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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 MATLAB library which
computes elements of a Hammersley quasirandom sequence.
</p>
<p>
<b>HAMMERSLEY</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
DIM_NUM-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:DIM_NUM) + STEP * LEAP(1:DIM_NUM).
</pre>
</p>
<p>
The arguments that the user may set include:
<ul>
<li>
DIM_NUM, the spatial dimension, <br>
default: DIM_NUM = 1, <br>
required: 1 <= DIM_NUM;
</li>
<li>
STEP, the subsequence index.<br>
default: STEP = 0,<br>
required: 0 <= STEP.
</li>
<li>
SEED(1:DIM_NUM), the Hammersley sequence index corresponding
to STEP = 0.<br>
default: SEED(1:DIM_NUM) = (0, 0, ... 0),<br>
required: 0 <= SEED(1:DIM_NUM);
</li>
<li>
LEAP(1:DIM_NUM), the succesive jumps in the Hammersley sequence.<br>
default: LEAP(1:DIM_NUM) = (1, 1, ..., 1).<br>
required: 1 <= LEAP(1:DIM_NUM).
</li>
<li>
BASE(1:DIM_NUM), the Hammersley bases.<br>
default: BASE(1:DIM_NUM) = (2, 3, 5, 7, 11... ),<br>
or (-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 DIM_NUM-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 = "../../m_src/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 DIM_NUM-dimensional
Hammersley sequence, you get the standard
<a href = "../../m_src/halton/halton.html">Halton sequence</a> of dimension
DIM_NUM-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 DIM_NUM = 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 = "../../m_src/cvt/cvt.html">
CVT</a>,
a MATLAB library which
computes elements of a Centroidal Voronoi Tessellation.
</p>
<p>
<a href = "../../m_src/faure/faure.html">
FAURE</a>,
a MATLAB library which
computes elements of a Faure quasirandom sequence.
</p>
<p>
<a href = "../../m_src/grid/grid.html">
GRID</a>,
a MATLAB library which
computes elements of a grid sequence.
</p>
<p>
<a href = "../../m_src/halton/halton.html">
HALTON</a>,
a MATLAB library which
computes elements of a Halton quasirandom sequence.
</p>
<p>
<a href = "../../m_src/hammersley_dataset/hammersley_dataset.html">
HAMMERSLEY_DATASET</a>,
a MATLAB program which
creates a Hammersley sequence and writes it to a file.
</p>
<p>
<a href = "../../m_src/hex_grid/hex_grid.html">
HEX_GRID</a>,
a MATLAB library which
computes elements of a hexagonal grid dataset.
</p>
<p>
<a href = "../../m_src/ihs/ihs.html">
IHS</a>,
a MATLAB library which
computes elements of an improved distributed Latin hypercube dataset.
</p>
<p>
<a href = "../../m_src/latin_center/latin_center.html">
LATIN_CENTER</a>,
a MATLAB library which
computes elements of a Latin Hypercube dataset, choosing center points.
</p>
<p>
<a href = "../../m_src/latin_edge/latin_edge.html">
LATIN_EDGE</a>,
a MATLAB library which
computes elements of a Latin Hypercube dataset, choosing edge points.
</p>
<p>
<a href = "../../m_src/latin_random/latin_random.html">
LATIN_RANDOM</a>,
a MATLAB library which
computes elements of a Latin Hypercube dataset, choosing points at random.
</p>
<p>
<a href = "../../m_src/lattice_rule/lattice_rule.html">
LATTICE_RULE</a>,
a MATLAB library which
approximates multidimensional integrals using lattice rules.
</p>
<p>
<a href = "../../m_src/lcvt/lcvt.html">
LCVT</a>,
a MATLAB library which
computes a latinized Centroidal Voronoi Tessellation.
</p>
<p>
<a href = "../../m_src/niederreiter2/niederreiter2.html">
NIEDERREITER2</a>,
a MATLAB library which
computes elements of a Niederreiter quasirandom sequence using base 2.
</p>
<p>
<a href = "../../m_src/normal/normal.html">
NORMAL</a>,
a MATLAB library which
computes elements of a sequence of pseudorandom normally distributed values.
</p>
<p>
<a href = "../../m_src/sobol/sobol.html">
SOBOL</a>,
a MATLAB library which
computes elements of a Sobol quasirandom sequence.
</p>
<p>
<a href = "../../m_src/uniform/uniform.html">
UNIFORM</a>,
a MATLAB library which
computes elements of a uniform pseudorandom sequence.
</p>
<p>
<a href = "../../m_src/van_der_corput/van_der_corput.html">
VAN_DER_CORPUT</a>,
a MATLAB library which
computes elements of a 1D van der Corput sequence.
</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 = "arc_cosine.m">arc_cosine.m</a>,
computes the arc cosine function, with argument truncation.
</li>
<li>
<a href = "atan4.m">atan4.m</a>,
computes the inverse tangent of the ratio Y / X.
</li>
<li>
<a href = "get_seed.m">get_seed.m</a>,
returns a random seed for the random number generator.
</li>
<li>
<a href = "halham_dim_num_check.m">halham_dim_num_check.m</a>,
checks DIM_NUM for a Halton or Hammersley sequence.
</li>
<li>
<a href = "halham_leap_check.m">halham_leap_check.m</a>,
checks LEAP for a Halton or Hammersley sequence.
</li>
<li>
<a href = "halham_n_check.m">halham_n_check.m</a>,
checks N for a Halton or Hammersley sequence.
</li>
<li>
<a href = "halham_seed_check.m">halham_seed_check.m</a>,
checks SEED for a Halton or Hammersley sequence.
</li>
<li>
<a href = "halham_step_check.m">halham_step_check.m</a>,
checks STEP for a Halton or Hammersley sequence.
</li>
<li>
<a href = "halham_write.m">halham_write.m</a>,
writes a Halton or Hammersley sequence to a file.
</li>
<li>
<a href = "hammersley.m">hammersley.m</a>,
computes the next element of the
Hammersley sequence in the current base.
</li>
<li>
<a href = "hammersley_base_check.m">hammersley_base_check.m</a>,
checks BASE for a Hammersley sequence.
</li>
<li>
<a href = "hammersley_base_get.m">hammersley_base_get.m</a>,
returns the current base.
</li>
<li>
<a href = "hammersley_base_set.m">hammersley_base_set.m</a>,
sets the current base.
</li>
<li>
<a href = "hammersley_dim_num_get.m">hammersley_dim_num_get.m</a>,
returns the current dimension.
</li>
<li>
<a href = "hammersley_dim_num_set.m">hammersley_dim_num_set.m</a>,
sets the current dimension.
</li>
<li>
<a href = "hammersley_leap_get.m">hammersley_leap_get.m</a>,
returns the leap vector.
</li>
<li>
<a href = "hammersley_leap_set.m">hammersley_leap_set.m</a>,
sets the leap vector.
</li>
<li>
<a href = "hammersley_seed_get.m">hammersley_seed_get.m</a>,
returns the current seed.
</li>
<li>
<a href = "hammersley_seed_set.m">hammersley_seed_set.m</a>,
sets the current seed.
</li>
<li>
<a href = "hammersley_sequence.m">hammersley_sequence.m</a>,
computes the next N elements of the Hammersley sequence,
starting at the current seed, in the current base.
</li>
<li>
<a href = "hammersley_step_get.m">hammersley_step_get.m</a>,
returns the step of the leaped Hammersley subsequence.
</li>
<li>
<a href = "hammersley_step_set.m">hammersley_step_set.m</a>,
sets the step of the leaped Hammersley subsequence.
</li>
<li>
<a href = "i4_to_hammersley.m">i4_to_hammersley.m</a>,
computes the SEED-th element of the
Hammersley sequence with the given BASE vector.
</li>
<li>
<a href = "i4_to_hammersley_sequence.m">
i4_to_hammersley_sequence.m</a>,
computes the next N elements, starting at SEED, of the
Hammersley sequence with the given BASE vector.
</li>
<li>
<a href = "i4vec_transpose_print.m">i4vec_transpose_print.m</a>,
prints an integer vector "transposed".
</li>
<li>
<a href = "prime.m">prime.m</a>,
returns any of the first PRIME_MAX prime numbers.
</li>
<li>
<a href = "s_len_trim.m">s_len_trim.m</a>,
returns the length of a string to the last nonblank.
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>,
prints the current YMDHMS date as a timestamp.
</li>
<li>
<a href = "u1_to_sphere_unit_2d.m">u1_to_sphere_unit_2d.m</a>,
maps a point in the unit interval onto the unit sphere in 2D.
</li>
<li>
<a href = "u2_to_ball_unit_2d.m">u2_to_ball_unit_2d.m</a>,
maps a point in the unit box to the unit ball in 2D.
</li>
<li>
<a href = "u2_to_sphere_unit_3d.m">u2_to_sphere_unit_3d.m</a>,
maps a point in the unit box onto the unit sphere in 3D.
</li>
<li>
<a href = "u3_to_ball_unit_3d.m">u3_to_ball_unit_3d.m</a>,
maps a point in the unit box to the unit ball in 3D.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "hammersley_test.m">
hammersley_test.m</a>, runs all the tests.
</li>
<li>
<a href = "hammersley_test_output.txt">
hammersley_test_output.txt</a>,
the output file.
</li>
<li>
<a href = "hammersley_test01.m">
hammersley_test01.m</a>, tests I4_TO_HAMMERSLEY_SEQUENCE.
</li>
<li>
<a href = "hammersley_test02.m">
hammersley_test02.m</a>, tests I4_TO_HAMMERSLEY_SEQUENCE.
</li>
<li>
<a href = "hammersley_test03.m">
hammersley_test03.m</a>, tests I4_TO_HAMMERSLEY_SEQUENCE.
</li>
<li>
<a href = "hammersley_test04.m">
hammersley_test04.m</a>, tests I4_TO_HAMMERSLEY_SEQUENCE.
</li>
<li>
<a href = "hammersley_test05.m">
hammersley_test05.m</a>, tests I4_TO_HAMMERSLEY_SEQUENCE.
</li>
<li>
<a href = "hammersley_test06.m">
hammersley_test06.m</a>, tests HAMMERSLEY_SEQUENCE.
</li>
<li>
<a href = "hammersley_test07.m">
hammersley_test07.m</a>, tests HAMMERSLEY_WRITE.
</li>
<li>
<a href = "hammersley_test08.m">
hammersley_test08.m</a>, tests I4_TO_HAMMERSLEY_SEQUENCE.
</li>
<li>
<a href = "../../datasets/hammersley/hammersley_04_00010.txt">
hammersley_04_00010.txt</a>, a Hammersley dataset written to
a file by one of the tests.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 05 May 2008.
</i>
<!-- John Burkardt -->
</body>
</html>