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<html>
<head>
<title>
HALTON - The Halton Quasirandom Sequence
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
HALTON <br> The Halton Quasirandom Sequence
</h1>
<hr>
<p>
<b>HALTON</b>
is a FORTRAN90 library which
computes elements of a Halton quasirandom sequence.
</p>
<p>
<b>HALTON</b> includes routines to make it easy to manipulate this
computation, to compute the next N entries, to compute a particular
entry, to restart the sequence at a particular point, or to compute
NDIM-dimensional versions of the sequence.
</p>
<p>
For the most straightforward use, try either
<ul>
<li>
<b>I4_TO_HALTON</b>, for one element of a sequence;
</li>
<li>
<b>I4_TO_HALTON_SEQUENCE</b>, for N elements of a sequence;
</li>
</ul>
Both of these routines require explicit input values for all
parameters.
</p>
<p>
For more convenience, there are two related routines with
almost no input arguments:
<ul>
<li>
<b>HALTON</b>, for one element of a sequence;
</li>
<li>
<b>HALTON_SEQUENCE</b>, for N elements of a sequence;
</li>
</ul>
These routines allow the user to either rely on the default
values of parameters, or to change a few of them by calling
appropriate routines.
</p>
<p>
The NDIM-dimensional Halton sequence is really NDIM separate
sequences, each generated by a particular base.
</p>
<p>
Routines in this library select elements of a "leaped" subsequence of
the Halton sequence. The subsequence elements are indexed by a
quantity called STEP, which starts at 0. The STEP-th subsequence
element is simply the Halton sequence element with index
<pre>
SEED(1:NDIM) + STEP * LEAP(1:NDIM).
</pre>
</p>
<p>
The arguments that the user may set include:
<ul>
<li>
NDIM, the spatial dimension, <br>
default: NDIM = 1, <br>
required: 1 <= NDIM;
</li>
<li>
STEP, the subsequence index.<br>
default: STEP = 0,<br>
required: 0 <= STEP.
</li>
<li>
SEED(1:NDIM), the Halton sequence index corresponding
to STEP = 0.<br>
default: SEED(1:NDIM) = (0, 0, ... 0),<br>
required: 0 <= SEED(1:NDIM);
</li>
<li>
LEAP(1:NDIM), the succesive jumps in the Halton sequence.<br>
default: LEAP(1:NDIM) = (1, 1, ..., 1).<br>
required: 1 <= LEAP(1:NDIM).
</li>
<li>
BASE(1:NDIM), the Halton bases.<br>
default: BASE(1:NDIM) = (2, 3, 5, 7, 11... ),<br>
required: 1 < BASE(1:NDIM).
</li>
</ul>
</p>
<p>
The NDIM-dimensional Halton sequence is derived from the 1-dimensional
<a href = "../van_der_corput/van_der_corput.html">
van der Corput sequence</a>. Each dimension typically uses a different
prime number as the base of the calculation.
</p>
<p>
The NDIM-dimensional Halton sequence is related to the NDIM+1 dimensional
<a href = "../hammersley/hammersley.html">Hammersley sequence</a>
of length NMAX. An NDIM+1 dimensional Hammersley
sequence of length NMAX becomes an NDIM-dimensional Halton sequence by
deleting the first dimension. An NDIM dimensional Halton sequence of NMAX
points becomes an NDIM+1 dimensional Hammersley sequence of length NMAX
by prefixing a first coordinate, and setting the value of this
first coordinate to I/NMAX for the I-th entry of the sequence.
</p>
<p>
While the Hammersley sequence has better dispersion properties
in technical measures such as the discrepancy, it suffers from the
problem that you must know, beforehand, the number of points you
are going to generate. Thus, if you have computed a Hammersley
sequence of length 100, and you want to compute a Hammersley sequence
of length 200, you must discard your current values and start over.
By contrast, you can compute 100 points of a Halton sequence, and
then 100 more, and this will be the same as computing the first 200
points of the Halton sequence in one calculation.
</p>
<p>
In low dimensions, the multidimensional Halton sequence quickly
"fills up" the space in a well-distributed pattern. However,
for higher dimensions (such as NDIM = 40) for instance, the initial
elements of the Halton sequence can be very poorly distributed;
it is only when N, the number of sequence elements, is large
enough relative to the spatial dimension, that the sequence is
properly behaved. Remedies for this problem were suggested
by Kocis and Whiten.
</p>
<p>
As an example of the use of Halton sequences, we also use them
to compute "random" points on or in the unit circle in 2D,
and the unit sphere in 3D.
</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>HALTON</b> is available in
<a href = "../../cpp_src/halton/halton.html">a C++ version</a> and
<a href = "../../f_src/halton/halton.html">a FORTRAN90 version</a> and
<a href = "../../m_src/halton/halton.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/cvt/cvt.html">
CVT</a>,
a FORTRAN90 library which
computes elements of a Centroidal Voronoi Tessellation.
</p>
<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 sequence.
</p>
<p>
<a href = "../../f_src/halton_dataset/halton_dataset.html">
HALTON_DATASET</a>,
a FORTRAN90 program which
creates a Halton sequence and writes it to a file.
</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/lattice_rule/lattice_rule.html">
LATTICE_RULE</a>,
a FORTRAN90 library which
approximates multidimensional integrals using lattice rules.
</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 sequence using base 2.
</p>
<p>
<a href = "../../f_src/normal/normal.html">
NORMAL</a>,
a FORTRAN90 library which
computes elements of a sequence of pseudorandom normally distributed values.
</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/toms647/toms647.html">
TOMS647</a>,
a FORTRAN90 library which
is a version of ACM TOMS algorithm 647,
for evaluating Faure, Halton and Sobol quasirandom sequences.
</p>
<p>
<a href = "../../f_src/uniform/uniform.html">
UNIFORM</a>,
a FORTRAN90 library which
computes elements of a uniform pseudorandom sequence.
</p>
<p>
<a href = "../../f_src/van_der_corput/van_der_corput.html">
VAN_DER_CORPUT</a>,
a FORTRAN90 library which
computes elements of a 1D van der Corput sequence.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
John Halton,<br>
On the efficiency of certain quasi-random sequences of points
in evaluating multi-dimensional integrals,<br>
Numerische Mathematik,<br>
Volume 2, 1960, pages 84-90.
</li>
<li>
John Halton, GB Smith,<br>
Algorithm 247: Radical-Inverse Quasi-Random Point Sequence,<br>
Communications of the ACM,<br>
Volume 7, 1964, pages 701-702.
</li>
<li>
Ladislav Kocis, William Whiten,<br>
Computational Investigations of Low-Discrepancy Sequences,<br>
ACM Transactions on Mathematical Software,<br>
Volume 23, Number 2, 1997, pages 266-294.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "halton.f90">halton.f90</a>, the source code.
</li>
<li>
<a href = "halton.sh">halton.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "halton_prb.f90">halton_prb.f90</a>, a sample problem.
</li>
<li>
<a href = "halton_prb.sh">halton_prb.sh</a>,
commands to compile, link and run the sample problem.
</li>
<li>
<a href = "halton_prb_output.txt">halton_prb_output.txt</a>,
the output file.
</li>
<li>
<a href = "../../datasets/halton/halton_02_00010.txt">
halton_02_00010.txt</a>,
a Halton datafile created by the sample problem.
</li>
<li>
<a href = "../../datasets/halton/halton_02_00010.png">
halton_02_00010.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
the dataset.
</li>
<li>
<a href = "../../datasets/halton/halton_02_00100.txt">
halton_02_00100.txt</a>,
a Halton datafile created by the sample problem.
</li>
<li>
<a href = "../../datasets/halton/halton_02_00100.png">
halton_02_00100.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
the dataset.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>ARC_COSINE</b> computes the arc cosine function, with argument truncation.
</li>
<li>
<b>ATAN4</b> computes the inverse tangent of the ratio Y / X.
</li>
<li>
<b>GET_SEED</b> returns a seed for the random number generator.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>HALHAM_LEAP_CHECK</b> checks LEAP for a Halton or Hammersley sequence.
</li>
<li>
<b>HALHAM_N_CHECK</b> checks N for a Halton or Hammersley sequence.
</li>
<li>
<b>HALHAM_DIM_NUM_CHECK</b> checks DIM_NUM for a Halton or Hammersley sequence.
</li>
<li>
<b>HALHAM_SEED_CHECK</b> checks SEED for a Halton or Hammersley sequence.
</li>
<li>
<b>HALHAM_STEP_CHECK</b> checks STEP for a Halton or Hammersley sequence.
</li>
<li>
<b>HALHAM_WRITE</b> writes a Halton or Hammersley subsequence to a file.
</li>
<li>
<b>HALTON</b> computes the next element in a leaped Halton subsequence.
</li>
<li>
<b>HALTON_BASE_GET</b> gets the base vector for a leaped Halton subsequence.
</li>
<li>
<b>HALTON_BASE_CHECK</b> checks BASE for a Halton sequence.
</li>
<li>
<b>HALTON_BASE_SET</b> sets the base vector for a leaped Halton subsequence.
</li>
<li>
<b>HALTON_LEAP_GET</b> gets the leap vector for a leaped Halton subsequence.
</li>
<li>
<b>HALTON_LEAP_SET</b> sets the leap vector for a leaped Halton subsequence.
</li>
<li>
<b>HALTON_MEMORY</b> holds data associated with a leaped Halton subsequence.
</li>
<li>
<b>HALTON_DIM_NUM_GET</b> gets the spatial dimension for a leaped Halton subsequence.
</li>
<li>
<b>HALTON_DIM_NUM_SET</b> sets the spatial dimension for a leaped Halton subsequence.
</li>
<li>
<b>HALTON_SEED_GET</b> gets the seed vector for a leaped Halton subsequence.
</li>
<li>
<b>HALTON_SEED_SET</b> sets the seed vector for a leaped Halton subsequence.
</li>
<li>
<b>HALTON_SEQUENCE</b> computes N elements of a leaped Halton subsequence.
</li>
<li>
<b>HALTON_STEP_GET</b> gets the "step" for a leaped Halton subsequence.
</li>
<li>
<b>HALTON_STEP_SET</b> sets the "step" for a leaped Halton subsequence.
</li>
<li>
<b>I4_TO_HALTON</b> computes one element of a leaped Halton subsequence.
</li>
<li>
<b>I4_TO_HALTON_SEQUENCE</b> computes N elements of a leaped Halton subsequence.
</li>
<li>
<b>I4VEC_TRANSPOSE_PRINT</b> prints an I4VEC "transposed".
</li>
<li>
<b>PRIME</b> returns any of the first PRIME_MAX prime numbers.
</li>
<li>
<b>U1_TO_SPHERE_UNIT_2D</b> maps a point in the unit interval to the unit circle.
</li>
<li>
<b>U2_TO_BALL_UNIT_2D</b> maps points from the unit box to the unit ball in 2D.
</li>
<li>
<b>U2_TO_SPHERE_UNIT_3D</b> maps a point in the unit box onto the unit sphere in 3D.
</li>
<li>
<b>U3_TO_BALL_UNIT_3D</b> maps points from the unit box to the unit ball in 3D.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</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 December 2010.
</i>
<!-- John Burkardt -->
</body>
</html>