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<html>
<head>
<title>
SPARSE_GRID_MIXED - Sparse Grids using Mixed Factors
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SPARSE_GRID_MIXED <br> Sparse Grids using Mixed Factors
</h1>
<hr>
<p>
<b>SPARSE_GRID_MIXED</b>
is a FORTRAN90 library which
can be used to construct a sparse grid whose factors are
possibly distinct 1D quadrature rules.
</p>
<p>
<b>SPARSE_GRID_MIXED</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>SPARSE_GRID_MIXED</b> library.
</p>
<p>
The 1D quadrature rules are designed to approximate an integral
of the form:
<blockquote><b>
Integral ( A < X < B ) F(X) W(X) dX
</b></blockquote>
where <b>W(X)</b> is a weight function, by the quadrature sum:
<blockquote><b>
Sum ( 1 <= I <= ORDER) F(X(I)) * W(I)
</b></blockquote>
where the set of <b>X</b> values are known as <i>abscissas</i> and
the set of <b>W</b> values are known as <i>weights</i>.
</p>
<p>
Note that the letter <b>W</b>, unfortunately, is used to denote both the
weight function in the original integral, and the vector of weight
values in the quadrature sum.
</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 Exponential Growth</td>
<td>CC</td>
<td>Exponential</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
<tr>
<td>2</td>
<td>Fejer Type 2, Exponential Growth</td>
<td>F2</td>
<td>Exponential</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
<tr>
<td>3</td>
<td>Gauss Patterson, Exponential Growth</td>
<td>GP</td>
<td>Exponential</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
<tr>
<td>4</td>
<td>Gauss-Legendre</td>
<td>GL</td>
<td>Linear</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
<tr>
<td>5</td>
<td>Gauss-Hermite</td>
<td>GH</td>
<td>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>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>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>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>Linear</td>
<td>[-1,+1]</td>
<td>(1-x)<sup>alpha</sup> (1+x)<sup>beta</sup></td>
</tr>
<tr>
<td>10</td>
<td>Golub-Welsch</td>
<td>GW</td>
<td>?</td>
<td>?</td>
<td>?</td>
</tr>
<tr>
<td>11</td>
<td>Clenshaw-Curtis, Slow Exponential Growth</td>
<td>CC_SE</td>
<td>Slow exponential</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
<tr>
<td>12</td>
<td>Fejer Type 2, Slow Exponential Growth</td>
<td>F2_SE</td>
<td>Slow exponential</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
<tr>
<td>13</td>
<td>Gauss Patterson, Slow Exponential Growth</td>
<td>GP_SE</td>
<td>Slow exponential</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
<tr>
<td>14</td>
<td>Clenshaw-Curtis, Moderate Exponential Growth</td>
<td>CC_ME</td>
<td>Moderate exponential</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
<tr>
<td>15</td>
<td>Fejer Type 2, Moderate Exponential Growth</td>
<td>F2_ME</td>
<td>Moderate exponential</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
<tr>
<td>16</td>
<td>Gauss Patterson, Moderate Exponential Growth</td>
<td>GP_ME</td>
<td>Moderate exponential</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
<tr>
<td>17</td>
<td>Clenshaw-Curtis Nested, linear growth</td>
<td>CCN</td>
<td>Linear (2*L+1)</td>
<td>[-1,+1]</td>
<td>1</td>
</tr>
</table>
</p>
<p>
A sparse grid is a quadrature rule for a multidimensional integral.
It is formed by taking a certain linear combination of lower-order
product rules. The product rules, in turn, are formed as direct products
of 1D quadrature rules. It is common to form a sparse grid in which the
1D component quadrature rules are the same. This package, however,
is intended to produce sparse grids based on sums of product rules for
which the rule chosen for each spatial dimension may be freely chosen
from the set listed above.
</p>
<p>
These sparse grids are still indexed by a number known as <b>level</b>,
and assembled as a sum of low order product rules. As the value of
<b>level</b> increases, the point growth becomes more complicated.
This is because the 1D rules have somewhat varying point growth patterns
to begin with, and the varying levels of nestedness have a dramatic
effect on the sparsity of the total grid.
</p>
<p>
Since a sparse grid is made up of a combination of product grids,
it is frequently the case that many of the product grids include the
same point. For efficiency, it is usually desirable to merge or
consolidate such duplicate points when expressing the resulting
sparse grid rule. It is possible to "logically" determine when
a duplicate point will be generated; however, this logic changes
depending on the specific 1-dimensional rules being used, and the
tests can become quite elaborate. Moreover, for rules which include
real parameters, the determination of duplication can require
a numerical tolerance.
</p>
<p>
In order to simplify the matter of the detection of duplicate points,
the codes presented begin by generating the entire "naive" set of
points. Then a user-specified tolerance <b>TOL</b> is used to determine when
two points are equal. If the maximum difference between any two components
is less than or equal to <b>TOL</b>, the points are declared to be equal.
</p>
<p>
A reasonable value for <b>TOL</b> might be the square root of the machine
precision. Setting <b>TOL</b> to zero means that only points which are
identical to the last significant digit are taken to be duplicates. Setting
<b>TOL</b> to a negative value means that no duplicate points will be eliminated -
in other words, this choice produces the full or "naive" grid.
</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_MIXED</b> is available in
<a href = "../../cpp_src/sparse_grid_mixed/sparse_grid_mixed.html">a C++ version</a> and
<a href = "../../f_src/sparse_grid_mixed/sparse_grid_mixed.html">a FORTRAN90 version</a> and
<a href = "../../m_src/sparse_grid_mixed/sparse_grid_mixed.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 = "../../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_gl/sparse_grid_gl.html">
SPARSE_GRID_GL</a>,
a FORTRAN90 library which
creates sparse grids based on Gauss-Legendre rules.
</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_laguerre/sparse_grid_laguerre.html">
SPARSE_GRID_LAGUERRE</a>,
a FORTRAN90 library which
creates sparse grids based on Gauss-Laguerre 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 = "../../f_src/sparse_grid_mixed_dataset/sparse_grid_mixed_dataset.html">
SPARSE_GRID_MIXED_DATASET</a>,
a FORTRAN90 program which
creates a sparse grid dataset based on a mixture of 1D rules.
</p>
<p>
<a href = "../../f_src/sparse_grid_open/sparse_grid_open.html">
SPARSE_GRID_OPEN</a>,
a FORTRAN90 library which
creates sparse grids based on
open rules (Fejer 2, Gauss-Patterson, Newton-Cotes-Open).
</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>
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>
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>
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 = "sparse_grid_mixed.f90">sparse_grid_mixed.f90</a>, the source code.
</li>
<li>
<a href = "sparse_grid_mixed.sh">sparse_grid_mixed.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>COMP_NEXT_PRB</b> tests COMP_NEXT.
<ul>
<li>
<a href = "comp_next_prb.f90">comp_next_prb.f90</a>,
the test program.
</li>
<li>
<a href = "comp_next_prb.sh">comp_next_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "comp_next_prb_output.txt">
comp_next_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>PRODUCT_MIXED_WEIGHT_PRB</b> tests PRODUCT_MIXED_WEIGHT.
<ul>
<li>
<a href = "product_mixed_weight_prb.f90">product_mixed_weight_prb.f90</a>,
the test program.
</li>
<li>
<a href = "product_mixed_weight_prb.sh">product_mixed_weight_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "product_mixed_weight_prb_output.txt">
product_mixed_weight_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>SPARSE_GRID_MIXED_INDEX_PRB</b> tests SPARSE_GRID_MIXED_INDEX.
<ul>
<li>
<a href = "sparse_grid_mixed_index_prb.f90">sparse_grid_mixed_index_prb.f90</a>,
the test program.
</li>
<li>
<a href = "sparse_grid_mixed_index_prb.sh">sparse_grid_mixed_index_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "sparse_grid_mixed_index_prb_output.txt">
sparse_grid_mixed_index_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>SPARSE_GRID_MIXED_POINT_PRB</b> tests SPARSE_GRID_MIXED_POINT.
<ul>
<li>
<a href = "sparse_grid_mixed_point_prb.f90">sparse_grid_mixed_point_prb.f90</a>,
the test program.
</li>
<li>
<a href = "sparse_grid_mixed_point_prb.sh">sparse_grid_mixed_point_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "sparse_grid_mixed_point_prb_output.txt">
sparse_grid_mixed_point_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>SPARSE_GRID_MIXED_SIZE_PRB</b> tests SPARSE_GRID_MIXED_SIZE and
SPARSE_GRID_MIXED_SIZE_TOTAL.
<ul>
<li>
<a href = "sparse_grid_mixed_size_prb.f90">sparse_grid_mixed_size_prb.f90</a>,
the test program.
</li>
<li>
<a href = "sparse_grid_mixed_size_prb.sh">sparse_grid_mixed_size_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "sparse_grid_mixed_size_prb_output.txt">
sparse_grid_mixed_size_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>SPARSE_GRID_MIXED_SIZE_TABLE</b> makes a point count table.
<ul>
<li>
<a href = "sparse_grid_mixed_size_table.f90">sparse_grid_mixed_size_table.f90</a>,
the test program.
</li>
<li>
<a href = "sparse_grid_mixed_size_table.sh">sparse_grid_mixed_size_table.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "sparse_grid_mixed_size_table_output.txt">
sparse_grid_mixed_size_table_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>SPARSE_GRID_MIXED_UNIQUE_INDEX_PRB</b> tests SPARSE_GRID_MIXED_UNIQUE_INDEX.
<ul>
<li>
<a href = "sparse_grid_mixed_unique_index_prb.f90">sparse_grid_mixed_unique_index_prb.f90</a>,
the test program.
</li>
<li>
<a href = "sparse_grid_mixed_unique_index_prb.sh">sparse_grid_mixed_unique_index_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "sparse_grid_mixed_unique_index_prb_output.txt">
sparse_grid_mixed_unique_index_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>SPARSE_GRID_MIXED_WEIGHT_PRB</b> tests SPARSE_GRID_MIXED_WEIGHT.
<ul>
<li>
<a href = "sparse_grid_mixed_weight_prb.f90">sparse_grid_mixed_weight_prb.f90</a>,
the test program.
</li>
<li>
<a href = "sparse_grid_mixed_weight_prb.sh">sparse_grid_mixed_weight_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "sparse_grid_mixed_weight_prb_output.txt">
sparse_grid_mixed_weight_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>SPARSE_GRID_MIXED_WRITE_PRB</b> tests SPARSE_GRID_MIXED_WRITE.
<ul>
<li>
<a href = "sparse_grid_mixed_write_prb.f90">sparse_grid_mixed_write_prb.f90</a>,
the test program.
</li>
<li>
<a href = "sparse_grid_mixed_write_prb.sh">sparse_grid_mixed_write_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "sparse_grid_mixed_write_prb_output.txt">
sparse_grid_mixed_write_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>SPARSE_GRID_MIXED_INDEX</b> indexes a sparse grid made from mixed 1D rules.
</li>
<li>
<b>SPARSE_GRID_MIXED_POINT</b> computes the points of a sparse grid rule.
</li>
<li>
<b>SPARSE_GRID_MIXED_SIZE</b> sizes a sparse grid, discounting duplicate points.
</li>
<li>
<b>SPARSE_GRID_MIXED_SIZE_TOTAL</b> sizes a sparse grid, counting duplicate points.
</li>
<li>
<b>SPARSE_GRID_MIXED_UNIQUE_INDEX</b> maps nonunique points to unique points.
</li>
<li>
<b>SPARSE_GRID_MIXED_WEIGHT</b> computes sparse grid weights for a mix of 1D rules.
</li>
<li>
<b>SPARSE_GRID_MIXED_WRITE</b> writes a sparse grid rule to five 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 21 February 2010.
</i>
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