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<html>
<head>
<title>
RANDOM_DATA - Generation of random data
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
RANDOM_DATA <br> Generation of random data
</h1>
<hr>
<p>
<b>RANDOM_DATA</b>
is a C++ library which
uses a random number generator (RNG) to sample points for
various probability distributions, spatial dimensions, and geometries,
including the M-dimensional cube, ellipsoid, simplex and sphere.
</p>
<p>
Most of these routines assume that there is an available source
of pseudorandom numbers, distributed uniformly in the unit
interval [0,1]. In this package, that role is played by the
routine <b>R8_UNIFORM_01</b>, which allows us some portability.
We can get the same results in C, FORTRAN or MATLAB, for instance.
In general, however, it would be more efficient to use the
language-specific random number generator for this purpose.
</p>
<p>
If we have a source of pseudorandom values in [0,1], it's trivial
to generate pseudorandom points in any line segment; it's easy to
take pairs of pseudorandom values to sample a square, or triples to
sample a cube. It's easy to see how to deal with square region that
is translated from the origin, or scaled by different amounts in
either axis, or given a rigid rotation. The same simple transformations
can be applied to higher dimensional cubes, without giving us any
concern.
</p>
<p>
For all these simple shapes, which are just generalizations of
a square, we can easily see how to generate sample points that
we can guarantee will lie inside the region; in most cases, we
can also guarantee that these points will tend to be <i>uniformly
distributed</i>, that is, every subregion can expect to contain
a number of points proportional to its share of the total area.
</p>
<p>
However, we will <b>not</b> achieve uniform distribution in the
simple case of a rectangle of nonequal sides <b>[0,A]</b> x <b>[0,B]</b>,
if we naively scale the random values <b>(u1,u2)</b> to
<b>(A*u1,B*u2)</b>. In that case, the expected point density of
a wide, short region will differ from that of a narrow tall region.
The absence of uniformity is most obvious if the points are plotted.
</p>
<p>
If you realize that uniformity is desirable, and easily lost,
it is possible to adjust the approach so that rectangles are
properly handled.
</p>
<p>
But rectangles are much too simple. We are interested in circles,
triangles, and other shapes. Once the geometry of the region
becomes more "interesting", there are two common ways to continue.
</p>
<p>
In the <i>acceptance-rejection method</i>,
uniform points are generated in a superregion that encloses the
region. Then, points that do not lie within the region are rejected.
More points are generated until enough have been accepted to satisfy the
needs. If a circle was the region of interest, for instance, we
could surround it with a box, generate points in the box, and throw
away those points that don't actually lie in the circle. The resulting
set of samples will be a uniform sampling of the circle.
</p>
<p>
In the <i>direct mapping</i> method, a formula or mapping
is determined so that each time a set of values is taken from
the pseudorandom number generator, it is guaranteed to correspond
to a point in the region. For the circle problem, we can use
one uniform random number to choose an angle between 0 and 2 PI,
the other to choose a radius. (The radius must be chosen in
an appropriate way to guarantee uniformity, however.) Thus,
every time we input two uniform random values, we get a pair
(R,T) that corresponds to a point in the circle.
</p>
<p>
The acceptance-rejection method can be simple to program, and
can handle arbitrary regions. The direct mapping method is
less sensitive to variations in the aspect ratio of a region
and other irregularities. However, direct mappings are only
known for certain common mathematical shapes.
</p>
<p>
Points may also be generated according to a nonuniform density.
This creates an additional complication in programming. However,
there are some cases in which it is possible to use direct mapping
to turn a stream of scalar uniform random values into a set of
multivariate data that is governed by a normal distribution.
</p>
<p>
Another way to generate points replaces the uniform pseudorandom number
generator by a <i>quasirandom number generator</i>. The main difference
is that successive elements of a quasirandom sequence may be highly
correlated (bad for certain Monte Carlo applications) but will tend
to cover the region in a much more regular way than pseudorandom
numbers. Any process that uses uniform random numbers to carry out
sampling can easily be modified to do the same sampling with
a quasirandom sequence like the Halton sequence, for instance.
</p>
<p>
The library includes a routine that can write the resulting
data points to a file.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files 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>RANDOM_DATA</b> is available in
<a href = "../../c_src/random_data/random_data.html">a C version</a> and
<a href = "../../cpp_src/random_data/random_data.html">a C++ version</a> and
<a href = "../../f77_src/random_data/random_data.html">a FORTRAN77 version</a> and
<a href = "../../f_src/random_data/random_data.html">a FORTRAN90 version</a> and
<a href = "../../m_src/random_data/random_data.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/asa183/asa183.html">
ASA183</a>,
a C++ library which
implements the Wichman-Hill pseudorandom number generator.
</p>
<p>
<a href = "../../cpp_src/discrete_pdf_sample_2d/discrete_pdf_sample_2d.html">
DISCRETE_PDF_SAMPLE_2D</a>,
a C++ program which
demonstrates how to construct a Probability Density Function (PDF)
from a table of sample data, and then to use that PDF to create new samples.
</p>
<p>
<a href = "../../c_src/rbox/rbox.html">
RBOX</a>,
a C program which
produces random data from a number of regions.
</p>
<p>
<a href = "../../cpp_src/rsites/rsites.html">
RSITES</a>,
a C++ program which
produces random data in an M-dimensional box.
</p>
<p>
<a href = "../../cpp_src/sphere_quad/sphere_quad.html">
SPHERE_QUAD</a>,
a C++ library which
approximates an integral over the surface of the unit sphere
by applying a triangulation to the surface;
</p>
<p>
<a href = "../../cpp_src/tetrahedron_monte_carlo/tetrahedron_monte_carlo.html">
TETRAHEDRON_MONTE_CARLO</a>,
a C++ program which
uses the Monte Carlo method to estimate integrals over a tetrahedron.
</p>
<p>
<a href = "../../datasets/tetrahedron_samples/tetrahedron_samples.html">
TETRAHEDRON_SAMPLES</a>,
a dataset directory which
contains examples of sets of sample points from the unit tetrahedron.
</p>
<p>
<a href = "../../cpp_src/triangle_histogram/triangle_histogram.html">
TRIANGLE_HISTOGRAM</a>,
a C++ program which
computes histograms of data on the unit triangle.
</p>
<p>
<a href = "../../cpp_src/triangle_monte_carlo/triangle_monte_carlo.html">
TRIANGLE_MONTE_CARLO</a>,
a C++ program which
uses the Monte Carlo method to estimate integrals over a triangle.
</p>
<p>
<a href = "../../datasets/triangle_samples/triangle_samples.html">
TRIANGLE_SAMPLES</a>,
a dataset directory which
contains examples of sets of sample points from the unit triangle.
</p>
<p>
<a href = "../../cpp_src/uniform/uniform.html">
UNIFORM</a>,
a C++ library which
samples the uniform random distribution.
</p>
<p>
<a href = "../../m_src/xyz_display/xyz_display.html">
XYZ_DISPLAY</a>,
a MATLAB program which
reads XYZ information defining points in 3D,
and displays an image in the MATLAB graphics window.
</p>
<p>
<a href = "../../cpp_src/xyz_display_opengl/xyz_display_opengl.html">
XYZ_DISPLAY_OPENGL</a>,
a C++ program which
reads XYZ information defining points in 3D,
and displays an image using OpenGL.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Milton Abramowitz, Irene Stegun,<br>
Handbook of Mathematical Functions,<br>
National Bureau of Standards, 1964,<br>
ISBN: 0-486-61272-4,<br>
LC: QA47.A34.
</li>
<li>
James Arvo,<br>
Stratified sampling of spherical triangles,<br>
Computer Graphics Proceedings, Annual Conference Series, <br>
ACM SIGGRAPH '95, pages 437-438, 1995.
</li>
<li>
Gerard Bashein, Paul Detmer,<br>
Centroid of a Polygon,<br>
in Graphics Gems IV,<br>
edited by Paul Heckbert,<br>
AP Professional, 1994,<br>
ISBN: 0123361559,<br>
LC: T385.G6974.
</li>
<li>
Paul Bratley, Bennett Fox, Linus Schrage,<br>
A Guide to Simulation,<br>
Second Edition,<br>
Springer, 1987,<br>
ISBN: 0387964673,<br>
LC: QA76.9.C65.B73.
</li>
<li>
Russell Cheng,<br>
Random Variate Generation,<br>
in Handbook of Simulation,<br>
edited by Jerry Banks,<br>
Wiley, 1998,<br>
ISBN: 0471134031,<br>
LC: T57.62.H37.
</li>
<li>
Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart,<br>
LINPACK User's Guide,<br>
SIAM, 1979,<br>
ISBN13: 978-0-898711-72-1,<br>
LC: QA214.L56.
</li>
<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, Number 1, December 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, Number 12, December 1964, pages 701-702.
</li>
<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, June 1997, pages 266-294.
</li>
<li>
Pierre LEcuyer,<br>
Random Number Generation,<br>
in Handbook of Simulation,<br>
edited by Jerry Banks,<br>
Wiley, 1998,<br>
ISBN: 0471134031,<br>
LC: T57.62.H37.
</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>
Reuven Rubinstein,<br>
Monte Carlo Optimization, Simulation and Sensitivity of
Queueing Networks,<br>
Krieger, 1992,<br>
ISBN: 0894647644,<br>
LC: QA298.R79.
</li>
<li>
Peter Shirley,<br>
Nonuniform Random Point Sets Via Warping,<br>
in Graphics Gems III,<br>
edited by David Kirk,<br>
Academic Press, 1992,<br>
ISBN: 0124096735,<br>
LC: T385.G6973
</li>
<li>
Greg Turk,<br>
Generating Random Points in a Triangle,<br>
in Graphics Gems I,<br>
edited by Andrew Glassner,<br>
AP Professional, 1990,<br>
ISBN: 0122861663,<br>
LC: T385.G697
</li>
<li>
Daniel Zwillinger, editor,<br>
CRC Standard Mathematical Tables and Formulae,<br>
30th Edition,<br>
CRC Press, 1996,<br>
ISBN: 0-8493-2479-3,<br>
LC: QA47.M315.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "random_data.cpp">random_data.cpp</a>,
the source code.
</li>
<li>
<a href = "random_data.hpp">random_data.hpp</a>,
the include file.
</li>
<li>
<a href = "random_data.sh">random_data.sh</a>,
commands to compile the source code
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "random_data_prb.cpp">random_data_prb.cpp</a>,
the sample calling program.
</li>
<li>
<a href = "random_data_prb.sh">random_data_prb.sh</a>,
commands to compile, link and run the sample calling program.
</li>
<li>
<a href = "random_data_prb_output.txt">random_data_prb_output.txt</a>,
output from the sample calling program.
</li>
</ul>
</p>
<p>
The sample calling program generates sets of points:
<ul>
<li>
<a href = "bad_in_tetrahedron.txt">bad_in_tetrahedron.txt</a>,
points in the unit tetrahedron, not uniformly chosen.
</li>
<li>
<a href = "bad_in_triangle.txt">bad_in_triangle.txt</a>,
points in the unit triangle, not uniformly chosen.
</li>
<li>
<a href = "brownian.txt">brownian.txt</a>, Brownian motion.
</li>
<li>
<a href = "grid_in_cube01.txt">grid_in_cube01.txt</a>,
grid points in the unit hypercube, with CENTER = 1.
</li>
<li>
<a href = "halton_in_circle01_accept.txt">
halton_in_circle01_accept.txt</a>,
Halton points in the unit circle, acceptance/rejection.
</li>
<li>
<a href = "halton_in_circle01_map.txt">
halton_in_circle01_map.txt</a>,
Halton points in the unit circle, direct mapping.
</li>
<li>
<a href = "halton_in_cube01.txt">halton_in_cube01.txt</a>,
Halton points in the unit hypercube.
</li>
<li>
<a href = "hammersley_in_cube01.txt">
hammersley_in_cube01.txt</a>, Hammersley points.
</li>
<li>
<a href = "normal.txt">normal.txt</a>, normal points, with
strong correlation between the two coordinates.
</li>
<li>
<a href = "normal_circular.txt">normal_circular.txt</a>,
circular normal points.
</li>
<li>
<a href = "normal_simple.txt">normal_simple.txt</a>,
normal points in which there is no correlation between the
X and Y coordinates.
</li>
<li>
<a href = "polygon_vertices.txt">polygon_vertices.txt</a>,
the vertices of a polygon to be filled by random points.
</li>
<li>
<a href = "uniform_in_annulus.txt">
uniform_in_annulus.txt</a>,
uniform points in an annulus, mapping.
</li>
<li>
<a href = "uniform_in_annulus_accept.txt">
uniform_in_annulus_accept.txt</a>,
uniform points in an annulus, acceptance/rejection.
</li>
<li>
<a href = "uniform_in_annulus_sector.txt">
uniform_in_annulus_sector.txt</a>,
uniform points in an annulus sector.
</li>
<li>
<a href = "uniform_in_cube01.txt">uniform_in_cube01.txt</a>,
uniform points in the unit hypercube.
</li>
<li>
<a href = "uniform_in_circle01_map.txt">
uniform_in_circle01_map.txt</a>,
uniform points in the unit circle.
</li>
<li>
<a href = "uniform_in_ellipsoid_map.txt">
uniform_in_ellipsoid_map.txt</a>,
uniform points in an ellipsoid.
</li>
<li>
<a href = "uniform_in_parallelogram_map.txt">
uniform_in_parallelogram_map.txt</a>,
uniform points in a parallelogram.
</li>
<li>
<a href = "uniform_in_polygon_map.txt">
uniform_in_polygon_map.txt</a>,
uniform points in a polygon (a star, in this case).
</li>
<li>
<a href = "uniform_in_sector_map.txt">
uniform_in_sector_map.txt</a>,
uniform points in a circular sector.
</li>
<li>
<a href = "uniform_in_simplex01_map.txt">
uniform_in_simplex01_map.txt</a>,
uniform points in the unit simplex.
</li>
<li>
<a href = "uniform_in_sphere01_map.txt">
uniform_in_sphere01_map.txt</a>,
uniform points in the unit sphere.
</li>
<li>
<a href = "uniform_in_triangle_map1.txt">
uniform_in_triangle_map2.txt</a>,
uniform points in an arbitrary triangle, Turk method 1.
</li>
<li>
<a href = "uniform_in_triangle_map2.txt">
uniform_in_triangle_map2.txt</a>,
uniform points in an arbitrary triangle, Turk method 2.
</li>
<li>
<a href = "uniform_in_triangle01_map.txt">
uniform_in_triangle01_map.txt</a>,
uniform points in the unit triangle.
</li>
<li>
<a href = "uniform_on_cube01.txt">
uniform_on_cube01.txt</a>,
uniform points on the unit cube.
</li>
<li>
<a href = "uniform_on_ellipsoid_map.txt">
uniform_on_ellipsoid_map.txt</a>,
uniform points on an ellipsoid.
</li>
<li>
<a href = "uniform_on_hemisphere01_phong.txt">
uniform_on_hemisphere01_phong.txt</a>,
uniform random points on a hemisphere, Phong distribution.
</li>
<li>
<a href = "uniform_on_simplex01_map.txt">
uniform_on_simplex01_map.txt</a>,
uniform points on the unit simplex.
</li>
<li>
<a href = "uniform_on_sphere01_map.txt">
uniform_on_sphere01_map.txt</a>,
uniform points on the unit sphere in 2D.
</li>
<li>
<a href = "uniform_on_sphere01_patch_tp.txt">
uniform_on_sphere01_patch_tp.txt</a>,
uniform random points on an TP (theta,phi) "patch" of the unit sphere in 3D.
</li>
<li>
<a href = "uniform_on_sphere01_patch_xyz.txt">
uniform_on_sphere01_patch_xyz.txt</a>,
uniform random points on an XYZ "patch" of the unit sphere in 3D.
</li>
<li>
<a href = "uniform_on_sphere01_triangle_xyz.txt">
uniform_on_sphere01_triangle_xyz.txt</a>,
uniform random points on a spherical triangle of the unit sphere in 3D
using XYZ coordinates.
</li>
<li>
<a href = "uniform_walk.txt">uniform_walk.txt</a>,
points on a uniform random walk.
</li>
</ul>
</p>
<p>
A file of commands is provided to simplify the use of PLOT_POINTS:
<ul>
<li>
<a href = "plot_points_input.txt">plot_points_input.txt</a>,
commands to PLOT_POINTS.
</li>
<li>
<a href = "plot_points_output.txt">plot_points_output.txt</a>,
printed output from PLOT_POINTS in response to the commands.
</li>
</ul>
</p>
<p>
PLOT_POINTS makes Encapsulated PostScript
images of the points, in
cases where the data is 2 dimensional. These EPS files are
converted to PNG images
for posting on this web page.
<ul>
<li>
<a href = "brownian.png">brownian.png</a>.
</li>
<li>
<a href = "grid_in_cube01.png">grid_in_cube01.png</a>
</li>
<li>
<a href = "halton_in_circle01_accept.png">
halton_in_circle01_accept.png</a>
</li>
<li>
<a href = "halton_in_circle01_map.png">
halton_in_circle01_map.png</a>
</li>
<li>
<a href = "halton_in_cube01.png">halton_in_cube01.png</a>
</li>
<li>
<a href = "hammersley_in_cube01.png">hammersley_in_cube01.png</a>
</li>
<li>
<a href = "normal.png">normal.png</a>
</li>
<li>
<a href = "normal_circular.png">normal_circular.png</a>
</li>
<li>
<a href = "normal_simple.png">normal_simple.png</a>
</li>
<li>
<a href = "polygon_vertices.png">polygon_vertices.png</a>,
</li>
<li>
<a href = "uniform_in_annulus.png">
uniform_in_annulus.png</a>,
</li>
<li>
<a href = "uniform_in_annulus_accept.png">
uniform_in_annulus_accept.png</a>,
</li>
<li>
<a href = "uniform_in_annulus_sector.png">
uniform_in_annulus_sector.png</a>,
</li>
<li>
<a href = "uniform_in_cube01.png">uniform_in_cube01.png</a>,
</li>
<li>
<a href = "uniform_in_circle01_map.png">
uniform_in_circle01_map.png</a>,
</li>
<li>
<a href = "uniform_in_ellipsoid_map.png">
uniform_in_ellipsoid_map.png</a>,
</li>
<li>
<a href = "uniform_in_parallelogram_map.png">
uniform_in_parallelogram_map.png</a>,
</li>
<li>
<a href = "uniform_in_polygon_map.png">
uniform_in_polygon_map.png</a>,
</li>
<li>
<a href = "uniform_in_sector_map.png">
uniform_in_sector_map.png</a>,
</li>
<li>
<a href = "uniform_in_simplex01_map.png">
uniform_in_simplex01_map.png</a>,
</li>
<li>
<a href = "uniform_in_sphere01_map.png">
uniform_in_sphere01_map.png</a>,
</li>
<li>
<a href = "uniform_in_triangle_map1.png">
uniform_in_triangle_map1.png</a>,
</li>
<li>
<a href = "uniform_in_triangle_map2.png">
uniform_in_triangle_map2.png</a>,
</li>
<li>
<a href = "uniform_in_triangle01_map.png">
uniform_in_triangle01_map.png</a>,
</li>
<li>
<a href = "uniform_on_ellipsoid_map.png">
uniform_on_ellipsoid_map.png</a>,
</li>
<li>
<a href = "uniform_on_simplex01_map.png">
uniform_on_simplex01_map.png</a>,
</li>
<li>
<a href = "uniform_on_sphere01_map.png">
uniform_on_sphere01_map.png</a>,
</li>
<li>
<a href = "uniform_on_sphere01_patch_tp.png">
uniform_on_sphere01_patch_tp.png</a>
</li>
<li>
<a href = "uniform_on_sphere01_patch_xyz.png">
uniform_on_sphere01_patch_xyz.png</a>
</li>
<li>
<a href = "uniform_walk.png">uniform_walk.png</a>,
</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>BAD_IN_SIMPLEX01</b> is a "bad" (nonuniform) sampling of the unit simplex.
</li>
<li>
<b>BROWNIAN</b> creates Brownian motion points.
</li>
<li>
<b>DAXPY</b> computes constant times a vector plus a vector.
</li>
<li>
<b>DDOT</b> forms the dot product of two vectors.
</li>
<li>
<b>DGE_MXV</b> multiplies a DGE matrix times a vector.
</li>
<li>
<b>DIRECTION_UNIFORM_ND</b> generates a random direction vector in ND.
</li>
<li>
<b>DPOFA</b> factors a real symmetric positive definite matrix.
</li>
<li>
<b>DPOSL</b> solves a linear system factored by DPOCO or DPOFA.
</li>
<li>
<b>DUT_MXV</b> multiplies an DUT matrix times a vector.
</li>
<li>
<b>GET_SEED</b> returns a random seed for the random number generator.
</li>
<li>
<b>GRID_IN_CUBE01</b> generates a grid dataset in the unit hypercube.
</li>
<li>
<b>GRID_SIDE</b> finds the smallest DIM_NUM dimensional grid containing at least N points.
</li>
<li>
<b>HALHAM_DIM_NUM_CHECK</b> checks DIM_NUM for a Halton or Hammersley sequence.
</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_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>HALTON_BASE_CHECK</b> is TRUE if BASE is legal.
</li>
<li>
<b>HALTON_IN_CIRCLE01_ACCEPT</b> accepts Halton points in a unit circle.
</li>
<li>
<b>HALTON_IN_CIRCLE01_MAP</b> maps Halton points into a unit circle.
</li>
<li>
<b>HALTON_IN_CUBE01</b> generates Halton points in the unit hypercube.
</li>
<li>
<b>HAMMERSLEY_BASE_CHECK</b> is TRUE if BASE is legal.
</li>
<li>
<b>HAMMERSLEY_IN_CUBE01</b> computes Hammersley points in the unit hypercube.
</li>
<li>
<b>I4_FACTORIAL</b> returns N!.
</li>
<li>
<b>I4_MAX</b> returns the maximum of two I4's.
</li>
<li>
<b>I4_MIN</b> returns the smaller of two I4's.
</li>
<li>
<b>I4_MODP</b> returns the nonnegative remainder of I4 division.
</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>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_UNIFORM_AB</b> returns a scaled pseudorandom I4 between A and B.
</li>
<li>
<b>I4VEC_TRANSPOSE_PRINT</b> prints an I4VEC "transposed".
</li>
<li>
<b>KSUB_RANDOM2</b> selects a random subset of size K from a set of size N.
</li>
<li>
<b>NORMAL</b> creates normally distributed points in DIM_NUM space.
</li>
<li>
<b>NORMAL_CIRCULAR</b> creates circularly normal points in 2 space.
</li>
<li>
<b>NORMAL_MULTIVARIATE</b> samples a multivariate normal distribution.
</li>
<li>
<b>NORMAL_SIMPLE</b> creates normally distributed points in DIM_NUM space.
</li>
<li>
<b>POLYGON_CENTROID_2D</b> computes the centroid of a polygon in 2D.
</li>
<li>
<b>PRIME</b> returns any of the first PRIME_MAX prime numbers.
</li>
<li>
<b>R4_ABS</b> returns the absolute value of an R4.
</li>
<li>
<b>R4_NINT</b> returns the nearest integer to an R4.
</li>
<li>
<b>R8_EPSILON</b> returns the round off unit for double precision arithmetic.
</li>
<li>
<b>R8_MAX</b> returns the maximum of two R8's.
</li>
<li>
<b>R8_MIN</b> returns the minimum of two R8's.
</li>
<li>
<b>R8_NINT</b> returns the nearest integer to a double precision real value.
</li>
<li>
<b>R8_NORMAL_01</b> samples the standard normal probability distribution.
</li>
<li>
<b>R8_PI</b> returns the value of PI to 16 digits.
</li>
<li>
<b>R8_UNIFORM_01</b> is a portable pseudorandom number generator.
</li>
<li>
<b>R8MAT_NORMAL_01_NEW</b> returns a unit pseudonormal R8MAT.
</li>
<li>
<b>R8MAT_PRINT</b> prints an R8MAT, with an optional title.
</li>
<li>
<b>R8MAT_PRINT_SOME</b> prints some of an R8MAT.
</li>
<li>
<b>R8MAT_UNIFORM_01_NEW</b> returns a new unit pseudorandom R8MAT.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file with no header.
</li>
<li>
<b>R8VEC_DOT_PRODUCT</b> computes the dot product of a pair of R8VEC's.
</li>
<li>
<b>R8VEC_NORM</b> returns the L2 norm of an R8VEC.
</li>
<li>
<b>R8VEC_NORMAL_01</b> samples the standard normal probability distribution.
</li>
<li>
<b>R8VEC_NORMAL_01_NEW</b> returns a unit pseudonormal R8VEC.
</li>
<li>
<b>R8VEC_PRINT</b> prints an R8VEC.
</li>
<li>
<b>R8VEC_SUM</b> returns the sum of an R8VEC.
</li>
<li>
<b>R8VEC_UNIFORM_01</b> fills a double precision vector with pseudorandom values.
</li>
<li>
<b>R8VEC_UNIFORM_01_NEW</b> returns a new unit pseudorandom R8VEC.
</li>
<li>
<b>R8VEC_ZERO_NEW</b> creates and zeroes an R8VEC.
</li>
<li>
<b>RANDOM_INITIALIZE</b> initializes the RANDOM random number generator.
</li>
<li>
<b>S_LEN_TRIM</b> returns the length of a string to the last nonblank.
</li>
<li>
<b>SCALE_FROM_SIMPLEX01</b> rescales data from a unit to non-unit simplex.
</li>
<li>
<b>SCALE_TO_BALL01</b> translates and rescales data to fit within the unit ball.
</li>
<li>
<b>SCALE_TO_BLOCK01</b> translates and rescales data to fit in the unit block.
</li>
<li>
<b>SCALE_TO_CUBE01</b> translates and rescales data to the unit hypercube.
</li>
<li>
<b>STRI_ANGLES_TO_AREA</b> computes the area of a spherical triangle.
</li>
<li>
<b>STRI_SIDES_TO_ANGLES</b> computes spherical triangle angles.
</li>
<li>
<b>STRI_VERTICES_TO_SIDES_3D</b> computes spherical triangle sides.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TRIANGLE_AREA_2D</b> computes the area of a triangle in 2D.
</li>
<li>
<b>TUPLE_NEXT_FAST</b> computes the next element of a tuple space, "fast".
</li>
<li>
<b>UNIFORM_IN_ANNULUS</b> samples a circular annulus.
</li>
<li>
<b>UNIFORM_IN_ANNULUS_ACCEPT</b> accepts points in an annulus.
</li>
<li>
<b>UNIFORM_IN_ANNULUS_SECTOR</b> samples an annular sector in 2D.
</li>
<li>
<b>UNIFORM_IN_CIRCLE01_MAP</b> maps uniform points into the unit circle.
</li>
<li>
<b>UNIFORM_IN_CUBE01</b> creates uniform points in the unit hypercube.
</li>
<li>
<b>UNIFORM_IN_ELLIPSOID_MAP</b> maps uniform points into an ellipsoid.
</li>
<li>
<b>UNIFORM_IN_PARALLELOGRAM_MAP</b> maps uniform points into a parallelogram.
</li>
<li>
<b>UNIFORM_IN_POLYGON_MAP</b> maps uniform points into a polygon.
</li>
<li>
<b>UNIFORM_IN_SECTOR_MAP</b> maps uniform points into a circular sector.
</li>
<li>
<b>UNIFORM_IN_SIMPLEX01</b> maps uniform points into the unit simplex.
</li>
<li>
<b>UNIFORM_IN_SPHERE01_MAP</b> maps uniform points into the unit sphere.
</li>
<li>
<b>UNIFORM_IN_TETRAHEDRON</b> returns uniform points in a tetrahedron.
</li>