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<html>
<head>
<title>
SPARSE_GRID_GL - Sparse Grids Based on Gauss-Legendre Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SPARSE_GRID_GL <br> Sparse Grids Based on Gauss-Legendre Rules
</h1>
<hr>
<p>
<b>SPARSE_GRID_GL</b>
is a FORTRAN90 library which
constructs sparse grids based on 1D Gauss-Legendre rules.
</p>
<p>
Sparse grids are more naturally constructed from a nested family
of quadrature rules. Gauss-Legendre rules are not nested, but
have higher accuracy. Thus, there is a tradeoff. If we compare
two sparse grids of the same "level", one using Gauss-Legendre rules
and the other, say, Clenshaw-Curtis rules, then the Gauss-Legendre
sparse grid will have higher accuracy...but also a significantly
greater number of points. When measuring efficiency, we really need
to balance the cost in quadrature points against the accuracy, and
so it is not immediately obvious which choice is best!
</p>
<p>
To slightly complicate matters, Gauss-Legendre rules are very weakly
nested, in that the rules of odd order all include the abscissa
value X=0.0. A sparse grid constructed from Gauss-Legendre rules
will thus have to keep track of this minor point as well.
</p>
<p>
Here is a table showing the number of points in a sparse grid based on
Gauss-Legendre rules, indexed by the spatial dimension, and by the
"level", which is simply an index for the family of sparse grids.
<table border = "1">
<tr>
<th>DIM:</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th>
</tr>
<tr>
<th>LEVEL_MAX</th><th> </th><th> </th><th> </th><th> </th><th> </th><th> </th>
</tr>
<tr>
<th>0</th><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td>
</tr>
<tr>
<th>1</th><td>3</td><td>5</td><td>7</td><td>9</td><td>11</td><td>13</td>
</tr>
<tr>
<th>2</th><td>9</td><td>21</td><td>37</td><td>57</td><td>81</td><td>109</td>
</tr>
<tr>
<th>3</th><td>23</td><td>73</td><td>159</td><td>289</td><td>471</td><td>713</td>
</tr>
<tr>
<th>4</th><td>53</td><td>225</td><td>597</td><td>1265</td><td>2341</td><td>3953</td>
</tr>
<tr>
<th>5</th><td>115</td><td>637</td><td>2031</td><td>4969</td><td>10363</td><td>19397</td>
</tr>
<tr>
<th>6</th><td>241</td><td>1693</td><td>6405</td><td>17945</td><td>41913</td><td>86517</td>
</tr>
</table>
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The code described and made available on this web page is distributed
under the
<a href = "gnu_lgpl.txt">GNU LGPL</a> license.
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>SPARSE_GRID_GL</b> is available in
<a href = "../../cpp_src/sparse_grid_gl/sparse_grid_gl.html">a C++ version</a> and
<a href = "../../f_src/sparse_grid_gl/sparse_grid_gl.html">a FORTRAN90 version</a> and
<a href = "../../m_src/sparse_grid_gl/sparse_grid_gl.html">a MATLAB version.</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../datasets/quadrature_rules/quadrature_rules.html">
QUADRATURE_RULES</a>,
a dataset directory which
defines quadrature rules;
a number of examples of sparse grid quadrature rules are included.
</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/sgmga/sgmga.html">
SGMGA</a>,
a FORTRAN90 library which
creates sparse grids based on a mixture of 1D quadrature rules,
allowing anisotropic weights for each dimension.
</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.
</p>
<p>
<a href = "../../f_src/sparse_grid_cc/sparse_grid_cc.html">
SPARSE_GRID_CC</a>,
a FORTRAN90 library which
define sparse grids based on 1D Clenshaw Curtis quadrature rules.
</p>
<p>
<a href = "../../datasets/sparse_grid_gl/sparse_grid_gl.html">
SPARSE_GRID_GL</a>,
a dataset directory which
contains sparse
grids based on a Gauss Legendre rule.
</p>
<p>
<a href = "../../f_src/sparse_grid_gl_dataset/sparse_grid_gl_dataset.html">
SPARSE_GRID_GL_DATASET</a>,
a FORTRAN90 program which
creates a Gauss-Legendre sparse grid from user input.
</p>
<p>
<a href = "../../datasets/sparse_grid_gp/sparse_grid_gp.html">
SPARSE_GRID_GP</a>,
a dataset directory which
contains sparse
grids based on a Gauss Patterson rule.
</p>
<p>
<a href = "../../f_src/sparse_grid_hermite/sparse_grid_hermite.html">
SPARSE_GRID_HERMITE</a>,
a FORTRAN90 library which
creates sparse grids based on Gauss-Hermite rules.
</p>
<p>
<a href = "../../f_src/sparse_grid_mixed/sparse_grid_mixed.html">
SPARSE_GRID_MIXED</a>,
a FORTRAN90 library which
constructs a sparse grid using different rules in each spatial dimension.
</p>
<p>
<a href = "../../datasets/sparse_grid_ncc/sparse_grid_ncc.html">
SPARSE_GRID_NCC</a>,
a dataset directory which
contains sparse
grids based on a 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 sparse
grids based on a Newton Cotes open rule.
</p>
<p>
<a href = "../../f_src/sparse_grid_open/sparse_grid_open.html">
SPARSE_GRID_OPEN</a>,
a FORTRAN90 library which
define sparse grids based on open nested quadrature 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>
Volker Barthelmann, Erich Novak, Klaus Ritter,<br>
High Dimensional Polynomial Interpolation on Sparse Grids,<br>
Advances in Computational Mathematics,<br>
Volume 12, Number 4, 2000, pages 273-288.
</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>
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>
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>
Dennis Stanton, Dennis White,<br>
Constructive Combinatorics,<br>
Springer, 1986,<br>
ISBN: 0387963472,<br>
LC: QA164.S79.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "sparse_grid_gl.f90">sparse_grid_gl.f90</a>,
the source code.
</li>
<li>
<a href = "sparse_grid_gl.sh">sparse_grid_gl.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "sparse_grid_gl_prb.f90">sparse_grid_gl_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "sparse_grid_gl_prb.sh">sparse_grid_gl_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "sparse_grid_gl_prb_output.txt">
sparse_grid_gl_prb_output.txt</a>,
the output file.
</li>
<li>
<a href = "../../datasets/sparse_grid_gl/gl_d2_level3_r.txt">gl_d2_level3_r.txt</a>,
the "R" file for a sparse grid quadrature rule for spatial dimension 2
and level 3.
</li>
<li>
<a href = "../../datasets/sparse_grid_gl/gl_d2_level3_w.txt">gl_d2_level3_w.txt</a>,
the "W" file for a sparse grid quadrature rule for spatial dimension 2
and level 3.
</li>
<li>
<a href = "../../datasets/sparse_grid_gl/gl_d2_level3_x.txt">gl_d2_level3_x.txt</a>,
the "X" file for a sparse grid quadrature rule for spatial dimension 2
and level 3.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>COMP_NEXT</b> computes the compositions of the integer N into K parts.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>GL_ABSCISSA</b> sets abscissas for multidimensional Gauss-Legendre quadrature.
</li>
<li>
<b>GL_WEIGHTS</b> returns weights for certain Gauss-Legendre quadrature rules.
</li>
<li>
<b>I4_CHOOSE</b> computes the binomial coefficient C(N,K).
</li>
<li>
<b>I4_HUGE</b> returns a "huge" I4.
</li>
<li>
<b>I4_LOG_2</b> returns the integer part of the logarithm base 2 of an I4.
</li>
<li>
<b>INDEX_LEVEL_GL:</b> determine first level at which given index is generated.
</li>
<li>
<b>LEVEL_TO_ORDER_OPEN</b> converts a level to an order for open rules.
</li>
<li>
<b>MONOMIAL_INT01</b> returns the integral of a monomial over the [0,1] hypercube.
</li>
<li>
<b>MONOMIAL_QUADRATURE</b> applies a quadrature rule to a monomial.
</li>
<li>
<b>MONOMIAL_VALUE</b> evaluates a monomial.
</li>
<li>
<b>MULTIGRID_INDEX_Z</b> returns an indexed multidimensional grid.
</li>
<li>
<b>PRODUCT_WEIGHT_GL:</b> weights for a product Gauss-Legendre rule.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file.
</li>
<li>
<b>R8VEC_DIRECT_PRODUCT2</b> creates a direct product of R8VEC's.
</li>
<li>
<b>S_BLANK_DELETE</b> removes blanks from a string, left justifying the remainder.
</li>
<li>
<b>SPARSE_GRID_GL</b> computes a sparse grid of Gauss-Legendre points.
</li>
<li>
<b>SPARSE_GRID_GL_INDEX</b> indexes the points forming a sparse grid of GL points.
</li>
<li>
<b>SPARSE_GRID_GL_SIZE</b> sizes a sparse grid of Gauss-Legendre points.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>VEC_COLEX_NEXT2</b> generates vectors in colex order.
</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 07 November 2009.
</i>
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