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<html>
<head>
<title>
HYPERCUBE_EXACTNESS - Exactness of Multidimensional Quadrature
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
HYPERCUBE_EXACTNESS <br> Exactness of Multidimensional Quadrature
</h1>
<hr>
<p>
<b>HYPERCUBE_EXACTNESS</b>
is a MATLAB program which
investigates the polynomial exactness of a quadrature rule
over the unit hypercube in M dimensions.
</p>
<p>
The polynomial exactness of a quadrature rule is defined as the
highest total degree <b>D</b> such that the quadrature rule is
guaranteed to integrate exactly all polynomials of total degree
<b>DEGREE_MAX</b> 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 <b>DEGREE</b> of
<blockquote><b>
x<sup>2</sup>y z<sup>5</sup>
</b></blockquote>
is 2+1+5=8.
</p>
<p>
To be thorough, the program starts at <b>DEGREE</b> = 0, and then
proceeds to <b>DEGREE</b> = 1, 2, and so on up to a maximum degree
<b>DEGREE_MAX</b> specified by the user. At each value of <b>DEGREE</b>,
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.
</p>
<p>
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.
</p>
<p>
Note that the three files that define the quadrature rule
are assumed to have related names, of the form
<ul>
<li>
<i>prefix</i>_<b>x.txt</b>
</li>
<li>
<i>prefix</i>_<b>w.txt</b>
</li>
<li>
<i>prefix</i>_<b>r.txt</b>
</li>
</ul>
When running the program, the user only enters the common <i>prefix</i>
part of the file names, which is enough information for the program
to find all three files.
</p>
<p>
The exactness results are written to an output file with the
corresponding name:
<ul>
<li>
<i>prefix</i>_<b>exact.txt</b>
</li>
</ul>
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>hypercube_exactness</b> ( <i>'prefix'</i>, <i>degree_max</i> )
</blockquote>
where
<ul>
<li>
<i>'prefix'</i> is the common prefix for the files containing the abscissa, weight
and region information of the quadrature rule;
</li>
<li>
<i>degree_max</i> 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!
</li>
</ul>
</p>
<p>
If the arguments are not supplied on the command line, the
program will prompt for them.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>HYPERCUBE_EXACTNESS</b> is available in
<a href = "../../c_src/hypercube_exactness/hypercube_exactness.html">a C version</a> and
<a href = "../../cpp_src/hypercube_exactness/hypercube_exactness.html">a C++ version</a> and
<a href = "../../f77_src/hypercube_exactness/hypercube_exactness.html">a FORTRAN90 version</a> and
<a href = "../../f_src/hypercube_exactness/hypercube_exactness.html">a FORTRAN90 version</a> and
<a href = "../../m_src/hypercube_exactness/hypercube_exactness.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/cube_exactness/cube_exactness.html">
CUBE_EXACTNESS</a>,
a MATLAB library which
investigates the polynomial exactness of quadrature rules
over the interior of a cube in 3D.
</p>
<p>
<a href = "../../m_src/hypercube_grid/hypercube_grid.html">
HYPERCUBE_GRID</a>,
a MATLAB library which
computes a grid of points
over the interior of a hypercube in M dimensions.
</p>
<p>
<a href = "../../m_src/pyramid_exactness/pyramid_exactness.html">
PYRAMID_EXACTNESS</a>,
a MATLAB program which
investigates the polynomial exactness of a quadrature rule
over the interior of the unit pyramid in 3D.
</p>
<p>
<a href = "../../m_src/sphere_exactness/sphere_exactness.html">
SPHERE_EXACTNESS</a>,
a MATLAB program which
tests the polynomial exactness of a quadrature rule
over the surface of the unit sphere in 3D.
</p>
<p>
<a href = "../../m_src/square_exactness/square_exactness.html">
SQUARE_EXACTNESS</a>,
a MATLAB library which
investigates the polynomial exactness of quadrature rules for f(x,y)
over the interior of a square (rectangle/quadrilateral) in 2D.
</p>
<p>
<a href = "../../m_src/tetrahedron_exactness/tetrahedron_exactness.html">
TETRAHEDRON_EXACTNESS</a>,
a MATLAB program which
investigates the polynomial exactness of a quadrature rule
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../m_src/triangle_exactness/triangle_exactness.html">
TRIANGLE_EXACTNESS</a>,
a MATLAB program which
investigates the monomial exactness quadrature rule
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/wedge_exactness/wedge_exactness.html">
WEDGE_EXACTNESS</a>,
a MATLAB program which
investigates the monomial exactness of a quadrature rule
over the interior of the unit wedge in 3D.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Philip Davis, Philip Rabinowitz,<br>
Methods of Numerical Integration,<br>
Second Edition,<br>
Dover, 2007,<br>
ISBN: 0486453391,<br>
LC: QA299.3.D28.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "hypercube_exactness.m">hypercube_exactness.m</a>,
is the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>CC_D1_O2</b> is a Clenshaw-Curtis order 2 rule for 1D.
<ul>
<li>
<a href = "../../datasets/quadrature_rules/cc_d1_o2_x.txt">cc_d1_o2_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/cc_d1_o2_w.txt">cc_d1_o2_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/cc_d1_o2_r.txt">cc_d1_o2_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "cc_d1_o2_exact.txt">cc_d1_o2_exact.txt</a>,
the results of the exactness test, up to degree 5.
</li>
</ul>
</p>
<p>
<b>CC_D1_O3</b> 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!
<ul>
<li>
<a href = "../../datasets/quadrature_rules/cc_d1_o3_x.txt">cc_d1_o3_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/cc_d1_o3_w.txt">cc_d1_o3_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/cc_d1_o3_r.txt">cc_d1_o3_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "cc_d1_o3_exact.txt">cc_d1_o3_exact.txt</a>,
the results of the exactness test, up to degree 5.
</li>
</ul>
</p>
<p>
<b>CC_D2_O3x3</b> is a Clenshaw-Curtis 3x3 product rule for 2D.
<ul>
<li>
<a href = "../../datasets/quadrature_rules/cc_d2_o3x3_x.txt">cc_d2_o3x3_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/cc_d2_o3x3_w.txt">cc_d2_o3x3_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/cc_d2_o3x3_r.txt">cc_d2_o3x3_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "cc_d2_o3x3_exact.txt">cc_d2_o3x3_exact.txt</a>,
the results of the exactness test, up to degree 8.
</li>
</ul>
</p>
<p>
<b>CC_D3_O3x3x3</b> is a Clenshaw-Curtis 3x3x3 product rule for 3D.
<ul>
<li>
<a href = "../../datasets/quadrature_rules/cc_d3_o3x3x3_x.txt">cc_d3_o3x3x3_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/cc_d3_o3x3x3_w.txt">cc_d3_o3x3x3_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/cc_d3_o3x3x3_r.txt">cc_d3_o3x3x3_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "cc_d3_o3x3x3_exact.txt">cc_d3_o3x3x3_exact.txt</a>,
the results of the exactness test, up to degree 5.
</li>
</ul>
</p>
<p>
<b>CCGL_D2_O3x2</b> is a product rule for 2D whose factors are
a Clenshaw-Curtis of order 3 and a Gauss-Legendre rule of order 2.
<ul>
<li>
<a href = "../../datasets/quadrature_rules/ccgl_d2_o3x2_x.txt">ccgl_d2_o3x2_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/ccgl_d2_o3x2_w.txt">ccgl_d2_o3x2_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/ccgl_d2_o3x2_r.txt">ccgl_d2_o3x2_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "ccgl_d2_o3x2_exact.txt">ccgl_d2_o3x2_exact.txt</a>,
the results of the exactness test, up to degree 5.
</li>
</ul>
</p>
<p>
<b>CC_D2_LEVEL0</b> is a Clenshaw Curtis sparse rule for 2D
of level 0 and order 1.
<ul>
<li>
<a href = "../../datasets/quadrature_rules/cc_d2_level0_x.txt">cc_d2_level0_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/cc_d2_level0_w.txt">cc_d2_level0_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/cc_d2_level0_r.txt">cc_d2_level0_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "cc_d2_level0_exact.txt">cc_d2_level0_exact.txt</a>,
the results of the exactness test, up to degree 4.
</li>
</ul>
</p>
<p>
<b>CC_D2_LEVEL1</b> is a Clenshaw Curtis sparse rule for 2D
of level 1 and order 5.
<ul>
<li>
<a href = "../../datasets/quadrature_rules/cc_d2_level1_x.txt">cc_d2_level1_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/cc_d2_level1_w.txt">cc_d2_level1_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/cc_d2_level1_r.txt">cc_d2_level1_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "cc_d2_level1_exact.txt">cc_d2_level1_exact.txt</a>,
the results of the exactness test, up to degree 4.
</li>
</ul>
</p>
<p>
<b>CC_D2_LEVEL2</b> is a Clenshaw Curtis sparse rule for 2D
of level 2 and order 13.
<ul>
<li>
<a href = "../../datasets/quadrature_rules/cc_d2_level2_x.txt">cc_d2_level2_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/cc_d2_level2_w.txt">cc_d2_level2_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/cc_d2_level2_r.txt">cc_d2_level2_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "cc_d2_level2_exact.txt">cc_d2_level2_exact.txt</a>,
the results of the exactness test, up to degree 6.
</li>
</ul>
</p>
<p>
<b>CC_D2_LEVEL3</b> is a Clenshaw Curtis sparse rule for 2D
of level 3 and order 25.
<ul>
<li>
<a href = "../../datasets/quadrature_rules/cc_d2_level3_x.txt">cc_d2_level3_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/cc_d2_level3_w.txt">cc_d2_level3_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/cc_d2_level3_r.txt">cc_d3_level2_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "cc_d2_level2_exact.txt">cc_d2_level2_exact.txt</a>,
the results of the exactness test, up to degree 9.
</li>
</ul>
</p>
<p>
<b>CC_D2_LEVEL4</b> is a Clenshaw Curtis sparse grid rule for 2D
of level 4 and order 65.
<ul>
<li>
<a href = "../../datasets/sparse_grid_cc/cc_d2_level4_x.txt">cc_d2_level4_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/sparse_grid_cc/cc_d2_level4_w.txt">cc_d2_level4_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/sparse_grid_cc/cc_d2_level4_r.txt">cc_d3_level4_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "cc_d2_level4_exact.txt">cc_d2_level4_exact.txt</a>,
the results of the exactness test, up to degree 17.
</li>
</ul>
</p>
<p>
<b>CCS_D2_LEVEL4</b> is a Clenshaw Curtis "Slow-Exponential-Growth" sparse grid rule for 2D
of level 4 and order 49.
<ul>
<li>
<a href = "../../datasets/sparse_grid_ccs/ccs_d2_level4_x.txt">ccs_d2_level4_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/sparse_grid_ccs/ccs_d2_level4_w.txt">ccs_d2_level4_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/sparse_grid_ccs/ccs_d2_level4_r.txt">ccs_d3_level4_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "ccs_d2_level4_exact.txt">ccs_d2_level4_exact.txt</a>,
the results of the exactness test, up to degree 17.
</li>
</ul>
</p>
<p>
<b>GL_D1_O3</b> is a Gauss-Legendre order 3 rule for 1D.
<ul>
<li>
<a href = "../../datasets/quadrature_rules/gl_d1_o3_x.txt">gl_d1_o3_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/gl_d1_o3_w.txt">gl_d1_o3_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/gl_d1_o3_r.txt">gl_d1_o3_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "gl_d1_o3_exact.txt">gl_d1_o3_exact.txt</a>,
the results of the exactness test, up to degree 5.
</li>
</ul>
</p>
<p>
<b>GL_D2_O3x3</b> is a Gauss-Legendre 3x3 product rule for 2D.
<ul>
<li>
<a href = "../../datasets/quadrature_rules/gl_d2_o3x3_x.txt">gl_d2_o3x3_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/gl_d2_o3x3_w.txt">gl_d2_o3x3_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/gl_d2_o3x3_r.txt">gl_d2_o3x3_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "gl_d2_o3x3_exact.txt">gl_d2_o3x3_exact.txt</a>,
the results of the exactness test, up to degree 5.
</li>
</ul>
</p>
<p>
<b>GL_D3_O3x3x3</b> is a Gauss-Legendre 3x3x3 product rule for 3D.
<ul>
<li>
<a href = "../../datasets/quadrature_rules/gl_d3_o3x3x3_x.txt">gl_d3_o3x3x3_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/gl_d3_o3x3x3_w.txt">gl_d3_o3x3x3_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/gl_d3_o3x3x3_r.txt">gl_d3_o3x3x3_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "gl_d3_o3x3x3_exact.txt">gl_d3_o3x3x3_exact.txt</a>,
the results of the exactness test, up to degree 5.
</li>
</ul>
</p>
<p>
<b>NCC_D1_O5</b> is a Newton-Cotes Closed order 5 rule for 1D.
<ul>
<li>
<a href = "../../datasets/quadrature_rules/ncc_d1_o5_x.txt">ncc_d1_o5_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/ncc_d1_o5_w.txt">ncc_d1_o5_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/ncc_d1_o5_r.txt">ncc_d1_o5_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "ncc_d1_o5_exact.txt">ncc_d1_o5_exact.txt</a>,
the results of the exactness test, up to degree 7.
</li>
</ul>
</p>
<p>
<b>NCC_D2_O5x5</b> is a Newton-Cotes Closed 5x5 product rule for 2D.
<ul>
<li>
<a href = "../../datasets/quadrature_rules/ncc_d2_o5x5_x.txt">ncc_d2_o5x5_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/ncc_d2_o5x5_w.txt">ncc_d2_o5x5_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/ncc_d2_o5x5_r.txt">ncc_d2_o5x5_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "ncc_d2_o5x5_exact.txt">ncc_d2_o5x5_exact.txt</a>,
the results of the exactness test, up to degree 7.
</li>
</ul>
</p>
<p>
<b>NCC_D3_O125</b> is a Newton-Cotes Closed 5x5x5 product rule for 3D.
<ul>
<li>
<a href = "../../datasets/quadrature_rules/ncc_d3_o5x5x5_x.txt">ncc_d3_o5x5x5_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/ncc_d3_o5x5x5_w.txt">ncc_d3_o5x5x5_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules/ncc_d3_o5x5x5_r.txt">ncc_d3_o5x5x5_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "ncc_d3_o5x5x5_exact.txt">ncc_d3_o5x5x5_exact.txt</a>,
the results of the exactness test, up to degree 7.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 16 May 2007.
</i>
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