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<html>
<head>
<title>
SPARSE_GRID_OPEN_DATASET - Sparse Grid from Open 1D Quadrature Rule
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SPARSE_GRID_OPEN_DATASET <br> Sparse Grid from Open 1D Quadrature Rule
</h1>
<hr>
<p>
<b>SPARSE_GRID_OPEN_DATASET</b>
is a FORTRAN90 program which
computes a sparse quadrature rule for
an arbitrary spatial dimension, associated with a particular
"level" of the Smolyak construction,
and based on an open fully nested 1D quadrature rule.
</p>
<p>
The program offers a choice of open 1D quadrature rules to be used:
<ul>
<li>
<b>2: F2</b>, the Fejer type 2 rule;
</li>
<li>
<b>3: GP</b>, the Gauss-Patterson rule;
</li>
<li>
<b>4: NCO</b>, the Newton-Cotes Open rule;
</li>
<li>
<b>5: TS</b>, the Tanh-Sinh rule;
</li>
</ul>
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>sparse_grid_open_dataset</b> <i>dim_num</i> <i>level_max</i> <i>rule</i>
</blockquote>
where
<dl>
<dt>
<b>dim_num</b>
</dt>
<dd>
the spatial dimension;
</dd>
<dt>
<b>level_max</b>
</dt>
<dd>
the level of the Smolyak construction;
</dd>
<dt>
<b>rule</b>
</dt>
<dd>
the index (2/3/4/5) of the 1D quadrature rule to use.
</dd>
</dl>
</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>SPARSE_GRID_OPEN_DATASET</b> is available in
<a href = "../../cpp_src/sparse_grid_open_dataset/sparse_grid_open_dataset.html">a C++ version</a> and
<a href = "../../f_src/sparse_grid_open_dataset/sparse_grid_open_dataset.html">a FORTRAN90 version</a> and
<a href = "../../m_src/sparse_grid_open_dataset/sparse_grid_open_dataset.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/cc_display/cc_display.html">
CC_DISPLAY</a>,
a MATLAB library which
can compute and display Clenshaw Curtis grids in two dimensions,
as well as sparse grids formed from sums of Clenshaw Curtis grids.
</p>
<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
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 = "../../datasets/sparse_grid_cc/sparse_grid_cc.html">
SPARSE_GRID_CC</a>,
a dataset directory which
contains 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 = "../../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_f2/sparse_grid_f2.html">
SPARSE_GRID_F2</a>,
a dataset directory which
contains sparse
grids based on a Fejer Type 2 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 = "../../f_src/sparse_grid_laguerre_dataset/sparse_grid_laguerre_dataset.html">
SPARSE_GRID_LAGUERRE_DATASET</a>,
a FORTRAN90 program which
creates a sparse grid dataset based on Gauss-Laguerrre 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 = "../../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
defines 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>
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, 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>
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>
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_open_dataset.f90">sparse_grid_open_dataset.f90</a>, the source code.
</li>
<li>
<a href = "sparse_grid_open_dataset.sh">sparse_grid_open_dataset.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>F2_D2_LEVEL2</b> is an example computation based on a Fejer type 2 rule
in two dimensions and level 2.
<ul>
<li>
<a href = "f2_d2_level2_output.txt">
f2_d2_level2_output.txt</a>,
the printed output from a run of the program for DIM_NUM=2
and LEVEL_MAX=2 and RULE=2.
</li>
<li>
<a href = "f2_d2_level2_x.txt">
f2_d2_level2_x.txt</a>,
the abscissas of the computed quadrature rule.
</li>
<li>
<a href = "f2_d2_level2_w.txt">
f2_d2_level2_w.txt</a>,
the weights of the computed quadrature rule.
</li>
<li>
<a href = "f2_d2_level2_r.txt">
f2_d2_level2_r.txt</a>,
the integration region of the computed quadrature rule.
</li>
</ul>
</p>
<p>
<b>GP_D2_LEVEL2</b> is an example computation based on a Gauss-Patterson
rule in two dimensions and level 2.
<ul>
<li>
<a href = "gp_d2_level2_output.txt">
gp_d2_level2_output.txt</a>,
the printed output from a run of the program for DIM_NUM=2
and LEVEL_MAX=2 and RULE=3.
</li>
<li>
<a href = "gp_d2_level2_x.txt">
gp_d2_level2_x.txt</a>,
the abscissas of the computed quadrature rule.
</li>
<li>
<a href = "gp_d2_level2_w.txt">
gp_d2_level2_w.txt</a>,
the weights of the computed quadrature rule.
</li>
<li>
<a href = "gp_d2_level2_r.txt">
gp_d2_level2_r.txt</a>,
the integration region of the computed quadrature rule.
</li>
</ul>
</p>
<p>
<b>NCO_D2_LEVEL2</b> is an example computation based on a
Newton-Cotes Open rule in two dimensions and level 2.
<ul>
<li>
<a href = "nco_d2_level2_output.txt">
nco_d2_level2_output.txt</a>,
the printed output from a run of the program for DIM_NUM=2
and LEVEL_MAX=2 and RULE=4.
</li>
<li>
<a href = "nco_d2_level2_x.txt">
nco_d2_level2_x.txt</a>,
the abscissas of the computed quadrature rule.
</li>
<li>
<a href = "nco_d2_level2_w.txt">
nco_d2_level2_w.txt</a>,
the weights of the computed quadrature rule.
</li>
<li>
<a href = "nco_d2_level2_r.txt">
nco_d2_level2_r.txt</a>,
the integration region of the computed quadrature rule.
</li>
</ul>
</p>
<p>
<b>TS_D2_LEVEL4</b> is an example computation based on a
tanh-sinh rule in two dimensions and level 4.
<ul>
<li>
<a href = "ts_d2_level4_output.txt">
ts_d2_level4_output.txt</a>, the output file.
</li>
<li>
<a href = "ts_d2_level4_x.txt">
ts_d2_level4_x.txt</a>, the abscissas.
</li>
<li>
<a href = "ts_d2_level4_w.txt">
ts_d2_level4_w.txt</a>, the weights.
</li>
<li>
<a href = "ts_d2_level4_r.txt">
ts_d2_level4_r.txt</a>, the region.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for SPARSE_GRID_OPEN_DATASET.
</li>
<li>
<b>ABSCISSA_LEVEL_OPEN_ND:</b> first level at which given abscissa is generated.
</li>
<li>
<b>CHOOSE</b> computes the binomial coefficient C(N,K).
</li>
<li>
<b>COMP_NEXT</b> computes the compositions of the integer N into K parts.
</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>GET_UNIT</b> returns a free FORTRAN unit number.
</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_MODP</b> returns the nonnegative remainder of I4 division.
</li>
<li>
<b>I4MAT_TRANSPOSE_PRINT_SOME</b> prints some of the transpose of an I4MAT.
</li>
<li>
<b>INDEX_TO_LEVEL_OPEN</b> determines the level of a point given its index.
</li>
<li>
<b>LEVEL_TO_ORDER_OPEN</b> converts a level to an order for open rules.
</li>
<li>
<b>MULTIGRID_INDEX1</b> returns an indexed multidimensional grid.
</li>
<li>
<b>MULTIGRID_SCALE_OPEN</b> renumbers a grid as a subgrid on a higher level.
</li>
<li>
<b>NCO_ABSCISSA</b> returns the I-th abscissa for the Newton Cotes Open rule.
</li>
<li>
<b>NCO_WEIGHTS</b> computes weights for a Newton Cotes Open rule.
</li>
<li>
<b>PRODUCT_WEIGHTS_OPEN:</b> weights for an open product rule.
</li>
<li>
<b>R8_HUGE</b> returns a very large R8.
</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_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 I4 from a string.
</li>
<li>
<b>SPARSE_GRID_OFN_SIZE</b> sizes a sparse grid using Open Fully Nested rules.
</li>
<li>
<b>SPGRID_OPEN_INDEX</b> computes open grids with 0 <= LEVEL <= LEVEL_MAX.
</li>
<li>
<b>SPGRID_OPEN_WEIGHTS</b> gathers the weights.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TS_ABSCISSA</b> returns the I-th abscissa for the tanh-sinh rule.
</li>
<li>
<b>TS_WEIGHTS</b> computes weights for a tanh-sinh rule.
</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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