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tetrahedron_exactness.cpp
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# include <cstdlib>
# include <iostream>
# include <iomanip>
# include <cmath>
# include <ctime>
# include <fstream>
# include <string>
using namespace std;
int main ( int argc, char *argv[] );
char ch_cap ( char ch );
bool ch_eqi ( char ch1, char ch2 );
int ch_to_digit ( char ch );
void comp_next ( int n, int k, int a[], bool *more, int *h, int *t );
int file_column_count ( string input_filename );
int file_row_count ( string input_filename );
double *monomial_value ( int dim_num, int point_num, double x[], int expon[] );
double *r8mat_data_read ( string input_filename, int m, int n );
double r8mat_det_4d ( double a[4*4] );
void r8mat_header_read ( string input_filename, int *m, int *n );
double r8vec_dot ( int n, double a1[], double a2[] );
int s_len_trim ( string s );
int s_to_i4 ( string s, int *last, bool *error );
double s_to_r8 ( string s, int *lchar, bool *error );
bool s_to_r8vec ( string s, int n, double rvec[] );
int s_word_count ( string s );
double tet01_monomial_integral ( int dim_num, int expon[] );
double tet01_monomial_quadrature ( int dim_num, int expon[], int point_num,
double x[], double weight[] );
void tetrahedron_order4_physical_to_reference ( double tetra[], int n,
double phy[], double ref[] );
double tetrahedron_volume ( double tetra[3*4] );
void timestamp ( );
//****************************************************************************80
int main ( int argc, char *argv[] )
//****************************************************************************80
//
// Purpose:
//
// MAIN is the main program for TETRAHEDRON_EXACTNESS.
//
// Discussion:
//
// This program investigates the polynomial exactness of a quadrature
// rule for the tetrahedron.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 07 October 2010
//
// Author:
//
// John Burkardt
//
{
int degree;
int degree_max;
int dim;
int dim_num;
int dim_num2;
int dim_num3;
bool error;
int *expon;
int h;
int last;
bool more;
int point;
int point_num;
int point_num2;
int point_num3;
double quad_error;
string quad_filename;
string quad_r_filename;
string quad_w_filename;
string quad_x_filename;
double *r;
int t;
double volume;
double *w;
double *x;
double *x_ref;
timestamp ( );
cout << "\n";
cout << "TETRAHEDRON_EXACTNESS\n";
cout << " C++ version\n";
cout << "\n";
cout << " Investigate the polynomial exactness of a quadrature\n";
cout << " rule for the tetrahedron by integrating all monomials\n";
cout << " of a given degree.\n";
cout << "\n";
cout << " The rule will be adjusted to the unit tetrahedron.\n";
//
// Get the quadrature file root name:
//
if ( 1 < argc )
{
quad_filename = argv[1];
}
else
{
cout << "\n";
cout << "TETRAHEDRON_EXACTNESS:\n";
cout << " Enter the \"root\" name of the quadrature files.\n";
cin >> quad_filename;
}
//
// Create the names of:
// the quadrature X file;
// the quadrature W file;
// the quadrature R file;
//
quad_r_filename = quad_filename + "_r.txt";
quad_w_filename = quad_filename + "_w.txt";
quad_x_filename = quad_filename + "_x.txt";
//
// The second command line argument is the maximum degree.
//
if ( 2 < argc )
{
degree_max = s_to_i4 ( argv[2], &last, &error );
}
else
{
cout << "\n";
cout << "TETRAHEDRON_EXACTNESS:\n";
cout << " Please enter the maximum total degree to check.\n";
cin >> degree_max;
}
//
// Summarize the input.
//
cout << "\n";
cout << "TETRAHEDRON_EXACTNESS: User input:\n";
cout << " Quadrature rule X file = \"" << quad_x_filename
<< "\".\n";
cout << " Quadrature rule W file = \"" << quad_w_filename
<< "\".\n";
cout << " Quadrature rule R file = \"" << quad_r_filename
<< "\".\n";
cout << " Maximum total degree to check = " << degree_max << "\n";
//
// Read the X file.
//
r8mat_header_read ( quad_x_filename, &dim_num, &point_num );
cout << "\n";
cout << " Spatial dimension = " << dim_num << "\n";
cout << " Number of points = " << point_num << "\n";
if ( dim_num != 3 )
{
cerr << "\n";
cerr << "TETRAHEDRON_EXACTNESS - Fatal error!\n";
cerr << " The quadrature abscissas must be 3 dimensional.\n";
exit ( 1 );
}
x = r8mat_data_read ( quad_x_filename, dim_num, point_num );
//
// Read the W file.
//
r8mat_header_read ( quad_w_filename, &dim_num2, &point_num2 );
if ( dim_num2 != 1 )
{
cerr << "\n";
cerr << "TETRAHEDRON_EXACTNESS - Fatal error!\n";
cerr << " The quadrature weight file should have exactly\n";
cerr << " one value on each line.\n";
exit ( 1 );
}
if ( point_num2 != point_num )
{
cerr << "\n";
cerr << "TETRAHEDRON_EXACTNESS - Fatal error!\n";
cerr << " The quadrature weight file should have exactly\n";
cerr << " the same number of lines as the abscissa file.\n";
exit ( 1 );
}
w = r8mat_data_read ( quad_w_filename, 1, point_num );
//
// Read the R file.
//
r8mat_header_read ( quad_r_filename, &dim_num3, &point_num3 );
if ( dim_num3 != dim_num )
{
cerr << "\n";
cerr << "TETRAHEDRON_EXACTNESS - Fatal error!\n";
cerr << " The quadrature region file should have the same\n";
cerr << " number of values on each line as the abscissa file\n";
cerr << " does.\n";
exit ( 1 );
}
if ( point_num3 != 4 )
{
cerr << "\n";
cerr << "TETRAHEDRON_EXACTNESS - Fatal error!\n";
cerr << " The quadrature region file should have 4 lines.\n";
exit ( 1 );
}
r = r8mat_data_read ( quad_r_filename, dim_num, 4 );
//
// Rescale the weights.
//
volume = tetrahedron_volume ( r );
for ( point = 0; point < point_num; point++ )
{
w[point] = ( 1.0 / 6.0 ) * w[point] / volume;
}
//
// Translate the abscissas.
//
x_ref = new double[dim_num*point_num];
tetrahedron_order4_physical_to_reference ( r, point_num, x, x_ref );
//
// Explore the monomials.
//
expon = new int[dim_num];
cout << "\n";
cout << " Error Degree Exponents\n";
cout << "\n";
for ( degree = 0; degree <= degree_max; degree++ )
{
more = false;
h = 0;
t = 0;
for ( ; ; )
{
comp_next ( degree, dim_num, expon, &more, &h, &t );
quad_error = tet01_monomial_quadrature ( dim_num, expon, point_num, x_ref,
w );
cout << " " << setw(12) << quad_error
<< " " << setw(2) << degree
<< " ";
for ( dim = 0; dim < dim_num; dim++ )
{
cout << setw(3) << expon[dim];
}
cout << "\n";
if ( !more )
{
break;
}
}
cout << "\n";
}
//
// Free memory.
//
delete [] expon;
delete [] r;
delete [] w;
delete [] x;
delete [] x_ref;
//
// Terminate.
//
cout << "\n";
cout << "TETRAHEDRON_EXACTNESS:\n";
cout << " Normal end of execution.\n";
cout << "\n";
timestamp ( );
return 0;
}
//****************************************************************************80
char ch_cap ( char ch )
//****************************************************************************80
//
// Purpose:
//
// CH_CAP capitalizes a single character.
//
// Discussion:
//
// This routine should be equivalent to the library "toupper" function.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 19 July 1998
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, char CH, the character to capitalize.
//
// Output, char CH_CAP, the capitalized character.
//
{
if ( 97 <= ch && ch <= 122 )
{
ch = ch - 32;
}
return ch;
}
//****************************************************************************80
bool ch_eqi ( char ch1, char ch2 )
//****************************************************************************80
//
// Purpose:
//
// CH_EQI is true if two characters are equal, disregarding case.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 13 June 2003
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, char CH1, CH2, the characters to compare.
//
// Output, bool CH_EQI, is true if the two characters are equal,
// disregarding case.
//
{
if ( 97 <= ch1 && ch1 <= 122 )
{
ch1 = ch1 - 32;
}
if ( 97 <= ch2 && ch2 <= 122 )
{
ch2 = ch2 - 32;
}
return ( ch1 == ch2 );
}
//****************************************************************************80
int ch_to_digit ( char ch )
//****************************************************************************80
//
// Purpose:
//
// CH_TO_DIGIT returns the integer value of a base 10 digit.
//
// Example:
//
// CH DIGIT
// --- -----
// '0' 0
// '1' 1
// ... ...
// '9' 9
// ' ' 0
// 'X' -1
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 13 June 2003
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, char CH, the decimal digit, '0' through '9' or blank are legal.
//
// Output, int CH_TO_DIGIT, the corresponding value. If the
// character was 'illegal', then DIGIT is -1.
//
{
int digit;
if ( '0' <= ch && ch <= '9' )
{
digit = ch - '0';
}
else if ( ch == ' ' )
{
digit = 0;
}
else
{
digit = -1;
}
return digit;
}
//****************************************************************************80
void comp_next ( int n, int k, int a[], bool *more, int *h, int *t )
//****************************************************************************80
//
// Purpose:
//
// COMP_NEXT computes the compositions of the integer N into K parts.
//
// Discussion:
//
// A composition of the integer N into K parts is an ordered sequence
// of K nonnegative integers which sum to N. The compositions (1,2,1)
// and (1,1,2) are considered to be distinct.
//
// The routine computes one composition on each call until there are no more.
// For instance, one composition of 6 into 3 parts is
// 3+2+1, another would be 6+0+0.
//
// On the first call to this routine, set MORE = FALSE. The routine
// will compute the first element in the sequence of compositions, and
// return it, as well as setting MORE = TRUE. If more compositions
// are desired, call again, and again. Each time, the routine will
// return with a new composition.
//
// However, when the LAST composition in the sequence is computed
// and returned, the routine will reset MORE to FALSE, signaling that
// the end of the sequence has been reached.
//
// This routine originally used a SAVE statement to maintain the
// variables H and T. I have decided that it is safer
// to pass these variables as arguments, even though the user should
// never alter them. This allows this routine to safely shuffle
// between several ongoing calculations.
//
//
// There are 28 compositions of 6 into three parts. This routine will
// produce those compositions in the following order:
//
// I A
// - ---------
// 1 6 0 0
// 2 5 1 0
// 3 4 2 0
// 4 3 3 0
// 5 2 4 0
// 6 1 5 0
// 7 0 6 0
// 8 5 0 1
// 9 4 1 1
// 10 3 2 1
// 11 2 3 1
// 12 1 4 1
// 13 0 5 1
// 14 4 0 2
// 15 3 1 2
// 16 2 2 2
// 17 1 3 2
// 18 0 4 2
// 19 3 0 3
// 20 2 1 3
// 21 1 2 3
// 22 0 3 3
// 23 2 0 4
// 24 1 1 4
// 25 0 2 4
// 26 1 0 5
// 27 0 1 5
// 28 0 0 6
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 02 July 2008
//
// Author:
//
// Original FORTRAN77 version by Albert Nijenhuis, Herbert Wilf.
// C++ version by John Burkardt.
//
// Reference:
//
// Albert Nijenhuis, Herbert Wilf,
// Combinatorial Algorithms for Computers and Calculators,
// Second Edition,
// Academic Press, 1978,
// ISBN: 0-12-519260-6,
// LC: QA164.N54.
//
// Parameters:
//
// Input, int N, the integer whose compositions are desired.
//
// Input, int K, the number of parts in the composition.
//
// Input/output, int A[K], the parts of the composition.
//
// Input/output, bool *MORE.
// Set MORE = FALSE on first call. It will be reset to TRUE on return
// with a new composition. Each new call returns another composition until
// MORE is set to FALSE when the last composition has been computed
// and returned.
//
// Input/output, int *H, *T, two internal parameters needed for the
// computation. The user should allocate space for these in the calling
// program, include them in the calling sequence, but never alter them!
//
{
int i;
if ( !( *more ) )
{
*t = n;
*h = 0;
a[0] = n;
for ( i = 1; i < k; i++ )
{
a[i] = 0;
}
}
else
{
if ( 1 < *t )
{
*h = 0;
}
*h = *h + 1;
*t = a[*h-1];
a[*h-1] = 0;
a[0] = *t - 1;
a[*h] = a[*h] + 1;
}
*more = ( a[k-1] != n );
return;
}
//****************************************************************************80
int file_column_count ( string filename )
//****************************************************************************80
//
// Purpose:
//
// FILE_COLUMN_COUNT counts the columns in the first line of a file.
//
// Discussion:
//
// The file is assumed to be a simple text file.
//
// Most lines of the file are presumed to consist of COLUMN_NUM words,
// separated by spaces. There may also be some blank lines, and some
// comment lines, which have a "#" in column 1.
//
// The routine tries to find the first non-comment non-blank line and
// counts the number of words in that line.
//
// If all lines are blanks or comments, it goes back and tries to analyze
// a comment line.
//
// Licensing:
//
// This code is distributed under the 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
double *monomial_value ( int dim_num, int point_num, double x[], int expon[] )
//****************************************************************************80
//
// Purpose:
//
// MONOMIAL_VALUE evaluates a monomial.
//
// Discussion:
//
// This routine evaluates a monomial of the form
//
// product ( 1 <= dim <= dim_num ) x(dim)^expon(dim)
//
// where the exponents are nonnegative integers. Note that
// if the combination 0^0 is encountered, it should be treated
// as 1.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 05 May 2007
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, int DIM_NUM, the spatial dimension.
//
// Input, int POINT_NUM, the number of points at which the
// monomial is to be evaluated.
//
// Input, double X[DIM_NUM*POINT_NUM], the point coordinates.
//
// Input, int EXPON[DIM_NUM], the exponents.
//
// Output, double MONOMIAL_VALUE[POINT_NUM], the value of the monomial.
//
{
int dim;
int point;
double *value;
value = new double[point_num];
for ( point = 0; point < point_num; point++ )
{
value[point] = 1.0;
}
for ( dim = 0; dim < dim_num; dim++ )
{
if ( 0 != expon[dim] )
{
for ( point = 0; point < point_num; point++ )
{
value[point] = value[point] * pow ( x[dim+point*dim_num], expon[dim] );
}
}
}
return value;
}
//****************************************************************************80
double *r8mat_data_read ( string input_filename, int m, int n )
//****************************************************************************80
//
// Purpose:
//
// R8MAT_DATA_READ reads the data from an R8MAT file.
//
// Discussion:
//
// The file is assumed to contain one record per line.
//
// Records beginning with '#' are comments, and are ignored.
// Blank lines are also ignored.
//
// Each line that is not ignored is assumed to contain exactly (or at least)
// M real numbers, representing the coordinates of a point.
//
// There are assumed to be exactly (or at least) N such records.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 23 February 2009
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, string INPUT_FILENAME, the name of the input file.
//
// Input, int M, the number of spatial dimensions.
//
// Input, int N, the number of points. The program
// will stop reading data once N values have been read.
//
// Output, double R8MAT_DATA_READ[M*N], the table data.
//
{
bool error;
ifstream input;
int i;
int j;
string line;
double *table;
double *x;
input.open ( input_filename.c_str ( ) );
if ( !input )
{
cerr << "\n";
cerr << "R8MAT_DATA_READ - Fatal error!\n";
cerr << " Could not open the input file: \"" << input_filename << "\"\n";
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
double r8mat_det_4d ( double a[4*4] )
//****************************************************************************80
//
// Purpose:
//
// R8MAT_DET_4D computes the determinant of a 4 by 4 R8MAT.
//
// Discussion:
//
// The two dimensional array is stored as a one dimensional vector,
// by COLUMNS.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 10 September 2003
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, double A[4*4], the matrix whose determinant is desired.
//
// Output, double R8MAT_DET_4D, the determinant of the matrix.
//
{
double det;
det =
a[0+0*4] * (
a[1+1*4] * ( a[2+2*4] * a[3+3*4] - a[2+3*4] * a[3+2*4] )
- a[1+2*4] * ( a[2+1*4] * a[3+3*4] - a[2+3*4] * a[3+1*4] )