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<html>
<head>
<title>
NINT_EXACTNESS_MIXED - Exactness of Multidimensional Quadrature Using Mixed Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
NINT_EXACTNESS_MIXED <br> Exactness of Multidimensional Quadrature Using Mixed Rules
</h1>
<hr>
<p>
<b>NINT_EXACTNESS_MIXED</b>
is a MATLAB program which
investigates the polynomial exactness of a multidimensional
quadrature rule based on a direct product of one-dimensional rules.
The one dimensional rules, in turn, may be any mixture of rules
for 6 standard quadrature problems.
</p>
<p>
The six standard 1d quadrature problems that may be used as factors
for the multidimensional problem are:
<ul>
<li>
<b>Legendre integration</b><br>
interval: [-1,+1],<br>
weight: w(x) = 1,<br>
rules: Clenshaw Curtis, Fejer Type 2, Gauss Legendre, Gauss Patterson.
</li>
<li>
<b>Jacobi integration</b>:<br>
interval: [-1,+1],<br>
weight: w(x) = (1-x)^alpha (1+x)^beta.<br>
rules: Gauss Jacobi.
</li>
<li>
<b>Laguerre integration</b> <br>
interval: [0,+oo),<br>
weight: w(x) = exp(-x).<br>
rules: Gauss Laguerre.
</li>
<li>
<b>Generalized Laguerre integration</b><br>
interval: [0,+oo),<br>
weight: w(x) = x^alpha exp(-x).<br>
rules: Gauss Laguerre.
</li>
<li>
<b>Hermite integration</b><br>
interval: (-oo,+oo),<br>
weight: w(x) = exp(-x*x).<br>
rules: Gauss Hermite.
</li>
<li>
<b>Generalized Hermite integration</b> <br>
interval: (-oo,+oo),<br>
weight: w(x) = |x|^alpha exp(-x*x).<br>
rules: generalized Gauss Hermite.
</li>
</ul>
</p>
<p>
The M-dimensional quadrature region R based on mixed factors is formed by the
direct product
<blockquote><b>
R = R1 x R2 x ... x Rm
</b></blockquote>
where each factor region Ri is the region associated with one of the six
rules. Thus, R is a sort of generalized hyperrectangle, with the understanding
that in some coordinate directions the region may be semi-infinite or
infinite.
</p>
<p>
The M-dimensional weight function W based on mixed factors is formed by the
dproduct
<blockquote><b>
w(x1,x2,...xm) = w1(x1) x w2(x2) x ... x wm(xm)
</b></blockquote>
where each factor weight wi(xi) is the weight function associated with one of the six
rules. Some weight functions include parameters alpha and beta, and these
parameters may be specified independently in each dimension.
</p>
<p>
For a quadrature region R based on mixed factors, the corresponding
monomial integrand has the form
<blockquote><b>
Mono(X,E) = X1^E1 x X2^E2 x ... x Xm^Em
</b></blockquote>
where each exponent Ei is a nonnegative integer.
</p>
<p>
The total degree of a monomial Mono(X,E) is:
<blockquote><b>
TotalDegree(Mono(X,E)) = Sum ( 1 <= I <= M ) E(I)
</b></blockquote>
</p>
<p>
Thus, for instance, the total degree of
<blockquote><b>
x1<sup>2</sup> * x2 * x3<sup>5</sup>
</b></blockquote>
is 2+1+5=8.
</p>
<p>
The corresponding monomial integral is:
<blockquote><b>
Integral ( X in R ) Mono(X,E) W(X) dX
</b></blockquote>
where each exponent Ei is a nonnegative integer.
</p>
<p>
The <b>monomial exactness</b> of a quadrature rule is the maximum number D
such that, for every monomial of total degree D or less, the quadrature
rule produces the exact value of the monomial integral.
</p>
<p>
The <b>polynomial exactness</b> of a quadrature rule is the maximum number D
such that, for every polynomial of total degree D or less, the quadrature
rule produces the exact value of the polynomial integral. The total degree
of a polynomial is simply the maximum of the total degrees of the monomials
that form the polynomial.
</p>
<p>
This program is given a quadrature rule based on mixed factors, and seeks
to determine the polynomial exactness of the rule. It does this simply
by applying the quadrature rule to all the monomials of a total degree
0 up to some limit specified by the user.
</p>
<p>
The program is very flexible and interactive. The quadrature rule
is defined by five files, to be read at input, and the
maximum degree is specified by the user as well.
</p>
<p>
The files that define the quadrature rule
are assumed to have related names, of the form
<ul>
<li>
<i>prefix</i>_<b>a.txt</b>, the "ALPHA" file;
</li>
<li>
<i>prefix</i>_<b>b.txt</b>, the "BETA" file;
</li>
<li>
<i>prefix</i>_<b>r.txt</b>, the "REGION" file;
</li>
<li>
<i>prefix</i>_<b>w.txt</b>, the "WEIGHT" file;
</li>
<li>
<i>prefix</i>_<b>x.txt</b>, the "ABSCISSA" file.
</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 the files.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>nint_exactness_mixed</b> ( <i>'prefix'</i>, <i>degree_max</i> )
</blockquote>
where
<ul>
<li>
<i>'prefix'</i> is the common prefix for the files containing the alpha, beta, region, weight
and abscissa 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>NINT_EXACTNESS_MIXED</b> is available in
<a href = "../../cpp_src/nint_exactness_mixed/nint_exactness_mixed.html">a C++ version</a> and
<a href = "../../f_src/nint_exactness_mixed/nint_exactness_mixed.html">a FORTRAN90 version</a> and
<a href = "../../m_src/nint_exactness_mixed/nint_exactness_mixed.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/int_exactness/int_exactness.html">
INT_EXACTNESS</a>,
a MATLAB program which
tests the polynomial exactness of one dimensional quadrature rules.
</p>
<p>
<a href = "../../m_src/nint_exactness/nint_exactness.html">
NINT_EXACTNESS</a>,
a MATLAB program which
tests the polynomial exactness of integration rules for the unit hypercube.
</p>
<p>
<a href = "../../m_src/nintlib/nintlib.html">
NINTLIB</a>,
a MATLAB library which
numerically estimates integrals in multiple 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 for the pyramid.
</p>
<p>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed.html">
SPARSE_GRID_MIXED</a>,
a dataset directory which
contains multidimensional Smolyak sparse grids
based on a mixed set of 1D factor rules.
</p>
<p>
<a href = "../../m_src/sparse_grid_mixed/sparse_grid_mixed.html">
SPARSE_GRID_MIXED</a>,
a MATLAB library which
defines multidimensional quadrature rules using sparse grid techniques,
based on a mixed set of 1D quadrature rules.
</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 for the unit sphere;
</p>
<p>
<a href = "../../m_src/test_nint/test_nint.html">
TEST_NINT</a>,
a MATLAB library which
defines integrand functions for testing
multidimensional quadrature routines.
</p>
<p>
<a href = "../../m_src/testpack/testpack.html">
TESTPACK</a>,
a MATLAB library which
defines a set of integrands used to test multidimensional quadrature.
</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 for the tetrahedron.
</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 = "nint_exactness_mixed.m">nint_exactness_mixed.m</a>,
is the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>SPARSE_GRID_MIXED_D2_L2_CCXCC</b> is a level 2 sparse grid quadrature rule for
dimension 2, based on 1D factors of [Clenshaw Curtis, Clenshaw Curtis].
<ul>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxcc_a.txt">
sparse_grid_mixed_d2_l2_ccxcc_a.txt</a>,
the A file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxcc_b.txt">
sparse_grid_mixed_d2_l2_ccxcc_b.txt</a>,
the B file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxcc_r.txt">
sparse_grid_mixed_d2_l2_ccxcc_r.txt</a>,
the R file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxcc_w.txt">
sparse_grid_mixed_d2_l2_ccxcc_w.txt</a>,
the W file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxcc_x.txt">
sparse_grid_mixed_d2_l2_ccxcc_x.txt</a>,
the X file.
</li>
<li>
<a href = "sparse_grid_mixed_d2_l2_ccxcc_exactness.txt">
sparse_grid_mixed_d2_l2_ccxcc_exactness.txt</a>,
the output file for DEGREE_MAX = 7.
</li>
</ul>
</p>
<p>
<b>SPARSE_GRID_MIXED_D2_L2_CCXGL</b> is a level 2 sparse grid quadrature rule for
dimension 2, based on 1D factors of [Clenshaw Curtis, Gauss Legendre].
<ul>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxgl_a.txt">
sparse_grid_mixed_d2_l2_ccxgl_a.txt</a>,
the A file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxgl_b.txt">
sparse_grid_mixed_d2_l2_ccxgl_b.txt</a>,
the B file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxgl_r.txt">
sparse_grid_mixed_d2_l2_ccxgl_r.txt</a>,
the R file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxgl_w.txt">
sparse_grid_mixed_d2_l2_ccxgl_w.txt</a>,
the W file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxgl_x.txt">
sparse_grid_mixed_d2_l2_ccxgl_x.txt</a>,
the X file.
</li>
<li>
<a href = "sparse_grid_mixed_d2_l2_ccxgl_exactness.txt">
sparse_grid_mixed_d2_l2_ccxgl_exactness.txt</a>,
the output file for DEGREE_MAX = 7.
</li>
</ul>
</p>
<p>
<b>SPARSE_GRID_MIXED_D2_L2_CCXGLG</b> is a level 2 sparse grid quadrature rule for
dimension 2, based on 1D factors of [Clenshaw Curtis, Generalized Gauss Laguerre].
The Generalized Gauss Laguerre rule uses ALPHA = 1.5.
<ul>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxglg_a.txt">
sparse_grid_mixed_d2_l2_ccxglg_a.txt</a>,
the A file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxglg_b.txt">
sparse_grid_mixed_d2_l2_ccxglg_b.txt</a>,
the B file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxglg_r.txt">
sparse_grid_mixed_d2_l2_ccxglg_r.txt</a>,
the R file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxglg_w.txt">
sparse_grid_mixed_d2_l2_ccxglg_w.txt</a>,
the W file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxglg_x.txt">
sparse_grid_mixed_d2_l2_ccxglg_x.txt</a>,
the X file.
</li>
<li>
<a href = "sparse_grid_mixed_d2_l2_ccxglg_exactness.txt">
sparse_grid_mixed_d2_l2_ccxglg_exactness.txt</a>,
the output file for DEGREE_MAX = 7.
</li>
</ul>
</p>
<p>
<b>SPARSE_GRID_MIXED_D2_L2_CCXLG</b> is a level 2 sparse grid quadrature rule for
dimension 2, based on 1D factors of [Clenshaw Curtis, Gauss Laguerre].
<ul>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxlg_a.txt">
sparse_grid_mixed_d2_l2_ccxlg_a.txt</a>,
the A file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxlg_b.txt">
sparse_grid_mixed_d2_l2_ccxlg_b.txt</a>,
the B file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxlg_r.txt">
sparse_grid_mixed_d2_l2_ccxlg_r.txt</a>,
the R file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxlg_w.txt">
sparse_grid_mixed_d2_l2_ccxlg_w.txt</a>,
the W file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_ccxlg_x.txt">
sparse_grid_mixed_d2_l2_ccxlg_x.txt</a>,
the X file.
</li>
<li>
<a href = "sparse_grid_mixed_d2_l2_ccxlg_exactness.txt">
sparse_grid_mixed_d2_l2_ccxlg_exactness.txt</a>,
the output file for DEGREE_MAX = 7.
</li>
</ul>
</p>
<p>
<b>SPARSE_GRID_MIXED_D2_L2_F2XGJ</b> is a level 2 sparse grid quadrature rule for
dimension 2, based on 1D factors of [Fejer Type 2, Gauss Jacobi]. The
Gauss Jacobi rule uses ALPHA = 0.5, BETA = 1.5.
<ul>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_f2xgj_a.txt">
sparse_grid_mixed_d2_l2_f2xgj_a.txt</a>,
the A file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_f2xgj_b.txt">
sparse_grid_mixed_d2_l2_f2xgj_b.txt</a>,
the B file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_f2xgj_r.txt">
sparse_grid_mixed_d2_l2_f2xgj_r.txt</a>,
the R file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_f2xgj_w.txt">
sparse_grid_mixed_d2_l2_f2xgj_w.txt</a>,
the W file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l2_f2xgj_x.txt">
sparse_grid_mixed_d2_l2_f2xgj_x.txt</a>,
the X file.
</li>
<li>
<a href = "sparse_grid_mixed_d2_l2_f2xgj_exactness.txt">
sparse_grid_mixed_d2_l2_f2xgj_exactness.txt</a>,
the output file for DEGREE_MAX = 7.
</li>
</ul>
</p>
<p>
<b>SPARSE_GRID_MIXED_D2_L3_CCXGP</b> is a level 3 sparse grid quadrature rule for
dimension 2, based on 1D factors of [Clenshaw Curtis, Gauss Patterson].
<ul>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l3_ccxgp_a.txt">
sparse_grid_mixed_d2_l3_ccxgp_a.txt</a>,
the A file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l3_ccxgp_b.txt">
sparse_grid_mixed_d2_l3_ccxgp_b.txt</a>,
the B file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l3_ccxgp_r.txt">
sparse_grid_mixed_d2_l3_ccxgp_r.txt</a>,
the R file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l3_ccxgp_w.txt">
sparse_grid_mixed_d2_l3_ccxgp_w.txt</a>,
the W file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d2_l3_ccxgp_x.txt">
sparse_grid_mixed_d2_l3_ccxgp_x.txt</a>,
the X file.
</li>
<li>
<a href = "sparse_grid_mixed_d2_l3_ccxgp_exactness.txt">
sparse_grid_mixed_d2_l3_ccxgp_exactness.txt</a>,
the output file for DEGREE_MAX = 9.
</li>
</ul>
</p>
<p>
<b>SPARSE_GRID_MIXED_D3_L2_CCXF2XGH</b> is a level 2 sparse grid quadrature rule for
dimension 3, based on 1D factors of [Clenshaw Curtis, Fejer Type 2, Gauss Hermite].
<ul>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d3_l2_ccxf2xgh_a.txt">
sparse_grid_mixed_d3_l2_ccxf2xgh_a.txt</a>,
the A file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d3_l2_ccxf2xgh_b.txt">
sparse_grid_mixed_d3_l2_ccxf2xgh_b.txt</a>,
the B file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d3_l2_ccxf2xgh_r.txt">
sparse_grid_mixed_d3_l2_ccxf2xgh_r.txt</a>,
the R file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d3_l2_ccxf2xgh_w.txt">
sparse_grid_mixed_d3_l2_ccxf2xgh_w.txt</a>,
the W file.
</li>
<li>
<a href = "../../datasets/sparse_grid_mixed/sparse_grid_mixed_d3_l2_ccxf2xgh_x.txt">
sparse_grid_mixed_d3_l2_ccxf2xgh_x.txt</a>,
the X file.
</li>
<li>
<a href = "sparse_grid_mixed_d3_l2_ccxf2xgh_exactness.txt">
sparse_grid_mixed_d3_l2_ccxf2xgh_exactness.txt</a>,
the output file.
</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 17 October 2008.
</i>
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