boost_1_45_0/libs/math/doc/math.qbk - nest-learning-thermostat/5.0/boost - Git at Google

 [article Boost.Math
     [quickbook 1.5]
     [copyright 2006, 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow, Hubert Holin, Daryle Walker, Xiaogang Zhang, Bruno Lalande, Johan R&#xE5;de, Gautam Sewani and Thijs van den Berg]
     [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])
     ]
     [authors [Maddock, John], [Bristow, Paul A.], [Holin, Hubert], [Walker, Daryle], [Zhang, Xiaogang], [Lalande, Bruno], [R&#xE5;de, Johan], [Sewani, Gautam], [van den Berg, Thijs]]
     [last-revision $Date: 2010-10-04 11:17:21 -0400 (Mon, 04 Oct 2010) $]
 ]

 [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.]]

 ]
	[article Boost.Math
	[quickbook 1.5]
	[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]
	[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])
	]
	[authors [Maddock, John], [Bristow, Paul A.], [Holin, Hubert], [Walker, Daryle], [Zhang, Xiaogang], [Lalande, Bruno], [Råde, Johan], [Sewani, Gautam], [van den Berg, Thijs]]
	[last-revision $Date: 2010-10-04 11:17:21 -0400 (Mon, 04 Oct 2010) $]
	]

	[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 ('''α,β,γ,δ''')]
	[def __quat_formula ['[^q = '''α + βi + γj + δk''']]]
	[def __quat_complex_formula ['[^q = ('''α + βi) + (γ + δi)j''' ]]]
	[def __not_equal ['[^xy '''≠''' yx]]]
	[def __octulple ('''α,β,γ,δ,ε,ζ,η,θ''')]
	[def __oct_formula ['[^o = '''α + βi + γj + δk + εe' + ζi' + ηj' + θk' ''']]]
	[def __oct_complex_formula ['[^o = ('''α + βi) + (γ + δi)j + (ε + ζi)e' + (η - θi)j' ''']]]
	[def __oct_quat_formula ['[^o = ('''α + βi + γj + δk) + (ε + ζi + ηj - θj)e' ''']]]
	[def __oct_not_equal ['[^x(yz) '''≠''' (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 ['[^ii = jj = 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.]]

	]