HYPERCUBE_EXACTNESS
Exactness of Multidimensional Quadrature

HYPERCUBE_EXACTNESS is a FORTRAN90 program which investigates the polynomial exactness of a quadrature rule over the interior of the hypercube in M dimensions.

The polynomial exactness of a quadrature rule is defined as the highest total degree D such that the quadrature rule is guaranteed to integrate exactly all polynomials of total degree DEGREE_MAX or less, ignoring roundoff. The total degree of a polynomial is the maximum of the degrees of all its monomial terms. The degree of a monomial term is the sum of the exponents. Thus, for instance, the DEGREE of

x²y z⁵

To be thorough, the program starts at DEGREE = 0, and then proceeds to DEGREE = 1, 2, and so on up to a maximum degree DEGREE_MAX specified by the user. At each value of DEGREE, the program generates every possible monomial term, applies the quadrature rule to it, and determines the quadrature error. The program uses a scaling factor on each monomial so that the exact integral should always be 1; therefore, each reported error can be compared on a fixed scale.

The program is very flexible and interactive. The quadrature rule is defined by three files, to be read at input, and the maximum degree is specified by the user as well.

Note that the three files that define the quadrature rule are assumed to have related names, of the form

• prefix_x.txt
• prefix_w.txt
• prefix_r.txt

The exactness results are written to an output file with the corresponding name:

• prefix_exact.txt

Usage:

hypercube_exactness prefix degree_max

• prefix is the common prefix for the files containing the abscissa, weight and region information of the quadrature rule;
• degree_max is the maximum total monomial degree to check. This should be a relatively small nonnegative number, particularly if the spatial dimension is high. A value of 5 or 10 might be reasonable, but a value of 50 or 100 is probably never a good input!

If the arguments are not supplied on the command line, the program will prompt for them.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

HYPERCUBE_EXACTNESS is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CUBE_EXACTNESS, a FORTRAN90 library which investigates the polynomial exactness of quadrature rules over the interior of a cube in 3D.

HYPERCUBE_GRID, a FORTRAN90 library which computes a grid of points over the interior of a hypercube in M dimensions.

INT_EXACTNESS, a FORTRAN90 program which tests the polynomial exactness of one dimensional quadrature rules.

PYRAMID_EXACTNESS, a FORTRAN90 program which investigates the polynomial exactness of a quadrature rule over the interior of the unit pyramid in 3D.

SPHERE_EXACTNESS, a FORTRAN90 program which tests the polynomial exactness of a quadrature rule over the surface of the unit sphere in 3D.

SQUARE_EXACTNESS, a FORTRAN90 library which investigates the polynomial exactness of quadrature rules for f(x,y) over the interior of a rectangle in 2D.

TETRAHEDRON_EXACTNESS, a FORTRAN90 program which investigates the polynomial exactness of a quadrature rule for the tetrahedron.

TRIANGLE_EXACTNESS, a FORTRAN90 program which investigates the polynomial exactness of a quadrature rule over the interior of a triangle in 2D.

WEDGE_EXACTNESS, a FORTRAN90 program which investigates the monomial exactness of a quadrature rule over the interior of the unit wedge in 3D.

Reference:

1. Philip Davis, Philip Rabinowitz,
   Methods of Numerical Integration,
   Second Edition,
   Dover, 2007,
   ISBN: 0486453391,
   LC: QA299.3.D28.

Source Code:

• hypercube_exactness.f90, the source code.

Examples and Tests:

CC_D1_O2 is a Clenshaw-Curtis order 2 rule for 1D.

• cc_d1_o2_x.txt, the abscissas of the rule.
• cc_d1_o2_w.txt, the weights of the rule.
• cc_d1_o2_r.txt, defines the region for the rule.
• cc_d1_o2_exact.txt, the results of the exactness test, up to degree 5.

CC_D1_O3 is a Clenshaw-Curtis order 3 rule for 1D. If you are paying attention, you may be surprised to see that a Clenshaw Curtis rule of odd order has one more degree of accuracy than you'd expect!

• cc_d1_o3_x.txt, the abscissas of the rule.
• cc_d1_o3_w.txt, the weights of the rule.
• cc_d1_o3_r.txt, defines the region for the rule.
• cc_d1_o3_exact.txt, the results of the exactness test, up to degree 5.

CC_D2_O3x3 is a Clenshaw-Curtis 3x3 product rule for 2D.

• cc_d2_o3x3_x.txt, the abscissas of the rule.
• cc_d2_o3x3_w.txt, the weights of the rule.
• cc_d2_o3x3_r.txt, defines the region for the rule.
• cc_d2_o3x3_exact.txt, the results of the exactness test, up to degree 8.

CC_D3_O3x3x3 is a Clenshaw-Curtis 3x3x3 product rule for 3D.

• cc_d3_o3x3x3_x.txt, the abscissas of the rule.
• cc_d3_o3x3x3_w.txt, the weights of the rule.
• cc_d3_o3x3x3_r.txt, defines the region for the rule.
• cc_d3_o3x3x3_exact.txt, the results of the exactness test, up to degree 5.

CCGL_D2_O3x2 is a product rule for 2D whose factors are a Clenshaw-Curtis of order 3 and a Gauss-Legendre rule of order 2.

• ccgl_d2_o3x2_x.txt, the abscissas of the rule.
• ccgl_d2_o3x2_w.txt, the weights of the rule.
• ccgl_d2_o3x2_r.txt, defines the region for the rule.
• ccgl_d2_o3x2_exact.txt, the results of the exactness test, up to degree 5.

CC_D2_LEVEL0 is a Clenshaw Curtis sparse grid rule for 2D of level 0 and order 1.

• cc_d2_level0_x.txt, the abscissas of the rule.
• cc_d2_level0_w.txt, the weights of the rule.
• cc_d2_level0_r.txt, defines the region for the rule.
• cc_d2_level0_exact.txt, the results of the exactness test, up to degree 4.

CC_D2_LEVEL1 is a Clenshaw Curtis sparse grid rule for 2D of level 1 and order 5.

• cc_d2_level1_x.txt, the abscissas of the rule.
• cc_d2_level1_w.txt, the weights of the rule.
• cc_d2_level1_r.txt, defines the region for the rule.
• cc_d2_level1_exact.txt, the results of the exactness test, up to degree 4.

CC_D2_LEVEL2 is a Clenshaw Curtis sparse grid rule for 2D of level 2 and order 13.

• cc_d2_level2_x.txt, the abscissas of the rule.
• cc_d2_level2_w.txt, the weights of the rule.
• cc_d2_level2_r.txt, defines the region for the rule.
• cc_d2_level2_exact.txt, the results of the exactness test, up to degree 6.

CC_D2_LEVEL3 is a Clenshaw Curtis sparse grid rule for 2D of level 3 and order 25.

• cc_d2_level3_x.txt, the abscissas of the rule.
• cc_d2_level3_w.txt, the weights of the rule.
• cc_d3_level3_r.txt, defines the region for the rule.
• cc_d2_level3_exact.txt, the results of the exactness test, up to degree 9.

CC_D2_LEVEL4 is a Clenshaw Curtis sparse grid rule for 2D of level 4 and order 65.

• cc_d2_level4_x.txt, the abscissas of the rule.
• cc_d2_level4_w.txt, the weights of the rule.
• cc_d3_level4_r.txt, defines the region for the rule.
• cc_d2_level4_exact.txt, the results of the exactness test, up to degree 17.

CC_D100_LEVEL1 is a Clenshaw Curtis sparse grid rule for 100D of level 1 and order 201.

• cc_d100_level1_x.txt, the abscissas of the rule.
• cc_d100_level1_w.txt, the weights of the rule.
• cc_d100_level1_r.txt, defines the region for the rule.
• cc_d100_level1_exact.txt, the results of the exactness test, up to degree 2, demonstrating the accurate integration of 10,101 separate monomial terms in 100D, using a 201 point rule.

CCS_D2_LEVEL4 is a Clenshaw Curtis "Slow-Exponential-Growth" sparse grid rule for 2D of level 4 and order 49.

• ccs_d2_level4_x.txt, the abscissas of the rule.
• ccs_d2_level4_w.txt, the weights of the rule.
• ccs_d3_level4_r.txt, defines the region for the rule.
• ccs_d2_level4_exact.txt, the results of the exactness test, up to degree 17.

GL_D1_O3 is a Gauss-Legendre order 3 rule for 1D.

• gl_d1_o3_x.txt, the abscissas of the rule.
• gl_d1_o3_w.txt, the weights of the rule.
• gl_d1_o3_r.txt, defines the region for the rule.
• gl_d1_o3_exact.txt, the results of the exactness test, up to degree 5.

GL_D2_O3x3 is a Gauss-Legendre 3x3 product rule for 2D.

• gl_d2_o3x3_x.txt, the abscissas of the rule.
• gl_d2_o3x3_w.txt, the weights of the rule.
• gl_d2_o3x3_r.txt, defines the region for the rule.
• gl_d2_o3x3_exact.txt, the results of the exactness test, up to degree 5.

GL_D3_O3x3x3 is a Gauss-Legendre 3x3x3 product rule for 3D.

• gl_d3_o3x3x3_x.txt, the abscissas of the rule.
• gl_d3_o3x3x3_w.txt, the weights of the rule.
• gl_d3_o3x3x3_r.txt, defines the region for the rule.
• gl_d3_o3x3x3_exact.txt, the results of the exactness test, up to degree 5.

NCC_D1_O5 is a Newton-Cotes Closed order 5 rule for 1D.

• ncc_d1_o5_x.txt, the abscissas of the rule.
• ncc_d1_o5_w.txt, the weights of the rule.
• ncc_d1_o5_r.txt, defines the region for the rule.
• ncc_d1_o5_exact.txt, the results of the exactness test, up to degree 7.

NCC_D2_O5x5 is a Newton-Cotes Closed 5x5 product rule for 2D.

• ncc_d2_o5x5_x.txt, the abscissas of the rule.
• ncc_d2_o5x5_w.txt, the weights of the rule.
• ncc_d2_o5x5_r.txt, defines the region for the rule.
• ncc_d2_o5x5_exact.txt, the results of the exactness test, up to degree 7.

NCC_D3_O5x5x5 is a Newton-Cotes Closed 5x5x5 product rule for 3D.

• ncc_d3_o5x5x5_x.txt, the abscissas of the rule.
• ncc_d3_o5x5x5_w.txt, the weights of the rule.
• ncc_d3_o5x5x5_r.txt, defines the region for the rule.
• ncc_d3_o5x5x5_exact.txt, the results of the exactness test, up to degree 7.

List of Routines:

• MAIN is the main program for HYPERCUBE_EXACTNESS.
• CH_CAP capitalizes a single character.
• CH_EQI is a case insensitive comparison of two characters for equality.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• COMP_NEXT computes the compositions of the integer N into K parts.
• FILE_COLUMN_COUNT counts the number of columns in the first line of a file.
• FILE_ROW_COUNT counts the number of row records in a file.
• GET_UNIT returns a free FORTRAN unit number.
• MONOMIAL_INT01 returns the integral of a monomial over the [0,1] hypercube.
• MONOMIAL_QUADRATURE applies a quadrature rule to a monomial.
• MONOMIAL_VALUE evaluates a monomial.
• R8MAT_DATA_READ reads data from an R8MAT file.
• R8MAT_HEADER_READ reads the header from an R8MAT file.
• S_TO_I4 reads an I4 from a string.
• S_TO_R8 reads an R8 from a string.
• S_TO_R8VEC reads an R8VEC from a string.
• S_WORD_COUNT counts the number of "words" in a string.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.