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<html>
<head>
<title>
SPARSE_GRID_MIXED_DATASET - Create Sparse Grid Using Mixed Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SPARSE_GRID_MIXED_DATASET <br> Create Sparse Grid Using Mixed Rules
</h1>
<hr>
<p>
<b>SPARSE_GRID_MIXED_DATASET</b>
is a C++ program which
creates a sparse grid dataset from a mixture of 1D rules, using
instructions from a user.
</p>
<p>
The available 1D quadrature rules are identified by a 2 or 3 letter code,
and include:
<ol>
<li>
<b>CC</b>, Clenshaw-Curtis:<br>
defined on [-1,+1], with w(x)=1,<br>
a closed, fully nested rule.
</li>
<li>
<b>F2</b>, Fejer Type 2: <br>
defined on (-1,+1), with w(x)=1,<br>
an open, fully nested rule.
</li>
<li>
<b>GP</b>, Gauss Patterson: <br>
defined on (-1,+1), with w(x)=1,<br>
a family of the midpoint rule, the 3 point
Gauss Legendre rule, and then successive Patterson refinements,<br>
an open, fully nested rule.
</li>
<li>
<b>GL</b>, Gauss Legendre: <br>
defined on (-1,+1), with w(x)=1,<br>
an open, weakly nested rule.
</li>
<li>
<b>GH</b>, Gauss Hermite: <br>
defined on (-oo,+oo), with w(x)=exp(-x*x),<br>
an open, weakly nested rule.
</li>
<li>
<b>GGH</b>, Generalized Gauss Hermite: <br>
defined on (-oo,+oo), with w(x)=|x|^alpha * exp(-x*x),<br>
an open, weakly nested rule.
</li>
<li>
<b>LG</b>, Gauss Laguerre: <br>
defined on (0,+oo) with w(x)=exp(-x),<br>
an open, non-nested rule.
</li>
<li>
<b>GLG</b>, Generalized Gauss Laguerre: <br>
defined on (0,+oo) with w(x)=x^alpha * exp(-x),<br>
an open, non-nested rule.
</li>
<li>
<b>GJ</b>, Gauss Jacobi: <br>
defined on [-1,+1] with w(x)=(1-x)^alpha (1+x)^beta<br>
an open, non-nested rule.
</li>
<li>
<b>GW</b>, Golub Welsch: <br>
a rule defined by the user based on the Golub-Welsch algorithm,<br>
with points and weights supplied by external routines,<br>
presumably an open, non-nested rule.<br>
<i>(not supported yet)</i>
</li>
<li>
<b>CCS</b>, Clenshaw-Curtis Slow-Growth:<br>
defined on [-1,+1], with w(x)=1,<br>
a closed, fully nested rule.
</li>
<li>
<b>F2S</b>, Fejer Type 2, Slow-Growth: <br>
defined on (-1,+1), with w(x)=1,<br>
an open, fully nested rule.
</li>
<li>
<b>GPS</b>, Gauss Patterson, Slow-Growth: <br>
defined on (-1,+1), with w(x)=1,<br>
a family of the midpoint rule, the 3 point
Gauss Legendre rule, and then successive Patterson refinements,<br>
an open, fully nested rule.
</li>
</ol>
</p>
<h3 align = "center">
Usage:
</h3>
<p>
The user first specifies:
<ul>
<li>
<b>M</b> the spatial dimension of the quadrature region,
</li>
<li>
<b>L</b> the level that defines the Smolyak grid.
</li>
</ul>
and these two values can actuallyb be specified on the command line.
</p>
<p>
Then the user specifies rules for each of the M dimensions.
A rule, when specified, may be used for one, or for multiple consecutive
dimensions.
<ul>
<li>
RULE identifies the 1D rule, and must be one of
"CC", "F2", "GP", "GL", "GH", "GGH", "LG", "GLG", "GJ", "GW",
"CCS", "F2S", "GPS".
</li>
<li>
the number of times the rule is to be used for consecutive dimensions;
</li>
<li>
ALPHA, a rule parameter, if needed. (Rules GGH, GLG and GJ);
</li>
<li>
BETA, a rule parameter, if needed. (Rule GJ only).
</li>
</ul>
</p>
<p>
Finally, the user specifies a "file identifier", that is, the main part
of a string to be used when creating the files.
</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>SPARSE_GRID_MIXED_DATASET</b> is available in
<a href = "../../cpp_src/sparse_grid_mixed_dataset/sparse_grid_mixed_dataset.html">a C++ version</a> and
<a href = "../../f_src/sparse_grid_mixed_dataset/sparse_grid_mixed_dataset.html">a FORTRAN90 version</a> and
<a href = "../../m_src/sparse_grid_mixed_dataset/sparse_grid_mixed_dataset.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/sandia_rules/sandia_rules.html">
SANDIA_RULES</a>,
a C++ 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 = "../../datasets/sparse_grid_cc/sparse_grid_cc.html">
SPARSE_GRID_CC</a>,
a dataset directory which
contains multidimensional Smolyak sparse grids
based on the 1D Clenshaw Curtis rule;
</p>
<p>
<a href = "../../cpp_src/sparse_grid_cc_dataset/sparse_grid_cc_dataset.html">
SPARSE_GRID_CC_DATASET</a>,
a C++ program which
creates a sparse grid dataset based on Clenshaw-Curtis rules.
</p>
<p>
<a href = "../../datasets/sparse_grid_ccs/sparse_grid_ccs.html">
SPARSE_GRID_CCS</a>,
a dataset directory which
contains multidimensional Smolyak sparse grids
based on the 1D Clenshaw Curtis rule with Slow Exponential Growth;
</p>
<p>
<a href = "../../datasets/sparse_grid_f2/sparse_grid_f2.html">
SPARSE_GRID_F2</a>,
a dataset directory which
contains multidimensional Smolyak sparse grids
based on the 1D Fejer type 2 rule;
</p>
<p>
<a href = "../../datasets/sparse_grid_gl/sparse_grid_gl.html">
SPARSE_GRID_GL</a>,
a dataset directory which
contains multidimensional Smolyak sparse grids
based on the 1D Gauss-Legendre rule;
</p>
<p>
<a href = "../../cpp_src/sparse_grid_gl_dataset/sparse_grid_gl_dataset.html">
SPARSE_GRID_GL_DATASET</a>,
a C++ program which
creates a sparse grid dataset based on Gauss-Legendre rules.
</p>
<p>
<a href = "../../datasets/sparse_grid_gp/sparse_grid_gp.html">
SPARSE_GRID_GP</a>,
a dataset directory which
contains multidimensional Smolyak sparse grids
based on the 1D Gauss-Patterson rule;
</p>
<p>
<a href = "../../datasets/sparse_grid_hermite/sparse_grid_hermite.html">
SPARSE_GRID_HERMITE</a>,
a dataset directory which
contains multidimensional Smolyak sparse grids
based on the 1D Gauss-Hermite rule;
</p>
<p>
<a href = "../../cpp_src/sparse_grid_hermite_dataset/sparse_grid_hermite_dataset.html">
SPARSE_GRID_HERMITE_DATASET</a>,
a C++ program which
creates a sparse grid dataset based on Gauss-Hermite rules.
</p>
<p>
<a href = "../../datasets/sparse_grid_laguerre/sparse_grid_laguerre.html">
SPARSE_GRID_LAGUERRE</a>,
a dataset directory which
contains multidimensional Smolyak sparse grids
based on the 1D Gauss-Laguerre rule;
</p>
<p>
<a href = "../../cpp_src/sparse_grid_laguerre_dataset/sparse_grid_laguerre_dataset.html">
SPARSE_GRID_LAGUERRE_DATASET</a>,
a C++ program which
creates a sparse grid dataset based on Gauss-Laguerrre rules.
</p>
<p>
<a href = "../../cpp_src/sparse_grid_mixed/sparse_grid_mixed.html">
SPARSE_GRID_MIXED</a>,
a C++ library which
creates a sparse grid dataset based on a mixed set of 1D factor rules.
</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 = "../../datasets/sparse_grid_ncc/sparse_grid_ncc.html">
SPARSE_GRID_NCC</a>,
a dataset directory which
contains multidimensional Smolyak sparse grids
based on the 1D Newton Cotes Closed rule;
</p>
<p>
<a href = "../../datasets/sparse_grid_nco/sparse_grid_nco.html">
SPARSE_GRID_NCO</a>,
a dataset directory which
contains multidimensional Smolyak sparse grids
based on the 1D Newton Cotes Open rule;
</p>
<p>
<a href = "../../cpp_src/sparse_grid_open_dataset/sparse_grid_open_dataset.html">
SPARSE_GRID_OPEN_DATASET</a>,
a C++ program which
creates a sparse grid dataset based on
open rules (Fejer 2, Gauss-Patterson, Newton-Cotes-Open).
</p>
<p>
<a href = "../../datasets/sparse_grid_tanh_sinh/sparse_grid_tanh_sinh.html">
SPARSE_GRID_TANH_SINH</a>,
a dataset directory which
contains multidimensional Smolyak sparse grids based on a 1D tanh-sinh rule;
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Thomas Gerstner, Michael Griebel,<br>
Numerical Integration Using Sparse Grids,<br>
Numerical Algorithms,<br>
Volume 18, Number 3-4, 1998, pages 209-232.
</li>
<li>
Fabio Nobile, Raul Tempone, Clayton Webster,<br>
A Sparse Grid Stochastic Collocation Method for Partial Differential
Equations with Random Input Data,<br>
SIAM Journal on Numerical Analysis,<br>
Volume 46, Number 5, 2008, pages 2309-2345.
</li>
<li>
Sergey Smolyak,<br>
Quadrature and Interpolation Formulas for Tensor Products of
Certain Classes of Functions,<br>
Doklady Akademii Nauk SSSR,<br>
Volume 4, 1963, pages 240-243.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "sparse_grid_mixed_dataset.cpp">sparse_grid_mixed_dataset.cpp</a>,
the source code.
</li>
<li>
<a href = "sparse_grid_mixed_dataset.sh">sparse_grid_mixed_dataset.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for SPARSE_GRID_MIXED_DATASET.
</li>
<li>
<b>CH_CAP</b> capitalizes a single character.
</li>
<li>
<b>RULE_STRING_TO_INDEX</b> converts a string identifying a rule to an index.
</li>
<li>
<b>S_EQI</b> is a case insensitive comparison of two strings for equality.
</li>
<li>
<b>SPARSE_GRID_MIXED_DATASET_HANDLE</b> handles the creation of the dataset.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 25 December 2009.
</i>
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