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

using namespace std;

int main ( int argc, char *argv[] );
void assemble_poisson_dsp ( int node_num, double node_xy[],
  int element_num, int element_node[], int quad_num, int nz_num, int ia[],
  int ja[], double a[], double f[] );
void ax ( double *a, int *ia, int *ja, double *x, double *w, int n,
  int nz_num );
void basis_one_t3 ( double t[2*3], int i, double p[2], double *qi,
  double *dqidx, double *dqidy );
char ch_cap ( char c );
bool ch_eqi ( char c1, char c2 );
int ch_to_digit ( char c );
void dirichlet_apply_dsp ( int node_num, double node_xy[], int node_condition[],
  int nz_num, int ia[], int ja[], double a[], double f[] );
void dirichlet_condition ( int node_num, double node_xy[], double node_rhs[] );
int dsp_ij_to_k ( int nz_num, int row[], int col[], int i, int j );
void dsp_print_some ( int m, int n, int nz_num, int row[], int col[],
  double a[], int ilo, int jlo, int ihi, int jhi, string title );
int file_column_count ( string input_filename );
int file_row_count ( string input_filename );
void h_coef ( int node_num, double node_xy[], double node_h[] );
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_some ( int m, int n, int a[], int ilo, int jlo,
  int ihi, int jhi, string title );
int i4vec2_compare ( int n, int a1[], int a2[], int i, int j );
void i4vec2_sort_a ( int n, int a1[], int a2[] );
void k_coef ( int node_num, double node_xy[], double node_k[] );
void mgmres ( double a[], int ia[], int ja[], double x[], double rhs[],
  int n, int nz_num, int itr_max, int mr, double tol_abs, double tol_rel );
void mult_givens ( double c, double s, int k, double g[] );
void quad_rule ( int quad_num, double quad_w[], double quad_xy[] );
double r8_abs ( double x );
double r8_huge ( void );
int r8_nint ( double x );
void r8mat_transpose_print_some ( int m, int n, double a[], int ilo, int jlo,
  int ihi, int jhi, string title );
double *r8mat_data_read ( string input_filename, int m, int n );
void r8mat_header_read ( string input_filename, int *m, int *n );
void r8mat_write ( string output_filename, int m, int n, double table[] );
double r8vec_amax ( int n, double a[] );
double r8vec_dot_product ( int n, double a1[], double a2[] );
void r8vec_print_some ( int n, double a[], int i_lo, int i_hi, string title );
double *r8vec_uniform_01 ( int n, int *seed );
void rhs ( int node_num, double node_xy[], double node_rhs[] );
void reference_to_physical_t3 ( double t[2*3], int n, double ref[],
  double phy[] );
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 solution_evaluate ( double xy[2], double t[2*3], double node_u[3],
  double *u, double *dudx, double *dudy );
void sort_heap_external ( int n, int *indx, int *i, int *j, int isgn );
void timestamp ( void );
double triangle_area_2d ( double t[2*3] );
int triangulation_order3_adj_count ( int node_num, int triangle_num,
  int triangle_node[], int triangle_neighbor[], int adj_col[] );
void triangulation_order3_adj_set2 ( int node_num, int triangle_num,
  int triangle_node[], int triangle_neighbor[], int adj_num, int adj_col[],
  int ia[], int ja[] );
bool *triangulation_order3_boundary_node ( int node_num, int element_num,
  int element_node[] );
void triangulation_order3_neighbor_triangles ( int triangle_num,
  int triangle_node[], int triangle_neighbor[] );

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

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

//****************************************************************************80
//
//  Purpose:
//
//    MAIN is the main program for FEM2D_POISSON_SPARSE.
//
//  Discussion:
//
//    This program uses a sparse matrix storage format and an iterative solver,
//    which allow it to solve larger problems faster.
//
//    This program solves the Poisson equation
//
//      -DEL H(X,Y) DEL U(X,Y) + K(X,Y) * U(X,Y) = F(X,Y)
//
//    in a triangulated region in the plane.
//
//    Along the boundary of the region, Dirichlet conditions
//    are imposed:
//
//      U(X,Y) = G(X,Y)
//
//    The code uses continuous piecewise linear basis functions on
//    triangles.
//
//  Problem specification:
//
//    The user defines the geometry by supplying two data files
//    which list the node coordinates, and list the nodes that make up
//    each element.
//
//    The user specifies the right hand side of the Dirichlet boundary
//    conditions by supplying a function
//
//      void dirichlet_condition ( int node_num, double node_xy[2*node_num],
//        double node_bc[node_num] )
//
//    The user specifies the coefficient function H(X,Y) of the Poisson
//    equation by supplying a routine of the form
//
//      void h_coef ( int node_num, double node_xy[2*node_num],
//        double node_h[node_num] )
//
//    The user specifies the coefficient function K(X,Y) of the Poisson
//    equation by supplying a routine of the form
//
//      void k_coef ( int node_num, double node_xy[2*node_num],
//        double node_k[node_num] )
//
//    The user specifies the right hand side of the Poisson equation
//    by supplying a routine of the form
//
//      void rhs ( int node_num, double node_xy[2*node_num],
//        double node_f[node_num] )
//
//  Usage:
//
//    fem2d_poisson_sparse prefix
//
//    where 'prefix' is the common filename prefix so that:
//
//    * prefix_nodes.txt contains the coordinates of the nodes;
//    * prefix_elements.txt contains the indices of nodes forming each element.
//
//    Files created include:
//
//    * prefix_values.txt, the value of the solution at every node.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    25 January 2013
//
//  Author:
//
//    John Burkardt
//
//  Local parameters:
//
//    Local, double A[NZ_NUM], the coefficient matrix.
//
//    Local, int ELEMENT_NODE[3*ELEMENT_NUM];
//    ELEMENT_NODE(I,J) is the global index of local node I in element J.
//
//    Local, int ELEMENT_NUM, the number of elements.
//
//    Local, integer ELEMENT_ORDER, the element order.
//
//    Local, double F[NODE_NUM], the right hand side.
//
//    Local, int IA[NZ_NUM], the row indices of the nonzero entries
//    of the coefficient matrix.
//
//    Local, int JA[NZ_NUM], the column indices of the nonzero entries
//    of the coefficient matrix.
//
//    Local, bool NODE_BOUNDARY[NODE_NUM], is TRUE if the node is
//    found to lie on the boundary of the region.
//
//    Local, int NODE_CONDITION[NODE_NUM],
//    indicates the condition used to determine the variable at a node.
//    0, there is no condition (and no variable) at this node.
//    1, a finite element equation is used;
//    2, a Dirichlet condition is used.
//    3, a Neumann condition is used.
//
//    Local, int NODE_NUM, the number of nodes.
//
//    Local, double NODE_U[NODE_NUM], the finite element coefficients.
//
//    Local, double NODE_XY[2*NODE_NUM], the coordinates of nodes.
//
//    Local, int NZ_NUM, the number of nonzero entries
//    in the coefficient matrix.
//
//    Local, integer QUAD_NUM, the number of quadrature points used for
//    assembly.  This is currently set to 3, the lowest reasonable value.
//    Legal values are 1, 3, 4, 6, 7, 9, 13, and for some problems, a value
//    of QUAD_NUM greater than 3 may be appropriate.
//
{
  double *a;
  int *adj_col;
  bool debug = false;
  int dim_num;
  string element_filename;
  int *element_neighbor;
  int *element_node;
  int element_num;
  int element_order;
  int element_show;
  double *f;
  int *ia;
  int ierr;
  int itr_max;
  int *ja;
  int job;
  int mr;
  int node;
  bool *node_boundary;
  int *node_condition;
  string node_filename;
  bool node_label;
  int node_num;
  int node_show;
  double *node_u;
  double *node_xy;
  int nz_num;
  string prefix;
  int quad_num = 3;
  int seed = 123456789;
  string solution_filename;
  double temp;
  double tol_abs;
  double tol_rel;

  timestamp ( );
  cout << "\n";
  cout << "FEM2D_POISSON_SPARSE:\n";
  cout << "  C++ version:\n";
  cout << "\n";
  cout << "  Compiled on " << __DATE__ << " at " << __TIME__ << ".\n";
  cout << "\n";
  cout << "  A finite element method solver for the Poisson problem\n";
  cout << "  in an arbitrary triangulated region in 2 dimensions,\n";
  cout << "  using sparse storage and an iterative solver.\n";
  cout << "\n";
  cout << "  - DEL H(x,y) DEL U(x,y) + K(x,y) * U(x,y) = F(x,y) in the region\n";
  cout << "\n";
  cout << "                                     U(x,y) = G(x,y) on the boundary.\n";
  cout << "\n";
  cout << "  The finite element method is used,\n";
  cout << "  with triangular elements,\n";
  cout << "  which must be a 3 node linear triangle.\n";
//
//  Get the filename prefix.
//
  if ( 1 <= argc )
  {
    prefix = argv[1];
  }
  else
  {
    cout << "\n";
    cout << "  Please enter the filename prefix:\n";

    cin >> prefix;
  }
//
//  Create the file names.
//
  node_filename = prefix + "_nodes.txt";
  element_filename = prefix + "_elements.txt";
  solution_filename = prefix + "_values.txt";

  cout << "\n";
  cout << "  Node file is \"" << node_filename << "\".\n";
  cout << "  Element file is \"" << element_filename << "\".\n";
//
//  Read the node coordinate file.
//
  r8mat_header_read ( node_filename, &dim_num, &node_num );

  cout << "  Number of nodes =          " << node_num << "\n";

  node_condition = new int[node_num];

  node_xy = r8mat_data_read ( node_filename, dim_num, node_num );

  r8mat_transpose_print_some ( dim_num, node_num, node_xy, 1, 1, 2, 10,
    "  First 10 nodes" );
//
//  Read the triangle description file.
//
  i4mat_header_read ( element_filename, &element_order, &element_num );

  cout << "\n";
  cout << "  Element order =            " << element_order << "\n";
  cout << "  Number of elements =       " << element_num << "\n";

  if ( element_order != 3 )
  {
    cout << "\n";
    cout << "FEM2D_POISSON_SPARSE - Fatal error!\n";
    cout << "  The input triangulation has order " << element_order << "\n";
    cout << "  However, a triangulation of order 3 is required.\n";
    exit ( 1 );
  }
  element_node = i4mat_data_read ( element_filename, element_order,
    element_num );

  i4mat_transpose_print_some ( 3, element_num,
    element_node, 1, 1, 3, 10, "  First 10 elements" );

  cout << "\n";
  cout << "  Quadrature order =          " << quad_num << "\n";
//
//  Determine which nodes are boundary nodes and which have a
//  finite element unknown.  Then set the boundary values.
//
  node_boundary = triangulation_order3_boundary_node ( node_num, element_num,
    element_node );
//
//  Determine the node conditions.
//  For now, we'll just assume all boundary nodes are Dirichlet.
//
  for ( node = 0; node < node_num; node++ )
  {
    if ( node_boundary[node] )
    {
      node_condition[node] = 2;
    }
    else
    {
      node_condition[node] = 1;
    }
  }
//
//  Determine the element neighbor array, just so we can estimate
//  the nonzeros.
//
  element_neighbor = new int[3*element_num];

  triangulation_order3_neighbor_triangles ( element_num, element_node,
    element_neighbor );
//
//  Count the number of nonzeros.
//
  adj_col = new int[node_num+1];

  nz_num = triangulation_order3_adj_count ( node_num, element_num,
    element_node, element_neighbor, adj_col );

  cout << "\n";
  cout << "  Number of nonzero coefficients NZ_NUM = " << nz_num << "\n";
//
//  Set up the sparse row and column index vectors.
//
  ia = new int[nz_num];
  ja = new int[nz_num];

  triangulation_order3_adj_set2 ( node_num, element_num, element_node,
    element_neighbor, nz_num, adj_col, ia, ja );

  delete [] adj_col;
  delete [] element_neighbor;
//
//  Allocate space for the coefficient matrix A and right hand side F.
//
  a = new double[nz_num];
  f = new double[node_num];
//
//  Assemble the finite element coefficient matrix A and the right-hand side F.
//
  assemble_poisson_dsp ( node_num, node_xy, element_num,
    element_node, quad_num, nz_num, ia, ja, a, f );
//
//  Print a portion of the matrix.
//
  if ( debug )
  {
    dsp_print_some ( node_num, node_num, nz_num, ia, ja, a, 1, 1, 10, 10,
      "  Part of Finite Element matrix A:" );

    r8vec_print_some ( node_num, f, 1, 10,
      "  Part of right hand side vector F:" );
  }
//
//  Adjust the linear system to account for Dirichlet boundary conditions.
//
  dirichlet_apply_dsp ( node_num, node_xy, node_condition, nz_num, ia, ja,
    a, f );

  if ( debug )
  {
    dsp_print_some ( node_num, node_num, nz_num, ia, ja, a, 1, 1, 10, 10,
      "  Part of finite Element matrix A after boundary adjustments:" );

    r8vec_print_some ( node_num, f, 1, 10,
      "  Part of right hand side vector F:" );
  }
//
//  Solve the linear system using an iterative solver.
//
  node_u = r8vec_uniform_01 ( node_num, &seed );

  itr_max = 20;
  mr = 20;
  tol_abs = 0.000001;
  tol_rel = 0.000001;

  mgmres ( a, ia, ja, node_u, f, node_num, nz_num, itr_max, mr, tol_abs,
    tol_rel );

  r8vec_print_some ( node_num, node_u, 1, 10,
    "  Part of the solution vector vector U:" );
//
//  Write an ASCII file that can be read into MATLAB.
//
  r8mat_write ( solution_filename, 1, node_num, node_u );

  cout << "\n";
  cout << "FEM2D_POISSON_SPARSE:\n";
  cout << "  Wrote an ASCII file\n";
  cout << "    \"" << solution_filename << "\".\n";
  cout << "  of the form\n";
  cout << "    U ( X(I), Y(I) )\n";
  cout << "  which can be used for plotting.\n";

  if ( debug )
  {
    r8vec_print_some ( node_num, node_u, 1, 10,
      "  Part of the solution vector:" );
  }
//
//  Free memory.
//
  delete [] a;
  delete [] element_node;
  delete [] f;
  delete [] ia;
  delete [] ja;
  delete [] node_boundary;
  delete [] node_condition;
  delete [] node_u;
  delete [] node_xy;
//
//  Terminate.
//
  cout << "\n";
  cout << "FEM2D_POISSON_SPARSE:\n";
  cout << "  Normal end of execution.\n";

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

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

void assemble_poisson_dsp ( int node_num, double node_xy[],
  int element_num, int element_node[], int quad_num, int nz_num, int ia[],
  int ja[], double a[], double f[] )

//****************************************************************************80
//
//  Purpose:
//
//    ASSEMBLE_POISSON_DSP assembles the system for the Poisson equation.
//
//  Discussion:
//
//    The matrix is sparse, and stored in the DSP or "sparse triple" format.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    13 July 2007
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int NODE_NUM, the number of nodes.
//
//    Input, double NODE_XY[2*NODE_NUM], the
//    coordinates of nodes.
//
//    Input, int ELEMENT_NUM, the number of triangles.
//
//    Input, int ELEMENT_NODE[3*ELEMENT_NUM];
//    ELEMENT_NODE(I,J) is the global index of local node I in triangle J.
//
//    Input, int QUAD_NUM, the number of quadrature points used in assembly.
//
//    Input, int NZ_NUM, the number of nonzero entries.
//
//    Input, int IA[NZ_NUM], JA[NZ_NUM], the row and column
//    indices of the nonzero entries.
//
//    Output, double A[NZ_NUM], the nonzero entries of the matrix.
//
//    Output, double F(NODE_NUM), the right hand side.
//
//  Local parameters:
//
//    Local, double BI, DBIDX, DBIDY, the value of some basis function
//    and its first derivatives at a quadrature point.
//
//    Local, double BJ, DBJDX, DBJDY, the value of another basis
//    function and its first derivatives at a quadrature point.
//
{
  double area;
  int basis;
  double bi;
  double bj;
  double dbidx;
  double dbidy;
  double dbjdx;
  double dbjdy;
  int element;
  int i;
  int j;
  int k;
  double k_value;
  int node;
  int nz;
  double p[2];
  double *phys_h;
  double *phys_k;
  double *phys_rhs;
  double *phys_xy;
  int quad;
  double *quad_w;
  double *quad_xy;
  double t3[2*3];
  int test;
  double *w;

  phys_h = new double[quad_num];
  phys_k = new double[quad_num];
  phys_rhs = new double[quad_num];
  phys_xy = new double[2*quad_num];

  quad_w = new double[quad_num];
  quad_xy = new double[2*quad_num];

  w = new double[quad_num];
//
//  Initialize the arrays to zero.
//
  for ( node = 0; node < node_num; node++ )
  {
    f[node] = 0.0;
  }
  for ( nz = 0; nz < nz_num; nz++ )
  {
    a[nz] = 0.0;
  }
//
//  Get the quadrature weights and nodes.
//
  quad_rule ( quad_num, quad_w, quad_xy );
//
//  Add up all quantities associated with the ELEMENT-th element.
//
  for ( element = 0; element < element_num; element++ )
  {
//
//  Make a copy of the element.
//
    for ( j = 0; j < 3; j++ )
    {
      for ( i = 0; i < 2; i++ )
      {
        t3[i+j*2] = node_xy[i+(element_node[j+element*3]-1)*2];
      }
    }
//
//  Map the quadrature points QUAD_XY to points XY in the physical element.
//
    reference_to_physical_t3 ( t3, quad_num, quad_xy, phys_xy );

    area = r8_abs ( triangle_area_2d ( t3 ) );

    for ( quad = 0; quad < quad_num; quad++ )
    {
      w[quad] = quad_w[quad] * area;
    }

    rhs ( quad_num, phys_xy, phys_rhs );
    h_coef ( quad_num, phys_xy, phys_h );
    k_coef ( quad_num, phys_xy, phys_k );
//
//  Consider the QUAD-th quadrature point.
//
    for ( quad = 0; quad < quad_num; quad++ )
    {
      p[0] = phys_xy[0+quad*2];
      p[1] = phys_xy[1+quad*2];
//
//  Consider the TEST-th test function.
//
//  We generate an integral for every node associated with an unknown.
//  But if a node is associated with a boundary condition, we do nothing.
//
      for ( test = 1; test <= 3; test++ )
      {
        i = element_node[test-1+element*3];

        basis_one_t3 ( t3, test, p, &bi, &dbidx, &dbidy );

        f[i-1] = f[i-1] + w[quad] * phys_rhs[quad] * bi;
//
//  Consider the BASIS-th basis function, which is used to form the
//  value of the solution function.
//
        for ( basis = 1; basis <= 3; basis++ )
        {
          j = element_node[basis-1+element*3];

          basis_one_t3 ( t3, basis, p, &bj, &dbjdx, &dbjdy );

          k = dsp_ij_to_k ( nz_num, ia, ja, i, j );

          a[k-1] = a[k-1] + w[quad] * (
              phys_h[quad] * ( dbidx * dbjdx + dbidy * dbjdy )
            + phys_k[quad] * bj * bi );
        }
      }
    }
  }
  delete [] phys_h;
  delete [] phys_k;
  delete [] phys_rhs;
  delete [] phys_xy;
  delete [] quad_w;
  delete [] quad_xy;
  delete [] w;

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

void ax ( double *a, int *ia, int *ja, double *x, double *w, int n, int nz_num )

//****************************************************************************80
//
//  Purpose:
//
//    AX computes A * X for a sparse matrix.
//
//  Discussion:
//
//    The matrix A is assumed to be sparse.  To save on storage, only
//    the nonzero entries of A are stored.  For instance, the K-th nonzero
//    entry in the matrix is stored by:
//
//      A(K) = value of entry,
//      IA(K) = row of entry,
//      JA(K) = column of entry.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    08 August 2006
//
//  Author:
//
//    Original C version by Lili Ju.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Richard Barrett, Michael Berry, Tony Chan, James Demmel,
//    June Donato, Jack Dongarra, Victor Eijkhout, Roidan Pozo,
//    Charles Romine, Henk van der Vorst,
//    Templates for the Solution of Linear Systems:
//    Building Blocks for Iterative Methods,
//    SIAM, 1994,
//    ISBN: 0898714710,
//    LC: QA297.8.T45.
//
//    Tim Kelley,
//    Iterative Methods for Linear and Nonlinear Equations,
//    SIAM, 2004,
//    ISBN: 0898713528,
//    LC: QA297.8.K45.
//
//    Yousef Saad,
//    Iterative Methods for Sparse Linear Systems,
//    Second Edition,
//    SIAM, 2003,
//    ISBN: 0898715342,
//    LC: QA188.S17.
//
//  Parameters:
//
//    Input, double A[NZ_NUM], the matrix values.
//
//    Input, int IA[NZ_NUM], JA[NZ_NUM], the row and column indices
//    of the matrix values.
//
//    Input, double X[N], the vector to be multiplied by A.
//
//    Output, double W[N], the value of A*X.
//
//    Input, int N, the order of the system.
//
//    Input, int NZ_NUM, the number of nonzeros.
//
{
  int i;
  int j;
  int k;

  for ( i = 0; i < n; i++ )
  {
    w[i] = 0.0;
  }

  for ( k = 0; k < nz_num; k++ )
  {
    i = ia[k] - 1;
    j = ja[k] - 1;
    w[i] = w[i] + a[k] * x[j];
  }
  return;
}
//****************************************************************************80

void basis_one_t3 ( double t[2*3], int i, double p[2], double *qi,
  double *dqidx, double *dqidy )

//****************************************************************************80
//
//  Purpose:
//
//    BASIS_ONE_T3 evaluates basis functions for a linear triangular element.
//
//  Discussion:
//
//    The routine is given the coordinates of the nodes of a triangle.
//
//           3
//          / .
//         /   .
//        /     .
//       1-------2
//
//    It evaluates the linear basis function Q(I)(X,Y) associated with
//    node I, which has the property that it is a linear function
//    which is 1 at node I and zero at the other two nodes.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    06 January 2006
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double T[2*3], the coordinates of the nodes.
//
//    Input, int I, the index of the desired basis function.
//    I should be between 1 and 3.
//
//    Input, double P[2], the coordinates of the point where
//    the basis function is to be evaluated.
//
//    Output, double *QI, *DQIDX, *DQIDY, the value of the I-th basis function
//    and its X and Y derivatives.
//
{
  double area;
  int ip1;
  int ip2;
  double temp;

  area = t[0+0*2] * ( t[1+1*2] - t[1+2*2] )
       + t[0+1*2] * ( t[1+2*2] - t[1+0*2] )
       + t[0+2*2] * ( t[1+0*2] - t[1+1*2] );

  if ( area == 0.0 )
  {
    cout << "\n";
    cout << "BASIS_ONE_T3 - Fatal error!\n";
    cout << "  Element has zero area.\n";
    cout << "  Area = " << area << "\n";
    cout << "\n";
    cout << "  Node 1: ( " << t[0+0*2] << ", " << t[1+0*2] << " )\n";
    cout << "  Node 2: ( " << t[0+1*2] << ", " << t[1+1*2] << " )\n";
    cout << "  Node 3: ( " << t[0+2*2] << ", " << t[1+2*2] << " )\n";
    exit ( 1 );
  }

  if ( i < 1 || 3 < i )
  {
    cout << "\n";
    cout << "BASIS_ONE_T3 - Fatal error!\n";
    cout << "  Basis index I is not between 1 and 3.\n";
    cout << "  I = " << i << "\n";
    exit ( 1 );
  }

  ip1 = i4_wrap ( i + 1, 1, 3 );
  ip2 = i4_wrap ( i + 2, 1, 3 );

  *qi = ( ( t[0+(ip2-1)*2] - t[0+(ip1-1)*2] )
        * ( p[1]           - t[1+(ip1-1)*2] )
        - ( t[1+(ip2-1)*2] - t[1+(ip1-1)*2] )
        * ( p[0]           - t[0+(ip1-1)*2] ) ) / area;

  *dqidx = - ( t[1+(ip2-1)*2] - t[1+(ip1-1)*2] ) / area;
  *dqidy =   ( t[0+(ip2-1)*2] - t[0+(ip1-1)*2] ) / area;

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

char ch_cap ( char c )

//****************************************************************************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 C, the character to capitalize.
//
//    Output, char CH_CAP, the capitalized character.
//
{
  if ( 97 <= c && c <= 122 )
  {
    c = c - 32;
  }

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

bool ch_eqi ( char c1, char c2 )

//****************************************************************************80
//
//  Purpose:
//
//    CH_EQI is true if two characters are equal, disregarding case.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    13 June 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, char C1, char C2, the characters to compare.
//
//    Output, bool CH_EQI, is true if the two characters are equal,
//    disregarding case.
//
{
  if ( 97 <= c1 && c1 <= 122 )
  {
    c1 = c1 - 32;
  }
  if ( 97 <= c2 && c2 <= 122 )
  {
    c2 = c2 - 32;
  }

  return ( c1 == c2 );
}
//****************************************************************************80

int ch_to_digit ( char c )

//****************************************************************************80
//
//  Purpose:
//
//    CH_TO_DIGIT returns the integer value of a base 10 digit.
//
//  Example:
//
//     C   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 C, the decimal digit, '0' through '9' or blank are legal.
//
//    Output, int CH_TO_DIGIT, the corresponding integer value.  If C was
//    'illegal', then DIGIT is -1.
//
{
  int digit;

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

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

void dirichlet_apply_dsp ( int node_num, double node_xy[], int node_condition[],
  int nz_num, int ia[], int ja[], double a[], double f[] )

//****************************************************************************80
//
//  Purpose:
//
//    DIRICHLET_APPLY_DSP accounts for Dirichlet boundary conditions.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    13 July 2007
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int NODE_NUM, the number of nodes.
//
//    Input, double NODE_XY[2*NODE_NUM], the coordinates of nodes.
//
//    Input, int NODE_CONDITION[NODE_NUM], reports the condition
//    used to set the unknown associated with the node.
//    0, unknown.
//    1, finite element equation.
//    2, Dirichlet condition;
//    3, Neumann condition.
//
//    Input, int NZ_NUM, the number of nonzero entries.
//
//    Input, int IA[NZ_NUM], JA[NZ_NUM], the row and column
//    indices of the nonzero entries.
//
//    Input/output, double A[NZ_NUM], the nonzero entries of the matrix.
//    On output, adjusted to account for Dirichlet boundary conditions.
//
//    Input/output, double F[NODE_NUM], the right hand side.
//    On output, adjusted to account for Dirichlet boundary conditions.
//
{
  int column;
  int DIRICHLET = 2;
  int node;
  double *node_bc;
  int nz;

  node_bc = new double[node_num];

  dirichlet_condition ( node_num, node_xy, node_bc );
//
//  Consider every matrix entry, NZ.
//
//  If the row I corresponds to a boundary node, then
//  zero out all off diagonal matrix entries, set the diagonal to 1,
//  and the right hand side to the Dirichlet boundary condition value.
//
  for ( nz = 0; nz < nz_num; nz++ )
  {
    node = ia[nz];

    if ( node_condition[node-1] == DIRICHLET )
    {
      column = ja[nz];

      if ( column == node )
      {
        a[nz] = 1.0;
        f[node-1] = node_bc[node-1];
      }
      else
      {
        a[nz] = 0.0;
      }
    }
  }

  delete [] node_bc;

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

int dsp_ij_to_k ( int nz_num, int row[], int col[], int i, int j )

//****************************************************************************80
//
//  Purpose:
//
//    DSP_IJ_TO_K seeks the compressed index of the (I,J) entry of A.
//
//  Discussion:
//
//    If A(I,J) is nonzero, then its value is stored in location K.
//
//    This routine searches the DSP storage structure for the index K
//    corresponding to (I,J), returning -1 if no such entry was found.
//
//    This routine assumes that the data structure has been sorted,
//    so that the entries of ROW are ascending sorted, and that the
//    entries of COL are ascending sorted, within the group of entries
//    that have a common value of ROW.
//
//    The DSP storage format stores the row, column and value of each nonzero
//    entry of a sparse matrix.
//
//    The DSP format is used by CSPARSE ("sparse triplet"), DLAP/SLAP
//    ("nonsymmetric SLAP triad"), by MATLAB, and by SPARSEKIT ("COO" format).
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    11 July 2007
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int NZ_NUM, the number of nonzero elements in
//    the matrix.
//
//    Input, int ROW[NZ_NUM], COL[NZ_NUM], the row and
//    column indices of the nonzero elements.
//
//    Input, int I, J, the row and column indices of the
//    matrix entry.
//
//    Output, int DSP_IJ_TO_K, the DSP index of the (I,J) entry.
//
{
  int hi;
  int k;
  int lo;
  int md;

  lo = 1;
  hi = nz_num;

  for ( ; ; )
  {
    if ( hi < lo )
    {
      k = -1;
      break;
    }

    md = ( lo + hi ) / 2;

    if ( row[md-1] < i || ( row[md-1] == i && col[md-1] < j ) )
    {
      lo = md + 1;
    }
    else if ( i < row[md-1] || ( row[md-1] == i && j < col[md-1] ) )
    {
      hi = md - 1;
    }
    else
    {
      k = md;
      break;
    }
  }

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

void dsp_print_some ( int m, int n, int nz_num, int row[], int col[],
  double a[], int ilo, int jlo, int ihi, int jhi, string title )

//****************************************************************************80
//
//  Purpose:
//
//    DSP_PRINT_SOME prints some of a DSP matrix.
//
//  Discussion:
//
//    This version of DSP_PRINT_SOME has been specifically modified to allow,
//    and correctly handle, the case in which a single matrix location
//    A(I,J) is referenced more than once by the sparse matrix structure.
//    In such cases, the routine prints out the sum of all the values.
//
//    The DSP storage format stores the row, column and value of each nonzero
//    entry of a sparse matrix.
//
//    It is possible that a pair of indices (I,J) may occur more than
//    once.  Presumably, in this case, the intent is that the actual value
//    of A(I,J) is the sum of all such entries.  This is not a good thing
//    to do, but I seem to have come across this in MATLAB.
//
//    The DSP format is used by CSPARSE ("sparse triplet"), DLAP/SLAP
//    (nonsymmetric case), by MATLAB, and by SPARSEKIT ("COO" format).
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    06 September 2006
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns of the matrix.
//
//    Input, int NZ_NUM, the number of nonzero elements in the matrix.
//
//    Input, int ROW[NZ_NUM], COL[NZ_NUM], the row and column indices
//    of the nonzero elements.
//
//    Input, double A[NZ_NUM], the nonzero elements of the matrix.
//
//    Input, int ILO, JLO, IHI, JHI, the first row and
//    column, and the last row and column to be printed.
//
//    Input, string TITLE, a title to print.
//
{
# define INCX 5

  double aij[INCX];
  int i;
  int i2hi;
  int i2lo;
  int inc;
  int j;
  int j2;
  int j2hi;
  int j2lo;
  int k;
  bool nonzero;

  cout << "\n";
  cout << title << "\n";
//
//  Print the columns of the matrix, in strips of 5.
//
  for ( j2lo = jlo; j2lo <= jhi; j2lo = j2lo + INCX )
  {
    j2hi = j2lo + INCX - 1;
    j2hi = i4_min ( j2hi, n );
    j2hi = i4_min ( j2hi, jhi );

    inc = j2hi + 1 - j2lo;

    cout << "\n";

    cout << "  Col:  ";
    for ( j = j2lo; j <= j2hi; j++ )
    {
      cout << setw(7) << j << "       ";
    }
    cout << "\n";
    cout << "  Row\n";
    cout << "  ---\n";
//
//  Determine the range of the rows in this strip.
//
    i2lo = i4_max ( ilo, 1 );
    i2hi = i4_min ( ihi, m );

    for ( i = i2lo; i <= i2hi; i++ )
    {
//
//  Print out (up to) 5 entries in row I, that lie in the current strip.
//
      nonzero = false;
      for ( j2 = 0; j2 < INCX; j2++ )
      {
        aij[j2] = 0.0;
      }

      for ( k = 1; k <= nz_num; k++ )
      {
        if ( i == row[k-1] && j2lo <= col[k-1] && col[k-1] <= j2hi )
        {
          j2 = col[k-1] - j2lo;

          if ( a[k-1] == 0.0 )
          {
            continue;
          }

          nonzero = true;
          aij[j2] = aij[j2] + a[k-1];
        }
      }

      if ( nonzero )
      {
        cout << setw(6) << i;
        for ( j2 = 0; j2 < inc; j2++ )
        {
          cout << setw(12) << aij[j2] << "  ";
        }
        cout << "\n";
      }
    }
  }

  return;
# undef INCX
}
//****************************************************************************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";
    exit ( 1 );
  }
//
//  Read one line, but skip blank lines and comment lines.
//
  got_one = false;

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

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

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

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

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

    input.open ( filename.c_str ( ) );

    for ( ; ; )
    {
      input >> text;

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

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

  input.close ( );

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

  column_num = s_word_count ( text );

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

int file_row_count ( string input_filename )

//****************************************************************************80
//
//  Purpose:
//
//    FILE_ROW_COUNT counts the number of row records in a file.
//
//  Discussion:
//
//    It does not count lines that are blank, or that begin with a
//    comment symbol '#'.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    23 February 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string INPUT_FILENAME, the name of the input file.
//
//    Output, int FILE_ROW_COUNT, the number of rows found.
//
{
  int bad_num;
  int comment_num;
  ifstream input;
  int i;
  string line;
  int record_num;
  int row_num;

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

  input.open ( input_filename.c_str ( ) );

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

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

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

    record_num = record_num + 1;

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

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

    row_num = row_num + 1;

  }

  input.close ( );

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

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 integer 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   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 integer to lie between given limits by wrapping.
//
//  Example:
//
//    ILO = 4, IHI = 8
//
//    I  I4_WRAP
//
//    -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:
//
//    An I4MAT is an array of I4's.
//
//    The file is assumed to contain one record per line.
//
//    Records beginning with '#' are comments, and are ignored.
//    Blank lines are also ignored.
//
//    Each line that is not ignored is assumed to contain exactly (or at least)
//    M real numbers, representing the coordinates of a point.
//
//    There are assumed to be exactly (or at least) N such records.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    23 February 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string INPUT_FILENAME, the name of the input file.
//
//    Input, int M, the number of spatial dimensions.
//
//    Input, int N, the number of points.  The program
//    will stop reading data once N values have been read.
//
//    Output, int I4MAT_DATA_READ[M*N], the 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";
    exit ( 1 );
  }

  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.
//
//  Discussion:
//
//    An I4MAT is an array of I4's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    23 February 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string INPUT_FILENAME, the name of the input file.
//
//    Output, int *M, the number of spatial dimensions.
//
//    Output, int *N, the number of points
//
{
  *m = file_column_count ( input_filename );

  if ( *m <= 0 )
  {
    cerr << "\n";
    cerr << "I4MAT_HEADER_READ - Fatal error!\n";
    cerr << "  FILE_COLUMN_COUNT failed.\n";
    exit ( 1 );
  }

  *n = file_row_count ( input_filename );

  if ( *n <= 0 )
  {
    cerr << "\n";
    cerr << "I4MAT_HEADER_READ - Fatal error!\n";
    cerr << "  FILE_ROW_COUNT failed.\n";
    exit ( 1 );
  }

  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.
//
//  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 for the matrix.
{
# 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

int i4vec2_compare ( int n, int a1[], int a2[], int i, int j )

//****************************************************************************80
//
//  Purpose:
//
//    I4VEC2_COMPARE compares pairs of integers stored in two vectors.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    11 September 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of data items.
//
//    Input, int A1[N], A2[N], contain the two components of each item.
//
//    Input, int I, J, the items to be compared.  These values will be
//    1-based indices for the arrays A1 and A2.
//
//    Output, int I4VEC2_COMPARE, the results of the comparison:
//    -1, item I < item J,
//     0, item I = item J,
//    +1, item J < item I.
//
{
  int isgn;

  isgn = 0;

       if ( a1[i-1] < a1[j-1] )
  {
    isgn = -1;
  }
  else if ( a1[i-1] == a1[j-1] )
  {
         if ( a2[i-1] < a2[j-1] )
    {
      isgn = -1;
    }
    else if ( a2[i-1] < a2[j-1] )
    {
      isgn = 0;
    }
    else if ( a2[j-1] < a2[i-1] )
    {
      isgn = +1;
    }
  }
  else if ( a1[j-1] < a1[i-1] )
  {
    isgn = +1;
  }

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

void i4vec2_sort_a ( int n, int a1[], int a2[] )

//****************************************************************************80
//
//  Purpose:
//
//    I4VEC2_SORT_A ascending sorts a vector of pairs of integers.
//
//  Discussion:
//
//    Each item to be sorted is a pair of integers (I,J), with the I
//    and J values stored in separate vectors A1 and A2.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    11 September 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of items of data.
//
//    Input/output, int A1[N], A2[N], the data to be sorted..
//
{
  int i;
  int indx;
  int isgn;
  int j;
  int temp;
//
//  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 )
    {
      temp    = a1[i-1];
      a1[i-1] = a1[j-1];
      a1[j-1] = temp;

      temp    = a2[i-1];
      a2[i-1] = a2[j-1];
      a2[j-1] = temp;
    }
//
//  Compare the I and J objects.
//
    else if ( indx < 0 )
    {
      isgn = i4vec2_compare ( n, a1, a2, i, j );
    }
    else if ( indx == 0 )
    {
      break;
    }

  }

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

void mgmres ( double a[], int ia[], int ja[], double x[], double rhs[],
  int n, int nz_num, int itr_max, int mr, double tol_abs, double tol_rel )

//****************************************************************************80
//
//  Purpose:
//
//    MGMRES applies the restarted GMRES iteration to a linear system.
//
//  Discussion:
//
//    The linear system A*X=B is solved iteratively.
//
//    The matrix A is assumed to be sparse.  To save on storage, only
//    the nonzero entries of A are stored.  For instance, the K-th nonzero
//    entry in the matrix is stored by:
//
//      A(K) = value of entry,
//      IA(K) = row of entry,
//      JA(K) = column of entry.
//
//    The "matrices" H and V are treated as one-dimensional vectors
//    which store the matrix data in row major form.
//
//    This requires that references to H[I][J] be replaced by references
//    to H[I+J*(MR+1)] and references to V[I][J] by V[I+J*N].
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    13 July 2007
//
//  Author:
//
//    Original C version by Lili Ju.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Richard Barrett, Michael Berry, Tony Chan, James Demmel,
//    June Donato, Jack Dongarra, Victor Eijkhout, Roidan Pozo,
//    Charles Romine, Henk van der Vorst,
//    Templates for the Solution of Linear Systems:
//    Building Blocks for Iterative Methods,
//    SIAM, 1994,
//    ISBN: 0898714710,
//    LC: QA297.8.T45.
//
//    Tim Kelley,
//    Iterative Methods for Linear and Nonlinear Equations,
//    SIAM, 2004,
//    ISBN: 0898713528,
//    LC: QA297.8.K45.
//
//    Yousef Saad,
//    Iterative Methods for Sparse Linear Systems,
//    Second Edition,
//    SIAM, 2003,
//    ISBN: 0898715342,
//    LC: QA188.S17.
//
//  Parameters:
//
//    Input, double A[NZ_NUM], the matrix values.
//
//    Input, int IA[NZ_NUM], JA[NZ_NUM], the row and column indices
//    of the matrix values.
//
//    Input/output, double X[N]; on input, an approximation to
//    the solution.  On output, an improved approximation.
//
//    Input, double RHS[N], the right hand side of the linear system.
//
//    Input, int N, the order of the linear system.
//
//    Input, int NZ_NUM, the number of nonzero matrix values.
//
//    Input, int ITR_MAX, the maximum number of (outer) iterations to take.
//
//    Input, int MR, the maximum number of (inner) iterations to take.
//    MR must be less than N.
//
//    Input, double TOL_ABS, an absolue tolerance applied to the
//    current residual.
//
//    Input, double TOL_REL, a relative tolerance comparing the
//    current residual to the initial residual.
//
{
  double av;
  double *c;
  double delta = 1.0e-03;
  double *g;
  double *h;
  double htmp;
  int i;
  int itr;
  int itr_used;
  int j;
  int k;
  int k_copy;
  double mu;
  double *r;
  double rho;
  double rho_tol;
  double *s;
  double *v;
  bool verbose = true;
  double *y;

  c = new double[mr];
  g = new double[mr+1];
  h = new double[(mr+1)*mr];
  r = new double[n];
  s = new double[mr];
  v = new double[n*(mr+1)];
  y = new double[mr+1];

  itr_used = 0;

  if ( n < mr )
  {
    cout << "\n";
    cout << "MGMRES - Fatal error!\n";
    cout << "  N < MR.\n";
    cout << "  N = " << n << "\n";
    cout << "  MR = " << mr << "\n";
    exit ( 1 );
  }

  for ( itr = 1; itr <= itr_max; itr++ )
  {
    ax ( a, ia, ja, x, r, n, nz_num );

    for ( i = 0; i < n; i++ )
    {
      r[i] = rhs[i] - r[i];
    }

    rho = sqrt ( r8vec_dot_product ( n, r, r ) );

    if ( verbose )
    {
      cout << "  ITR = " << itr << "  Residual = " << rho << "\n";
    }

    if ( itr == 1 )
    {
      rho_tol = rho * tol_rel;
    }

    for ( i = 0; i < n; i++)
    {
      v[i+0*n] = r[i] / rho;
    }

    g[0] = rho;
    for ( i = 1; i <= mr; i++ )
    {
      g[i] = 0.0;
    }

    for ( i = 0; i < mr+1; i++ )
    {
      for ( j = 0; j < mr; j++ )
      {
        h[i+j*(mr+1)] = 0.0;
      }
    }

    for ( k = 1; k <= mr; k++ )
    {
      k_copy = k;

      ax ( a, ia, ja, v+(k-1)*n, v+k*n, n, nz_num );

      av = sqrt ( r8vec_dot_product ( n, v+k*n, v+k*n ) );

      for ( j = 1; j <= k; j++ )
      {
        h[(j-1)+(k-1)*(mr+1)] = r8vec_dot_product ( n, v+k*n, v+(j-1)*n );
        for ( i = 0; i < n; i++ )
        {
          v[i+k*n] = v[i+k*n] - h[(j-1)+(k-1)*(mr+1)] * v[i+(j-1)*n];
        }
      }

      h[k+(k-1)*(mr+1)] = sqrt ( r8vec_dot_product ( n, v+k*n, v+k*n ) );

      if ( ( av + delta * h[k+(k-1)*(mr+1)] ) == av )
      {
         for ( j = 1; j <= k; j++ )
         {
           htmp = r8vec_dot_product ( n, v+k*n, v+(j-1)*n );
           h[(j-1)+(k-1)*(mr+1)] = h[(j-1)+(k-1)*(mr+1)] + htmp;
           for ( i = 0; i < n; i++ )
           {
             v[i+k*n] = v[i+k*n] - htmp * v[i+(j-1)*n];
           }
         }
         h[k+(k-1)*(mr+1)] = sqrt ( r8vec_dot_product ( n, v+k*n, v+k*n ) );
      }

      if ( h[k+(k-1)*(mr+1)] != 0.0 )
      {
        for ( i = 0; i < n; i++ )
        {
          v[i+k*n] = v[i+k*n] / h[k+(k-1)*(mr+1)];
        }
      }

      if ( 1 < k )
      {
        for ( i = 1; i <= k+1; i++ )
        {
          y[i-1] = h[(i-1)+(k-1)*(mr+1)];
        }
        for ( j = 1; j <= k - 1; j++ )
        {
          mult_givens ( c[j-1], s[j-1], j-1, y );
        }
        for ( i = 1; i <= k+1; i++ )
        {
          h[i-1+(k-1)*(mr+1)] = y[i-1];
        }
      }
      mu = sqrt ( pow ( h[(k-1)+(k-1)*(mr+1)], 2 )
                + pow ( h[ k   +(k-1)*(mr+1)], 2 ) );
      c[k-1] =  h[(k-1)+(k-1)*(mr+1)] / mu;
      s[k-1] = -h[ k   +(k-1)*(mr+1)] / mu;
      h[(k-1)+(k-1)*(mr+1)] = c[k-1] * h[(k-1)+(k-1)*(mr+1)]
                            - s[k-1] * h[ k   +(k-1)*(mr+1)];
      h[k+(k-1)*(mr+1)] = 0;
      mult_givens ( c[k-1], s[k-1], k-1, g );

      rho = fabs ( g[k] );

      itr_used = itr_used + 1;

      if ( verbose )
      {
        cout << "  K =   " << k << "  Residual = " << rho << "\n";
      }

      if ( rho <= rho_tol && rho <= tol_abs )
      {
        break;
      }
    }

    k = k_copy - 1;
    y[k] = g[k] / h[k+k*(mr+1)];

    for ( i = k; 1 <= i; i-- )
    {
      y[i-1] = g[i-1];
      for ( j = i+1; j <= k+1; j++ )
      {
        y[i-1] = y[i-1] - h[(i-1)+(j-1)*(mr+1)] * y[j-1];
      }
      y[i-1] = y[i-1] / h[(i-1)+(i-1)*(mr+1)];
    }

    for ( i = 1; i <= n; i++ )
    {
      for ( j = 1; j <= k + 1; j++ )
      {
        x[i-1] = x[i-1] + v[(i-1)+(j-1)*n] * y[j-1];
      }
    }

    if ( rho <= rho_tol && rho <= tol_abs )
    {
      break;
    }
  }

  if ( verbose )
  {
    cout << "\n";
    cout << "MGMRES\n";
    cout << "  Number of iterations = " << itr_used << "\n";
    cout << "  Final residual = " << rho << "\n";
  }

  delete [] c;
  delete [] g;
  delete [] h;
  delete [] r;
  delete [] s;
  delete [] v;
  delete [] y;

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

void mult_givens ( double c, double s, int k, double g[] )

//****************************************************************************80
//
//  Purpose:
//
//    MULT_GIVENS applies a Givens rotation to two successive entries of a vector.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    08 August 2006
//
//  Author:
//
//    Original C version by Lili Ju.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Richard Barrett, Michael Berry, Tony Chan, James Demmel,
//    June Donato, Jack Dongarra, Victor Eijkhout, Roidan Pozo,
//    Charles Romine, Henk van der Vorst,
//    Templates for the Solution of Linear Systems:
//    Building Blocks for Iterative Methods,
//    SIAM, 1994,
//    ISBN: 0898714710,
//    LC: QA297.8.T45.
//
//    Tim Kelley,
//    Iterative Methods for Linear and Nonlinear Equations,
//    SIAM, 2004,
//    ISBN: 0898713528,
//    LC: QA297.8.K45.
//
//    Yousef Saad,
//    Iterative Methods for Sparse Linear Systems,
//    Second Edition,
//    SIAM, 2003,
//    ISBN: 0898715342,
//    LC: QA188.S17.
//
//  Parameters:
//
//    Input, double C, S, the cosine and sine of a Givens
//    rotation.
//
//    Input, int K, indicates the location of the first vector entry.
//
//    Input/output, double G[K+2], the vector to be modified.  On output,
//    the Givens rotation has been applied to entries G(K) and G(K+1).
//
{
  double g1;
  double g2;

  g1 = c * g[k] - s * g[k+1];
  g2 = s * g[k] + c * g[k+1];

  g[k]   = g1;
  g[k+1] = g2;

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

void quad_rule ( int quad_num, double quad_w[], double quad_xy[] )

//****************************************************************************80
//
//  Purpose:
//
//    QUAD_RULE sets the quadrature rule for assembly.
//
//  Discussion:
//
//    The quadrature rule is given for a reference element.
//
//      0 <= X,
//      0 <= Y, and
//      X + Y <= 1.
//
//      ^
//    1 | *
//      | |.
//    Y | | .
//      | |  .
//    0 | *---*
//      +------->
//        0 X 1
//
//    The rules have the following precision:
//
//    QUAD_NUM  Precision
//
//     1        1
//     3        2
//     4        3
//     6        4
//     7        5
//     9        6
//    13        7
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    06 January 2006
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int QUAD_NUM, the number of quadrature nodes.
//
//    Output, double QUAD_W[QUAD_NUM], the quadrature weights.
//
//    Output, double QUAD_XY[2*QUAD_NUM],
//    the coordinates of the quadrature nodes.
//
{
  double a;
  double b;
  double c;
  double d;
  double e;
  double f;
  double g;
  double h;
  double t;
  double u;
  double v;
  double w;

  if ( quad_num == 1 )
  {
    quad_xy[0+0*2] = 1.0 / 3.0;
    quad_xy[1+0*2] = 1.0 / 3.0;

    quad_w[0] = 1.0;
  }
  else if ( quad_num == 3 )
  {
    quad_xy[0+0*2] = 0.5;
    quad_xy[1+0*2] = 0.0;
    quad_xy[0+1*2] = 0.5;
    quad_xy[1+1*2] = 0.5;
    quad_xy[0+2*2] = 0.0;
    quad_xy[1+2*2] = 0.5;

    quad_w[0] = 1.0 / 3.0;
    quad_w[1] = 1.0 / 3.0;
    quad_w[2] = 1.0 / 3.0;
  }
  else if ( quad_num == 4 )
  {
    a =   6.0 / 30.0;
    b =  10.0 / 30.0;
    c =  18.0 / 30.0;

    d =  25.0 / 48.0;
    e = -27.0 / 48.0;

    quad_xy[0+0*2] = b;
    quad_xy[1+0*2] = b;
    quad_xy[0+1*2] = c;
    quad_xy[1+1*2] = a;
    quad_xy[0+2*2] = a;
    quad_xy[1+2*2] = c;
    quad_xy[0+3*2] = a;
    quad_xy[1+3*2] = a;

    quad_w[0] = e;
    quad_w[1] = d;
    quad_w[2] = d;
    quad_w[3] = d;
  }
  else if ( quad_num == 6 )
  {
    a = 0.816847572980459;
    b = 0.091576213509771;
    c = 0.108103018168070;
    d = 0.445948490915965;
    v = 0.109951743655322;
    w = 0.223381589678011;

    quad_xy[0+0*2] = a;
    quad_xy[1+0*2] = b;
    quad_xy[0+1*2] = b;
    quad_xy[1+1*2] = a;
    quad_xy[0+2*2] = b;
    quad_xy[1+2*2] = b;
    quad_xy[0+3*2] = c;
    quad_xy[1+3*2] = d;
    quad_xy[0+4*2] = d;
    quad_xy[1+4*2] = c;
    quad_xy[0+5*2] = d;
    quad_xy[1+5*2] = d;

    quad_w[0] = v;
    quad_w[1] = v;
    quad_w[2] = v;
    quad_w[3] = w;
    quad_w[4] = w;
    quad_w[5] = w;
  }
  else if ( quad_num == 7 )
  {
    a = 1.0 / 3.0;
    b = ( 9.0 + 2.0 * sqrt ( 15.0 ) ) / 21.0;
    c = ( 6.0 -       sqrt ( 15.0 ) ) / 21.0;
    d = ( 9.0 - 2.0 * sqrt ( 15.0 ) ) / 21.0;
    e = ( 6.0 +       sqrt ( 15.0 ) ) / 21.0;
    u = 0.225;
    v = ( 155.0 - sqrt ( 15.0 ) ) / 1200.0;
    w = ( 155.0 + sqrt ( 15.0 ) ) / 1200.0;

    quad_xy[0+0*2] = a;
    quad_xy[1+0*2] = a;
    quad_xy[0+1*2] = b;
    quad_xy[1+1*2] = c;
    quad_xy[0+2*2] = c;
    quad_xy[1+2*2] = b;
    quad_xy[0+3*2] = c;
    quad_xy[1+3*2] = c;
    quad_xy[0+4*2] = d;
    quad_xy[1+4*2] = e;
    quad_xy[0+5*2] = e;
    quad_xy[1+5*2] = d;
    quad_xy[0+6*2] = e;
    quad_xy[1+6*2] = e;

    quad_w[0] = u;
    quad_w[1] = v;
    quad_w[2] = v;
    quad_w[3] = v;
    quad_w[4] = w;
    quad_w[5] = w;
    quad_w[6] = w;
  }
  else if ( quad_num == 9 )
  {
    a = 0.124949503233232;
    b = 0.437525248383384;
    c = 0.797112651860071;
    d = 0.165409927389841;
    e = 0.037477420750088;

    u = 0.205950504760887;
    v = 0.063691414286223;

    quad_xy[0+0*2] = a;
    quad_xy[1+0*2] = b;
    quad_xy[0+1*2] = b;
    quad_xy[1+1*2] = a;
    quad_xy[0+2*2] = b;
    quad_xy[1+2*2] = b;
    quad_xy[0+3*2] = c;
    quad_xy[1+3*2] = d;
    quad_xy[0+4*2] = c;
    quad_xy[1+4*2] = e;
    quad_xy[0+5*2] = d;
    quad_xy[1+5*2] = c;
    quad_xy[0+6*2] = d;
    quad_xy[1+6*2] = e;
    quad_xy[0+7*2] = e;
    quad_xy[1+7*2] = c;
    quad_xy[0+8*2] = e;
    quad_xy[1+8*2] = d;

    quad_w[0] = u;
    quad_w[1] = u;
    quad_w[2] = u;
    quad_w[3] = v;
    quad_w[4] = v;
    quad_w[5] = v;
    quad_w[6] = v;
    quad_w[7] = v;
    quad_w[8] = v;
  }
  else if ( quad_num == 13 )
  {
    h = 1.0 / 3.0;
    a = 0.479308067841923;
    b = 0.260345966079038;
    c = 0.869739794195568;
    d = 0.065130102902216;
    e = 0.638444188569809;
    f = 0.312865496004875;
    g = 0.048690315425316;

    w = -0.149570044467670;
    t =  0.175615257433204;
    u =  0.053347235608839;
    v =  0.077113760890257;

    quad_xy[0+ 0*2] = h;
    quad_xy[1+ 0*2] = h;
    quad_xy[0+ 1*2] = a;
    quad_xy[1+ 1*2] = b;
    quad_xy[0+ 2*2] = b;
    quad_xy[1+ 2*2] = a;
    quad_xy[0+ 3*2] = b;
    quad_xy[1+ 3*2] = b;

    quad_xy[0+ 4*2] = c;
    quad_xy[1+ 4*2] = d;
    quad_xy[0+ 5*2] = d;
    quad_xy[1+ 5*2] = c;
    quad_xy[0+ 6*2] = d;
    quad_xy[1+ 6*2] = d;

    quad_xy[0+ 7*2] = e;
    quad_xy[1+ 7*2] = f;
    quad_xy[0+ 8*2] = e;
    quad_xy[1+ 8*2] = g;
    quad_xy[0+ 9*2] = f;
    quad_xy[1+ 9*2] = e;

    quad_xy[0+10*2] = f;
    quad_xy[1+10*2] = g;
    quad_xy[0+11*2] = g;
    quad_xy[1+11*2] = e;
    quad_xy[0+12*2] = g;
    quad_xy[1+12*2] = f;

    quad_w[ 0] = w;
    quad_w[ 1] = t;
    quad_w[ 2] = t;
    quad_w[ 3] = t;
    quad_w[ 4] = u;
    quad_w[ 5] = u;
    quad_w[ 6] = u;
    quad_w[ 7] = v;
    quad_w[ 8] = v;
    quad_w[ 9] = v;
    quad_w[10] = v;
    quad_w[11] = v;
    quad_w[12] = v;
  }
  else
  {
    cout << "\n";
    cout << "QUAD_RULE - Fatal error!\n";
    cout << "  No rule is available of order QUAD_NUM = " << quad_num << "\n";
    exit ( 1 );
  }
  return;
}
//****************************************************************************80

double r8_abs ( double x )

//****************************************************************************80
//
//  Purpose:
//
//    R8_ABS returns the absolute value of an R8.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    02 April 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double X, the quantity whose absolute value is desired.
//
//    Output, double R8_ABS, the absolute value of X.
//
{
  if ( 0.0 <= x )
  {
    return x;
  }
  else
  {
    return ( -x );
  }
}
//****************************************************************************80

double r8_huge ( void )

//****************************************************************************80
//
//  Purpose:
//
//    R8_HUGE returns a "huge" R8.
//
//  Discussion:
//
//    HUGE_VAL is the largest representable legal double precision number, 
//    and is usually defined in math.h, or sometimes in stdlib.h.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    31 August 2004
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Output, double R8_HUGE, a "huge" R8.
//
{
  return HUGE_VAL;
}
//****************************************************************************80

int r8_nint ( double x )

//****************************************************************************80
//
//  Purpose:
//
//    R8_NINT returns the nearest integer to an R8.
//
//  Example:
//
//        X         R8_NINT
//
//      1.3         1
//      1.4         1
//      1.5         1 or 2
//      1.6         2
//      0.0         0
//     -0.7        -1
//     -1.1        -1
//     -1.6        -2
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    26 August 2004
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double X, the value.
//
//    Output, int R8_NINT, the nearest integer to X.
//
{
  int s;
  int value;

  if ( x < 0.0 )
  {
    s = -1;
  }
  else
  {
    s = 1;
  }
  value = s * ( int ) ( fabs ( x ) + 0.5 );

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

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

//****************************************************************************80
//
//  Purpose:
//
//    R8MAT_DATA_READ reads the data from an R8MAT file.
//
//  Discussion:
//
//    An R8MAT is an array of R8's.
//
//    The file is assumed to contain one record per line.
//
//    Records beginning with '#' are comments, and are ignored.
//    Blank lines are also ignored.
//
//    Each line that is not ignored is assumed to contain exactly (or at least)
//    M real numbers, representing the coordinates of a point.
//
//    There are assumed to be exactly (or at least) N such records.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    23 February 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string INPUT_FILENAME, the name of the input file.
//
//    Input, int M, the number of spatial dimensions.
//
//    Input, int N, the number of points.  The program
//    will stop reading data once N values have been read.
//
//    Output, double R8MAT_DATA_READ[M*N], the 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";
    exit ( 1 );
  }

  table = new double[m*n];

  x = new double[m];

  j = 0;

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

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

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

    error = s_to_r8vec ( line, m, x );

    if ( error )
    {
      continue;
    }

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

  }

  input.close ( );

  delete [] x;

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

void r8mat_header_read ( string input_filename, int *m, int *n )

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

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

  *n = file_row_count ( input_filename );

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

  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 double precision values, which
//    may be stored as a vector in column-major order.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    11 August 2004
//
//  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

void r8mat_write ( string output_filename, int m, int n, double table[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8MAT_WRITE writes an R8MAT file.
//
//  Discussion:
//
//    An R8MAT is an array of R8's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    29 June 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string OUTPUT_FILENAME, the output filename.
//
//    Input, int M, the spatial dimension.
//
//    Input, int N, the number of points.
//
//    Input, double TABLE[M*N], the data.
//
{
  int i;
  int j;
  ofstream output;
//
//  Open the file.
//
  output.open ( output_filename.c_str ( ) );

  if ( !output )
  {
    cerr << "\n";
    cerr << "R8MAT_WRITE - Fatal error!\n";
    cerr << "  Could not open the output file \"" << output_filename << "\".\n";
    exit ( 1 );
  }
//
//  Write the data.
//
  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      output << "  " << setw(24) << setprecision(16) << table[i+j*m];
    }
    output << "\n";
  }
//
//  Close the file.
//
  output.close ( );

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

double r8vec_amax ( int n, double a[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_AMAX returns the maximum absolute value in an R8VEC.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    18 September 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries in the array.
//
//    Input, double A[N], the array.
//
//    Output, double AMAX, the value of the entry
//    of largest magnitude.
//
{
  double amax;
  int i;

  amax = 0.0;
  for ( i = 0; i < n; i++ )
  {
    if ( amax < fabs ( a[i] ) )
    {
      amax = fabs ( a[i] );
    }
  }

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

double r8vec_dot_product ( int n, double a1[], double a2[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_DOT_PRODUCT computes the dot product of a pair of R8VEC's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    03 July 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries in the vectors.
//
//    Input, double A1[N], A2[N], the two vectors to be considered.
//
//    Output, double R8VEC_DOT_PRODUCT, the dot product of the vectors.
//
{
  int i;
  double value;

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

void r8vec_print_some ( int n, double a[], int i_lo, int i_hi, string title )

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_PRINT_SOME prints "some" of an R8VEC.
//
//  Discussion:
//
//    An R8VEC is a vector of R8 values.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    16 October 2006
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries of the vector.
//
//    Input, double A[N], the vector to be printed.
//
//    Input, integer I_LO, I_HI, the first and last indices to print.
//    The routine expects 1 <= I_LO <= I_HI <= N.
//
//    Input, string TITLE, a title.
//
{
  int i;

  cout << "\n";
  cout << title << "\n";
  cout << "\n";
  for ( i = i4_max ( 1, i_lo ); i <= i4_min ( n, i_hi ); i++ )
  {
    cout << "  " << setw(8)  << i       << "  "
         << "  " << setw(14) << a[i-1]  << "\n";
  }

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

double *r8vec_uniform_01 ( int n, int *seed )

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_UNIFORM_01 returns a unit pseudorandom R8VEC.
//
//  Discussion:
//
//    This routine implements the recursion
//
//      seed = 16807 * seed mod ( 2^31 - 1 )
//      unif = seed / ( 2^31 - 1 )
//
//    The integer arithmetic never requires more than 32 bits,
//    including a sign bit.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    19 August 2004
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Paul Bratley, Bennett Fox, Linus Schrage,
//    A Guide to Simulation,
//    Springer Verlag, pages 201-202, 1983.
//
//    Bennett Fox,
//    Algorithm 647:
//    Implementation and Relative Efficiency of Quasirandom
//    Sequence Generators,
//    ACM Transactions on Mathematical Software,
//    Volume 12, Number 4, pages 362-376, 1986.
//
//    Peter Lewis, Allen Goodman, James Miller,
//    A Pseudo-Random Number Generator for the System/360,
//    IBM Systems Journal,
//    Volume 8, pages 136-143, 1969.
//
//  Parameters:
//
//    Input, int N, the number of entries in the vector.
//
//    Input/output, int *SEED, a seed for the random number generator.
//
//    Output, double R8VEC_UNIFORM_01[N], the vector of pseudorandom values.
//
{
  int i;
  int k;
  double *r;

  if ( *seed == 0 )
  {
    cerr << "\n";
    cerr << "R8VEC_UNIFORM_01 - Fatal error!\n";
    cerr << "  Input value of SEED = 0.\n";
    exit ( 1 );
  }

  r = new double[n];

  for ( i = 0; i < n; i++ )
  {
    k = *seed / 127773;

    *seed = 16807 * ( *seed - k * 127773 ) - k * 2836;

    if ( *seed < 0 )
    {
      *seed = *seed + 2147483647;
    }

    r[i] = ( double ) ( *seed ) * 4.656612875E-10;
  }

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

void reference_to_physical_t3 ( double t[2*3], int n, double ref[],
  double phy[] )

//****************************************************************************80
//
//  Purpose:
//
//    REFERENCE_TO_PHYSICAL_T3 maps reference points to physical points.
//
//  Discussion:
//
//    Given the vertices of an order 3 physical triangle and a point
//    (XSI,ETA) in the reference triangle, the routine computes the value
//    of the corresponding image point (X,Y) in physical space.
//
//    Note that this routine may also be appropriate for an order 6
//    triangle, if the mapping between reference and physical space
//    is linear.  This implies, in particular, that the sides of the
//    image triangle are straight and that the "midside" nodes in the
//    physical triangle are literally halfway along the sides of
//    the physical triangle.
//
//  Reference Element T3:
//
//    |
//    1  3
//    |  |.
//    |  | .
//    S  |  .
//    |  |   .
//    |  |    .
//    0  1-----2
//    |
//    +--0--R--1-->
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    24 June 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double T[2*3], the coordinates of the vertices.
//    The vertices are assumed to be the images of (0,0), (1,0) and
//    (0,1) respectively.
//
//    Input, int N, the number of objects to transform.
//
//    Input, double REF[2*N], points in the reference triangle.
//
//    Output, double PHY[2*N], corresponding points in the
//    physical triangle.
//
{
  int i;
  int j;

  for ( i = 0; i < 2; i++ )
  {
    for ( j = 0; j < n; j++ )
    {
      phy[i+j*2] = t[i+0*2] * ( 1.0 - ref[0+j*2] - ref[1+j*2] )
                 + t[i+1*2] *       + ref[0+j*2]
                 + t[i+2*2] *                    + ref[1+j*2];
    }
  }

  return;
}
//****************************************************************************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_l ( string s )

//****************************************************************************80
//
//  Purpose:
//
//    S_TO_L reads an L from a string.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    03 December 2010
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string S, the string to be read.
//
//    Output, bool S_TO_L, the value of the L.
//
{
  int i;
  bool l;
  int length;

  length = s.length ( );

  if ( length < 1 )
  {
    cerr << "\n";
    cerr << "S_TO_L - Fatal error!\n";
    cerr << "  Input string is empty.\n";
    exit ( 1 );
  }

  for ( i = 0; i < length; i++ )
  {
    if ( s[i] == '0' ||
         s[i] == 'f' ||
         s[i] == 'F' )
    {
      l = false;
      return l;
    }
    else if ( s[i] == '1' ||
              s[i] == 't' ||
              s[i] == 'T' )
    {
      l = true;
      return l;
    }
  }
  cerr << "\n";
  cerr << "S_TO_L - Fatal error!\n";
  cerr << "  Input did not contain boolean data.\n";
  exit ( 1 );
}
//****************************************************************************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 solution_evaluate ( double xy[2], double t[2*3], double node_u[3],
  double *u, double *dudx, double *dudy )

//****************************************************************************80
//
//  Purpose:
//
//    SOLUTION_EVALUATE evaluates the solution at a point in a triangle.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    06 January 2006
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double XY[2], the point where the solution is
//    to be evaluated.
//
//    Input, double T[2*3], the coordinates of the vertices
//    of the triangle which contains XY.
//
//    Input, double NODE_U[3], the value of the solution
//    at the nodes of the triangle.
//
//    Output, double U, DUDX, DUDY, the solution and its X and
//    Y derivatives at XY.
//
{
  double b;
  double dbdx;
  double dbdy;
  int i;

  *u = 0.0;
  *dudx = 0.0;
  *dudy = 0.0;

  for ( i = 1; i <= 3; i++ )
  {
    basis_one_t3 ( t, i, xy, &b, &dbdx, &dbdy );

    *u    = *u    + node_u[i-1] * b;
    *dudx = *dudx + node_u[i-1] * dbdx;
    *dudy = *dudy + node_u[i-1] * dbdy;
  }
  return;
}
//****************************************************************************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 and 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 ( void )

//****************************************************************************80
//
//  Purpose:
//
//    TIMESTAMP prints the current YMDHMS date as a time stamp.
//
//  Example:
//
//    May 31 2001 09:45:54 AM
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    02 October 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    None
//
{
# define TIME_SIZE 40

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

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

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

  cout << time_buffer << "\n";

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

double triangle_area_2d ( double t[2*3] )

//****************************************************************************80
//
//  Purpose:
//
//    TRIANGLE_AREA_2D computes the area of a triangle in 2D.
//
//  Discussion:
//
//    If the triangle's vertices are given in counterclockwise order,
//    the area will be positive.  If the triangle's vertices are given
//    in clockwise order, the area will be negative!
//
//    An earlier version of this routine always returned the absolute
//    value of the computed area.  I am convinced now that that is
//    a less useful result!  For instance, by returning the signed
//    area of a triangle, it is possible to easily compute the area
//    of a nonconvex polygon as the sum of the (possibly negative)
//    areas of triangles formed by node 1 and successive pairs of vertices.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    17 October 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double T[2*3], the vertices of the triangle.
//
//    Output, double TRIANGLE_AREA_2D, the area of the triangle.
//
{
  double area;

  area = 0.5 * (
    t[0+0*2] * ( t[1+1*2] - t[1+2*2] ) +
    t[0+1*2] * ( t[1+2*2] - t[1+0*2] ) +
    t[0+2*2] * ( t[1+0*2] - t[1+1*2] ) );

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

int triangulation_order3_adj_count ( int node_num, int triangle_num,
  int triangle_node[], int triangle_neighbor[], int adj_col[] )

//****************************************************************************80
//
//  Purpose:
//
//    TRIANGULATION_ORDER3_ADJ_COUNT counts adjacencies in a triangulation.
//
//  Discussion:
//
//    This routine is called to count the adjacencies, so that the
//    appropriate amount of memory can be set aside for storage when
//    the adjacency structure is created.
//
//    The triangulation is assumed to involve 3-node triangles.
//
//    Two nodes are "adjacent" if they are both nodes in some triangle.
//    Also, a node is considered to be adjacent to itself.
//
//  Diagram:
//
//       3
//    s  |\
//    i  | \
//    d  |  \
//    e  |   \  side 2
//       |    \
//    3  |     \
//       |      \
//       1-------2
//
//         side 1
//
//    The local node numbering
//
//
//   21-22-23-24-25
//    |\ |\ |\ |\ |
//    | \| \| \| \|
//   16-17-18-19-20
//    |\ |\ |\ |\ |
//    | \| \| \| \|
//   11-12-13-14-15
//    |\ |\ |\ |\ |
//    | \| \| \| \|
//    6--7--8--9-10
//    |\ |\ |\ |\ |
//    | \| \| \| \|
//    1--2--3--4--5
//
//    A sample grid.
//
//
//    Below, we have a chart that summarizes the adjacency relationships
//    in the sample grid.  On the left, we list the node, and its neighbors,
//    with an asterisk to indicate the adjacency of the node to itself
//    (in some cases, you want to count this self adjacency and in some
//    you don't).  On the right, we list the number of adjancencies to
//    lower-indexed nodes, to the node itself, to higher-indexed nodes,
//    the total number of adjacencies for this node, and the location
//    of the first and last entries required to list this set of adjacencies
//    in a single list of all the adjacencies.
//
//    N   Adjacencies                Below  Self   Above   Total First  Last
//
//   --  -- -- -- -- -- -- --           --    --      --      --   ---     0
//    1:  *  2  6                        0     1       2       3     1     3
//    2:  1  *  3  6  7                  1     1       3       5     4     8
//    3:  2  *  4  7  8                  1     1       3       5     9    13
//    4:  3  *  5  8  9                  1     1       3       5    14    18
//    5:  4  *  9 10                     1     1       2       4    19    22
//    6:  1  2  *  7 11                  2     1       2       5    23    27
//    7:  2  3  6  *  8 11 12            3     1       3       7    28    34
//    8:  3  4  7  *  9 12 13            3     1       3       7    35    41
//    9:  4  5  8  * 10 13 14            3     1       3       7    42    48
//   10:  5  9  * 14 15                  2     1       2       5    49    53
//   11:  6  7  * 12 16                  2     1       2       5    54    58
//   12:  7  8 11  * 13 16 17            3     1       3       7    59    65
//   13:  8  9 12  * 14 17 18            3     1       3       7    66    72
//   14:  9 10 13  * 15 18 19            3     1       3       7    73    79
//   15: 10 14  * 19 20                  2     1       2       5    80    84
//   16: 11 12  * 17 21                  2     1       2       5    85    89
//   17: 12 13 16  * 18 21 22            3     1       3       7    90    96
//   18: 13 14 17  * 19 22 23            3     1       3       7    97   103
//   19: 14 15 18  * 20 23 24            3     1       3       7   104   110
//   20: 15 19  * 24 25                  2     1       2       5   111   115
//   21: 16 17  * 22                     2     1       1       4   116   119
//   22: 17 18 21  * 23                  3     1       1       5   120   124
//   23: 18 19 22  * 24                  3     1       1       5   125   129
//   24: 19 20 23  * 25                  3     1       1       5   130   134
//   25: 20 24  *                        2     1       0       3   135   137
//   --  -- -- -- -- -- -- --           --    --      --      --   138   ---
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    25 August 2006
//
//  Author:
//
//    John Burkardt
//
//  Parameters
//
//    Input, int NODE_NUM, the number of nodes.
//
//    Input, int TRIANGLE_NUM, the number of triangles.
//
//    Input, int TRIANGLE_NODE[3*TRIANGLE_NUM], lists the nodes that
//    make up each triangle, in counterclockwise order.
//
//    Input, int TRIANGLE_NEIGHBOR[3*TRIANGLE_NUM], for each side of
//    a triangle, lists the neighboring triangle, or -1 if there is
//    no neighbor.
//
//    Output, TRIANGULATION_ORDER3_ADJ_COUNT, the number of adjacencies.
//
//    Output, int ADJ_COL[NODE_NUM+1].  Information about column J is stored
//    in entries ADJ_COL(J) through ADJ_COL(J+1)-1 of ADJ.
//
{
  int adj_num;
  int i;
  int n1;
  int n2;
  int n3;
  int node;
  int triangle;
  int element_order = 3;
  int triangle2;

  adj_num = 0;
//
//  Set every node to be adjacent to itself.
//
  for ( node = 0; node < node_num; node++ )
  {
    adj_col[node] = 1;
  }
//
//  Examine each triangle.
//
  for ( triangle = 0; triangle < triangle_num; triangle++ )
  {
    n1 = triangle_node[0+triangle*element_order];
    n2 = triangle_node[1+triangle*element_order];
    n3 = triangle_node[2+triangle*element_order];
//
//  Add edge (1,2) if this is the first occurrence,
//  that is, if the edge (1,2) is on a boundary (TRIANGLE2 <= 0)
//  or if this triangle is the first of the pair in which the edge
//  occurs (TRIANGLE < TRIANGLE2).
//
    triangle2 = triangle_neighbor[0+triangle*3];

    if ( triangle2 < 0 || triangle < triangle2 )
    {
      adj_col[n1-1] = adj_col[n1-1] + 1;
      adj_col[n2-1] = adj_col[n2-1] + 1;
    }
//
//  Add edge (2,3).
//
    triangle2 = triangle_neighbor[1+triangle*3];

    if ( triangle2 < 0 || triangle < triangle2 )
    {
      adj_col[n2-1] = adj_col[n2-1] + 1;
      adj_col[n3-1] = adj_col[n3-1] + 1;
    }
//
//  Add edge (3,1).
//
    triangle2 = triangle_neighbor[2+triangle*3];

    if ( triangle2 < 0 || triangle < triangle2 )
    {
      adj_col[n1-1] = adj_col[n1-1] + 1;
      adj_col[n3-1] = adj_col[n3-1] + 1;
    }
  }
//
//  We used ADJ_COL to count the number of entries in each column.
//  Convert it to pointers into the ADJ array.
//
  for ( node = node_num; 1 <= node; node-- )
  {
    adj_col[node] = adj_col[node-1];
  }
  adj_col[0] = 1;
  for ( i = 1; i <= node_num; i++ )
  {
    adj_col[i]= adj_col[i-1] + adj_col[i];
  }

  adj_num = adj_col[node_num] - 1;

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

void triangulation_order3_adj_set2 ( int node_num, int triangle_num,
  int triangle_node[], int triangle_neighbor[], int adj_num, int adj_col[],
  int ia[], int ja[] )

//****************************************************************************80
//
//  Purpose:
//
//    TRIANGULATION_ORDER3_ADJ_SET2 sets adjacencies in a triangulation.
//
//  Discussion:
//
//    This routine is called to set up the arrays IA and JA that
//    record which nodes are adjacent in a triangulation.
//
//    The triangulation is assumed to involve 3-node triangles.
//
//    Two nodes are "adjacent" if they are both nodes in some triangle.
//    Also, a node is considered to be adjacent to itself.
//
//    This routine can be used to create the compressed column storage
//    for a linear triangle finite element discretization of
//    Poisson's equation in two dimensions.
//
//  Diagram:
//
//       3
//    s  |\
//    i  | \
//    d  |  \
//    e  |   \  side 2
//       |    \
//    3  |     \
//       |      \
//       1-------2
//
//         side 1
//
//    The local node numbering
//
//
//   21-22-23-24-25
//    |\ |\ |\ |\ |
//    | \| \| \| \|
//   16-17-18-19-20
//    |\ |\ |\ |\ |
//    | \| \| \| \|
//   11-12-13-14-15
//    |\ |\ |\ |\ |
//    | \| \| \| \|
//    6--7--8--9-10
//    |\ |\ |\ |\ |
//    | \| \| \| \|
//    1--2--3--4--5
//
//    A sample grid
//
//
//    Below, we have a chart that summarizes the adjacency relationships
//    in the sample grid.  On the left, we list the node, and its neighbors,
//    with an asterisk to indicate the adjacency of the node to itself
//    (in some cases, you want to count this self adjacency and in some
//    you don't).  On the right, we list the number of adjancencies to
//    lower-indexed nodes, to the node itself, to higher-indexed nodes,
//    the total number of adjacencies for this node, and the location
//    of the first and last entries required to list this set of adjacencies
//    in a single list of all the adjacencies.
//
//    N   Adjacencies                Below  Self    Above  Total First  Last
//
//   --  -- -- -- -- -- -- --           --    --      --      --   ---     0
//    1:  *  2  6                        0     1       2       3     1     3
//    2:  1  *  3  6  7                  1     1       3       5     4     8
//    3:  2  *  4  7  8                  1     1       3       5     9    13
//    4:  3  *  5  8  9                  1     1       3       5    14    18
//    5:  4  *  9 10                     1     1       2       4    19    22
//    6:  1  2  *  7 11                  2     1       2       5    23    27
//    7:  2  3  6  *  8 11 12            3     1       3       7    28    34
//    8:  3  4  7  *  9 12 13            3     1       3       7    35    41
//    9:  4  5  8  * 10 13 14            3     1       3       7    42    48
//   10:  5  9  * 14 15                  2     1       2       5    49    53
//   11:  6  7  * 12 16                  2     1       2       5    54    58
//   12:  7  8 11  * 13 16 17            3     1       3       7    59    65
//   13:  8  9 12  * 14 17 18            3     1       3       7    66    72
//   14:  9 10 13  * 15 18 19            3     1       3       7    73    79
//   15: 10 14  * 19 20                  2     1       2       5    80    84
//   16: 11 12  * 17 21                  2     1       2       5    85    89
//   17: 12 13 16  * 18 21 22            3     1       3       7    90    96
//   18: 13 14 17  * 19 22 23            3     1       3       7    97   103
//   19: 14 15 18  * 20 23 24            3     1       3       7   104   110
//   20: 15 19  * 24 25                  2     1       2       5   111   115
//   21: 16 17  * 22                     2     1       1       4   116   119
//   22: 17 18 21  * 23                  3     1       1       5   120   124
//   23: 18 19 22  * 24                  3     1       1       5   125   129
//   24: 19 20 23  * 25                  3     1       1       5   130   134
//   25: 20 24  *                        2     1       0       3   135   137
//   --  -- -- -- -- -- -- --           --    --      --      --   138   ---
//
//    For this example, the initial portion of the IA and JA arrays will be:
//
//      (1,1), (1,2), (1,6),
//      (2,1), (2,2), (2,3), (2,6), (2,7),
//      (3,2), (3,3), (3,4), (3,7), (3,8),
//     ...
//      (25,20), (25,24), (25,25)
//
//    for a total of 137 pairs of values.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    15 July 2007
//
//  Author:
//
//    John Burkardt
//
//  Parameters
//
//    Input, int NODE_NUM, the number of nodes.
//
//    Input, int TRIANGLE_NUM, the number of triangles.
//
//    Input, int TRIANGLE_NODE[3*TRIANGLE_NUM], lists the nodes that
//    make up each triangle in counterclockwise order.
//
//    Input, int TRIANGLE_NEIGHBOR[3*TRIANGLE_NUM], for each side of
//    a triangle, lists the neighboring triangle, or -1 if there is
//    no neighbor.
//
//    Input, int ADJ_NUM, the number of adjacencies.
//
//    Input, int ADJ_COL[NODE_NUM+1].  Information about column J is stored
//    in entries ADJ_COL(J) through ADJ_COL(J+1)-1 of ADJ.
//
//    Output, int IA[ADJ_NUM], JA[ADJ_NUM], the adjacency information.
//
{
  int adj;
  int *adj_copy;
  int k;
  int k1;
  int k2;
  int n1;
  int n2;
  int n3;
  int node;
  int triangle;
  int triangle2;
  int element_order = 3;

  for ( adj = 0; adj < adj_num; adj++ )
  {
    ia[adj] = -1;
  }

  for ( adj = 0; adj < adj_num; adj++ )
  {
    ja[adj] = -1;
  }

  adj_copy = new int[node_num];
  for ( node = 0; node < node_num; node++ )
  {
    adj_copy[node] = adj_col[node];
  }
//
//  Set every node to be adjacent to itself.
//
  for ( node = 1; node <= node_num; node++ )
  {
    ia[adj_copy[node-1]-1] = node;
    ja[adj_copy[node-1]-1] = node;
    adj_copy[node-1] = adj_copy[node-1] + 1;
  }
//
//  Examine each triangle.
//
  for ( triangle = 0; triangle < triangle_num; triangle++ )
  {
    n1 = triangle_node[0+triangle*element_order];
    n2 = triangle_node[1+triangle*element_order];
    n3 = triangle_node[2+triangle*element_order];
//
//  Add edge (1,2) if this is the first occurrence,
//  that is, if the edge (1,2) is on a boundary (TRIANGLE2 <= 0)
//  or if this triangle is the first of the pair in which the edge
//  occurs (TRIANGLE < TRIANGLE2).
//
    triangle2 = triangle_neighbor[0+triangle*3];

    if ( triangle2 < 0 || triangle < triangle2 )
    {
      ia[adj_copy[n1-1]-1] = n1;
      ja[adj_copy[n1-1]-1] = n2;
      adj_copy[n1-1] = adj_copy[n1-1] + 1;

      ia[adj_copy[n2-1]-1] = n2;
      ja[adj_copy[n2-1]-1] = n1;
      adj_copy[n2-1] = adj_copy[n2-1] + 1;
    }
//
//  Add edge (2,3).
//
    triangle2 = triangle_neighbor[1+triangle*3];

    if ( triangle2 < 0 || triangle < triangle2 )
    {
      ia[adj_copy[n2-1]-1] = n2;
      ja[adj_copy[n2-1]-1] = n3;
      adj_copy[n2-1] = adj_copy[n2-1] + 1;

      ia[adj_copy[n3-1]-1] = n3;
      ja[adj_copy[n3-1]-1] = n2;
      adj_copy[n3-1] = adj_copy[n3-1] + 1;
    }
//
//  Add edge (3,1).
//
    triangle2 = triangle_neighbor[2+triangle*3];

    if ( triangle2 < 0 || triangle < triangle2 )
    {
      ia[adj_copy[n1-1]-1] = n1;
      ja[adj_copy[n1-1]-1] = n3;
      adj_copy[n1-1] = adj_copy[n1-1] + 1;

      ia[adj_copy[n3-1]-1] = n3;
      ja[adj_copy[n3-1]-1] = n1;
      adj_copy[n3-1] = adj_copy[n3-1] + 1;
    }
  }
//
//  Lexically sort the IA, JA values.
//
  i4vec2_sort_a ( adj_num, ia, ja );

  delete [] adj_copy;

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

bool *triangulation_order3_boundary_node ( int node_num, int element_num,
  int element_node[] )

//****************************************************************************80
//
//  Purpose:
//
//    TRIANGULATION_ORDER3_BOUNDARY_NODE indicates nodes on the boundary.
//
//  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:
//
//    25 January 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int NODE_NUM, the number of nodes.
//
//    Input, int ELEMENT_NUM, the number of triangles.
//
//    Input, int ELEMENT_NODE[3*ELEMENT_NUM], the nodes that make up the
//    triangles.  These should be listed in counterclockwise order.
//
//    Output, bool TRIANGULATION_ORDER3_BOUNDARY_NODE[NODE_NUM],
//    is TRUE if the node is on a boundary edge.
//
{
  int e1;
  int e2;
  int *edge;
  bool equal;
  int i;
  int j;
  int m;
  int n;
  bool *node_boundary;

  m = 2;
  n = 3 * element_num;
//
//  Set up the edge array.
//
  edge = new int[m*n];

  for ( j = 0; j < element_num; j++ )
  {
    edge[0+(j               )*m] = element_node[0+j*3];
    edge[1+(j               )*m] = element_node[1+j*3];
    edge[0+(j+  element_num)*m] = element_node[1+j*3];
    edge[1+(j+  element_num)*m] = element_node[2+j*3];
    edge[0+(j+2*element_num)*m] = element_node[2+j*3];
    edge[1+(j+2*element_num)*m] = element_node[0+j*3];
  }
//
//  In each column, force the smaller entry to appear first.
//
  for ( j = 0; j < n; j++ )
  {
    e1 = i4_min ( edge[0+j*m], edge[1+j*m] );
    e2 = i4_max ( edge[0+j*m], edge[1+j*m] );
    edge[0+j*m] = e1;
    edge[1+j*m] = e2;
  }
//
//  Ascending sort the column array.
//
  i4col_sort_a ( m, n, edge );
//
//  Records which appear twice are internal edges and can be ignored.
//
  node_boundary = new bool[node_num];

  for ( i = 0; i < node_num; i++ )
  {
    node_boundary[i] = false;
  }

  j = 0;

  while ( j < 3 * element_num )
  {
    j = j + 1;

    if ( j == 3 * element_num )
    {
      for ( i = 0; i < m; i++ )
      {
        node_boundary[edge[i+(j-1)*m]-1] = true;
      }
      break;
    }

    equal = true;

    for ( i = 0; i < m; i++ )
    {
      if ( edge[i+(j-1)*m] != edge[i+j*m] )
      {
        equal = false;
      }
    }

    if ( equal )
    {
      j = j + 1;
    }
    else
    {
      for ( i = 0; i < m; i++ )
      {
        node_boundary[edge[i+(j-1)*m]-1] = true;
      }
    }

  }

  delete [] edge;

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

void triangulation_order3_neighbor_triangles ( int triangle_num,
  int triangle_node[], int triangle_neighbor[] )

//****************************************************************************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:
//
//    11 February 2007
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int TRIANGLE_NUM, the number of triangles.
//
//    Input, int TRIANGLE_NODE[3*TRIANGLE_NUM], the nodes that make up each
//    triangle.
//
//    Output, int 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.
//
{
  int *col;
  int i;
  int icol;
  int j;
  int k;
  int side1;
  int side2;
  int tri;
  int element_order = 3;
  int tri1;
  int tri2;

  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,1,T) or (J,I,1,T),
//    (J,K,2,T) or (K,J,2,T),
//    (K,I,3,T) or (I,K,3,T)
//
//  where we choose (I,J,1,T) if I < J, or else (J,I,1,T)
//
  for ( tri = 0; tri < triangle_num; tri++ )
  {
    i = triangle_node[0+tri*element_order];
    j = triangle_node[1+tri*element_order];
    k = triangle_node[2+tri*element_order];

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