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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, 2010 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="id877658"></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> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"><p><small>Last revised: September 01, 2010 at 19:10:43 +0100</small></p></td> |
| <td align="right"><div class="copyright-footer"></div></td> |
| </tr></table> |
| <hr> |
| <div class="spirit-nav"></div> |
| </body> |
| </html> |
