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<html>
<head>
<title>
SANDIA_CUBATURE - Numerical Integration in M Dimensions
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SANDIA_CUBATURE <br> Numerical Integration<br> in M Dimensions
</h1>
<hr>
<p>
<b>SANDIA_CUBATURE</b>
is a MATLAB library which
implements quadrature rules for certain multidimensional regions and weight functions.
</p>
<p>
We consider the following integration regions:
<ul>
<li>
<b>CN_GEG</b>, the N dimensional hypercube [-1,+1]^N, with the Gegenbauer
weight function:<br>
w(alpha;x) = product ( 1 <= i <= n ) ( 1 - x(i)^2 )^alpha;
</li>
<li>
<b>CN_JAC</b>, the N dimensional hypercube [-1,+1]^N, with the Beta or
Jacobi weight function:<br>
w(alpha,beta;x) = product ( 1 <= i <= n ) ( 1 - x(i) )^alpha * ( 1 + x(i) )^beta;
</li>
<li>
<b>CN_LEG</b>, the N dimensional hypercube [-1,+1]^N, with the Legendre
weight function:<br>
w(x) = 1;
</li>
<li>
<b>EN_HER</b>, the N-dimensional product space (-oo,+oo)^N,
with the Hermite weight function:<br>
w(x) = product ( 1 <= i <= n ) exp ( - x(i)^2 );
</li>
<li>
<b>EPN_GLG</b>, the positive product space [0,+oo)^N, with the generalized
Laguerre weight function:<br>
w(alpha;x) = product ( 1 <= i <= n ) x(i)^alpha exp ( - x(i) );
</li>
<li>
<b>EPN_LAG</b>, the positive product space [0,+oo)^N, with the exponential or
Laguerre weight function:<br>
w(x) = product ( 1 <= i <= n ) exp ( - x(i) );
</li>
</ul>
</p>
<p>
The available rules for region <b>EN_HER</b> all have odd precision, ranging
from 1 to 11. Some of these rules are valid for any spatial dimension <b>N</b>.
However, many of these rules are restricted to a limited range, such as
<b>2 <= N < 6</b>. Some of the rules have two forms; in that case,
the particular form is selectable by setting an input argument <b>OPTION</b>
to 1 or 2. Finally, note that in multidimensional integration, the dependence
of the order <b>O</b> (number of abscissas) on the spatial dimension <b>N</b>
is critical. Rules for which the order is a multiple of <b>2^N</b> are not
practical for large values of <b>N</b>. The source code for each rule lists its
formula for the order as a function of <b>N</b>.
</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>SANDIA_CUBATURE</b> is available in
<a href = "../../cpp_src/sandia_cubature/sandia_cubature.html">a C++ version</a> and
<a href = "../../f_src/sandia_cubature/sandia_cubature.html">a FORTRAN90 version</a> and
<a href = "../../m_src/sandia_cubature/sandia_cubature.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/sandia_rules/sandia_rules.html">
SANDIA_RULES</a>,
a MATLAB library which
produces 1D quadrature rules of
Chebyshev, Clenshaw Curtis, Fejer 2, Gegenbauer, generalized Hermite,
generalized Laguerre, Hermite, Jacobi, Laguerre, Legendre and Patterson types.
</p>
<p>
<a href = "../../m_src/stroud/stroud.html">
STROUD</a>,
a MATLAB library which
defines quadrature rules for a variety of multidimensional reqions.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Arthur Stroud,<br>
Approximate Calculation of Multiple Integrals,<br>
Prentice Hall, 1971,<br>
ISBN: 0130438936,<br>
LC: QA311.S85.
</li>
<li>
Arthur Stroud, Don Secrest,<br>
Gaussian Quadrature Formulas,<br>
Prentice Hall, 1966,<br>
LC: QA299.4G3S7.
</li>
<li>
Dongbin Xiu,<br>
Numerical integration formulas of degree two,<br>
Applied Numerical Mathematics,<br>
Volume 58, 2008, pages 1515-1520.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "c1_geg_monomial_integral.m">c1_geg_monomial_integral.m</a>
integral of a monomial with Gegenbauer weight over C1.
</li>
<li>
<a href = "c1_jac_monomial_integral.m">c1_jac_monomial_integral.m</a>
integral of a monomial with Jacobi weight over C1.
</li>
<li>
<a href = "c1_leg_monomial_integral.m">c1_leg_monomial_integral.m</a>
integral of a monomial with Legendre weight over C1.
</li>
<li>
<a href = "cn_geg_00_1.m">cn_geg_00_1.m</a>
implements the midpoint rule for region CN_GEG.
</li>
<li>
<a href = "cn_geg_01_1.m">cn_geg_01_1.m</a>
implements a precision 1 rule for region CN_GEG.
</li>
<li>
<a href = "cn_geg_02_xiu.m">cn_geg_02_xiu.m</a>
implements the Xiu rule for region CN_GEG.
</li>
<li>
<a href = "cn_geg_03_xiu.m">cn_geg_03_xiu.m</a>
implements the Xiu rule for region CN_GEG.
</li>
<li>
<a href = "cn_geg_monomial_integral.m">cn_geg_monoial_integral.m</a>
integral of a monomial with Gegenbauer weight over CN.
</li>
<li>
<a href = "cn_jac_00_1.m">cn_jac_00_1.m</a>
implements the midpoint rule for region CN_JAC.
</li>
<li>
<a href = "cn_jac_01_1.m">cn_jac_01_1.m</a>
implements a precision 1 rule for region CN_JAC.
</li>
<li>
<a href = "cn_jac_02_xiu.m">cn_jac_02_xiu.m</a>
implements the Xiu rule for region CN_JAC.
</li>
<li>
<a href = "cn_jac_monomial_integral.m">cn_jac_monoial_integral.m</a>
integral of a monomial with Jacobi weight over CN.
</li>
<li>
<a href = "cn_leg_01_1.m">cn_leg_01_1.m</a>
implements the midpoint rule for region CN_LEG.
</li>
<li>
<a href = "cn_leg_02_xiu.m">cn_leg_02_xiu.m</a>
implements the Xiu rule for region CN_LEG.
</li>
<li>
<a href = "cn_leg_03_1.m">cn_leg_03_1.m</a>
implements the Stroud CN:3-1 rule for region CN_LEG.
</li>
<li>
<a href = "cn_leg_03_xiu.m">cn_leg_03_xiu.m</a>
implements the Xiu rule for region CN_LEG.
</li>
<li>
<a href = "cn_leg_05_1.m">cn_leg_05_1.m</a>
implements the Stroud CN:5-1 rule for region CN_LEG.
</li>
<li>
<a href = "cn_leg_05_2.m">cn_leg_05_2.m</a>
implements the Stroud CN:5-2 rule for region CN_LEG.
</li>
<li>
<a href = "cn_leg_monomial_integral.m">cn_leg_monoial_integral.m</a>
integral of a monomial with Legendre weight over CN.
</li>
<li>
<a href = "en_her_01_1.m">en_her_01_1.m</a>
a precision 1 rule for EN_HER.
</li>
<li>
<a href = "en_her_02_xiu.m">en_her_02_xiu.m</a>
a precision 2 rule for EN_HER.
</li>
<li>
<a href = "en_her_03_1.m">en_her_03_1.m</a>
a precision 3 rule for EN_HER.
</li>
<li>
<a href = "en_her_03_2.m">en_her_03_2.m</a>
a precision 3 rule for EN_HER.
</li>
<li>
<a href = "en_her_03_xiu.m">en_her_03_xiu.m</a>
a precision 3 rule for EN_HER.
</li>
<li>
<a href = "en_her_05_1.m">en_her_05_1.m</a>
a precision 5 rule for EN_HER.
</li>
<li>
<a href = "en_her_05_2.m">en_her_05_2.m</a>
a precision 5 rule for EN_HER.
</li>
<li>
<a href = "en_her_05_3.m">en_her_05_3.m</a>
a precision 5 rule for EN_HER.
</li>
<li>
<a href = "en_her_05_4.m">en_her_05_4.m</a>
a precision 5 rule for EN_HER.
</li>
<li>
<a href = "en_her_05_5.m">en_her_05_5.m</a>
a precision 5 rule for EN_HER.
</li>
<li>
<a href = "en_her_05_6.m">en_her_05_6.m</a>
a precision 5 rule for EN_HER.
</li>
<li>
<a href = "en_her_07_1.m">en_her_07_1.m</a>
a precision 7 rule for EN_HER.
</li>
<li>
<a href = "en_her_07_2.m">en_her_07_2.m</a>
a precision 7 rule for EN_HER.
</li>
<li>
<a href = "en_her_07_3.m">en_her_07_3.m</a>
a precision 7 rule for EN_HER.
</li>
<li>
<a href = "en_her_09_1.m">en_her_09_1.m</a>
a precision 9 rule for EN_HER.
</li>
<li>
<a href = "en_her_11_1.m">en_her_11_1.m</a>
a precision 11 rule for EN_HER.
</li>
<li>
<a href = "en_her_monomial_integral.m">en_her_monomial_integral.m</a>
evaluates the exact integral of a monomial in Stroud's EN_HER region.
</li>
<li>
<a href = "ep1_glg_monomial_integral.m">ep1_glg_monomial_integral.m</a>
integral of monomial with generalized Laguerre weight on EP1.
</li>
<li>
<a href = "ep1_lag_monomial_integral.m">ep1_lag_monomial_integral.m</a>
integral of monomial with Laguerre weight on EP1.
</li>
<li>
<a href = "epn_glg_00_1.m">epn_glg_001.m</a>
implements the midpoint rule for region EPN_GLG.
</li>
<li>
<a href = "epn_glg_01_1.m">epn_glg_01_1.m</a>
implements a precision 1 rule for region EPN_GLG.
</li>
<li>
<a href = "epn_glg_02_xiu.m">epn_glg_02_xiu.m</a>
implements the Xiu rule for region EPN_GLG.
</li>
<li>
<a href = "epn_glg_monomial_integral.m">epn_glg_monomial_integral.m</a>
integral of monomial with generalized Laguerre weight on EPN.
</li>
<li>
<a href = "epn_lag_00_1.m">epn_lag_001.m</a>
implements the midpoint rule for region EPN_LAG.
</li>
<li>
<a href = "epn_lag_01_1.m">epn_lag_01_1.m</a>
implements a precision 1 rule for region EPN_LAG.
</li>
<li>
<a href = "epn_lag_02_xiu.m">epn_lag_02_xiu.m</a>
implements the Xiu rule for region EPN_LAG.
</li>
<li>
<a href = "epn_lag_monomial_integral.m">epn_lag_monomial_integral.m</a>
integral of monomial with Laguerre weight on EPN.
</li>
<li>
<a href = "gw_02_xiu.m">gw_02_xiu.m</a>
implements the Golub-Welsch version of the Xiu rule.
</li>
<li>
<a href = "monomial_value.m">
monomial_value.m</a>,
evaluates a monomial given the exponents.
</li>
<li>
<a href = "r8_factorial.m">r8_factorial.m</a>
evaluates the factorial function.
</li>
<li>
<a href = "r8_hyper_2f1.m">r8_hyper_2f1.m</a>
evaluates the hypergeometric function 2F1(A,B,C,X).
</li>
<li>
<a href = "r8_mop.m">
r8_mop.m</a>
returns the I-th power of -1 as an R8 value.
</li>
<li>
<a href = "r8_psi.m">r8_psi.m</a>
evaluates the function Psi(X).
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>,
prints the YMDHMS date as a timestamp;
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "sandia_cubature_test.m">sandia_cubature_test.m</a>,
runs all the tests.
</li>
<li>
<a href = "sandia_cubature_test_output.txt">sandia_cubature_test_output.txt</a>,
the output file.
</li>
<li>
<a href = "cn_geg_tests.m">cn_geg_tests.m</a>,
tests the CN_GEG rules for various monomials.
</li>
<li>
<a href = "cn_geg_test.m">cn_geg_test.m</a>,
tests the CN_GEG rules for a particular monomial.
</li>
<li>
<a href = "cn_jac_tests.m">cn_jac_tests.m</a>,
tests the CN_JAC rules for various monomials.
</li>
<li>
<a href = "cn_jac_test.m">cn_jac_test.m</a>,
tests the CN_JAC rules for a particular monomial.
</li>
<li>
<a href = "cn_leg_tests.m">cn_leg_tests.m</a>,
tests the CN_LEG rules for various monomials.
</li>
<li>
<a href = "cn_leg_test.m">cn_leg_test.m</a>,
tests the CN_LEG rules for a particular monomial.
</li>
<li>
<a href = "en_her_tests.m">en_her_tests.m</a>,
tests the EN_HER rules for various monomials.
</li>
<li>
<a href = "en_her_test.m">en_her_test.m</a>,
tests the EN_HER rules for a particular monomial.
</li>
<li>
<a href = "epn_glg_tests.m">epn_glg_tests.m</a>,
tests the EPN_GLG rules for various monomials.
</li>
<li>
<a href = "epn_glg_test.m">epn_glg_test.m</a>,
tests the EPN_GLG rules for a particular monomial.
</li>
<li>
<a href = "epn_lag_tests.m">epn_lag_tests.m</a>,
tests the EPN_LAG rules for various monomials.
</li>
<li>
<a href = "epn_lag_test.m">epn_lag_test.m</a>,
tests the EPN_LAG rules for a particular monomial.
</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 03 March 2010.
</i>
<!-- John Burkardt -->
</body>
</html>