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<div><h2 class="title">
<a name="boost_math"></a>Boost.Math</h2></div>
<div><div class="authorgroup">
<div class="author"><h3 class="author">
<span class="firstname">John</span> <span class="surname">Maddock</span>
</h3></div>
<div class="author"><h3 class="author">
<span class="firstname">Paul A.</span> <span class="surname">Bristow</span>
</h3></div>
<div class="author"><h3 class="author">
<span class="firstname">Hubert</span> <span class="surname">Holin</span>
</h3></div>
<div class="author"><h3 class="author">
<span class="firstname">Daryle</span> <span class="surname">Walker</span>
</h3></div>
<div class="author"><h3 class="author">
<span class="firstname">Xiaogang</span> <span class="surname">Zhang</span>
</h3></div>
<div class="author"><h3 class="author">
<span class="firstname">Bruno</span> <span class="surname">Lalande</span>
</h3></div>
<div class="author"><h3 class="author">
<span class="firstname">Johan</span> <span class="surname">Råde</span>
</h3></div>
<div class="author"><h3 class="author">
<span class="firstname">Gautam</span> <span class="surname">Sewani</span>
</h3></div>
<div class="author"><h3 class="author">
<span class="firstname">Thijs</span> <span class="surname">van den Berg</span>
</h3></div>
</div></div>
<div><p class="copyright">Copyright © 2006 , 2007, 2008, 2009 John Maddock, Paul A. Bristow,
Hubert Holin, Daryle Walker, Xiaogang Zhang, Bruno Lalande, Johan Råde,
Gautam Sewani and Thijs van den Berg</p></div>
<div><div class="legalnotice">
<a name="id759784"></a><p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
</p>
</div></div>
</div>
<hr>
</div>
<p>
The following libraries are present in Boost.Math:
</p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Library
</p>
</th>
<th>
<p>
Description
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
Complex Number Inverse Trigonometric Functions
</p>
<p>
<a href="../complex/html/index.html" target="_top">HTML Docs</a>
</p>
<p>
<a href="http://svn.boost.org/svn/boost/sandbox/pdf/math/release/complex-tr1.pdf" target="_top">PDF
Docs</a>
</p>
</td>
<td>
<p>
These complex number algorithms are the inverses of trigonometric functions
currently present in the C++ standard. Equivalents to these functions
are part of the C99 standard, and are part of the <a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2005/n1836.pdf" target="_top">Technical
Report on C++ Library Extensions</a>.
</p>
</td>
</tr>
<tr>
<td>
<p>
Greatest Common Divisor and Least Common Multiple
</p>
<p>
<a href="../gcd/html/index.html" target="_top">HTML Docs</a>
</p>
<p>
<a href="http://svn.boost.org/svn/boost/sandbox/pdf/math/release/math-gcd.pdf" target="_top">PDF
Docs</a>
</p>
</td>
<td>
<p>
The class and function templates in <boost/math/common_factor.hpp>
provide run-time and compile-time evaluation of the greatest common
divisor (GCD) or least common multiple (LCM) of two integers. These
facilities are useful for many numeric-oriented generic programming
problems.
</p>
</td>
</tr>
<tr>
<td>
<p>
Octonions
</p>
<p>
<a href="../octonion/html/index.html" target="_top">HTML Docs</a>
</p>
<p>
<a href="http://svn.boost.org/svn/boost/sandbox/pdf/math/release/octonion.pdf" target="_top">PDF
Docs</a>
</p>
</td>
<td>
<p>
Octonions, like <a href="../quaternion/html/index.html" target="_top">quaternions</a>,
are a relative of complex numbers.
</p>
<p>
Octonions see some use in theoretical physics.
</p>
<p>
In practical terms, an octonion is simply an octuple of real numbers
(α,β,γ,δ,ε,ζ,η,θ), which we can write in the form <span class="emphasis"><em><code class="literal">o = α + βi + γj + δk + εe' + ζi' + ηj' + θk'</code></em></span>,
where <span class="emphasis"><em><code class="literal">i</code></em></span>, <span class="emphasis"><em><code class="literal">j</code></em></span>
and <span class="emphasis"><em><code class="literal">k</code></em></span> are the same objects
as for quaternions, and <span class="emphasis"><em><code class="literal">e'</code></em></span>,
<span class="emphasis"><em><code class="literal">i'</code></em></span>, <span class="emphasis"><em><code class="literal">j'</code></em></span>
and <span class="emphasis"><em><code class="literal">k'</code></em></span> are distinct objects
which play essentially the same kind of role as <span class="emphasis"><em><code class="literal">i</code></em></span>
(or <span class="emphasis"><em><code class="literal">j</code></em></span> or <span class="emphasis"><em><code class="literal">k</code></em></span>).
</p>
<p>
Addition and a multiplication is defined on the set of octonions, which
generalize their quaternionic counterparts. The main novelty this time
is that <span class="bold"><strong>the multiplication is not only not commutative,
is now not even associative</strong></span> (i.e. there are quaternions
<span class="emphasis"><em><code class="literal">x</code></em></span>, <span class="emphasis"><em><code class="literal">y</code></em></span>
and <span class="emphasis"><em><code class="literal">z</code></em></span> such that <span class="emphasis"><em><code class="literal">x(yz)
≠ (xy)z</code></em></span>). A way of remembering things is by using
the following multiplication table:
</p>
<p>
<span class="inlinemediaobject"><img src="../../octonion/graphics/octonion_blurb17.jpeg" alt="octonion_blurb17"></span>
</p>
<p>
Octonions (and their kin) are described in far more details in this
other <a href="../../quaternion/TQE.pdf" target="_top">document</a> (with
<a href="../../quaternion/TQE_EA.pdf" target="_top">errata and addenda</a>).
</p>
<p>
Some traditional constructs, such as the exponential, carry over without
too much change into the realms of octonions, but other, such as taking
a square root, do not (the fact that the exponential has a closed form
is a result of the author, but the fact that the exponential exists
at all for octonions is known since quite a long time ago).
</p>
</td>
</tr>
<tr>
<td>
<p>
Special Functions
</p>
<p>
<a href="../sf_and_dist/html/index.html" target="_top">HTML Docs</a>
</p>
<p>
<a href="http://svn.boost.org/svn/boost/sandbox/pdf/math/release/math.pdf" target="_top">PDF
Docs</a>
</p>
</td>
<td>
<p>
Provides a number of high quality special functions, initially these
were concentrated on functions used in statistical applications along
with those in the Technical Report on C++ Library Extensions.
</p>
<p>
The function families currently implemented are the gamma, beta &
erf functions along with the incomplete gamma and beta functions (four
variants of each) and all the possible inverses of these, plus digamma,
various factorial functions, Bessel functions, elliptic integrals,
sinus cardinals (along with their hyperbolic variants), inverse hyperbolic
functions, Legrendre/Laguerre/Hermite polynomials and various special
power and logarithmic functions.
</p>
<p>
All the implementations are fully generic and support the use of arbitrary
"real-number" types, although they are optimised for use
with types with known-about significand (or mantissa) sizes: typically
float, double or long double.
</p>
</td>
</tr>
<tr>
<td>
<p>
Statistical Distributions
</p>
<p>
<a href="../sf_and_dist/html/index.html" target="_top">HTML Docs</a>
</p>
<p>
<a href="http://svn.boost.org/svn/boost/sandbox/pdf/math/release/math.pdf" target="_top">PDF
Docs</a>
</p>
</td>
<td>
<p>
Provides a reasonably comprehensive set of statistical distributions,
upon which higher level statistical tests can be built.
</p>
<p>
The initial focus is on the central univariate distributions. Both
continuous (like normal & Fisher) and discrete (like binomial &
Poisson) distributions are provided.
</p>
<p>
A comprehensive tutorial is provided, along with a series of worked
examples illustrating how the library is used to conduct statistical
tests.
</p>
</td>
</tr>
<tr>
<td>
<p>
Quaternions
</p>
<p>
<a href="../quaternion/html/index.html" target="_top">HTML Docs</a>
</p>
<p>
<a href="http://svn.boost.org/svn/boost/sandbox/pdf/math/release/quaternion.pdf" target="_top">PDF
Docs</a>
</p>
</td>
<td>
<p>
Quaternions are a relative of complex numbers.
</p>
<p>
Quaternions are in fact part of a small hierarchy of structures built
upon the real numbers, which comprise only the set of real numbers
(traditionally named <span class="emphasis"><em><span class="bold"><strong>R</strong></span></em></span>),
the set of complex numbers (traditionally named <span class="emphasis"><em><span class="bold"><strong>C</strong></span></em></span>), the set of quaternions (traditionally
named <span class="emphasis"><em><span class="bold"><strong>H</strong></span></em></span>) and
the set of octonions (traditionally named <span class="emphasis"><em><span class="bold"><strong>O</strong></span></em></span>),
which possess interesting mathematical properties (chief among which
is the fact that they are <span class="emphasis"><em>division algebras</em></span>,
<span class="emphasis"><em>i.e.</em></span> where the following property is true: if
<span class="emphasis"><em><code class="literal">y</code></em></span> is an element of that algebra
and is <span class="bold"><strong>not equal to zero</strong></span>, then <span class="emphasis"><em><code class="literal">yx
= yx'</code></em></span>, where <span class="emphasis"><em><code class="literal">x</code></em></span>
and <span class="emphasis"><em><code class="literal">x'</code></em></span> denote elements of that
algebra, implies that <span class="emphasis"><em><code class="literal">x = x'</code></em></span>).
Each member of the hierarchy is a super-set of the former.
</p>
<p>
One of the most important aspects of quaternions is that they provide
an efficient way to parameterize rotations in <span class="emphasis"><em><span class="bold"><strong>R<sup>3</sup></strong></span></em></span>
(the usual three-dimensional space) and <span class="emphasis"><em><span class="bold"><strong>R<sup>4</sup></strong></span></em></span>.
</p>
<p>
In practical terms, a quaternion is simply a quadruple of real numbers
(α,β,γ,δ), which we can write in the form <span class="emphasis"><em><code class="literal">q = α + βi + γj + δk</code></em></span>,
where <span class="emphasis"><em><code class="literal">i</code></em></span> is the same object
as for complex numbers, and <span class="emphasis"><em><code class="literal">j</code></em></span>
and <span class="emphasis"><em><code class="literal">k</code></em></span> are distinct objects
which play essentially the same kind of role as <span class="emphasis"><em><code class="literal">i</code></em></span>.
</p>
<p>
An addition and a multiplication is defined on the set of quaternions,
which generalize their real and complex counterparts. The main novelty
here is that <span class="bold"><strong>the multiplication is not commutative</strong></span>
(i.e. there are quaternions <span class="emphasis"><em><code class="literal">x</code></em></span>
and <span class="emphasis"><em><code class="literal">y</code></em></span> such that <span class="emphasis"><em><code class="literal">xy
≠ yx</code></em></span>). A good mnemotechnical way of remembering
things is by using the formula <span class="emphasis"><em><code class="literal">i*i = j*j = k*k =
-1</code></em></span>.
</p>
<p>
Quaternions (and their kin) are described in far more details in this
other <a href="../../quaternion/TQE.pdf" target="_top">document</a> (with
<a href="../../quaternion/TQE_EA.pdf" target="_top">errata and addenda</a>).
</p>
<p>
Some traditional constructs, such as the exponential, carry over without
too much change into the realms of quaternions, but other, such as
taking a square root, do not.
</p>
</td>
</tr>
</tbody>
</table></div>
<p>
The following Boost libraries are also mathematically oriented:
</p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Library
</p>
</th>
<th>
<p>
Description
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
<a href="../../../integer/index.html" target="_top">Integer</a>
</p>
</td>
<td>
<p>
Headers to ease dealing with integral types.
</p>
</td>
</tr>
<tr>
<td>
<p>
<a href="../../../numeric/interval/doc/interval.htm" target="_top">Interval</a>
</p>
</td>
<td>
<p>
As implied by its name, this library is intended to help manipulating
mathematical intervals. It consists of a single header <boost/numeric/interval.hpp>
and principally a type which can be used as interval<T>.
</p>
</td>
</tr>
<tr>
<td>
<p>
<a href="../../../multi_array/doc/index.html" target="_top">Multi Array</a>
</p>
</td>
<td>
<p>
Boost.MultiArray provides a generic N-dimensional array concept definition
and common implementations of that interface.
</p>
</td>
</tr>
<tr>
<td>
<p>
<a href="../../../numeric/conversion/index.html" target="_top">Numeric.Conversion</a>
</p>
</td>
<td>
<p>
The Boost Numeric Conversion library is a collection of tools to describe
and perform conversions between values of different numeric types.
</p>
</td>
</tr>
<tr>
<td>
<p>
<a href="../../../utility/operators.htm" target="_top">Operators</a>
</p>
</td>
<td>
<p>
The header <boost/operators.hpp> supplies several sets of class
templates (in namespace boost). These templates define operators at
namespace scope in terms of a minimal number of fundamental operators
provided by the class.
</p>
</td>
</tr>
<tr>
<td>
<p>
<a href="../../../random/index.html" target="_top">Random</a>
</p>
</td>
<td>
<p>
Random numbers are useful in a variety of applications. The Boost Random
Number Library (Boost.Random for short) provides a vast variety of
generators and distributions to produce random numbers having useful
properties, such as uniform distribution.
</p>
</td>
</tr>
<tr>
<td>
<p>
<a href="../../../rational/index.html" target="_top">Rational</a>
</p>
</td>
<td>
<p>
The header rational.hpp provides an implementation of rational numbers.
The implementation is template-based, in a similar manner to the standard
complex number class.
</p>
</td>
</tr>
<tr>
<td>
<p>
<a href="../../../numeric/ublas/doc/index.htm" target="_top">uBLAS</a>
</p>
</td>
<td>
<p>
uBLAS is a C++ template class library that provides BLAS level 1, 2,
3 functionality for dense, packed and sparse matrices. The design and
implementation unify mathematical notation via operator overloading
and efficient code generation via expression templates.
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
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