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<html>
<head>
<title>
SPARSE_GRID_HERMITE_DATASET - Create Gauss-Hermite Sparse Grid Dataset
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SPARSE_GRID_HERMITE_DATASET <br> Create Gauss-Hermite Sparse Grid Dataset
</h1>
<hr>
<p>
<b>SPARSE_GRID_HERMITE_DATASET</b>
is a FORTRAN90 program which
creates a Gauss-Hermite sparse grid dataset.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>sparse_grid_hermite_dataset</b> <i>dim_num</i> <i>level_max</i>
</blockquote>
where
<ul>
<li>
<i>dim_num</i> is the spatial dimension, typically between 2 and 10;
</li>
<li>
<i>level_max</i> is the sparse grid level, typically between 0 and 6,
which controls the number of points
in the grid. The 1D rules used will have order 2^(level_max)+1.
</li>
</ul>
</p>
<p>
Each sparse grid is stored using the "quadrature rule" format,
that is, as three files:
<ul>
<li>
an "R" or "region" file, which lists two points that bound the region;
</li>
<li>
a "W" or "weight" file, which lists the weight for each abscissa;
</li>
<li>
an "X" or "abscissa" file, which lists the abscissas of the rule.
</li>
</ul>
</p>
<p>
The abscissas are ordered to respect the natural nesting of the
sparse grids by level. That is, the file of level 3 points begins by
listing the points in the level 2 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_HERMITE_DATASET</b> is available in
<a href = "../../cpp_src/sparse_grid_hermite_dataset/sparse_grid_hermite_dataset.html">a C++ version</a> and
<a href = "../../f_src/sparse_grid_hermite_dataset/sparse_grid_hermite_dataset.html">a FORTRAN90 version</a> and
<a href = "../../m_src/sparse_grid_hermite_dataset/sparse_grid_hermite_dataset.html">a MATLAB version.</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/monte_carlo_rule/monte_carlo_rule.html">
MONTE_CARLO_RULE</a>,
a FORTRAN90 program which
generates a dataset of N random M-dimensional points,
regards it as a quadrature rule for the unit hypercube,
and writes out three files of information.
</p>
<p>
<a href = "../../datasets/quadrature_rules/quadrature_rules.html">
QUADRATURE_RULES</a>,
a dataset directory which
define 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 = "../../datasets/sparse_grid_cc/sparse_grid_cc.html">
SPARSE_GRID_CC</a>,
a dataset directory which
contains the abscissas of sparse
grids based on a 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
creates a sparse grid dataset based on Clenshaw-Curtis rules.
</p>
<p>
<a href = "../../datasets/sparse_grid_f2/sparse_grid_f2.html">
SPARSE_GRID_F2</a>,
a dataset directory which
contains the abscissas of sparse
grids based on a Fejer Type 2 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 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 the abscissas of sparse
grids based on a Gauss Patterson rule.
</p>
<p>
<a href = "../../f_src/sparse_grid_hermite_dataset/sparse_grid_hermite_dataset.html">
SPARSE_GRID_HERMITE_DATASET</a>,
a FORTRAN90 program which
creates a sparse grid dataset based on Gauss-Hermite rules.
</p>
<p>
<a href = "../../datasets/sparse_grid_hermite/sparse_grid_hermite.html">
SPARSE_GRID_HERMITE</a>,
a dataset directory which
contains Gauss-Hermite sparse grids.
</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 = "../../datasets/sparse_grid_ncc/sparse_grid_ncc.html">
SPARSE_GRID_NCC</a>,
a dataset directory which
contains the abscissas of 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 the abscissas of sparse
grids based on a Newton Cotes open rule.
</p>
<p>
<a href = "../../f_src/sparse_grid_open_dataset/sparse_grid_open_dataset.html">
SPARSE_GRID_OPEN_DATASET</a>,
a FORTRAN90 program which
creates a sparse grid dataset 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>
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_hermite_dataset.f90">sparse_grid_hermite_dataset.f90</a>, the source code.
</li>
<li>
<a href = "sparse_grid_hermite_dataset.sh">sparse_grid_hermite_dataset.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
Here are the three quadrature files created by the command
<blockquote><b>
sparse_grid_hermite_dataset 2 3
</b></blockquote>
<ul>
<li>
<a href = "sparse_grid_hermite_dataset_output.txt">
sparse_grid_hermite_dataset_output.txt</a>,
printed output from the command.
</li>
<li>
<a href = "gh_d2_level3_r.txt">gh_d2_level3_r.txt</a>,
the "R" or "region" file.
</li>
<li>
<a href = "gh_d2_level3_w.txt">gh_d2_level3_w.txt</a>,
the "W" or "weight" file.
</li>
<li>
<a href = "gh_d2_level3_x.txt">gh_d2_level3_x.txt</a>,
the "X" or "abscissa" file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for SPARSE_GRID_HERMITE_DATASET.
</li>
<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>HERMITE_COMPUTE</b> computes a Gauss-Hermite quadrature rule.
</li>
<li>
<b>HERMITE_COMPUTE_POINTS</b> computes points of a Hermite quadrature rule.
</li>
<li>
<b>HERMITE_COMPUTE_WEIGHTS</b> computes weights of a Hermite quadrature rule.
</li>
<li>
<b>IMTQLX</b> diagonalizes a symmetric tridiagonal matrix.
</li>
<li>
<b>LEVEL_TO_ORDER_HERMITE:</b> default growth for Hermite sparse grids.
</li>
<li>
<b>MONOMIAL_INTEGRAL_HERMITE</b> integrates a Hermite monomial.
</li>
<li>
<b>MONOMIAL_QUADRATURE_HERMITE</b> applies a quadrature rule to a monomial.
</li>
<li>
<b>MONOMIAL_VALUE</b> evaluates a monomial.
</li>
<li>
<b>PRODUCT_HERMITE_WEIGHT</b> computes the weights of a Hermite product rule.
</li>
<li>
<b>R8_CHOOSE</b> computes the binomial coefficient C(N,K) as an R8.
</li>
<li>
<b>R8_FACTORIAL2</b> computes the double factorial function.
</li>
<li>
<b>R8_GAMMA</b> evaluates Gamma(X) for a real argument.
</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.
</li>
<li>
<b>R8COL_COMPARE</b> compares columns in an R8COL.
</li>
<li>
<b>R8COL_SORT_HEAP_A</b> ascending heapsorts an R8COL.
</li>
<li>
<b>R8COL_SORT_HEAP_INDEX_A</b> does an indexed heap ascending sort of an R8COL.
</li>
<li>
<b>R8COL_SORTED_UNIQUE_COUNT</b> counts unique elements in a sorted R8COL.
</li>
<li>
<b>R8COL_SWAP</b> swaps columns I and J of an R8COL.
</li>
<li>
<b>R8COL_TOL_UNIQUE_COUNT</b> counts tolerably unique entries in an R8COL.
</li>
<li>
<b>R8COL_UNDEX</b> returns unique sorted indexes for an R8COL.
</li>
<li>
<b>R8MAT_PRINT</b> prints an R8MAT.
</li>
<li>
<b>R8MAT_PRINT_SOME</b> prints some of an R8MAT.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT</b> prints an R8MAT, transposed.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT_SOME</b> prints some of an R8MAT, transposed.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file.
</li>
<li>
<b>R8VEC_COMPARE</b> compares two R8VEC's.
</li>
<li>
<b>R8VEC_DIRECT_PRODUCT2</b> creates a direct product of R8VEC's.
</li>
<li>
<b>R8VEC_PRINT_SOME</b> prints "some" of an R8VEC.
</li>
<li>
<b>S_BLANK_DELETE</b> removes blanks from a string, left justifying the remainder.
</li>
<li>
<b>S_TO_I4</b> reads an integer value from a string.
</li>
<li>
<b>SORT_HEAP_EXTERNAL</b> externally sorts a list of items into ascending order.
</li>
<li>
<b>SPARSE_GRID_HERMITE</b> computes a Hermite sparse grid.
</li>
<li>
<b>SPARSE_GRID_HERMITE_INDEX</b> indexes a Hermite sparse grid.
</li>
<li>
<b>SPARSE_GRID_HERMITE_POINT</b> computes the points of a Hermite sparse grid.
</li>
<li>
<b>SPARSE_GRID_HERMITE_SIZE</b> sizes a Hermite sparse grid, discounting duplicates.
</li>
<li>
<b>SPARSE_GRID_HERMITE_SIZE_TOTAL</b> Hermite sparse grid size counting duplicates.
</li>
<li>
<b>SPARSE_GRID_HERMITE_UNIQUE_INDEX</b> maps nonunique points to unique points.
</li>
<li>
<b>SPARSE_GRID_HERMITE_WEIGHT</b> computes Hermite sparse grid weights.
</li>
<li>
<b>SPARSE_GRID_HERMITE_WRITE</b> writes a Hermite sparse grid rule to files.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>VEC_COLEX_NEXT3</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 16 May 2012.
</i>
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