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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 C++ 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 = "../../cpp_src/faure/faure.html">
FAURE</a>,
a C++ library which
computes elements of a Faure quasirandom sequence.
</p>
<p>
<a href = "../../cpp_src/grid/grid.html">
GRID</a>,
a C++ library which
computes elements of a grid dataset.
</p>
<p>
<a href = "../../cpp_src/halton/halton.html">
HALTON</a>,
a C++ library which
computes elements of a Halton quasirandom sequence.
</p>
<p>
<a href = "../../cpp_src/hammersley/hammersley.html">
HAMMERSLEY</a>,
a C++ library which
computes elements of a Hammersley quasirandom sequence.
</p>
<p>
<a href = "../../cpp_src/hex_grid/hex_grid.html">
HEX_GRID</a>,
a C++ library which
computes elements of a hexagonal grid dataset.
</p>
<p>
<a href = "../../cpp_src/ihs/ihs.html">
IHS</a>,
a C++ library which
computes elements of an improved distributed Latin hypercube dataset.
</p>
<p>
<a href = "../../cpp_src/latin_center/latin_center.html">
LATIN_CENTER</a>,
a C++ library which
computes elements of a Latin Hypercube dataset, choosing center points.
</p>
<p>
<a href = "../../cpp_src/latin_edge/latin_edge.html">
LATIN_EDGE</a>,
a C++ library which
computes elements of a Latin Hypercube dataset, choosing edge points.
</p>
<p>
<a href = "../../cpp_src/latin_random/latin_random.html">
LATIN_RANDOM</a>,
a C++ library which
computes elements of a Latin Hypercube dataset, choosing
points at random.
</p>
<p>
<a href = "../../cpp_src/lcvt/lcvt.html">
LCVT</a>,
a C++ library which
computes a latinized
Centroidal Voronoi Tessellation.
</p>
<p>
<a href = "../../cpp_src/niederreiter2/niederreiter2.html">
NIEDERREITER2</a>,
a C++ library which
computes elements of a Niederreiter quasirandom sequence with base 2.
</p>
<p>
<a href = "../../cpp_src/nint_exactness/nint_exactness.html">
NINT_EXACTNESS</a>,
a C++ program which
measures the polynomial exactness of a multidimensional quadrature rule.
</p>
<p>
<a href = "../../cpp_src/nintlib/nintlib.html">
NINTLIB</a>,
a C++ library which
contains routines for numerical estimation of integrals in multiple dimensions.
</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 = "../../cpp_src/quadrule/quadrule.html">
QUADRULE</a>,
a C++ library which
defines a number of quadrature rules.
</p>
<p>
<a href = "../../cpp_src/sobol/sobol.html">
SOBOL</a>,
a C++ library which
computes elements of a Sobol quasirandom sequence.
</p>
<p>
<a href = "../../cpp_src/stroud/stroud.html">
STROUD</a>,
a C++ library which
contains quadrature
rules for a variety of unusual areas, surfaces and volumes in 2D,
3D and N-dimensions.
</p>
<p>
<a href = "../../cpp_src/test_nint/test_nint.html">
TEST_NINT</a>,
a C++ library which
defines test functions for multi-dimensional quadrature routines.
</p>
<p>
<a href = "../../cpp_src/uniform/uniform.html">
UNIFORM</a>,
a C++ 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.cpp">lattice_rule.cpp</a>, the source code.
</li>
<li>
<a href = "lattice_rule.hpp">lattice_rule.hpp</a>, the include file.
</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.cpp">lattice_rule_prb.cpp</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 PNG 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 PNG 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>I4_HUGE</b> returns a "huge" I4.
</li>
<li>
<b>I4_MAX</b> returns the maximum of two I4's.
</li>
<li>
<b>I4_MIN</b> returns the minimum of two I4's.
</li>
<li>
<b>I4_POWER</b> returns the value of I^J.
</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>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R8_HUGE</b> returns a "huge" R8.
</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>R8VEC_UNIFORM_01</b> returns a unit pseudorandom R8VEC.
</li>
<li>
<b>S_LEN_TRIM</b> returns the length of a string to the last nonblank.
</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 = "../fcpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 23 November 2008.
</i>
<!-- John Burkardt -->
</body>
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</html>