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<html>
<head>
<title>
LATTICE_RULE - Lattice Rules for Multiple Integration
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
LATTICE_RULE <br> Lattice Rules for Multiple Integration
</h1>
<hr>
<p>
<b>LATTICE_RULE</b>
is a FORTRAN90 library which
approximates integrals in multiple
dimensions using lattice rules.
</p>
<p>
The lattice rules are defined on the unit square, or in higher
dimensions, on the unit hypercube.
</p>
<p>
Lattice rules are more suitable to integrands that are periodic,
with period 1, in all variables. However, there are techniques
that may be used when the integrand does not satisfy this requirement.
</p>
<p>
The performance of a lattice rule depends heavily on the choice
of the generator vectors. Once the spatial dimension and the
number of points have been chosen, there are techniques for
finding a good generator vector.
</p>
<p>
The simplest lattice rules are called "rank 1" rules, and
use a lattice generated by multiples of a single generator vector.
More elaborate rules can be generated by using more generator vectors,
up to the maximum, which is the dimension of the space.
</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>LATTICE_RULE</b> is available in
<a href = "../../cpp_src/lattice_rule/lattice_rule.html">a C++ version</a> and
<a href = "../../f_src/lattice_rule/lattice_rule.html">a FORTRAN90 version</a> and
<a href = "../../m_src/lattice_rule/lattice_rule.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Programs:
</h3>
<p>
<a href = "../../f_src/faure/faure.html">
FAURE</a>,
a FORTRAN90 library which
computes elements of a Faure quasirandom sequence.
</p>
<p>
<a href = "../../f_src/grid/grid.html">
GRID</a>,
a FORTRAN90 library which
computes elements of a grid dataset.
</p>
<p>
<a href = "../../f_src/halton/halton.html">
HALTON</a>,
a FORTRAN90 library which
computes elements of a Halton quasirandom sequence.
</p>
<p>
<a href = "../../f_src/hammersley/hammersley.html">
HAMMERSLEY</a>,
a FORTRAN90 library which
computes elements of a Hammersley quasirandom sequence.
</p>
<p>
<a href = "../../f_src/hex_grid/hex_grid.html">
HEX_GRID</a>,
a FORTRAN90 library which
computes elements of a hexagonal grid dataset.
</p>
<p>
<a href = "../../f_src/hex_grid_angle/hex_grid_angle.html">
HEX_GRID_ANGLE</a>,
a FORTRAN90 library which
computes elements of an angled hexagonal grid dataset.
</p>
<p>
<a href = "../../f_src/ihs/ihs.html">
IHS</a>,
a FORTRAN90 library which
computes elements of an improved distributed Latin hypercube dataset.
</p>
<p>
<a href = "../../f_src/latin_center/latin_center.html">
LATIN_CENTER</a>,
a FORTRAN90 library which
computes elements of a Latin Hypercube dataset, choosing center points.
</p>
<p>
<a href = "../../f_src/latin_edge/latin_edge.html">
LATIN_EDGE</a>,
a FORTRAN90 library which
computes elements of a Latin Hypercube dataset, choosing edge points.
</p>
<p>
<a href = "../../f_src/latin_random/latin_random.html">
LATIN_RANDOM</a>,
a FORTRAN90 library which
computes elements of a Latin Hypercube dataset, choosing
points at random.
</p>
<p>
<a href = "../../f_src/lcvt/lcvt.html">
LCVT</a>,
a FORTRAN90 library which
computes a latinized
Centroidal Voronoi Tessellation.
</p>
<p>
<a href = "../../f_src/niederreiter2/niederreiter2.html">
NIEDERREITER2</a>,
a FORTRAN90 library which
computes elements of a Niederreiter quasirandom sequence with base 2.
</p>
<p>
<a href = "../../f_src/product_rule/product_rule.html">
PRODUCT_RULE</a>,
a FORTRAN90 program which
can create a multidimensional quadrature rule as a product of
one dimensional rules.
</p>
<p>
<a href = "../../datasets/quadrature_rules/quadrature_rules.html">
QUADRATURE_RULES</a>,
a dataset directory which
contains sets of files that define quadrature
rules over various 1D intervals or multidimensional hypercubes.
</p>
<p>
<a href = "../../f_src/quadrule/quadrule.html">
QUADRULE</a>,
a FORTRAN90 library which
defines a number of quadrature rules.
</p>
<p>
<a href = "../../f_src/sobol/sobol.html">
SOBOL</a>,
a FORTRAN90 library which
computes elements of a Sobol quasirandom sequence.
</p>
<p>
<a href = "../../f_src/stroud/stroud.html">
STROUD</a>,
a FORTRAN90 library which
contains quadrature
rules for a variety of unusual areas, surfaces and volumes in 2D,
3D and N-dimensions.
</p>
<p>
<a href = "../../f_src/test_nint/test_nint.html">
TEST_NINT</a>,
a FORTRAN90 library which
defines test functioins for multi-dimensional quadrature routines.
</p>
<p>
<a href = "../../f_src/uniform/uniform.html">
UNIFORM</a>,
a FORTRAN90 library which
computes elements of a uniform pseudorandom sequence.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Seymour Haber,<br>
Parameters for Integrating Periodic Functions of
Several Variables,<br>
Mathematics of Computation,<br>
Volume 41, Number 163, July 1983, pages 115-129.
</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>
Ian Sloan, Stephen Joe,<br>
Lattice Methods for Multiple Integration,<br>
Oxford University Press, 1994,<br>
ISBN: 0198534728,<br>
LC: QA311.S56
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "lattice_rule.f90">lattice_rule.f90</a>, the source code.
</li>
<li>
<a href = "lattice_rule.sh">lattice_rule.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "lattice_rule_prb.f90">lattice_rule_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "lattice_rule_prb.sh">lattice_rule_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "lattice_rule_prb_output.txt">lattice_rule_prb_output.txt</a>,
the output file.
</li>
<li>
<a href = "f21.txt">f21.txt</a>,
the 21 points of a Fibonacci lattice for K = 8, M = 21.
</li>
<li>
<a href = "f21.png">f21.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
a Fibonacci lattice.
</li>
<li>
<a href = "f144.txt">f144.txt</a>,
the 144 points of a Fibonacci lattice for K = 12, M = 144.
</li>
<li>
<a href = "f144.png">f144.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
a Fibonacci lattice.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>E_01_2D</b> is the exact integral of 2d test function #1.
</li>
<li>
<b>F_01_2D</b> is the 2D test function #1.
</li>
<li>
<b>F2</b> evaluates a function of a scalar used in defining P2(Q).
</li>
<li>
<b>F20_S</b> evaluates a function of a vector used in defining P2(Q).
</li>
<li>
<b>FIBONACCI</b> returns the Fibonacci number of given index.
</li>
<li>
<b>FIBONACCI_LATTICE_B</b> applies an optimal Fibonacci lattice integration rule in 2D.
</li>
<li>
<b>FIBONACCI_LATTICE_Q</b> applies a Fibonacci lattice integration rule in 2D.
</li>
<li>
<b>FIBONACCI_LATTICE_Q_NODES</b> returns Fibonacci lattice nodes in 2D.
</li>
<li>
<b>FIBONACCI_LATTICE_Q1</b> applies a Fibonacci lattice integration rule in 2D.
</li>
<li>
<b>FIBONACCI_LATTICE_Q2</b> applies a Fibonacci lattice integration rule in 2D.
</li>
<li>
<b>FIBONACCI_LATTICE_Q3</b> applies a Fibonacci lattice integration rule in 2D.
</li>
<li>
<b>FIBONACCI_LATTICE_T</b> applies a symmetric Fibonacci lattice integration rule in 2D.
</li>
<li>
<b>FIND_Z20</b> finds the appropriate Z vector to minimize P2(QS).
</li>
<li>
<b>GRAY_NEXT</b> generates the next Gray code by switching one item at a time.
</li>
<li>
<b>I4VEC_PRINT</b> prints an I4VEC.
</li>
<li>
<b>LATTICE</b> applies a lattice integration rule.
</li>
<li>
<b>LATTICE_NP0</b> applies a lattice integration rule to a nonperiodic function.
</li>
<li>
<b>LATTICE_NP1</b> applies a lattice integration rule to a nonperiodic function.
</li>
<li>
<b>LATTICE_PRINT</b> prints the points in a lattice rule.
</li>
<li>
<b>MONTE_CARLO</b> applies a Monte Carlo integration rule.
</li>
<li>
<b>PRIME</b> returns any of the first PRIME_MAX prime numbers.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT</b> prints a R8MAT, transposed.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT_SOME</b> prints some of a R8MAT, transposed.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TUPLE_NEXT</b> computes the next element of a tuple space.
</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 27 November 2006.
</i>
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