been adding 1!  29 November 2012.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license. 
%
%  Modified:
%
%    29 November 2012
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, integer NODE_NUM, the number of nodes.
%
%    Input, integer ELEMENT_ORDER, the order of the elements.
%
%    Input, integer ELEMENT_NUM, the number of elements.
%
%    Input/output, integer ELEMENT_NODE(ELEMENT_ORDE,ELEMENT_NUM), the element
%    definitions.
%
  node_min = min ( min ( element_node(1:element_order,1:element_num) ) );
  node_max = max ( max ( element_node(1:element_order,1:element_num) ) );

  if ( node_min == 0 && node_max == node_num - 1 )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'MESH_BASE_ONE:\n' );
    fprintf ( 1, '  The element indexing appears to be 0-based!\n' );
    fprintf ( 1, '  This will be converted to 1-based.\n' );
    element_node(1:element_order,1:element_num) = ...
      element_node(1:element_order,1:element_num) + 1;
  elseif ( node_min == 1 && node_max == node_num )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'MESH_BASE_ONE:\n' );
    fprintf ( 1, '  The element indexing appears to be 1-based!\n' );
    fprintf ( 1, '  No conversion is necessary.\n' );
  else
    fprintf ( 1, '\n' );
    fprintf ( 1, 'MESH_BASE_ONE - Warning!\n' );
    fprintf ( 1, '  The element indexing is not of a recognized type.\n' );
    fprintf ( 1, '  NODE_MIN = %d\n', node_min );
    fprintf ( 1, '  NODE_MAX = %d\n', node_max );
    fprintf ( 1, '  NODE_NUM = %d\n', node_num );
  end

  return
end
function table = r8mat_data_read ( input_filename, m, n )

%*****************************************************************************80
%
%% R8MAT_DATA_READ reads data from an R8MAT file.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    27 January 2006
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, string INPUT_FILENAME, the name of the input file.
%
%    Input, integer M, N, the number of rows and columns of data.
%
%    Output, real TABLE(M,N), the point coordinates.
%
  table = zeros ( m, n );
%
%  Build up the format string for reading M real numbers.
%
  string = ' ';

  for i = 0 : m
    string = strcat ( string, ' %f' );
  end

  input_unit = fopen ( input_filename );

  if ( input_unit < 0 ) 
    fprintf ( 1, '\n' );
    fprintf ( 1, 'R8MAT_DATA_READ - Error!\n' );
    fprintf ( 1, '  Could not open the file.\n' );
    error ( 'R8MAT_DATA_READ - Error!' );
  end

  i = 0;

  while ( i < n )

    line = fgets ( input_unit );

    if ( line == -1 )
      break;
    end

    if ( line(1) == '#' )

    elseif ( s_len_trim ( line ) == 0 )
      
    else

      [ x, count ] = sscanf ( line, string );

      if ( count == m )
        i = i + 1;
        table(1:m,i) = x(1:m);
      end

    end

  end

  fclose ( input_unit );

  return
end
function [ m, n ] = r8mat_header_read ( input_filename )

%*****************************************************************************80
%
%% R8MAT_HEADER_READ reads the header from an R8MAT file.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    22 October 2004
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, string INPUT_FILENAME, the name of the input file.
%
%    Output, integer M, the spatial dimension.
%
%    Output, integer N, the number of points.
%
  m = file_column_count ( input_filename );

  if ( m <= 0 )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'R8MAT_HEADER_READ - Fatal error!\n' );
    fprintf ( 1, '  There was some kind of I/O problem while trying\n' );
    fprintf ( 1, '  to count the number of data columns in\n' );
    fprintf ( 1, '  the file %s.\n', input_filename );
  end

  n = file_row_count ( input_filename );

  if ( n <= 0 )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'R8MAT_HEADER_READ - Fatal error!\n' );
    fprintf ( 1, '  There was some kind of I/O problem while trying\n' );
    fprintf ( 1, '  to count the number of data rows in\n' );
    fprintf ( 1, '  the file %s\n', input_filename );
  end

  return
end
function r8mat_transpose_print ( m, n, a, title )

%*****************************************************************************80
%
%% R8MAT_TRANSPOSE_PRINT prints an R8MAT, transposed.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    10 August 2004
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, integer M, N, the number of rows and columns.
%
%    Input, real 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
end
function r8mat_transpose_print_some ( m, n, a, ilo, jlo, ihi, jhi, title )

%*****************************************************************************80
%
%% R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    23 May 2005
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, integer M, N, the number of rows and columns.
%
%    Input, real A(M,N), an M by N matrix to be printed.
%
%    Input, integer ILO, JLO, the first row and column to print.
%
%    Input, integer IHI, JHI, the last row and column to print.
%
%    Input, string TITLE, a title.
%
  incx = 5;

  fprintf ( 1, '\n' );
  fprintf ( 1, '%s\n', title );

  for i2lo = max ( ilo, 1 ) : incx : min ( ihi, m )

    i2hi = i2lo + incx - 1;
    i2hi = min ( i2hi, m );
    i2hi = min ( i2hi, ihi );

    inc = i2hi + 1 - i2lo;
    
    fprintf ( 1, '\n' );
    fprintf ( 1, '  Row: ' );
    for i = i2lo : i2hi
      fprintf ( 1, '%7d       ', i );
    end
    fprintf ( 1, '\n' );
    fprintf ( 1, '  Col\n' );

    j2lo = max ( jlo, 1 );
    j2hi = min ( jhi, n );

    for j = j2lo : j2hi

      fprintf ( 1, '%5d:', j );
      for i2 = 1 : inc
        i = i2lo - 1 + i2;
        fprintf ( 1, '%12f', a(i,j) );
      end
      fprintf ( 1, '\n' );

    end

  end

  return
end
function r8mat_write ( output_filename, m, n, table )

%*****************************************************************************80
%
%% R8MAT_WRITE writes an R8MAT file.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    09 August 2009
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, string OUTPUT_FILENAME, the output filename.
%
%    Input, integer M, the spatial dimension.
%
%    Input, integer N, the number of points.
%
%    Input, real TABLE(M,N), the points.
%

%
%  Open the file.
%
  output_unit = fopen ( output_filename, 'wt' );

  if ( output_unit < 0 ) 
    fprintf ( 1, '\n' );
    fprintf ( 1, 'R8MAT_WRITE - Error!\n' );
    fprintf ( 1, '  Could not open the output file.\n' );
    error ( 'R8MAT_WRITE - Error!' );
  end
%
%  Write the data.
%
%  For greater precision, try:
%
%     fprintf ( output_unit, '  %24,16f', table(i,j) );
%
  for j = 1 : n
    for i = 1 : m
      fprintf ( output_unit, '  %14f', table(i,j) );
    end
    fprintf ( output_unit, '\n' );
  end
%
%  Close the file.
%
  fclose ( output_unit );

  return
end
function len = s_len_trim ( s )

%*****************************************************************************80
%
%% S_LEN_TRIM returns the length of a character string to the last nonblank.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    14 June 2003
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, string S, the string to be measured.
%
%    Output, integer LEN, the length of the string up to the last nonblank.
%
  len = length ( s );

  while ( 0 < len )
    if ( s(len) ~= ' ' )
      return
    end
    len = len - 1;
  end

  return
end
function word_num = s_word_count ( s )

%*****************************************************************************80
%
%% S_WORD_COUNT counts the number of "words" in a string.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    30 January 2006
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, string S, the string to be examined.
%
%    Output, integer WORD_NUM, the number of "words" in the string.
%    Words are presumed to be separated by one or more blanks.
%
  FALSE = 0;
  TRUE = 1;

  word_num = 0;
  s_length = length ( s );

  if ( s_length <= 0 )
    return;
  end

  blank = TRUE;

  for i = 1 : s_length

    if ( s(i) == ' ' )
      blank = TRUE;
    elseif ( blank == TRUE )
      word_num = word_num + 1;
      blank = FALSE;
    end

  end

  return
end
function [ indx, i, j ] = sort_heap_external ( n, indx, isgn )

%*****************************************************************************80
%
%% SORT_HEAP_EXTERNAL externally sorts a list of items into ascending order.
%
%  Discussion:
%
%    The actual list of data is not passed to the routine.  Hence this
%    routine may be used to sort integers, reals, numbers, names,
%    dates, shoe sizes, and so on.  After each call, the routine asks
%    the user to compare or interchange two items, until a special
%    return value signals that the sorting is completed.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    05 February 2004
%
%  Author:
%
%    Original FORTRAN77 version by Albert Nijenhuis, Herbert Wilf.
%    MATLAB version by John Burkardt
%
%  Reference:
%
%    Albert Nijenhuis, Herbert Wilf.
%    Combinatorial Algorithms,
%    Academic Press, 1978, second edition,
%    ISBN 0-12-519260-6.
%
%  Parameters:
%
%    Input, integer N, the number of items to be sorted.
%
%    Input, integer INDX, the main communication signal.
%    The user must set INDX to 0 before the first call.
%    Thereafter, the user should set the input value of INDX
%    to the output value from the previous call.
%
%    Input, integer ISGN, results of comparison of elements I and J.
%    (Used only when the previous call returned INDX less than 0).
%    ISGN <= 0 means I is less than or equal to J;
%    0 <= ISGN means I is greater than or equal to J.
%
%    Output, integer INDX, the main communication signal.
%    If INDX is
%
%      greater than 0, the user should:
%      * interchange items I and J;
%      * call again.
%
%      less than 0, the user should:
%      * compare items I and J;
%      * set ISGN = -1 if I < J, ISGN = +1 if J < I;
%      * call again.
%
%      equal to 0, the sorting is done.
%
%    Output, integer I, J, the indices of two items.
%    On return with INDX positive, elements I and J should be interchanged.
%    On return with INDX negative, elements I and J should be compared, and
%    the result reported in ISGN on the next call.
%
  persistent i_save;
  persistent j_save;
  persistent k;
  persistent k1;
  persistent n1;
%
%  INDX = 0: This is the first call.
%
  if ( indx == 0 )
      
    i_save = -1;
    j_save = -1;
    k = floor ( n / 2 );
    k1 = k;
    n1 = n;
%
%  INDX < 0: The user is returning the results of a comparison.
%
  elseif ( indx < 0 )

    if ( indx == -2 )

      if ( isgn < 0 )
        i_save = i_save + 1;
      end

      j_save = k1;
      k1 = i_save;
      indx = -1;
      i = i_save;
      j = j_save;
      return;
    end

    if ( 0 < isgn )
      indx = 2;
      i = i_save;
      j = j_save;
      return;
    end

    if ( k <= 1 )

      if ( n1 == 1 )
        i_save = 0;
        j_save = 0;
        indx = 0;
      else
        i_save = n1;
        n1 = n1 - 1;
        j_save = 1;
        indx = 1;
      end

      i = i_save;
      j = j_save;
      return;

    end

    k = k - 1;
    k1 = k;
%
%  0 < INDX, the user was asked to make an interchange.
%
  elseif ( indx == 1 )

    k1 = k;

  end

  while ( 1 )

    i_save = 2 * k1;

    if ( i_save == n1 )
      j_save = k1;
      k1 = i_save;
      indx = -1;
      i = i_save;
      j = j_save;
      return;
    elseif ( i_save <= n1 )
      j_save = i_save + 1;
      indx = -2;
      i = i_save;
      j = j_save;
      return;
    end

    if ( k <= 1 )
      break;
    end

    k = k - 1;
    k1 = k;

  end

  if ( n1 == 1 )
    i_save = 0;
    j_save = 0;
    indx = 0;
    i = i_save;
    j = j_save;
  else
    i_save = n1;
    n1 = n1 - 1;
    j_save = 1;
    indx = 1;
    i = i_save;
    j = j_save;
  end

  return
end
function timestamp ( )

%*****************************************************************************80
%
%% TIMESTAMP prints the current YMDHMS date as a timestamp.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    14 February 2003
%
%  Author:
%
%    John Burkardt
%
  t = now;
  c = datevec ( t );
  s = datestr ( c, 0 );
  fprintf ( 1, '%s\n', s );

  return
end
function triangle_neighbor = triangulation_neighbor_triangles ( ...
  triangle_order, triangle_num, triangle_node )

%*****************************************************************************80
%
%% TRIANGULATION_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 use columns instead of 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, integer TRIANGLE_ORDER, the order of the triangles.
%
%    Input, integer TRIANGLE_NUM, the number of triangles.
%
%    Input, integer TRIANGLE_NODE(TRIANGLE_ORDER,TRIANGLE_NUM), the nodes that 
%    make up each triangle.
%
%    Output, integer TRIANGLE_NEIGHBOR(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.
%

%
%  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)
%
  col = zeros ( 4, triangle_order * triangle_num );
  
  for tri = 1 : triangle_num

    i = triangle_node(1,tri);
    j = triangle_node(2,tri);
    k = triangle_node(3,tri);

    if ( i < j )
      col(1:4,1+3*(tri-1)) = [ i, j, 3, tri ]';
    else
      col(1:4,1+3*(tri-1)) = [ j, i, 3, tri ]';
    end

    if ( j < k )
      col(1:4,2+3*(tri-1)) = [ j, k, 1, tri ]';
    else
      col(1:4,2+3*(tri-1)) = [ k, j, 1, tri ]';
    end

    if ( k < i )
      col(1:4,3+3*(tri-1)) = [ k, i, 2, tri ]';
    else
      col(1:4,3+3*(tri-1)) = [ i, k, 2, tri ]';
    end

  end
%
%  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.
%
  col = 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.
%
  triangle_neighbor(1:3,1:triangle_num) = -1;

  icol = 1;

  while ( 1 )

    if ( 3 * triangle_num <= icol )
      break
    end

    if ( col(1,icol) ~= col(1,icol+1) || col(2,icol) ~= col(2,icol+1) )
      icol = icol + 1;
      continue
    end

    side1 = col(3,icol);
    tri1 =  col(4,icol);
    side2 = col(3,icol+1);
    tri2 =  col(4,icol+1);

    triangle_neighbor(side1,tri1) = tri2;
    triangle_neighbor(side2,tri2) = tri1;

    icol = icol + 2;

  end

  return
end
function boundary_edge_num = triangulation_order3_boundary_edge_count ( ...
  triangle_num, triangle_node )

%*****************************************************************************80
%
%% TRIANGULATION_ORDER3_BOUNDARY_EDGE_COUNT counts the boundary edges.
%
%  Discussion:
%
%    This routine is given a triangulation, an abstract list of triples
%    of nodes.  It is assumed that the nodes in each triangle are listed
%    in a counterclockwise order, although the routine should work 
%    if the nodes are consistently listed in a clockwise order as well.
%
%    It is assumed that each edge of the triangulation is either 
%    * an INTERIOR edge, which is listed twice, once with positive
%      orientation and once with negative orientation, or;
%    * a BOUNDARY edge, which will occur only once.
%
%    This routine should work even if the region has holes - as long
%    as the boundary of the hole comprises more than 3 edges!
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    12 June 2005
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, integer TRIANGLE_NUM, the number of triangles.
%
%    Input, integer TRIANGLE_NODE(3,TRIANGLE_NUM), the nodes that make up the
%    triangles.  These should be listed in counterclockwise order.
%
%    Output, integer BOUNDARY_EDGE_NUM, the number of boundary edges.
%
  m = 2;
  n = 3 * triangle_num;
%
%  Set up the edge array.
%
  edge = zeros ( 2, 3 * triangle_num );

  edge(1:2,               1:  triangle_num) = triangle_node(1:2,1:triangle_num);
  edge(1:2,  triangle_num+1:2*triangle_num) = triangle_node(2:3,1:triangle_num);
  edge(1  ,2*triangle_num+1:3*triangle_num) = triangle_node(3,  1:triangle_num);
  edge(2  ,2*triangle_num+1:3*triangle_num) = triangle_node(1,  1:triangle_num);
%
%  In each column, force the smaller entry to appear first.
%
  e1(1:n) = min ( edge(1:2,1:n) );
  e2(1:n) = max ( edge(1:2,1:n) );

  edge(1,1:n) = e1(1:n);
  edge(2,1:n) = e2(1:n);
%
%  Ascending sort the column array.
%
  edge = i4col_sort_a ( m, n, edge );
%
%  Get the number of unique columns in EDGE.
%
  unique_num = i4col_sorted_unique_count ( m, n, edge );

  interior_edge_num = 3 * triangle_num - unique_num;

  boundary_edge_num = 3 * triangle_num - 2 * interior_edge_num;

  return
end