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<html>
<head>
<title>
HAMMERSLEY - The Hammersley Quasirandom Sequence
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
HAMMERSLEY <br> The Hammersley Quasirandom Sequence
</h1>
<hr>
<p>
<b>HAMMERSLEY</b>
is a C++ library which
computes the Hammersley quasirandom sequence.
</p>
<p>
<b>HAMMERSLEY</b> includes several subroutines 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_HAMMERSLEY</b>, for one element of a sequence;
</li>
<li>
<b>I4_TO_HAMMERSLEY_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>HAMMERSLEY</b>, for one element of a sequence;
</li>
<li>
<b>HAMMERSLEY_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>
Routines in this library select elements of a "leaped" subsequence of
the Hammersley sequence. The subsequence elements are indexed by a
quantity called STEP, which starts at 0. The STEP-th subsequence
element is simply the Hammersley 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 Hammersley 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 Hammersley sequence.<br>
default: LEAP(1:NDIM) = (1, 1, ..., 1).<br>
required: 1 <= LEAP(1:NDIM).
</li>
<li>
BASE(1:NDIM), the Hammersley bases.<br>
default: BASE(1:NDIM) = (2, 3, 5, 7, 11... ), or<br>
(-N, 2, 3, 5, 7, 11,...) if <b>N</b> is known,<br>
required: 1 < BASE(I) for any van der Corput dimension I, or
BASE(I) < 0 to generate the fractional sequence J/|BASE(I)|.
</li>
</ul>
</p>
<p>
In the standard NDIM-dimensional Hammersley sequence, it is assumed
that <b>N</b>, the number of values to be generated, is known
beforehand. The first dimension of entries in the sequence
will have the form <b>J/N</b> for J from 1 to N. The remaining
dimensions are computed using the 1-dimensional
<a href = "../van_der_corput/van_der_corput.html">
van der Corput sequence</a>, using successive primes as bases.
</p>
<p>
In a generalized Hammersley sequence, each coordinate is allowed
to be a fractional or van der Corput sequence. For any fractional
sequence, the denominator is arbitrary. However, it is extremely
desirable that the values in all coordinates stay between 0 and 1.
This happens automatically for any van der Corput sequence, but
for fractional sequences, this criterion is enforced using an
appropriate <i>modulus</i> function. The consequence is that if
you specify a small "base" for a fractional sequence, your sequence
will soon wrap around and you will get repeated values.
</p>
<p>
If you drop the first dimension of the standard NDIM-dimensional
Hammersley sequence, you get the standard
<a href = "../halton/halton.html">Halton sequence</a> of dimension NDIM-1.
</p>
<p>
The standard Hammersley sequence has slightly better dispersion
properties than the standard Halton sequence. However, 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 <b>N</b> = 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 Hammersley 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 Hammersley 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>
<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>HAMMERSLEY</b> is available in
<a href = "../../cpp_src/hammersley/hammersley.html">a C++ version</a> and
<a href = "../../f_src/hammersley/hammersley.html">a FORTRAN90 version</a> and
<a href = "../../m_src/hammersley/hammersley.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/box_behnken/box_behnken.html">
BOX_BEHNKEN</a>,
a C++ library which
computes a Box-Behnken design,
that is, a set of arguments to sample the behavior
of a function of multiple parameters;
</p>
<p>
<a href = "../../cpp_src/cvt/cvt.html">
CVT</a>,
a C++ library which
computes points in
a Centroidal Voronoi Tessellation.
</p>
<p>
<a href = "../../cpp_src/faure/faure.html">
FAURE</a>,
a C++ library which
computes Faure
sequences.
</p>
<p>
<a href = "../../cpp_src/grid/grid.html">
GRID</a>,
a C++ library which
computes points on a grid.
</p>
<p>
<a href = "../../cpp_src/halton/halton.html">
HALTON</a>,
a C++ library which
computes Halton
sequences.
</p>
<p>
<a href = "../../cpp_src/hammersley_dataset/hammersley_dataset.html">
HAMMERSLEY_DATASET</a>,
a C++ program which
computes Hammersley datasets.
</p>
<p>
<a href = "../../cpp_src/hex_grid/hex_grid.html">
HEX_GRID</a>,
a C++ library which
computes sets of points in a 2D hexagonal grid.
</p>
<p>
<a href = "../../cpp_src/ihs/ihs.html">
IHS</a>,
a C++ library which
computes improved Latin Hypercube datasets.
</p>
<p>
<a href = "../../cpp_src/latin_center/latin_center.html">
LATIN_CENTER</a>,
a C++ library which
computes Latin square data choosing the center value.
</p>
<p>
<a href = "../../cpp_src/latin_edge/latin_edge.html">
LATIN_EDGE</a>,
a C++ library which
computes Latin square data choosing the edge value.
</p>
<p>
<a href = "../../cpp_src/latin_random/latin_random.html">
LATIN_RANDOM</a>,
a C++ library which
computes Latin square data choosing a random value in the square.
</p>
<p>
<a href = "../../cpp_src/niederreiter2/niederreiter2.html">
NIEDERREITER2</a>,
a C++ library which
computes Niederreiter sequences with base 2.
</p>
<p>
<a href = "../../cpp_src/sobol/sobol.html">
SOBOL</a>,
a C++ library which
computes Sobol sequences.
</p>
<p>
<a href = "../../cpp_src/uniform/uniform.html">
UNIFORM</a>,
a C++ library which
computes uniform random values.
</p>
<p>
<a href = "../../cpp_src/van_der_corput/van_der_corput.html">
VAN_DER_CORPUT</a>,
a C++ library which
computes van der Corput sequences.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
John Hammersley,<br>
Monte Carlo methods for solving multivariable problems,<br>
Proceedings of the New York Academy of Science,<br>
Volume 86, 1960, pages 844-874.
</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 = "hammersley.cpp">hammersley.cpp</a>, the source code.
</li>
<li>
<a href = "hammersley.hpp">hammersley.hpp</a>, the include file.
</li>
<li>
<a href = "hammersley.sh">hammersley.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "hammersley_prb.cpp">hammersley_prb.cpp</a>,
a sample problem.
</li>
<li>
<a href = "hammersley_prb.sh">hammersley_prb.sh</a>,
commands to compile, link and run the sample problem.
</li>
<li>
<a href = "hammersley_prb_output.txt">hammersley_prb_output.txt</a>,
the output file.
</li>
<li>
<a href = "../../datasets/hammersley/hammersley_04_00010.txt">
hammersley_04_00010.txt</a>,
a Hammersley datafile created by the sample problem.
</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>DIGIT_TO_CH</b> returns the base 10 digit character corresponding to a digit.
</li>
<li>
<b>GET_SEED</b> returns a random seed for the random number generator.
</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 dataset to a file.
</li>
<li>
<b>HAMMERSLEY</b> computes the next element in a leaped Hammersley subsequence.
</li>
<li>
<b>HAMMERSLEY_BASE_CHECK</b> checks BASE for a Hammersley sequence.
</li>
<li>
<b>HAMMERSLEY_BASE_GET</b> gets the base vector for a leaped Hammersley subsequence.
</li>
<li>
<b>HAMMERSLEY_BASE_SET</b> sets the base vector for a leaped Hammersley subsequence.
</li>
<li>
<b>HAMMERSLEY_DIM_NUM_GET</b> gets the spatial dimension for a leaped Hammersley subsequence.
</li>
<li>
<b>HAMMERSLEY_DIM_NUM_SET</b> sets the spatial dimension for a leaped Hammersley subsequence.
</li>
<li>
<b>HAMMERSLEY_LEAP_GET</b> gets the leap vector for a leaped Hammersley subsequence.
</li>
<li>
<b>HAMMERSLEY_LEAP_SET</b> sets the leap vector for a leaped Hammersley subsequence.
</li>
<li>
<b>HAMMERSLEY_SEED_GET</b> gets the seed vector for a leaped Hammersley subsequence.
</li>
<li>
<b>HAMMERSLEY_SEED_SET</b> sets the seed vector for a leaped Hammersley subsequence.
</li>
<li>
<b>HAMMERSLEY_SEQUENCE</b> computes N elements in an DIM_NUM-dimensional Hammersley sequence.
</li>
<li>
<b>HAMMERSLEY_STEP_GET</b> gets the step for the leaped Hammersley subsequence.
</li>
<li>
<b>HAMMERSLEY_STEP_SET</b> sets the step for a leaped Hammersley subsequence.
</li>
<li>
<b>I4_LOG_10</b> returns the whole part of the logarithm base 10 of an integer.
</li>
<li>
<b>I4_MIN</b> returns the smaller of two I4's.
</li>
<li>
<b>I4_TO_HAMMERSLEY</b> computes one element of a leaped Hammersley subsequence.
</li>
<li>
<b>I4_TO_HAMMERSLEY_SEQUENCE</b> computes N elements of a leaped Hammersley subsequence.
</li>
<li>
<b>I4_TO_S</b> converts an integer to a string.
</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>R8_EPSILON</b> returns the R8 roundoff unit.
</li>
<li>
<b>R8VEC_DOT_PRODUCT</b> returns the dot product of two R8VEC's.
</li>
<li>
<b>R8VEC_NORM_L2</b> returns the L2 norm of an 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>TIMESTRING</b> returns the current YMDHMS date as a string.
</li>
<li>
<b>U1_TO_SPHERE_UNIT_2D</b> maps a point in the unit interval onto the circle in 2D.
</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 to 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>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 20 October 2006.
</i>
<!-- John Burkardt -->
</body>
</html>