triangle_symq_rule_prb.cpp

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

using namespace std;

# include "triangle_symq_rule.hpp"

int main ( );
void test01 ( );
void test02 ( int degree, int numnodes, double vert1[], double vert2[], 
  double vert3[] );
void test03 ( int degree, int numnodes, double vert1[], double vert2[], 
  double vert3[], string header );
void test04 ( int degree, int numnodes, double vert1[], double vert2[], 
  double vert3[], string header );
void test05 ( int degree, int numnodes, double vert1[], double vert2[], 
  double vert3[] );

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

int main ( )

//****************************************************************************80
//
//  Purpose:
//
//    MAIN is the main program for TRIANGLE_SYMQ_RULE_PRB.
//
//  Discussion:
//
//    TRIANGLE_SYMQ_RULE_PRB tests the TRIANGLE_SYMQ_RULE library.
//
//  Licensing:
//
//    This code is distributed under the GNU GPL license.
//
//  Modified:
//
//    30 June 2014
//
//  Author:
//
//    Original FORTRAN77 version by Hong Xiao, Zydrunas Gimbutas.
//    This C++ version by John Burkardt.
//
//  Reference:
//
//    Hong Xiao, Zydrunas Gimbutas,
//    A numerical algorithm for the construction of efficient quadrature
//    rules in two and higher dimensions,
//    Computers and Mathematics with Applications,
//    Volume 59, 2010, pages 663-676.
//
{
  int degree;
  string header;
  int itype;
  int numnodes;
  double vert1[2];
  double vert2[2];
  double vert3[2];

  timestamp ( );
  cout << "\n";
  cout << "TRIANGLE_SYMQ_RULE_PRB\n";
  cout << "  C++ version\n";
  cout << "  Test the TRIANGLE_SYMQ_RULE library.\n";

  test01 ( );

  for ( itype = 0; itype <= 2; itype++ )
  {
    if ( itype == 0 )
    {
      cout << "\n";
      cout << "  Region is user-defined triangle.\n";
      vert1[0] = 1.0;
      vert1[1] = 0.0;
      vert2[0] = 4.0;
      vert2[1] = 4.0;
      vert3[0] = 0.0;
      vert3[1] = 3.0;
      header = "user08";
      degree = 8;
    }
    else if ( itype == 1 )
    {
      cout << "\n";
      cout << "  Region is standard equilateral triangle.\n";
      vert1[0] = -1.0;
      vert1[1] = -1.0 / sqrt ( 3.0 );
      vert2[0] = +1.0;
      vert2[1] = -1.0 / sqrt ( 3.0 );
      vert3[0] =  0.0;
      vert3[1] =  2.0 / sqrt ( 3.0 );
      header = "equi08";
      degree = 8;
    }
    else if ( itype == 2 )
    {
      cout << "\n";
      cout << "  Region is the simplex (0,0),(1,0),(0,1).\n";
      vert1[0] = 0.0;
      vert1[1] = 0.0;
      vert2[0] = 1.0;
      vert2[1] = 0.0;
      vert3[0] = 0.0;
      vert3[1] = 1.0;
      header = "simp08";
      degree = 8;
    }

    cout << "\n";
    cout << "  Triangle:\n";
    cout << "\n";
    cout << vert1[0] << "  " << vert1[1] << "\n";
    cout << vert2[0] << "  " << vert2[1] << "\n";
    cout << vert3[0] << "  " << vert3[1] << "\n";
//
//  Determine the size of the rule.
//
    numnodes = rule_full_size ( degree );
//
//  Retrieve a rule and print it.
//
    test02 ( degree, numnodes, vert1, vert2, vert3 );
//
//  Get a rule, and write data files that gnuplot can use to plot the points.
//
    test03 ( degree, numnodes, vert1, vert2, vert3, header );

    test04 ( degree, numnodes, vert1, vert2, vert3, header );

    test05 ( degree, numnodes, vert1, vert2, vert3 );
  }
//
//  Terminate.
//
  cout << "\n";
  cout << "TRIANGLE_SYMQ_RULE_PRB\n";
  cout << "  Normal end of execution.\n";
  cout << "\n";
  timestamp ( );

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

void test01 ( )

//****************************************************************************80
//
//  Purpose:
//
//    TEST01 tests TRIANGLE_TO_SIMPLEX, TRIANGLE_TO_REF, REF_TO_TRIANGLE, SIMPLEX_TO_TRIANGLE.
//
//  Licensing:
//
//    This code is distributed under the GNU GPL license.
//
//  Modified:
//
//    30 June 2014
//
//  Author:
//
//    John Burkardt
//
{
  int i;
  double *rp1;
  double rv1[2];
  double rv2[2];
  double rv3[2];
  int seed;
  double *sp1;
  double *sp2;
  double sv1[2];
  double sv2[2];
  double sv3[2];
  double *tp1;
  double *tp2;
  double tv1[2];
  double tv2[2];
  double tv3[2];

  cout << "\n";
  cout << "TEST01\n";
  cout << "  Map points from one triangle to another.\n";
  cout << "\n";
  cout << "  R = reference triangle\n";
  cout << "  S = simplex\n";
  cout << "  T = user-defined triangle.\n";
  cout << "  REF_TO_TRIANGLE:     R => T\n";
  cout << "  SIMPLEX_TO_TRIANGLE: S => T\n";
  cout << "  TRIANGLE_TO_REF:     T => R\n";
  cout << "  TRIANGLE_TO_SIMPLEX: T => S\n";
//
//  Reference triangle
//
  rv1[0] = -1.0;
  rv1[1] = -1.0 / sqrt ( 3.0 );
  rv2[0] = +1.0;
  rv2[1] = -1.0 / sqrt ( 3.0 );
  rv3[0] =  0.0;
  rv3[1] =  2.0 / sqrt ( 3.0 );
//
//  Simplex
//
  sv1[0] = 0.0;
  sv1[1] = 0.0;
  sv2[0] = 1.0;
  sv2[1] = 0.0;
  sv3[0] = 0.0;
  sv3[1] = 1.0;
//
//  User triangle.
//
  tv1[0] = 1.0;
  tv1[1] = 0.0;
  tv2[0] = 4.0;
  tv2[1] = 4.0;
  tv3[0] = 0.0;
  tv3[1] = 3.0;

  seed = 123456789;

  for ( i = 1; i <= 5; i++ )
  {
    sp1 = r8vec_uniform_01_new ( 2, seed );

    if ( 1.0 < sp1[0] + sp1[1] )
    {
      sp1[0] = 1.0 - sp1[0];
      sp1[1] = 1.0 - sp1[1];
    }

    tp1 = simplex_to_triangle ( tv1, tv2, tv3, sp1 );
    rp1 = triangle_to_ref ( tv1, tv2, tv3, tp1 );
    tp2 = ref_to_triangle ( tv1, tv2, tv3, rp1 );
    sp2 = triangle_to_simplex ( tv1, tv2, tv3, tp2 );

    cout << "\n";
    cout << "  SP1: " << sp1[0] << "  " << sp1[1] << "\n";
    cout << "  TP1: " << tp1[0] << "  " << tp1[1] << "\n";
    cout << "  RP1: " << rp1[0] << "  " << rp1[1] << "\n";
    cout << "  TP2: " << tp2[0] << "  " << tp2[1] << "\n";
    cout << "  SP2: " << sp2[0] << "  " << sp2[1] << "\n";

    delete [] rp1;
    delete [] sp1;
    delete [] sp2;
    delete [] tp1;
    delete [] tp2;
  }

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

void test02 ( int degree, int numnodes, double vert1[], double vert2[], 
  double vert3[] )

//****************************************************************************80
//
//  Purpose:
//
//    TEST02 calls TRIASYMQ for a quadrature rule of given order and region.
//
//  Licensing:
//
//    This code is distributed under the GNU GPL license.
//
//  Modified:
//
//    28 June 2014
//
//  Author:
//
//    Original FORTRAN77 version by Hong Xiao, Zydrunas Gimbutas.
//    This C++ version by John Burkardt.
//
//  Reference:
//
//    Hong Xiao, Zydrunas Gimbutas,
//    A numerical algorithm for the construction of efficient quadrature
//    rules in two and higher dimensions,
//    Computers and Mathematics with Applications,
//    Volume 59, 2010, pages 663-676.
//
//  Parameters:
//
//    Input, int DEGREE, the desired total polynomial degree exactness
//    of the quadrature rule.  0 <= DEGREE <= 50.
//
//    Input, int NUMNODES, the number of nodes to be used by the rule.
//
//    Input, double VERT1[2], VERT2[2], VERT3[2], the
//    vertices of the triangle.
//
{
  double area;
  double d;
  int j;
  double *rnodes;
  double *weights;

  cout << "\n";
  cout << "TEST02\n";
  cout << "  Symmetric quadrature rule for a triangle.\n";
  cout << "  Polynomial exactness degree DEGREE = " << degree << "\n";

  area = triangle_area ( vert1, vert2, vert3 );
//
//  Retrieve and print a symmetric quadrature rule.
//
  rnodes = new double[2*numnodes];
  weights = new double[numnodes];

  triasymq ( degree, vert1, vert2, vert3, rnodes, weights, numnodes );

  cout << "\n";
  cout << "  NUMNODES = " << numnodes << "\n";

  cout << "\n";
  cout << "     J      W               X               Y\n";
  cout << "\n";
  for ( j = 0; j < numnodes; j++ )
  {
    cout << j << "  "
         << weights[j] << "  "
         << rnodes[0+j*2] << "  "
         << rnodes[1+j*2] << "\n";
  }

  d = r8vec_sum ( numnodes, weights );

  cout << "   Sum  " << d << "\n";
  cout << "  Area  " << area << "\n";

  delete [] rnodes;
  delete [] weights;

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

void test03 ( int degree, int numnodes, double vert1[], double vert2[], 
  double vert3[], string header )

//****************************************************************************80
//
//  Purpose:
//
//    TEST03 calls TRIASYMQ_GNUPLOT to generate graphics files.
//
//  Licensing:
//
//    This code is distributed under the GNU GPL license.
//
//  Modified:
//
//    30 June 2014
//
//  Author:
//
//    Original FORTRAN77 version by Hong Xiao, Zydrunas Gimbutas.
//    This C++ version by John Burkardt.
//
//  Reference:
//
//    Hong Xiao, Zydrunas Gimbutas,
//    A numerical algorithm for the construction of efficient quadrature
//    rules in two and higher dimensions,
//    Computers and Mathematics with Applications,
//    Volume 59, 2010, pages 663-676.
//
//  Parameters:
//
//    Input, int DEGREE, the desired total polynomial degree exactness
//    of the quadrature rule.  0 <= DEGREE <= 50.
//
//    Input, int NUMNODES, the number of nodes to be used by the rule.
//
//    Input, double VERT1[2], VERT2[2], VERT3[2], the
//    vertices of the triangle.
//
//    Input, string HEADER, an identifier for the graphics filenames.
//
{
  double *rnodes;
  double *weights;

  cout << "\n";
  cout << "TEST03\n";
  cout << "  TRIASYMQ_GNUPLOT creates gnuplot graphics files.\n";
  cout << "  Polynomial exactness degree DEGREE = " << degree << "\n";

  rnodes = new double[2*numnodes];
  weights = new double[numnodes];

  triasymq ( degree, vert1, vert2, vert3, rnodes, weights, numnodes );

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

  triasymq_gnuplot ( vert1, vert2, vert3, numnodes, rnodes, header );

  delete [] rnodes;
  delete [] weights;

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

void test04 ( int degree, int numnodes, double vert1[], double vert2[], 
  double vert3[], string header )

//****************************************************************************80
//
//  Purpose:
//
//    TEST04 gets a rule and writes it to a file.
//
//  Licensing:
//
//    This code is distributed under the GNU GPL license.
//
//  Modified:
//
//    30 June 2014
//
//  Author:
//
//    Original FORTRAN77 version by Hong Xiao, Zydrunas Gimbutas.
//    This C++ version by John Burkardt.
//
//  Reference:
//
//    Hong Xiao, Zydrunas Gimbutas,
//    A numerical algorithm for the construction of efficient quadrature
//    rules in two and higher dimensions,
//    Computers and Mathematics with Applications,
//    Volume 59, 2010, pages 663-676.
//
//  Parameters:
//
//    Input, int DEGREE, the desired total polynomial degree exactness
//    of the quadrature rule.  0 <= DEGREE <= 50.
//
//    Input, int NUMNODES, the number of nodes to be used by the rule.
//
//    Input, double VERT1[2], VERT2[2], VERT3[2], the
//    vertices of the triangle.
//
//    Input, string HEADER, an identifier for the filenames.
//
{
  int j;
  double *rnodes;
  ofstream rule_unit;
  string rule_filename;
  double *weights;

  cout << "\n";
  cout << "TEST04\n";
  cout << "  Get a quadrature rule for a triangle.\n";
  cout << "  Then write it to a file.\n";
  cout << "  Polynomial exactness degree DEGREE = " << degree << "\n";
//
//  Retrieve a symmetric quadrature rule.
//
  rnodes = new double[2*numnodes];
  weights = new double[numnodes];

  triasymq ( degree, vert1, vert2, vert3, rnodes, weights, numnodes );
//
//  Write the points and weights to a file.
//
  rule_filename = header + ".txt";
 
  rule_unit.open ( rule_filename.c_str ( ) );
  for ( j = 0; j < numnodes; j++ )
  {
    rule_unit << rnodes[0+j*2] << "  "
              << rnodes[1+j*2] << "  "
              << weights[j] << "\n";
  }
  rule_unit.close ( );
  cout << "\n";
  cout << "  Quadrature rule written to file '" << rule_filename << "'\n";

  delete [] rnodes;
  delete [] weights;

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

void test05 ( int degree, int numnodes, double vert1[], double vert2[], 
  double vert3[] )

//****************************************************************************80
//
//  Purpose:
//
//    TEST05 calls TRIASYMQ for a quadrature rule of given order and region.
//
//  Licensing:
//
//    This code is distributed under the GNU GPL license.
//
//  Modified:
//
//    28 June 2014
//
//  Author:
//
//    Original FORTRAN77 version by Hong Xiao, Zydrunas Gimbutas.
//    This C++ version by John Burkardt.
//
//  Reference:
//
//    Hong Xiao, Zydrunas Gimbutas,
//    A numerical algorithm for the construction of efficient quadrature
//    rules in two and higher dimensions,
//    Computers and Mathematics with Applications,
//    Volume 59, 2010, pages 663-676.
//
//  Parameters:
//
//    Input, int DEGREE, the desired total polynomial degree 
//    exactness of the quadrature rule.  0 <= DEGREE <= 50.
//
//    Input, int NUMNODES, the number of nodes to be used by the rule.
//
//    Input, double VERT1[2], VERT2[2], VERT3[2], the
//    vertices of the triangle.
//
{
  double area;
  double d;
  int i;
  int j;
  int npols;
  double *pols;
  double *r;
  double *rints;
  double *rnodes;
  double scale;
  double *weights;
  double z[2];

  cout << "\n";
  cout << "TEST05\n";
  cout << "  Compute a quadrature rule for a triangle.\n";
  cout << "  Check it by integrating orthonormal polynomials.\n";
  cout << "  Polynomial exactness degree DEGREE = " << degree << "\n";

  area = triangle_area ( vert1, vert2, vert3 );
//
//  Retrieve a symmetric quadrature rule.
//
  rnodes = new double[2*numnodes];
  weights = new double[numnodes];

  triasymq ( degree, vert1, vert2, vert3, rnodes, weights, numnodes );
//
//  Construct the matrix of values of the orthogonal polynomials
//  at the user-provided nodes
//
  npols = ( degree + 1 ) * ( degree + 2 ) / 2;
  rints = new double[npols];

  for ( j = 0; j < npols; j++ )
  {
    rints[j] = 0.0;
  }

  for ( i = 0; i < numnodes; i++ )
  {
    z[0] = rnodes[0+i*2];
    z[1] = rnodes[1+i*2];
    r = triangle_to_ref ( vert1, vert2, vert3, z );
    pols = ortho2eva ( degree, r );
    for ( j = 0; j < npols; j++ )
    {
      rints[j] = rints[j] + weights[i] * pols[j];
    }
    delete [] pols;
    delete [] r;
  }

  scale = sqrt ( sqrt ( 3.0 ) ) / sqrt ( area );

  for ( j = 0; j < npols; j++ )
  {
    rints[j] = rints[j] * scale;
  }

  d = pow ( rints[0] - sqrt ( area ), 2 );
  for ( j = 1; j < npols; j++ )
  {
    d = d + rints[j] * rints[j];
  }
  d = sqrt ( d ) / ( double ) ( npols );

  cout << "\n";
  cout << "  RMS integration error = " << d << "\n";

  delete [] rints;
  delete [] rnodes;
  delete [] weights;

  return;
}