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<html>
<head>
<title>
SANDIA_SPARSE - Sparse Grids for Sandia
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SANDIA_SPARSE <br> Sparse Grids for Sandia
</h1>
<hr>
<p>
<b>SANDIA_SPARSE</b>
is a FORTRAN90 library which
can be used to compute the points and weights of a Smolyak sparse
grid, based on a variety of 1-dimensional quadrature rules.
</p>
<p>
The sparse grids that can be created may be based on any one of
a variety of 1D quadrature rules. However, only <b>isotropic</b>
grids are generated, that is, the same 1D quadrature rule is used
in each dimension, and the same maximum order is used in each dimension.
This is a limitation of this library, and not an inherent limitation
of the sparse grid method.
</p>
<p>
The 1D quadrature rules that can be used to construct sparse grids include:
<ul>
<li>
<b>CFN</b>, Closed Fully Nested rules:
<ul>
<li>
<b>CC</b>, Clenshaw-Curtis:<br>
defined on [-1,+1], with w(x)=1.
</li>
</ul>
</li>
<li>
<b>OFN</b>, Open Fully Nested rules:
<ul>
<li>
<b>F1</b>, Fejer Type 1: <br>
defined on (-1,+1), with w(x)=1.
</li>
<li>
<b>F2</b>, Fejer Type 2: <br>
defined on (-1,+1), with w(x)=1.
</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.
</li>
</ul>
</li>
<li>
<b>OWN</b>, Open Weakly Nested rules:
<ul>
<li>
<b>GL</b>, Gauss Legendre: <br>
defined on (-1,+1), with w(x)=1.
</li>
<li>
<b>GH</b>, Gauss Hermite: <br>
defined on (-oo,+oo), with w(x)=exp(-x*x).
</li>
</ul>
</li>
<li>
<b>ONN</b>, Open Non-Nested rules:
<ul>
<li>
<b>LG</b>, Gauss Laguerre: <br>
defined on (0,+oo) with w(x)=exp(-x).
</li>
</ul>
</li>
</ul>
</p>
<h3 align = "center">
Point Growth of 1D Rules
</h3>
<p>
A major advantage of sparse grids is that they can achieve accuracy that
is comparable to a corresponding product rule, while using far fewer points,
that is, evaluations of the function that is to be integrated. We will leave
aside the issue of comparing accuracy for now, and simply focus on the pattern
of point growth.
</p>
<p>
A sparse grid is essentially a linear combination of lower order product
grids. One way point growth is controlled is to only use product grids
based on a set of factors that are nested. In other words, the underlying
1D rules are selected so that, when we increase the order of such a rule,
all the points of the current rule are included in the new one.
</p>
<p>
The exact details of how this works depend on the particular 1D rule being
used and the nesting behavior it satisfies. We classify the cases as follows:
<ul>
<li>
<b>CFN</b>, "Closed, Fully Nested", based on Clenshaw Curtis.
</li>
<li>
<b>OFN</b>, "Open, Fully Nested", based on Fejer Type 1, Fejer Type 2, or
Gauss Patterson.
</li>
<li>
<b>OWN</b>, "Open, Weakly Nested", based on Gauss Legendre or
Gauss Hermite rules.
</li>
<li>
<b>ONN</b>, "Open, Non-Nested", based on Gauss Laguerre rules;
</li>
</ul>
</p>
<p>
For <b>CFN</b> rules we have the following relationship between the level (index of
the grid) and the 1D order (the number of points in the 1D rule.)
<pre><b>
order = 2<sup>level</sup> + 1
</b></pre>
except that for the special case of <b>level=0</b> we assign
<b>order=1</b>.
</p>
<p>
For <b>OFN</b>, <b>OWN</b> and <b>ONN</b> rules, the relationship between level
and 1D order is:
<pre><b>
order = 2<sup>level+1</sup> - 1
</b></pre>
</p>
<p>
Thus, as we allow <b>level</b> to grow, the <b>order</b> of the 1D closed
and open rules behaves as follows:
<table border = "1" align = "center">
<tr><th>Level</th><th>CFN</th><th>OFN/OWN/ONN</th></tr>
<tr align = "right"><td> 0</td><td> 1</th><td> 1</th></tr>
<tr align = "right"><td> 1</td><td> 3</th><td> 3</th></tr>
<tr align = "right"><td> 2</td><td> 5</th><td> 7</th></tr>
<tr align = "right"><td> 3</td><td> 9</th><td> 15</th></tr>
<tr align = "right"><td> 4</td><td> 17</th><td> 31</th></tr>
<tr align = "right"><td> 5</td><td> 33</th><td> 63</th></tr>
<tr align = "right"><td> 6</td><td> 65</th><td> 127</th></tr>
<tr align = "right"><td> 7</td><td> 129</th><td> 255</th></tr>
<tr align = "right"><td> 8</td><td> 257</th><td> 511</th></tr>
<tr align = "right"><td> 9</td><td> 513</th><td> 1,023</th></tr>
<tr align = "right"><td> 10</td><td> 1,025</th><td> 2,057</th></tr>
</table>
</p>
<p>
When we move to multiple dimensions, the counting becomes more complicated. This is because
a multidimensional sparse grid is made up of a logical sum of product grids. A multidimensional
sparse grid has a multidimensional level, which is a single number. Each product grid
that forms part of this sparse grid has a multidimensional level which is the sum of the
1D levels of its factors. A sparse grid whose multidimensional level is represented by
<b>LEVEL</b> includes all product grids whose level ranges <b>LEVEL</b>+1-<b>DIM</b>
and <b>LEVEL</b>.
</p>
<p>
Thus, as one example, if <b>DIM</b> is 2, the sparse grid of level 3, formed from
a CFN rule, will be formed from the following product rules.
<table border = "1" align = "center">
<tr><th>level</th><th>level 1</th><th>level 2</th><th>order 1</th><th>order 2</th><th>order</th></tr>
<tr align = "right"><td>1</td><td>0</td><td>1</td><td>1</td><td>3</td><td>3</td></tr>
<tr align = "right"><td>1</td><td>1</td><td>0</td><td>3</td><td>1</td><td>3</td></tr>
<tr align = "right"><td>2</td><td>0</td><td>2</td><td>1</td><td>5</td><td>5</td></tr>
<tr align = "right"><td>2</td><td>1</td><td>1</td><td>3</td><td>3</td><td>9</td></tr>
<tr align = "right"><td>2</td><td>2</td><td>0</td><td>5</td><td>1</td><td>5</td></tr>
</table>
Because of the nesting pattern for <b>CFN</b> rules, instead of 25 points (the sum of the orders),
we will actually have just 13 unique points.
<p>
<p>
For a <b>CFN</b> sparse grid, here is the pattern of growth in the number of points,
as a function of spatial dimension and grid level:
<table border = "1" align = "center">
<tr><th>DIM </th><th> 1</th><th> 2</th><th> 3</th><th> 4</th><th> 5</th></tr>
<tr><th>Level</th><td> </td><td> </td><td> </td><td> </td><td> </td></tr>
<tr align = "right"><td> 0</td><td> 1</th><td> 1</th><td> 1</td><td> 1</td><td> 1</td></tr>
<tr align = "right"><td> 1</td><td> 3</th><td> 5</th><td> 7</td><td> 9</td><td> 11</td></tr>
<tr align = "right"><td> 2</td><td> 5</th><td> 13</th><td> 25</td><td> 41</td><td> 61</td></tr>
<tr align = "right"><td> 3</td><td> 9</th><td> 29</th><td> 69</td><td> 137</td><td> 241</td></tr>
<tr align = "right"><td> 4</td><td> 17</th><td> 65</th><td> 177</td><td> 401</td><td> 801</td></tr>
<tr align = "right"><td> 5</td><td> 33</th><td> 145</th><td> 441</td><td> 1,105</td><td> 2,433</td></tr>
<tr align = "right"><td> 6</td><td> 65</th><td> 321</th><td>1,073</td><td> 2,929</td><td> 6,993</td></tr>
<tr align = "right"><td> 7</td><td>129</th><td> 705</th><td>2,561</td><td> 7,537</td><td>19,313</td></tr>
<tr align = "right"><td> 8</td><td>257</th><td>1,537</th><td>6,017</td><td>18,945</td><td>51,713</td></tr>
</table>
</p>
<p>
For an <b>OFN</b> sparse grid, here is the pattern of growth in the number of points,
as a function of spatial dimension and grid level:
<table border = "1" align = "center">
<tr><th>DIM </th><th> 1</th><th> 2</th><th> 3</th><th> 4</th><th> 5</th></tr>
<tr><th>Level</th><td> </td><td> </td><td> </td><td> </td><td> </td></tr>
<tr align = "right"><td> 0</td><td> 1</th><td> 1</th><td> 1</td><td> 1</td><td> 1</td></tr>
<tr align = "right"><td> 1</td><td> 3</th><td> 5</th><td> 7</td><td> 9</td><td> 11</td></tr>
<tr align = "right"><td> 2</td><td> 7</th><td> 17</th><td> 31</td><td> 49</td><td> 71</td></tr>
<tr align = "right"><td> 3</td><td> 15</th><td> 49</th><td> 111</td><td> 209</td><td> 351</td></tr>
<tr align = "right"><td> 4</td><td> 31</th><td> 129</th><td> 351</td><td> 769</td><td> 1,471</td></tr>
<tr align = "right"><td> 5</td><td> 63</th><td> 321</th><td> 1,023</td><td> 2,561</td><td> 5,503</td></tr>
<tr align = "right"><td> 6</td><td>127</th><td> 769</th><td> 2,815</td><td> 7,937</td><td> 18,943</td></tr>
<tr align = "right"><td> 7</td><td>255</th><td>1,793</th><td> 7,423</td><td>23,297</td><td> 61,183</td></tr>
<tr align = "right"><td> 8</td><td>511</th><td>4,097</th><td>18,943</td><td>65,537</td><td>187,903</td></tr>
</table>
</p>
<p>
For an <b>OWN</b> sparse grid, here is the pattern of growth in the number of points,
as a function of spatial dimension and grid level:
<table border = "1" align = "center">
<tr><th>DIM </th><th> 1</th><th> 2</th><th> 3</th><th> 4</th><th> 5</th></tr>
<tr><th>Level</th><td> </td><td> </td><td> </td><td> </td><td> </td></tr>
<tr align = "right"><td> 0</td><td> 1</th><td> 1</th><td> 1</td><td> 1</td><td> 1</td></tr>
<tr align = "right"><td> 1</td><td> 3</th><td> 5</th><td> 7</td><td> 9</td><td> 11</td></tr>
<tr align = "right"><td> 2</td><td> 7</th><td> 21</th><td> 37</td><td> 57</td><td> 81</td></tr>
<tr align = "right"><td> 3</td><td> 15</th><td> 73</th><td> 159</td><td> 289</td><td> 471</td></tr>
<tr align = "right"><td> 4</td><td> 31</th><td> 225</th><td> 597</td><td> 1,265</td><td> 2,341</td></tr>
<tr align = "right"><td> 5</td><td> 63</th><td> 637</th><td> 2,031</td><td> 4,969</td><td> 10,363</td></tr>
<tr align = "right"><td> 6</td><td>127</th><td> 1,693</th><td> 6,405</td><td> 17,945</td><td> 41,913</td></tr>
<tr align = "right"><td> 7</td><td>255</th><td> 4,289</th><td>19,023</td><td> 60,577</td><td>157,583</td></tr>
<tr align = "right"><td> 8</td><td>511</th><td>10,473</th><td>53,829</td><td>193,457</td><td>557,693</td></tr>
</table>
</p>
<p>
For an <b>ONN</b> sparse grid, here is the pattern of growth in the number of points,
as a function of spatial dimension and grid level:
<table border = "1" align = "center">
<tr><th>DIM </th><th> 1</th><th> 2</th><th> 3</th><th> 4</th><th> 5</th></tr>
<tr><th>Level</th><td> </td><td> </td><td> </td><td> </td><td> </td></tr>
<tr align = "right"><td> 0</td><td> 1</th><td> 1</th><td> 1</td><td> 1</td><td> 1</td></tr>
<tr align = "right"><td> 1</td><td> 3</th><td> 7</th><td> 10</td><td> 13</td><td> 16</td></tr>
<tr align = "right"><td> 2</td><td> 7</th><td> 29</th><td> 58</td><td> 95</td><td> 141</td></tr>
<tr align = "right"><td> 3</td><td> 15</th><td> 95</th><td> 255</td><td> 515</td><td> 906</td></tr>
<tr align = "right"><td> 4</td><td> 31</th><td> 273</th><td> 945</td><td> 2,309</td><td> 4,746</td></tr>
<tr align = "right"><td> 5</td><td> 63</th><td> 723</th><td> 3,120</td><td> 9,065</td><td> 21,503</td></tr>
<tr align = "right"><td> 6</td><td>127</th><td> 1,813</th><td> 9,484</td><td> 32,259</td><td> 87,358</td></tr>
<tr align = "right"><td> 7</td><td>255</th><td> 4,375</th><td>27,109</td><td>106,455</td><td> 325,943</td></tr>
<tr align = "right"><td> 8</td><td>511</th><td>10,265</th><td>73,915</td><td>330,985</td><td>1,135,893</td></tr>
</table>
</p>
<h3 align = "center">
Usage:
</h3>
<p>
To integrate a function <b>f(x)</b> over a multidimensional cube [-1,+1]^DIM using
a sparse grid based on a Clenshaw Curtis rule, we might use a program something
like the following:
</p>
<pre>
dim = 2
level = 3
rule = 1
call levels_index_size ( dim, level, rule, point_num )
allocate ( w(1:point_num) )
allocate ( x(1:dim,1:point_num) )
call sparse_grid ( dim, level, rule, point_num, w, x )
quad = 0.0
do j = 1, point_num
quad = quad + w(j) * f ( x(1:dim,j) )
end do
</pre>
<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>SANDIA_SPARSE</b> is available in
<a href = "../../cpp_src/sandia_sparse/sandia_sparse.html">a C++ version</a> and
<a href = "../../f_src/sandia_sparse/sandia_sparse.html">a FORTRAN90 version</a> and
<a href = "../../m_src/sandia_sparse/sandia_sparse.html">a MATLAB version.</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<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/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,
by Knut Petras.
</p>
<p>
<a href = "../../f_src/sparse_grid_cc/sparse_grid_cc.html">
SPARSE_GRID_CC</a>,
a FORTRAN90 library which
can define a multidimensional sparse grid based on a 1D Clenshaw Curtis rule.
</p>
<p>
<a href = "../../f_src/sparse_grid_cc_dataset/sparse_grid_cc_dataset.html">
SPARSE_GRID_CC_DATASET</a>,
a FORTRAN90 program which
reads user input, creates a multidimensional sparse grid based on a
1D Clenshaw Curtis rule and writes it to three files that define a
quadrature rule.
</p>
<p>
<a href = "../../m_src/sparse_grid_display/sparse_grid_display.html">
SPARSE_GRID_DISPLAY</a>,
a MATLAB program which
can display a 2D or 3D sparse grid.
</p>
<p>
<a href = "../../m_src/spinterp/spinterp.html">
SPINTERP</a>,
a MATLAB library which
uses a sparse grid to perform multilinear hierarchical interpolation,
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>
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>
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>
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>
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 = "sandia_sparse.f90">sandia_sparse.f90</a>, the source code.
</li>
<li>
<a href = "sandia_sparse.sh">sandia_sparse.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "sandia_sparse_prb.f90">sandia_sparse_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "sandia_sparse_prb.sh">sandia_sparse_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "sandia_sparse_prb_output.txt">sandia_sparse_prb_output.txt</a>,
the output from a run of the sample program.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>ABSCISSA_LEVEL_CLOSED_ND:</b> first level at which given abscissa is generated.
</li>
<li>
<b>ABSCISSA_LEVEL_OPEN_ND:</b> first level at which given abscissa is generated.
</li>
<li>
<b>CC_ABSCISSA</b> returns the I-th abscissa for the Clenshaw Curtis rule.
</li>
<li>
<b>CC_WEIGHTS</b> computes Clenshaw Curtis weights.
</li>
<li>
<b>COMP_NEXT</b> computes the compositions of the integer N into K parts.
</li>
<li>
<b>F1_ABSCISSA</b> returns the I-th abscissa for the Fejer type 1 rule.
</li>
<li>
<b>F1_WEIGHTS</b> computes weights for a Fejer type 1 rule.
</li>
<li>
<b>F2_ABSCISSA</b> returns the I-th abscissa for the Fejer type 2 rule.
</li>
<li>
<b>F2_WEIGHTS</b> computes weights for a Fejer type 2 rule.
</li>
<li>
<b>GH_ABSCISSA</b> sets abscissas for multidimensional Gauss Hermite quadrature.
</li>
<li>
<b>GH_WEIGHTS</b> returns weights for certain Gauss Hermite quadrature rules.
</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>GP_ABSCISSA</b> returns the I-th abscissa for a Gauss Patterson rule.
</li>
<li>
<b>GP_WEIGHTS</b> sets weights for a Gauss Patterson rule.
</li>
<li>
<b>I4_LOG_2</b> returns the integer part of the logarithm base 2 of an I4.
</li>
<li>
<b>I4_MODP</b> returns the nonnegative remainder of I4 division.
</li>
<li>
<b>INDEX_LEVEL_OWN:</b> determine first level at which given index is generated.
</li>
<li>
<b>INDEX_TO_LEVEL_CLOSED</b> determines the level of a point given its index.
</li>
<li>
<b>INDEX_TO_LEVEL_OPEN</b> determines the level of a point given its index.
</li>
<li>
<b>LEVEL_TO_ORDER_CLOSED</b> converts a level to an order for closed rules.
</li>
<li>
<b>LEVEL_TO_ORDER_OPEN</b> converts a level to an order for open rules.
</li>
<li>
<b>LEVELS_INDEX</b> indexes a sparse grid.
</li>
<li>
<b>LEVELS_INDEX_CFN</b> indexes a sparse grid made from CFN 1D rules.
</li>
<li>
<b>LEVELS_INDEX_OFN</b> indexes a sparse grid made from OFN 1D rules.
</li>
<li>
<b>LEVELS_INDEX_ONN</b> indexes a sparse grid made from ONN 1D rules.
</li>
<li>
<b>LEVELS_INDEX_OWN</b> indexes a sparse grid made from OWN 1D rules.
</li>
<li>
<b>LEVELS_INDEX_SIZE</b> sizes a sparse grid.
</li>
<li>
<b>LEVELS_INDEX_SIZE_ONN</b> sizes a sparse grid made from ONN 1D rules.
</li>
<li>
<b>LEVELS_INDEX_SIZE_OWN</b> sizes a sparse grid made from OWN 1D rules.
</li>
<li>
<b>LG_ABSCISSA</b> sets abscissas for multidimensional Gauss Laguerre quadrature.
</li>
<li>
<b>LG_WEIGHTS</b> returns weights for certain Gauss Laguerre quadrature rules.
</li>
<li>
<b>MONOMIAL_INTEGRAL_HERMITE</b> integrates a Hermite monomial.
</li>
<li>
<b>MONOMIAL_INTEGRAL_LAGUERRE</b> integrates a Laguerre monomial.
</li>
<li>
<b>MONOMIAL_INTEGRAL_LEGENDRE</b> integrates a Legendre monomial.
</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_CFN</b> indexes a sparse grid based on CFN 1D rules.
</li>
<li>
<b>MULTIGRID_INDEX_OFN</b> indexes a sparse grid based on OFN 1D rules.
</li>
<li>
<b>MULTIGRID_INDEX_ONN</b> indexes a sparse grid based on ONN 1D rules.
</li>
<li>
<b>MULTIGRID_INDEX_OWN</b> indexes a sparse grid based on OWN 1D rules.
</li>
<li>
<b>MULTIGRID_SCALE_CLOSED</b> renumbers a grid as a subgrid on a higher level.
</li>
<li>
<b>MULTIGRID_SCALE_OPEN</b> renumbers a grid as a subgrid on a higher level.
</li>
<li>
<b>PRODUCT_WEIGHTS</b> computes the weights of a product rule.
</li>
<li>
<b>R8_CHOOSE</b> computes the binomial coefficient C(N,K) as an R8.
</li>
<li>
<b>R8_FACTORIAL</b> computes the factorial function.
</li>
<li>
<b>R8_FACTORIAL2</b> computes the double factorial function.
</li>
<li>
<b>R8_HUGE</b> returns a very large R8.
</li>
<li>
<b>R8_MOP</b> returns the I-th power of -1 as an R8 value.
</li>
<li>
<b>R8VEC_DIRECT_PRODUCT2</b> creates a direct product of R8VEC's.
</li>
<li>
<b>SPARSE_GRID</b> computes a sparse grid.
</li>
<li>
<b>SPARSE_GRID_CC_SIZE</b> sizes a sparse grid using Clenshaw Curtis rules.
</li>
<li>
<b>SPARSE_GRID_CFN</b> computes a sparse grid based on a CFN 1D rule.
</li>
<li>
<b>SPARSE_GRID_OFN</b> computes a sparse grid based on an OFN 1D rule.
</li>
<li>
<b>SPARSE_GRID_OFN_SIZE</b> sizes a sparse grid using Open Fully Nested rules.
</li>
<li>
<b>SPARSE_GRID_ONN</b> computes a sparse grid based on a ONN 1D rule.
</li>
<li>
<b>SPARSE_GRID_OWN</b> computes a sparse grid based on an OWN 1D rule.
</li>
<li>
<b>SPARSE_GRID_WEIGHTS_CFN</b> computes sparse grid weights based on a CFN 1D rule.
</li>
<li>
<b>SPARSE_GRID_WEIGHTS_OFN</b> computes sparse grid weights based on a OFN 1D rule.
</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 23 December 2009.
</i>
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