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<html>
<head>
<title>
SGMGA - Sparse Grid Mixed Growth Anisotropic Rules.
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SGMGA <br> Sparse Grid Mixed Growth Anisotropic Rules.
</h1>
<hr>
<p>
<b>SGMGA</b>
is a FORTRAN90 library which
implements a family of sparse grid rules. These rules are "mixed",
in that a different 1D quadrature rule can be specified for each dimension.
Moreover, each 1D quadrature rule comes in a family of increasing size
whose growth rate (typically linear or exponential) is chosen by the user.
Finally, the user may also specify different weights for each dimension,
resulting in anisotropic rules.
</p>
<p>
<b>SGMGA</b> calls many routines from the <b>SANDIA_RULES</b>
library. Source code or compiled copies of <i>both</i> libraries must
be available when a program wishes to use the <b>SGMGA</b> library.
</p>
<p>
Thanks to Drew Kouri, who pointed out a discrepancy in the computation of
the variable <b>level_1d_max</b> which meant that certain sparse grids
requested the generation of a 1D rule of a level that was higher than necessary
by 1. In particular, if the Gauss-Patterson rule was involved, sparse grids
that actually only needed rules of level 7 would ask also for level 8, resulting
in the computation being terminated. This problem was corrected on 25 April 2011.
</p>
<p>
<table border=1>
<tr>
<th>Index</th>
<th>Name</th>
<th>Abbreviation</th>
<th>Default Growth Rule</th>
<th>Interval</th>
<th>Weight function</th>
</tr>
<tr>
<td>1</td>
<td>Clenshaw-Curtis</td>
<td>CC</td>
<td>Moderate Exponential</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
<tr>
<td>2</td>
<td>Fejer Type 2</td>
<td>F2</td>
<td>Moderate Exponential</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
<tr>
<td>3</td>
<td>Gauss Patterson</td>
<td>GP</td>
<td>Moderate Exponential</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
<tr>
<td>4</td>
<td>Gauss-Legendre</td>
<td>GL</td>
<td>Moderate Linear</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
<tr>
<td>5</td>
<td>Gauss-Hermite</td>
<td>GH</td>
<td>Moderate Linear</td>
<td>(-oo,+oo)</td>
<td>e<sup>-x*x</sup></td>
</tr>
<tr>
<td>6</td>
<td>Generalized Gauss-Hermite</td>
<td>GGH</td>
<td>Moderate Linear</td>
<td>(-oo,+oo)</td>
<td>|x|<sup>alpha</sup> e<sup>-x*x</sup></td>
</tr>
<tr>
<td>7</td>
<td>Gauss-Laguerre</td>
<td>LG</td>
<td>Moderate Linear</td>
<td>[0,+oo)</td>
<td>e<sup>-x</sup></td>
</tr>
<tr>
<td>8</td>
<td>Generalized Gauss-Laguerre</td>
<td>GLG</td>
<td>Moderate Linear</td>
<td>[0,+oo)</td>
<td>x<sup>alpha</sup> e<sup>-x</sup></td>
</tr>
<tr>
<td>9</td>
<td>Gauss-Jacobi</td>
<td>GJ</td>
<td>Moderate Linear</td>
<td>[-1,+1]</td>
<td>(1-x)<sup>alpha</sup> (1+x)<sup>beta</sup></td>
</tr>
<tr>
<td>10</td>
<td>Hermite Genz-Keister</td>
<td>HGK</td>
<td>Moderate Exponential</td>
<td>(-oo,+oo)</td>
<td>e<sup>-x*x</sup></td>
</tr>
<tr>
<td>11</td>
<td>User Supplied Open</td>
<td>UO</td>
<td>Moderate Linear</td>
<td>?</td>
<td>?</td>
</tr>
<tr>
<td>12</td>
<td>User Supplied Closed</td>
<td>UC</td>
<td>Moderate Linear</td>
<td>?</td>
<td>?</td>
</tr>
</table>
</p>
<p>
For a given family, a growth rule can be prescribed, which determines
the orders O of the sequence of rules selected from the family. The
selected rules are indexed by L, which starts at 0. The polynomial precision P
of the rule is sometimes used to determine the appropriate order O.
<table border=1>
<tr>
<th>Index</th>
<th>Name</th>
<th>Order Formula</th>
</tr>
<tr>
<td>0</td>
<td>Default</td>
<td>"DF", moderate exponential or moderate linear</td>
</tr>
<tr>
<td>1</td>
<td>"SL", Slow linear</td>
<td>O=L+1</td>
</tr>
<tr>
<td>2</td>
<td>"SO", Slow Linear Odd</td>
<td>O=1+2*((L+1)/2)</td>
</tr>
<tr>
<td>3</td>
<td>"ML", Moderate Linear</td>
<td>O=2L+1</td>
</tr>
<tr>
<td>4</td>
<td>"SE", Slow Exponential</td>
<td>select smallest exponential order O so that 2L+1 <= P</td>
</tr>
<tr>
<td>5</td>
<td>"ME", Moderate Exponential</td>
<td>select smallest exponential order O so that 4L+1 <= P</td>
</tr>
<tr>
<td>6</td>
<td>"FE", Full Exponential</td>
<td>O=2^L+1 for Clenshaw Curtis, O=2^(L+1)-1 otherwise.</td>
</tr>
</table>
</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>SGMGA</b> is available in
<a href = "../../c_src/sgmga/sgmga.html">a C version</a> and
<a href = "../../cpp_src/sgmga/sgmga.html">a C++ version</a> and
<a href = "../../f_src/sgmga/sgmga.html">a FORTRAN90 version</a> and
<a href = "../../m_src/sgmga/sgmga.html">a MATLAB version.</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/nint_exactness_mixed/nint_exactness_mixed.html">
NINT_EXACTNESS_MIXED</a>,
a FORTRAN90 program which
measures the polynomial exactness of a multidimensional quadrature rule
based on a mixture of 1D quadrature rule factors.
</p>
<p>
<a href = "../../f_src/quadrule/quadrule.html">
QUADRULE</a>,
a FORTRAN90 library which
defines quadrature rules for various intervals and weight functions.
</p>
<p>
<a href = "../../f_src/sandia_rules/sandia_rules.html">
SANDIA_RULES</a>,
a FORTRAN90 library which
generates Gauss quadrature rules of various orders and types.
</p>
<p>
<a href = "../../f_src/sandia_sparse/sandia_sparse.html">
SANDIA_SPARSE</a>,
a FORTRAN90 library which
computes the points and weights of a Smolyak sparse
grid, based on a variety of 1-dimensional quadrature rules.
</p>
<p>
<a href = "../../datasets/sgmga/sgmga.html">
SGMGA</a>,
a dataset directory which
contains SGMGA files (Sparse Grid Mixed Growth Anisotropic), that is,
multidimensional Smolyak sparse grids
based on a mixture of 1D rules, and with a choice of exponential and linear
growth rates for the 1D rules and anisotropic weights for the dimensions.
</p>
<p>
<a href = "../../c_src/smolpack/smolpack.html">
SMOLPACK</a>,
a C library which
implements Novak and Ritter's method for estimating the integral
of a function over a multidimensional hypercube using sparse grids,
by Knut Petras.
</p>
<p>
<a href = "../../f_src/sparse_grid_mixed/sparse_grid_mixed.html">
SPARSE_GRID_MIXED</a>,
a FORTRAN90 library which
creates a sparse grid dataset based on a mixed set of 1D factor rules.
</p>
<p>
<a href = "../../m_src/toms847/toms847.html">
TOMS847</a>,
a MATLAB program which
uses sparse grids to carry out multilinear hierarchical interpolation.
It is commonly known as SPINTERP, and is by Andreas Klimke.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Milton Abramowitz, Irene Stegun,<br>
Handbook of Mathematical Functions,<br>
National Bureau of Standards, 1964,<br>
ISBN: 0-486-61272-4,<br>
LC: QA47.A34.
</li>
<li>
Charles Clenshaw, Alan Curtis,<br>
A Method for Numerical Integration on an Automatic Computer,<br>
Numerische Mathematik,<br>
Volume 2, Number 1, December 1960, pages 197-205.
</li>
<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>
<li>
Michael Eldred, John Burkardt,<br>
Comparison of Non-Intrusive Polynomial Chaos and Stochastic
Collocation Methods for Uncertainty Quantification,<br>
American Institute of Aeronautics and Astronautics,<br>
Paper 2009-0976,<br>
47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition,<br>
5 - 8 January 2009, Orlando, Florida.
</li>
<li>
Walter Gautschi,<br>
Numerical Quadrature in the Presence of a Singularity,<br>
SIAM Journal on Numerical Analysis,<br>
Volume 4, Number 3, September 1967, pages 357-362.
</li>
<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>
Gene Golub, John Welsch,<br>
Calculation of Gaussian Quadrature Rules,<br>
Mathematics of Computation,<br>
Volume 23, Number 106, April 1969, pages 221-230.
</li>
<li>
Prem Kythe, Michael Schaeferkotter,<br>
Handbook of Computational Methods for Integration,<br>
Chapman and Hall, 2004,<br>
ISBN: 1-58488-428-2,<br>
LC: QA299.3.K98.
</li>
<li>
Albert Nijenhuis, Herbert Wilf,<br>
Combinatorial Algorithms for Computers and Calculators,<br>
Second Edition,<br>
Academic Press, 1978,<br>
ISBN: 0-12-519260-6,<br>
LC: QA164.N54.
</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>
Fabio Nobile, Raul Tempone, Clayton Webster,<br>
An Anisotropic 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 2411-2442.
</li>
<li>
Thomas Patterson,<br>
The Optimal Addition of Points to Quadrature Formulae,<br>
Mathematics of Computation,<br>
Volume 22, Number 104, October 1968, pages 847-856.
</li>
<li>
Knut Petras,<br>
Smolyak Cubature of Given Polynomial Degree with Few Nodes
for Increasing Dimension,<br>
Numerische Mathematik,<br>
Volume 93, Number 4, February 2003, pages 729-753.
</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>
<li>
Arthur Stroud, Don Secrest,<br>
Gaussian Quadrature Formulas,<br>
Prentice Hall, 1966,<br>
LC: QA299.4G3S7.
</li>
<li>
Joerg Waldvogel,<br>
Fast Construction of the Fejer and Clenshaw-Curtis
Quadrature Rules,<br>
BIT Numerical Mathematics,<br>
Volume 43, Number 1, 2003, pages 1-18.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "sgmga.f90">sgmga.f90</a>, the source code.
</li>
<li>
<a href = "sgmga.sh">sgmga.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>SGMGA_ANISO_NORMALIZE_PRB</b> tests <b>SGMGA_ANISO_NORMALIZE</b> and
<b>SGMGA_IMPORTANCE_TO_ANISO</b>.
<ul>
<li>
<a href = "sgmga_aniso_normalize_prb.f90">sgmga_aniso_normalize_prb.f90</a>,
the program.
</li>
<li>
<a href = "sgmga_aniso_normalize_prb.sh">sgmga_aniso_normalize_prb.sh</a>,
commands to compile and run the program.
</li>
<li>
<a href = "sgmga_aniso_normalize_prb_output.txt">sgmga_aniso_normalize_prb_output.txt</a>,
the output file.
</li>
</ul>
<p>
<p>
<b>SGMGA_INDEX_PRB</b> tests <b>SGMGA_INDEX</b>.
<ul>
<li>
<a href = "sgmga_index_prb.f90">sgmga_index_prb.f90</a>,
the program.
</li>
<li>
<a href = "sgmga_index_prb.sh">sgmga_index_prb.sh</a>,
commands to compile and run the program.
</li>
<li>
<a href = "sgmga_index_prb_output.txt">sgmga_index_prb_output.txt</a>,
the output file.
</li>
</ul>
<p>
<p>
<b>SGMGA_POINT_PRB</b> tests <b>SGMGA_POINT</b>.
<ul>
<li>
<a href = "sgmga_point_prb.f90">sgmga_point_prb.f90</a>,
the program.
</li>
<li>
<a href = "sgmga_point_prb.sh">sgmga_point_prb.sh</a>,
commands to compile and run the program.
</li>
<li>
<a href = "sgmga_point_prb_output.txt">sgmga_point_prb_output.txt</a>,
the output file.
</li>
</ul>
<p>
<p>
<b>SGMGA_PRODUCT_WEIGHT_PRB</b> tests <b>SGMGA_PRODUCT_WEIGHT</b>.
<ul>
<li>
<a href = "sgmga_product_weight_prb.f90">sgmga_product_weight_prb.f90</a>,
the program.
</li>
<li>
<a href = "sgmga_product_weight_prb.sh">sgmga_product_weight_prb.sh</a>,
commands to compile and run the program.
</li>
<li>
<a href = "sgmga_product_weight_prb_output.txt">sgmga_product_weight_prb_output.txt</a>,
the output file.
</li>
</ul>
<p>
<p>
<b>SGMGA_SIZE_PRB</b> tests <b>SGMGA_SIZE</b> and <b>SGMGA_SIZE_TOTAL</b>.
<ul>
<li>
<a href = "sgmga_size_prb.f90">sgmga_size_prb.f90</a>,
the program.
</li>
<li>
<a href = "sgmga_size_prb.sh">sgmga_size_prb.sh</a>,
commands to compile and run the program.
</li>
<li>
<a href = "sgmga_size_prb_output.txt">sgmga_size_prb_output.txt</a>,
the output file.
</li>
</ul>
<p>
<p>
<b>SGMGA_SIZE_TABLE</b> tabulates the point counts from <b>SGMGA_SIZE</b>
for an isotropic rule over a range of dimensions and levels.
<ul>
<li>
<a href = "sgmga_size_table.f90">sgmga_size_table.f90</a>,
the program.
</li>
<li>
<a href = "sgmga_size_table.sh">sgmga_size_table.sh</a>,
commands to compile and run the program.
</li>
<li>
<a href = "sgmga_size_table_output.txt">sgmga_size_table_output.txt</a>,
the output file.
</li>
</ul>
<p>
<p>
<b>SGMGA_UNIQUE_INDEX_PRB</b> tests <b>SGMGA_UNIQUE_INDEX</b>.
<ul>
<li>
<a href = "sgmga_unique_index_prb.f90">sgmga_unique_index_prb.f90</a>,
the program.
</li>
<li>
<a href = "sgmga_unique_index_prb.sh">sgmga_unique_index_prb.sh</a>,
commands to compile and run the program.
</li>
<li>
<a href = "sgmga_unique_index_prb_output.txt">sgmga_unique_index_prb_output.txt</a>,
the output file.
</li>
</ul>
<p>
<p>
<b>SGMGA_VCN_PRB</b> tests <b>SGMGA_VCN</b> and <b>SGMGA_VCN_ORDERED</b>:
<ul>
<li>
<a href = "sgmga_vcn_prb.f90">sgmga_vcn_prb.f90</a>,
the program.
</li>
<li>
<a href = "sgmga_vcn_prb.sh">sgmga_vcn_prb.sh</a>,
commands to compile and run the program.
</li>
<li>
<a href = "sgmga_vcn_prb_output.txt">sgmga_vcn_prb_output.txt</a>,
the output file.
</li>
</ul>
<p>
<p>
<b>SGMGA_VCN_COEF_PRB</b> tests <b>SGMGA_VCN_COEF</b>.
<ul>
<li>
<a href = "sgmga_vcn_coef_prb.f90">sgmga_vcn_coef_prb.f90</a>,
the program.
</li>
<li>
<a href = "sgmga_vcn_coef_prb.sh">sgmga_vcn_coef_prb.sh</a>,
commands to compile and run the program.
</li>
<li>
<a href = "sgmga_vcn_coef_prb_output.txt">sgmga_vcn_coef_prb_output.txt</a>,
the output file.
</li>
</ul>
<p>
<p>
<b>SGMGA_WEIGHT_PRB</b> tests <b>SGMGA_WEIGHT</b>.
<ul>
<li>
<a href = "sgmga_weight_prb.f90">sgmga_weight_prb.f90</a>,
the program.
</li>
<li>
<a href = "sgmga_weight_prb.sh">sgmga_weight_prb.sh</a>,
commands to compile and run the program.
</li>
<li>
<a href = "sgmga_weight_prb_output.txt">sgmga_weight_prb_output.txt</a>,
the output file.
</li>
</ul>
<p>
<p>
<b>SGMGA_WRITE_PRB</b> tests <b>SGMGA_WRITE</b>.
<ul>
<li>
<a href = "sgmga_write_prb.f90">sgmga_write_prb.f90</a>,
the program.
</li>
<li>
<a href = "sgmga_write_prb.sh">sgmga_write_prb.sh</a>,
commands to compile and run the program.
</li>
<li>
<a href = "sgmga_write_prb_output.txt">sgmga_write_prb_output.txt</a>,
the output file.
</li>
</ul>
<p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>SGMGA_ANISO_BALANCE</b> "balances" an anisotropic weight vector.
</li>
<li>
<b>SGMGA_ANISO_NORMALIZE</b> normalizes the SGMGA anisotropic weight vector.
</li>
<li>
<b>SGMGA_IMPORTANCE_TO_ANISO:</b> importance vector to anisotropic weight vector.
</li>
<li>
<b>SGMGA_INDEX</b> indexes the unique points in an SGMGA grid.
</li>
<li>
<b>SGMGA_POINT</b> computes the points of an SGMGA rule.
</li>
<li>
<b>SGMGA_PRODUCT_WEIGHT</b> computes the weights of a mixed product rule.
</li>
<li>
<b>SGMGA_SIZE</b> sizes an SGMGA grid, discounting duplicate points.
</li>
<li>
<b>SGMGA_SIZE_TOTAL</b> sizes an SGMGA grid, counting duplicate points.
</li>
<li>
<b>SGMGA_UNIQUE_INDEX</b> maps nonunique to unique points of an SGMGA grid.
</li>
<li>
<b>SGMGA_VCN</b> returns the next constrained vector.
</li>
<li>
<b>SGMGA_VCN_COEF</b> returns the "next" constrained vector's coefficient.
</li>
<li>
<b>SGMGA_VCN_COEF_NAIVE</b> returns the "next" constrained vector's coefficient.
</li>
<li>
<b>SGMGA_VCN_NAIVE</b> returns the next constrained vector.
</li>
<li>
<b>SGMGA_VCN_ORDERED</b> returns the "next" constrained vector, with ordering.
</li>
<li>
<b>SGMGA_VCN_ORDERED_NAIVE</b> returns the "next" constrained vector, with ordering.
</li>
<li>
<b>SGMGA_WEIGHT</b> computes weights for an SGMGA rule.
</li>
<li>
<b>SGMGA_WRITE</b> writes an SGMGA rule to several files.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 29 July 2010.
</i>
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