[article Boost.Math
    [quickbook 1.4]
    [copyright 2007 Paul A. Bristow, Hubert Holin, John Maddock, Daryle Walker and Xiaogang Zhang]
    [license
        Distributed under the Boost Software License, Version 1.0.
        (See accompanying file LICENSE_1_0.txt or copy at
        [@http://www.boost.org/LICENSE_1_0.txt http://www.boost.org/LICENSE_1_0.txt])
    ]
    [category math]
    [last-revision $Date: 2008-07-15 12:30:27 -0400 (Tue, 15 Jul 2008) $]
]

[def __R ['[*R]]]
[def __C ['[*C]]]
[def __H ['[*H]]]
[def __O ['[*O]]]
[def __R3 ['[*'''R<superscript>3</superscript>''']]]
[def __R4 ['[*'''R<superscript>4</superscript>''']]]
[def __quadrulple ('''&#x03B1;,&#x03B2;,&#x03B3;,&#x03B4;''')]
[def __quat_formula ['[^q = '''&#x03B1; + &#x03B2;i + &#x03B3;j + &#x03B4;k''']]]
[def __quat_complex_formula ['[^q = ('''&#x03B1; + &#x03B2;i) + (&#x03B3; + &#x03B4;i)j''' ]]]
[def __not_equal ['[^xy '''&#x2260;''' yx]]]
[def __octulple ('''&#x03B1;,&#x03B2;,&#x03B3;,&#x03B4;,&#x03B5;,&#x03B6;,&#x03B7;,&#x03B8;''')]
[def __oct_formula ['[^o = '''&#x03B1; + &#x03B2;i + &#x03B3;j + &#x03B4;k + &#x03B5;e' + &#x03B6;i' + &#x03B7;j' + &#x03B8;k' ''']]]
[def __oct_complex_formula ['[^o = ('''&#x03B1; + &#x03B2;i) + (&#x03B3; + &#x03B4;i)j + (&#x03B5; + &#x03B6;i)e' + (&#x03B7; - &#x03B8;i)j' ''']]]
[def __oct_quat_formula ['[^o = ('''&#x03B1; + &#x03B2;i + &#x03B3;j + &#x03B4;k) + (&#x03B5; + &#x03B6;i + &#x03B7;j - &#x03B8;j)e' ''']]]
[def __oct_not_equal ['[^x(yz) '''&#x2260;''' (xy)z]]]
[template tr1[] [@http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2005/n1836.pdf Technical Report on C++ Library Extensions]]

The following libraries are present in Boost.Math:

[table
[[Library][Description]]

   [[Complex Number Inverse Trigonometric Functions
   
   [@../complex/html/index.html HTML Docs]
   
   [@http://svn.boost.org/svn/boost/sandbox/pdf/math/release/complex-tr1.pdf 
   PDF Docs]]
   [
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 [tr1].
   ]]
   
   [[Greatest Common Divisor and Least Common Multiple
   
   [@../gcd/html/index.html HTML Docs]
   
   [@http://svn.boost.org/svn/boost/sandbox/pdf/math/release/math-gcd.pdf 
   PDF Docs]]
   [
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.
   ]]

[[Octonions

[@../octonion/html/index.html HTML Docs]

[@http://svn.boost.org/svn/boost/sandbox/pdf/math/release/octonion.pdf 
PDF Docs]]
   [
Octonions, like [@../quaternion/html/index.html quaternions], are a relative of complex numbers.

Octonions see some use in theoretical physics.

In practical terms, an octonion is simply an octuple of real numbers __octulple, 
which we can write in the form __oct_formula, where ['[^i]], ['[^j]] and ['[^k]] 
are the same objects as for quaternions, and ['[^e']], ['[^i']], ['[^j']] and ['[^k']] 
are distinct objects which play essentially the same kind of role as ['[^i]] (or ['[^j]] or ['[^k]]).

Addition and a multiplication is defined on the set of octonions, 
which generalize their quaternionic counterparts. The main novelty this time 
is that [*the multiplication is not only not commutative, is now not even 
associative] (i.e. there are quaternions ['[^x]], ['[^y]] and ['[^z]] such that __oct_not_equal). 
A way of remembering things is by using the following multiplication table:

[$../../octonion/graphics/octonion_blurb17.jpeg]

Octonions (and their kin) are described in far more details in this other 
[@../../quaternion/TQE.pdf document] 
(with [@../../quaternion/TQE_EA.pdf errata and addenda]).

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).
   ]]
   
[[Special Functions

[@../sf_and_dist/html/index.html HTML Docs]

[@http://svn.boost.org/svn/boost/sandbox/pdf/math/release/math.pdf 
PDF Docs]]

   [ 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.

   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.

   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.  ]]

[[Statistical Distributions

[@../sf_and_dist/html/index.html HTML Docs]

[@http://svn.boost.org/svn/boost/sandbox/pdf/math/release/math.pdf 
PDF Docs]]

   [Provides a reasonably comprehensive set of statistical distributions, 
   upon which higher level statistical tests can be built.

   The initial focus is on the central univariate distributions. 
   Both continuous (like normal & Fisher) and discrete (like binomial 
   & Poisson) distributions are provided.

   A comprehensive tutorial is provided, along with a series of worked 
   examples illustrating how the library is used to conduct statistical tests. ]]

[[Quaternions

[@../quaternion/html/index.html HTML Docs]

[@http://svn.boost.org/svn/boost/sandbox/pdf/math/release/quaternion.pdf 
PDF Docs]]

   [
Quaternions are a relative of complex numbers.

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 __R), the set of complex numbers (traditionally named __C), 
the set of quaternions (traditionally named __H) and the set of octonions 
(traditionally named __O), which possess interesting mathematical properties 
(chief among which is the fact that they are ['division algebras], 
['i.e.] where the following property is true: if ['[^y]] is an element of that 
algebra and is [*not equal to zero], then ['[^yx = yx']], where ['[^x]] and ['[^x']] 
denote elements of that algebra, implies that ['[^x = x']]). 
Each member of the hierarchy is a super-set of the former.

One of the most important aspects of quaternions is that they provide an 
efficient way to parameterize rotations in __R3 (the usual three-dimensional space) 
and __R4.

In practical terms, a quaternion is simply a quadruple of real numbers __quadrulple, 
which we can write in the form __quat_formula, where ['[^i]] is the same object as for complex numbers, 
and ['[^j]] and ['[^k]] are distinct objects which play essentially the same kind of role as ['[^i]].

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 [*the multiplication is not commutative] (i.e. there are 
quaternions ['[^x]] and ['[^y]] such that __not_equal). A good mnemotechnical way of remembering 
things is by using the formula ['[^i*i = j*j = k*k = -1]].

Quaternions (and their kin) are described in far more details in this 
other [@../../quaternion/TQE.pdf document] 
(with [@../../quaternion/TQE_EA.pdf errata and addenda]).

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.
    ]]
    
]

The following Boost libraries are also mathematically oriented:

[table
[[Library][Description]]

   [[[@../../../integer/index.html Integer]]
   [Headers to ease dealing with integral types.]]

   [[[@../../../numeric/interval/doc/interval.htm Interval]]
   [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>. ]]
   
   [[[@../../../multi_array/doc/index.html Multi Array]]
   [Boost.MultiArray provides a generic N-dimensional array concept 
   definition and common implementations of that interface.]]

   [[[@../../../numeric/conversion/index.html Numeric.Conversion]]
   [The Boost Numeric Conversion library is a collection of tools to 
   describe and perform conversions between values of different numeric types.]]
   
   [[[@../../../utility/operators.htm Operators]]
   [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.]]

   [[[@../../../random/index.html Random]]
   [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.]]
   
   [[[@../../../rational/index.html Rational]]
   [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.]]
   
   [[[@../../../numeric/ublas/doc/index.htm uBLAS]]
   [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.]]

]

