[article Boost.Octonions
    [quickbook 1.3]
    [copyright 2001-2003 Hubert Holin]
    [purpose Octonions]
    [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 [Holin, Hubert]]
    [category math]
    [last-revision $Date: 2006-12-29 11:08:32 +0000 (Fri, 29 Dec 2006) $]
]

[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 __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]]]

[def __asinh [link boost_quaternions.math_special_functions.asinh asinh]]
[def __acosh [link boost_quaternions.math_special_functions.acosh acosh]]
[def __atanh [link boost_quaternions.math_special_functions.atanh atanh]]
[def __sinc_pi [link boost_quaternions.math_special_functions.sinc_pi sinc_pi]]
[def __sinhc_pi [link boost_quaternions.math_special_functions.sinhc_pi sinhc_pi]]

[def __log1p [link boost_quaternions.math_special_functions.log1p log1p]]
[def __expm1 [link boost_quaternions.math_special_functions.expm1 expm1]]
[def __hypot [link boost_quaternions.math_special_functions.hypot hypot]]

This manual is also available in 
[@http:svn.boost.org/svn/boost/sandbox/pdf/math/release/octonion.pdf 
printer friendly PDF format].

[section Octonions]

[section Overview]

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 octonions ['[^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).

[endsect]

[section Header File]

The interface and implementation are both supplied by the header file 
[@../../../../../boost/math/octonion.hpp octonion.hpp].

[endsect]

[section Synopsis]

   namespace boost{ namespace math{

   template<typename T> class ``[link boost_octonions.octonions.template_class_octonion octonion]``;
   template<>           class ``[link boost_octonions.octonions.octonion_specializations octonion<float>]``;
   template<>           class ``[link boost_octonions.octonion_double octonion<double>]``; 
   template<>           class ``[link boost_octonions.octonion_long_double octonion<long double>]``; 

   // operators

   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_addition_operators operator +]`` (T const & lhs, octonion<T> const & rhs);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_addition_operators operator +]`` (octonion<T> const & lhs, T const & rhs);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_addition_operators operator +]`` (::std::complex<T> const & lhs, octonion<T> const & rhs);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_addition_operators operator +]`` (octonion<T> const & lhs, ::std::complex<T> const & rhs);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_addition_operators operator +]`` (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_addition_operators operator +]`` (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_addition_operators operator +]`` (octonion<T> const & lhs, octonion<T> const & rhs);

   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_subtraction_operators operator -]`` (T const & lhs, octonion<T> const & rhs);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_subtraction_operators operator -]`` (octonion<T> const & lhs, T const & rhs);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_subtraction_operators operator -]`` (::std::complex<T> const & lhs, octonion<T> const & rhs);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_subtraction_operators operator -]`` (octonion<T> const & lhs, ::std::complex<T> const & rhs);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_subtraction_operators operator -]`` (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_subtraction_operators operator -]`` (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_subtraction_operators operator -]`` (octonion<T> const & lhs, octonion<T> const & rhs);

   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_multiplication_operators operator *]`` (T const & lhs, octonion<T> const & rhs);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_multiplication_operators operator *]`` (octonion<T> const & lhs, T const & rhs);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_multiplication_operators operator *]`` (::std::complex<T> const & lhs, octonion<T> const & rhs);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_multiplication_operators operator *]`` (octonion<T> const & lhs, ::std::complex<T> const & rhs);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_multiplication_operators operator *]`` (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_multiplication_operators operator *]`` (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_multiplication_operators operator *]`` (octonion<T> const & lhs, octonion<T> const & rhs);

   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_division_operators operator /]`` (T const & lhs, octonion<T> const & rhs);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_division_operators operator /]`` (octonion<T> const & lhs, T const & rhs);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_division_operators operator /]`` (::std::complex<T> const & lhs, octonion<T> const & rhs);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_division_operators operator /]`` (octonion<T> const & lhs, ::std::complex<T> const & rhs);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_division_operators operator /]`` (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_division_operators operator /]`` (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.binary_division_operators operator /]`` (octonion<T> const & lhs, octonion<T> const & rhs); 

   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.unary_plus_and_minus_operators operator +]`` (octonion<T> const & o);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.non_mem.unary_plus_and_minus_operators operator -]`` (octonion<T> const & o); 

   template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_equality_operators operator ==]`` (T const & lhs, octonion<T> const & rhs);
   template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_equality_operators operator ==]`` (octonion<T> const & lhs, T const & rhs);
   template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_equality_operators operator ==]`` (::std::complex<T> const & lhs, octonion<T> const & rhs);
   template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_equality_operators operator ==]`` (octonion<T> const & lhs, ::std::complex<T> const & rhs);
   template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_equality_operators operator ==]`` (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
   template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_equality_operators operator ==]`` (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
   template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_equality_operators operator ==]`` (octonion<T> const & lhs, octonion<T> const & rhs);

   template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_inequality_operators operator !=]`` (T const & lhs, octonion<T> const & rhs);
   template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_inequality_operators operator !=]`` (octonion<T> const & lhs, T const & rhs);
   template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_inequality_operators operator !=]`` (::std::complex<T> const & lhs, octonion<T> const & rhs);
   template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_inequality_operators operator !=]`` (octonion<T> const & lhs, ::std::complex<T> const & rhs);
   template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_inequality_operators operator !=]`` (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
   template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_inequality_operators operator !=]`` (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
   template<typename T> bool ``[link boost_octonions.octonions.non_mem.binary_inequality_operators operator !=]`` (octonion<T> const & lhs, octonion<T> const & rhs); 

   template<typename T, typename charT, class traits>
   ::std::basic_istream<charT,traits> & ``[link boost_octonions.octonions.non_mem.stream_extractor operator >>]`` (::std::basic_istream<charT,traits> & is, octonion<T> & o);

   template<typename T, typename charT, class traits>
   ::std::basic_ostream<charT,traits> & ``[link boost_octonions.octonions.non_mem.stream_inserter operator <<]`` (::std::basic_ostream<charT,traits> & os, octonion<T> const & o);

   // values

   template<typename T> T           ``[link boost_octonions.octonions.octonion_value_operations.real_and_unreal real]``(octonion<T> const & o);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.octonion_value_operations.real_and_unreal unreal]``(octonion<T> const & o);

   template<typename T> T           ``[link boost_octonions.octonions.octonion_value_operations.sup sup]``(octonion<T> const & o);
   template<typename T> T           ``[link boost_octonions.octonions.octonion_value_operations.l1 l1]``(octonion<T>const & o); 
   template<typename T> T           ``[link boost_octonions.octonions.octonion_value_operations.abs abs]``(octonion<T> const & o);
   template<typename T> T           ``[link boost_octonions.octonions.octonion_value_operations.norm norm]``(octonion<T>const  & o);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.octonion_value_operations.conj conj]``(octonion<T> const & o);

   template<typename T> octonion<T> ``[link boost_octonions.octonions.oct_create spherical]``(T const & rho, T const & theta, T const & phi1, T const & phi2, T const & phi3, T const & phi4, T const & phi5, T const & phi6);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.oct_create multipolar]``(T const & rho1, T const & theta1, T const & rho2, T const & theta2, T const & rho3, T const & theta3, T const & rho4, T const & theta4);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.oct_create cylindrical]``(T const & r, T const & angle, T const & h1, T const & h2, T const & h3, T const & h4, T const & h5, T const & h6);

   // transcendentals

   template<typename T> octonion<T> ``[link boost_octonions.octonions.octonions_transcendentals.exp exp]``(octonion<T> const & o);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.octonions_transcendentals.cos cos]``(octonion<T> const & o);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.octonions_transcendentals.sin sin]``(octonion<T> const & o);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.octonions_transcendentals.tan tan]``(octonion<T> const & o);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.octonions_transcendentals.cosh cosh]``(octonion<T> const & o);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.octonions_transcendentals.sinh sinh]``(octonion<T> const & o);
   template<typename T> octonion<T> ``[link boost_octonions.octonions.octonions_transcendentals.tanh tanh]``(octonion<T> const & o);

   template<typename T> octonion<T> ``[link boost_octonions.octonions.octonions_transcendentals.pow pow]``(octonion<T> const & o, int n);

   }  } // namespaces
   
[endsect]

[section Template Class octonion]

   namespace boost{ namespace math {
 
   template<typename T>
   class octonion
   {
   public:
      typedef T value_type;

      explicit  ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(T const & requested_a = T(), T const & requested_b = T(), T const & requested_c = T(), T const & requested_d = T(), T const & requested_e = T(), T const & requested_f = T(), T const & requested_g = T(), T const & requested_h = T());
      explicit  ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(::std::complex<T> const & z0, ::std::complex<T> const & z1 = ::std::complex<T>(), ::std::complex<T> const & z2 = ::std::complex<T>(), ::std::complex<T> const & z3 = ::std::complex<T>());
      explicit  ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(::boost::math::quaternion<T> const & q0, ::boost::math::quaternion<T> const & q1 = ::boost::math::quaternion<T>());
      template<typename X> 
      explicit  ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(octonion<X> const & a_recopier);

      T                             ``[link boost_octonions.octonions.octonion_member_functions.real_and_unreal_parts real]``() const;
      octonion<T>                   ``[link boost_octonions.octonions.octonion_member_functions.real_and_unreal_parts unreal]``() const;

      T                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_1]``() const;
      T                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_2]``() const;
      T                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_3]``() const;
      T                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_4]``() const;
      T                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_5]``() const;
      T                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_6]``() const;
      T                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_7]``() const;
      T                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_8]``() const;

      ::std::complex<T>             ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_1]``() const;
      ::std::complex<T>             ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_2]``() const;
      ::std::complex<T>             ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_3]``() const;
      ::std::complex<T>             ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_4]``() const;

      ::boost::math::quaternion<T>  ``[link boost_octonions.octonions.octonion_member_functions.individual_quaternion_components H_component_1]``() const;
      ::boost::math::quaternion<T>  ``[link boost_octonions.octonions.octonion_member_functions.individual_quaternion_components H_component_2]``() const;

      octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion<T> const  & a_affecter);
      template<typename X> 
      octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion<X> const  & a_affecter);
      octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (T const  & a_affecter);
      octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (::std::complex<T> const & a_affecter);
      octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (::boost::math::quaternion<T> const & a_affecter);

      octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (T const & rhs);
      octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (::std::complex<T> const & rhs);
      octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (::boost::math::quaternion<T> const & rhs);
      template<typename X> 
      octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (octonion<X> const & rhs);

      octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (T const & rhs);
      octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (::std::complex<T> const & rhs);
      octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (::boost::math::quaternion<T> const & rhs);
      template<typename X> 
      octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (octonion<X> const & rhs);

      octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (T const & rhs);
      octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (::std::complex<T> const & rhs);
      octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (::boost::math::quaternion<T> const & rhs);
      template<typename X> 
      octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (octonion<X> const & rhs);

      octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (T const & rhs);
      octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (::std::complex<T> const & rhs);
      octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (::boost::math::quaternion<T> const & rhs);
      template<typename X> 
      octonion<T> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (octonion<X> const & rhs);
   };

   } } // namespaces
   
[endsect]

[section Octonion Specializations]

   namespace boost{ namespace math{
   
   template<>
   class octonion<float>
   {
   public:
      typedef float value_type;

      explicit  ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(float const & requested_a = 0.0f, float const & requested_b = 0.0f, float const & requested_c = 0.0f, float const & requested_d = 0.0f, float const & requested_e = 0.0f, float const & requested_f = 0.0f, float const & requested_g = 0.0f, float const & requested_h = 0.0f);
      explicit  ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(::std::complex<float> const & z0, ::std::complex<float> const & z1 = ::std::complex<float>(), ::std::complex<float> const & z2 = ::std::complex<float>(), ::std::complex<float> const & z3 = ::std::complex<float>());
      explicit  ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(::boost::math::quaternion<float> const & q0, ::boost::math::quaternion<float> const & q1 = ::boost::math::quaternion<float>());
      explicit  ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(octonion<double> const & a_recopier);
      explicit  ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(octonion<long double> const & a_recopier);

      float                             ``[link boost_octonions.octonions.octonion_member_functions.real_and_unreal_parts real]``() const;
      octonion<float>                   ``[link boost_octonions.octonions.octonion_member_functions.real_and_unreal_parts unreal]``() const;

      float                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_1]``() const;
      float                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_2]``() const;
      float                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_3]``() const;
      float                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_4]``() const;
      float                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_5]``() const;
      float                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_6]``() const;
      float                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_7]``() const;
      float                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_8]``() const;

      ::std::complex<float>             ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_1]``() const;
      ::std::complex<float>             ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_2]``() const;
      ::std::complex<float>             ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_3]``() const;
      ::std::complex<float>             ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_4]``() const;

      ::boost::math::quaternion<float>  ``[link boost_octonions.octonions.octonion_member_functions.individual_quaternion_components H_component_1]``() const;
      ::boost::math::quaternion<float>  ``[link boost_octonions.octonions.octonion_member_functions.individual_quaternion_components H_component_2]``() const;

      octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion<float> const  & a_affecter);
      template<typename X> 
      octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion<X> const  & a_affecter);
      octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (float const  & a_affecter);
      octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (::std::complex<float> const & a_affecter);
      octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (::boost::math::quaternion<float> const & a_affecter);

      octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (float const & rhs);
      octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (::std::complex<float> const & rhs);
      octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (::boost::math::quaternion<float> const & rhs);
      template<typename X> 
      octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (octonion<X> const & rhs);

      octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (float const & rhs);
      octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (::std::complex<float> const & rhs);
      octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (::boost::math::quaternion<float> const & rhs);
      template<typename X> 
      octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (octonion<X> const & rhs);

      octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (float const & rhs);
      octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (::std::complex<float> const & rhs);
      octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (::boost::math::quaternion<float> const & rhs);
      template<typename X> 
      octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (octonion<X> const & rhs);

      octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (float const & rhs);
      octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (::std::complex<float> const & rhs);
      octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (::boost::math::quaternion<float> const & rhs);
      template<typename X> 
      octonion<float> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (octonion<X> const & rhs);
   };

[#boost_octonions.octonion_double]

   template<>
   class octonion<double>
   {
   public:
      typedef double value_type;

      explicit  ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(double const & requested_a = 0.0, double const & requested_b = 0.0, double const & requested_c = 0.0, double const & requested_d = 0.0, double const & requested_e = 0.0, double const & requested_f = 0.0, double const & requested_g = 0.0, double const & requested_h = 0.0);
      explicit  ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(::std::complex<double> const & z0, ::std::complex<double> const & z1 = ::std::complex<double>(), ::std::complex<double> const & z2 = ::std::complex<double>(), ::std::complex<double> const & z3 = ::std::complex<double>());
      explicit  ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(::boost::math::quaternion<double> const & q0, ::boost::math::quaternion<double> const & q1 = ::boost::math::quaternion<double>());
      explicit  ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(octonion<float> const & a_recopier);
      explicit  ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(octonion<long double> const & a_recopier);

      double                             ``[link boost_octonions.octonions.octonion_member_functions.real_and_unreal_parts real]``() const;
      octonion<double>                   ``[link boost_octonions.octonions.octonion_member_functions.real_and_unreal_parts unreal]``() const;

      double                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_1]``() const;
      double                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_2]``() const;
      double                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_3]``() const;
      double                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_4]``() const;
      double                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_5]``() const;
      double                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_6]``() const;
      double                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_7]``() const;
      double                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_8]``() const;

      ::std::complex<double>             ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_1]``() const;
      ::std::complex<double>             ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_2]``() const;
      ::std::complex<double>             ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_3]``() const;
      ::std::complex<double>             ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_4]``() const;

      ::boost::math::quaternion<double>  ``[link boost_octonions.octonions.octonion_member_functions.individual_quaternion_components H_component_1]``() const;
      ::boost::math::quaternion<double>  ``[link boost_octonions.octonions.octonion_member_functions.individual_quaternion_components H_component_2]``() const;

      octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion<double> const  & a_affecter);
      template<typename X> 
      octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion<X> const  & a_affecter);
      octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (double const  & a_affecter);
      octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (::std::complex<double> const & a_affecter);
      octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (::boost::math::quaternion<double> const & a_affecter);

      octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (double const & rhs);
      octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (::std::complex<double> const & rhs);
      octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (::boost::math::quaternion<double> const & rhs);
      template<typename X> 
      octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (octonion<X> const & rhs);

      octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (double const & rhs);
      octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (::std::complex<double> const & rhs);
      octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (::boost::math::quaternion<double> const & rhs);
      template<typename X> 
      octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (octonion<X> const & rhs);

      octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (double const & rhs);
      octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (::std::complex<double> const & rhs);
      octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (::boost::math::quaternion<double> const & rhs);
      template<typename X> 
      octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (octonion<X> const & rhs);

      octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (double const & rhs);
      octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (::std::complex<double> const & rhs);
      octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (::boost::math::quaternion<double> const & rhs);
      template<typename X> 
      octonion<double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (octonion<X> const & rhs);
   };

[#boost_octonions.octonion_long_double]

   template<>
   class octonion<long double>
   {
   public:
      typedef long double value_type;

      explicit   ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(long double const & requested_a = 0.0L, long double const & requested_b = 0.0L, long double const & requested_c = 0.0L, long double const & requested_d = 0.0L, long double const & requested_e = 0.0L, long double const & requested_f = 0.0L, long double const & requested_g = 0.0L, long double const & requested_h = 0.0L);
      explicit   ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``( ::std::complex<long double> const & z0, ::std::complex<long double> const & z1 = ::std::complex<long double>(), ::std::complex<long double> const & z2 = ::std::complex<long double>(), ::std::complex<long double> const & z3 = ::std::complex<long double>());
      explicit   ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``( ::boost::math::quaternion<long double> const & q0, ::boost::math::quaternion<long double> const & z1 = ::boost::math::quaternion<long double>());
      explicit   ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(octonion<float> const & a_recopier);
      explicit   ``[link boost_octonions.octonions.octonion_member_functions.constructors octonion]``(octonion<double> const & a_recopier);

      long double                             ``[link boost_octonions.octonions.octonion_member_functions.real_and_unreal_parts real]``() const;
      octonion<long double>                   ``[link boost_octonions.octonions.octonion_member_functions.real_and_unreal_parts unreal]``() const;

      long double                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_1]``() const;
      long double                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_2]``() const;
      long double                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_3]``() const;
      long double                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_4]``() const;
      long double                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_5]``() const;
      long double                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_6]``() const;
      long double                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_7]``() const;
      long double                             ``[link boost_octonions.octonions.octonion_member_functions.individual_real_components R_component_8]``() const;

      ::std::complex<long double>             ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_1]``() const;
      ::std::complex<long double>             ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_2]``() const;
      ::std::complex<long double>             ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_3]``() const;
      ::std::complex<long double>             ``[link boost_octonions.octonions.octonion_member_functions.individual_complex_components C_component_4]``() const;

      ::boost::math::quaternion<long double>  ``[link boost_octonions.octonions.octonion_member_functions.individual_quaternion_components H_component_1]``() const;
      ::boost::math::quaternion<long double>  ``[link boost_octonions.octonions.octonion_member_functions.individual_quaternion_components H_component_2]``() const;

      octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion<long double> const  & a_affecter);
      template<typename X> 
      octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (octonion<X> const  & a_affecter);
      octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (long double const  & a_affecter);
      octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (::std::complex<long double> const & a_affecter);
      octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.assignment_operators operator =]`` (::boost::math::quaternion<long double> const & a_affecter);

      octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (long double const & rhs);
      octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (::std::complex<long double> const & rhs);
      octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (::boost::math::quaternion<long double> const & rhs);
      template<typename X> 
      octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator +=]`` (octonion<X> const & rhs);

      octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (long double const & rhs);
      octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (::std::complex<long double> const & rhs);
      octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (::boost::math::quaternion<long double> const & rhs);
      template<typename X> 
      octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator -=]`` (octonion<X> const & rhs);

      octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (long double const & rhs);
      octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (::std::complex<long double> const & rhs);
      octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (::boost::math::quaternion<long double> const & rhs);
      template<typename X> 
      octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator *=]`` (octonion<X> const & rhs);

      octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (long double const & rhs);
      octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (::std::complex<long double> const & rhs);
      octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (::boost::math::quaternion<long double> const & rhs);
      template<typename X> 
      octonion<long double> & ``[link boost_octonions.octonions.octonion_member_functions.other_member_operators operator /=]`` (octonion<X> const & rhs);
   };

   } } // namespaces
   
[endsect]

[section Octonion Member Typedefs]

[*value_type]

Template version:

   typedef T value_type;

Float specialization version:

   typedef float value_type;
   
Double specialization version:

   typedef double value_type;

Long double specialization version:

   typedef long double value_type;

These provide easy acces to the type the template is built upon.

[endsect]

[section Octonion Member Functions]

[h3 Constructors]

Template version:

   explicit  octonion(T const & requested_a = T(), T const & requested_b = T(), T const & requested_c = T(), T const & requested_d = T(), T const & requested_e = T(), T const & requested_f = T(), T const & requested_g = T(), T const & requested_h = T());
   explicit  octonion(::std::complex<T> const & z0, ::std::complex<T> const & z1 = ::std::complex<T>(), ::std::complex<T> const & z2 = ::std::complex<T>(), ::std::complex<T> const & z3 = ::std::complex<T>());
   explicit  octonion(::boost::math::quaternion<T> const & q0, ::boost::math::quaternion<T> const & q1 = ::boost::math::quaternion<T>());
   template<typename X> 
   explicit octonion(octonion<X> const & a_recopier);

Float specialization version:

   explicit  octonion(float const & requested_a = 0.0f, float const & requested_b = 0.0f, float const & requested_c = 0.0f, float const & requested_d = 0.0f, float const & requested_e = 0.0f, float const & requested_f = 0.0f, float const & requested_g = 0.0f, float const & requested_h = 0.0f);
   explicit  octonion(::std::complex<float> const & z0, ::std::complex<float> const & z1 = ::std::complex<float>(), ::std::complex<float> const & z2 = ::std::complex<float>(), ::std::complex<float> const & z3 = ::std::complex<float>());
   explicit  octonion(::boost::math::quaternion<float> const & q0, ::boost::math::quaternion<float> const & q1 = ::boost::math::quaternion<float>());
   explicit  octonion(octonion<double> const & a_recopier); 
   explicit  octonion(octonion<long double> const & a_recopier);

Double specialization version:

   explicit  octonion(double const & requested_a = 0.0, double const & requested_b = 0.0, double const & requested_c = 0.0, double const & requested_d = 0.0, double const & requested_e = 0.0, double const & requested_f = 0.0, double const & requested_g = 0.0, double const & requested_h = 0.0);
   explicit  octonion(::std::complex<double> const & z0, ::std::complex<double> const & z1 = ::std::complex<double>(), ::std::complex<double> const & z2 = ::std::complex<double>(), ::std::complex<double> const & z3 = ::std::complex<double>());
   explicit  octonion(::boost::math::quaternion<double> const & q0, ::boost::math::quaternion<double> const & q1 = ::boost::math::quaternion<double>());
   explicit  octonion(octonion<float> const & a_recopier);
   explicit  octonion(octonion<long double> const & a_recopier);

Long double specialization version:

   explicit  octonion(long double const & requested_a = 0.0L, long double const & requested_b = 0.0L, long double const & requested_c = 0.0L, long double const & requested_d = 0.0L, long double const & requested_e = 0.0L, long double const & requested_f = 0.0L, long double const & requested_g = 0.0L, long double const & requested_h = 0.0L);
   explicit  octonion( ::std::complex<long double> const & z0, ::std::complex<long double> const & z1 = ::std::complex<long double>(), ::std::complex<long double> const & z2 = ::std::complex<long double>(), ::std::complex<long double> const & z3 = ::std::complex<long double>());
   explicit  octonion(::boost::math::quaternion<long double> const & q0, ::boost::math::quaternion<long double> const & q1 = ::boost::math::quaternion<long double>());
   explicit  octonion(octonion<float> const & a_recopier);
   explicit  octonion(octonion<double> const & a_recopier);

A default constructor is provided for each form, which initializes each component 
to the default values for their type (i.e. zero for floating numbers). 
This constructor can also accept one to eight base type arguments. 
A constructor is also provided to build octonions from one to four complex numbers 
sharing the same base type, and another taking one or two quaternions
sharing the same base type. The unspecialized template also sports a 
templarized copy constructor, while the specialized forms have copy 
constructors from the other two specializations, which are explicit 
when a risk of precision loss exists. For the unspecialized form, 
the base type's constructors must not throw.

Destructors and untemplated copy constructors (from the same type) 
are provided by the compiler. Converting copy constructors make use 
of a templated helper function in a "detail" subnamespace.

[h3 Other member functions]

[h4 Real and Unreal Parts]

   T            real()   const;
   octonion<T>  unreal() const;

Like complex number, octonions do have a meaningful notion of "real part", 
but unlike them there is no meaningful notion of "imaginary part". 
Instead there is an "unreal part" which itself is a octonion, 
and usually nothing simpler (as opposed to the complex number case). 
These are returned by the first two functions.

[h4 Individual Real Components]

   T   R_component_1() const;
   T   R_component_2() const;
   T   R_component_3() const;
   T   R_component_4() const;
   T   R_component_5() const;
   T   R_component_6() const;
   T   R_component_7() const;
   T   R_component_8() const;

A octonion having eight real components, these are returned by 
these eight functions. Hence real and R_component_1 return the same value.

[h4 Individual Complex Components]

   ::std::complex<T> C_component_1() const;
   ::std::complex<T> C_component_2() const;
   ::std::complex<T> C_component_3() const;
   ::std::complex<T> C_component_4() const;

A octonion likewise has four complex components. Actually, octonions 
are indeed a (left) vector field over the complexes, but beware, as 
for any octonion __oct_formula we also have __oct_complex_formula 
(note the [*minus] sign in the last factor). 
What the C_component_n functions return, however, are the complexes 
which could be used to build the octonion using the constructor, and 
[*not] the components of the octonion on the basis ['[^(1, j, e', j')]].

[h4 Individual Quaternion Components]

   ::boost::math::quaternion<T> H_component_1() const;
   ::boost::math::quaternion<T> H_component_2() const;

Likewise, for any octonion __oct_formula we also have __oct_quat_formula, though there 
is no meaningful vector-space-like structure based on the quaternions. 
What the H_component_n functions return are the quaternions which 
could be used to build the octonion using the constructor.

[h3 Octonion Member Operators]
[h4 Assignment Operators]

   octonion<T> & operator = (octonion<T> const & a_affecter);
   template<typename X> 
   octonion<T> & operator = (octonion<X> const & a_affecter);
   octonion<T> & operator = (T const & a_affecter);
   octonion<T> & operator = (::std::complex<T> const & a_affecter);
   octonion<T> & operator = (::boost::math::quaternion<T> const & a_affecter);

These perform the expected assignment, with type modification if 
necessary (for instance, assigning from a base type will set the 
real part to that value, and all other components to zero). 
For the unspecialized form, the base type's assignment operators must not throw.

[h4 Other Member Operators]

   octonion<T> & operator += (T const & rhs)
   octonion<T> & operator += (::std::complex<T> const & rhs);
   octonion<T> & operator += (::boost::math::quaternion<T> const & rhs);
   template<typename X> 
   octonion<T> & operator += (octonion<X> const & rhs);

These perform the mathematical operation `(*this)+rhs` and store the result in 
`*this`. The unspecialized form has exception guards, which the specialized 
forms do not, so as to insure exception safety. For the unspecialized form, 
the base type's assignment operators must not throw.

   octonion<T> & operator -= (T const & rhs)
   octonion<T> & operator -= (::std::complex<T> const & rhs);
   octonion<T> & operator -= (::boost::math::quaternion<T> const & rhs);
   template<typename X> 
   octonion<T> & operator -= (octonion<X> const & rhs);

These perform the mathematical operation `(*this)-rhs` and store the result 
in `*this`. The unspecialized form has exception guards, which the 
specialized forms do not, so as to insure exception safety. 
For the unspecialized form, the base type's assignment operators must not throw.

   octonion<T> & operator *= (T const & rhs)
   octonion<T> & operator *= (::std::complex<T> const & rhs);
   octonion<T> & operator *= (::boost::math::quaternion<T> const & rhs);
   template<typename X> 
   octonion<T> & operator *= (octonion<X> const & rhs);

These perform the mathematical operation `(*this)*rhs` in this order 
(order is important as multiplication is not commutative for octonions) 
and store the result in `*this`. The unspecialized form has exception guards, 
which the specialized forms do not, so as to insure exception safety. 
For the unspecialized form, the base type's assignment operators must 
not throw. Also, for clarity's sake, you should always group the 
factors in a multiplication by groups of two, as the multiplication is 
not even associative on the octonions (though there are of course cases 
where this does not matter, it usually does).

   octonion<T> & operator /= (T const & rhs)
   octonion<T> & operator /= (::std::complex<T> const & rhs);
   octonion<T> & operator /= (::boost::math::quaternion<T> const & rhs);
   template<typename X> 
   octonion<T> & operator /= (octonion<X> const & rhs);

These perform the mathematical operation `(*this)*inverse_of(rhs)` 
in this order (order is important as multiplication is not commutative 
for octonions) and store the result in `*this`. The unspecialized form 
has exception guards, which the specialized forms do not, so as to 
insure exception safety. For the unspecialized form, the base 
type's assignment operators must not throw. As for the multiplication, 
remember to group any two factors using parenthesis.

[endsect]

[section:non_mem Octonion Non-Member Operators]

[h4 Unary Plus and Minus Operators]

   template<typename T> octonion<T> operator + (octonion<T> const & o);

This unary operator simply returns o.

   template<typename T> octonion<T> operator - (octonion<T> const & o);

This unary operator returns the opposite of o.

[h4 Binary Addition Operators]

   template<typename T> octonion<T> operator + (T const & lhs, octonion<T> const & rhs);
   template<typename T> octonion<T> operator + (octonion<T> const & lhs, T const & rhs);
   template<typename T> octonion<T> operator + (::std::complex<T> const & lhs, octonion<T> const & rhs);
   template<typename T> octonion<T> operator + (octonion<T> const & lhs, ::std::complex<T> const & rhs);
   template<typename T> octonion<T> operator + (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
   template<typename T> octonion<T> operator + (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
   template<typename T> octonion<T> operator + (octonion<T> const & lhs, octonion<T> const & rhs);

These operators return `octonion<T>(lhs) += rhs`.

[h4 Binary Subtraction Operators]

   template<typename T> octonion<T> operator - (T const & lhs, octonion<T> const & rhs);
   template<typename T> octonion<T> operator - (octonion<T> const & lhs, T const & rhs);
   template<typename T> octonion<T> operator - (::std::complex<T> const & lhs, octonion<T> const & rhs);
   template<typename T> octonion<T> operator - (octonion<T> const & lhs, ::std::complex<T> const & rhs);
   template<typename T> octonion<T> operator - (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
   template<typename T> octonion<T> operator - (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
   template<typename T> octonion<T> operator - (octonion<T> const & lhs, octonion<T> const & rhs);

These operators return `octonion<T>(lhs) -= rhs`.

[h4 Binary Multiplication Operators]

   template<typename T> octonion<T> operator * (T const & lhs, octonion<T> const & rhs);
   template<typename T> octonion<T> operator * (octonion<T> const & lhs, T const & rhs);
   template<typename T> octonion<T> operator * (::std::complex<T> const & lhs, octonion<T> const & rhs);
   template<typename T> octonion<T> operator * (octonion<T> const & lhs, ::std::complex<T> const & rhs);
   template<typename T> octonion<T> operator * (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
   template<typename T> octonion<T> operator * (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
   template<typename T> octonion<T> operator * (octonion<T> const & lhs, octonion<T> const & rhs);

These operators return `octonion<T>(lhs) *= rhs`.

[h4 Binary Division Operators]

   template<typename T> octonion<T> operator / (T const & lhs, octonion<T> const & rhs);
   template<typename T> octonion<T> operator / (octonion<T> const & lhs, T const & rhs);
   template<typename T> octonion<T> operator / (::std::complex<T> const & lhs, octonion<T> const & rhs);
   template<typename T> octonion<T> operator / (octonion<T> const & lhs, ::std::complex<T> const & rhs);
   template<typename T> octonion<T> operator / (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
   template<typename T> octonion<T> operator / (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
   template<typename T> octonion<T> operator / (octonion<T> const & lhs, octonion<T> const & rhs);

These operators return `octonion<T>(lhs) /= rhs`. It is of course still an 
error to divide by zero...

[h4 Binary Equality Operators]

   template<typename T> bool operator == (T const & lhs, octonion<T> const & rhs);
   template<typename T> bool operator == (octonion<T> const & lhs, T const & rhs);
   template<typename T> bool operator == (::std::complex<T> const & lhs, octonion<T> const & rhs);
   template<typename T> bool operator == (octonion<T> const & lhs, ::std::complex<T> const & rhs);
   template<typename T> bool operator == (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
   template<typename T> bool operator == (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
   template<typename T> bool operator == (octonion<T> const & lhs, octonion<T> const & rhs);

These return true if and only if the four components of `octonion<T>(lhs)` 
are equal to their counterparts in `octonion<T>(rhs)`. As with any 
floating-type entity, this is essentially meaningless.

[h4 Binary Inequality Operators]

   template<typename T> bool operator != (T const & lhs, octonion<T> const & rhs);
   template<typename T> bool operator != (octonion<T> const & lhs, T const & rhs);
   template<typename T> bool operator != (::std::complex<T> const & lhs, octonion<T> const & rhs);
   template<typename T> bool operator != (octonion<T> const & lhs, ::std::complex<T> const & rhs);
   template<typename T> bool operator != (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
   template<typename T> bool operator != (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
   template<typename T> bool operator != (octonion<T> const & lhs, octonion<T> const & rhs);

These return true if and only if `octonion<T>(lhs) == octonion<T>(rhs)` 
is false. As with any floating-type entity, this is essentially meaningless.

[h4 Stream Extractor]

   template<typename T, typename charT, class traits>
   ::std::basic_istream<charT,traits> & operator >> (::std::basic_istream<charT,traits> & is, octonion<T> & o);

Extracts an octonion `o`. We accept any format which seems reasonable. 
However, since this leads to a great many ambiguities, decisions were made 
to lift these. In case of doubt, stick to lists of reals.

The input values must be convertible to T. If bad input is encountered, 
calls `is.setstate(ios::failbit)` (which may throw `ios::failure` (27.4.5.3)).

Returns `is`.

[h4 Stream Inserter]

   template<typename T, typename charT, class traits>
   ::std::basic_ostream<charT,traits> & operator << (::std::basic_ostream<charT,traits> & os, octonion<T> const & o);

Inserts the octonion `o` onto the stream `os` as if it were implemented as follows:

   template<typename T, typename charT, class traits>
   ::std::basic_ostream<charT,traits> & operator << ( ::std::basic_ostream<charT,traits> & os,
   octonion<T> const & o)
   {
      ::std::basic_ostringstream<charT,traits> s;

      s.flags(os.flags());
      s.imbue(os.getloc());
      s.precision(os.precision());

      s << '(' << o.R_component_1() << ','
         << o.R_component_2() << ','
         << o.R_component_3() << ','
         << o.R_component_4() << ','
         << o.R_component_5() << ','
         << o.R_component_6() << ','
         << o.R_component_7() << ','
         << o.R_component_8() << ')';

      return os << s.str();
   }
   
[endsect]

[section Octonion Value Operations]

[h4 Real and Unreal]

   template<typename T> T  real(octonion<T> const & o);
   template<typename T> octonion<T> unreal(octonion<T> const & o);

These return `o.real()` and `o.unreal()` respectively.

[h4 conj]

   template<typename T> octonion<T> conj(octonion<T> const & o);

This returns the conjugate of the octonion.

[h4 sup]

   template<typename T> T   sup(octonion<T> const & o);

This return the sup norm (the greatest among 
`abs(o.R_component_1())...abs(o.R_component_8()))` of the octonion.

[h4 l1]

   template<typename T> T   l1(octonion<T> const & o);

This return the l1 norm (`abs(o.R_component_1())+...+abs(o.R_component_8())`) 
of the octonion.

[h4 abs]

   template<typename T> T   abs(octonion<T> const & o);

This return the magnitude (Euclidian norm) of the octonion.

[h4 norm]

   template<typename T> T  norm(octonion<T>const  & o);

This return the (Cayley) norm of the octonion. The term "norm" might 
be confusing, as most people associate it with the Euclidian norm 
(and quadratic functionals). For this version of (the mathematical 
objects known as) octonions, the Euclidian norm (also known as 
magnitude) is the square root of the Cayley norm.

[endsect]

[section:oct_create Octonion Creation Functions]

   template<typename T> octonion<T> spherical(T const & rho, T const & theta, T const & phi1, T const & phi2, T const & phi3, T const & phi4, T const & phi5, T const & phi6);
   template<typename T> octonion<T> multipolar(T const & rho1, T const & theta1, T const & rho2, T const & theta2, T const & rho3, T const & theta3, T const & rho4, T const & theta4);
   template<typename T> octonion<T> cylindrical(T const & r, T const & angle, T const & h1, T const & h2, T const & h3, T const & h4, T const & h5, T const & h6);

These build octonions in a way similar to the way polar builds 
complex numbers, as there is no strict equivalent to 
polar coordinates for octonions.

`spherical` is a simple transposition of `polar`, it takes as inputs a 
(positive) magnitude and a point on the hypersphere, given 
by three angles. The first of these, ['theta] has a natural range of 
-pi to +pi, and the other two have natural ranges of 
-pi/2 to +pi/2 (as is the case with the usual spherical 
coordinates in __R3). Due to the many symmetries and periodicities, 
nothing untoward happens if the magnitude is negative or the angles are 
outside their natural ranges. The expected degeneracies (a magnitude of 
zero ignores the angles settings...) do happen however.

`cylindrical` is likewise a simple transposition of the usual 
cylindrical coordinates in __R3, which in turn is another derivative of 
planar polar coordinates. The first two inputs are the polar 
coordinates of the first __C component of the octonion. The third and 
fourth inputs are placed into the third and fourth __R components of the 
octonion, respectively.

`multipolar` is yet another simple generalization of polar coordinates. 
This time, both __C components of the octonion are given in polar coordinates.

In this version of our implementation of octonions, there is no 
analogue of the complex value operation arg as the situation is 
somewhat more complicated.

[endsect]

[section Octonions Transcendentals]

There is no `log` or `sqrt` provided for octonions in this implementation, 
and `pow` is likewise restricted to integral powers of the exponent. 
There are several reasons to this: on the one hand, the equivalent of 
analytic continuation for octonions ("branch cuts") remains to be 
investigated thoroughly (by me, at any rate...), and we wish to avoid 
the nonsense introduced in the standard by exponentiations of 
complexes by complexes (which is well defined, but not in the standard...). 
Talking of nonsense, saying that `pow(0,0)` is "implementation defined" is 
just plain brain-dead...

We do, however provide several transcendentals, chief among which is 
the exponential. That it allows for a "closed formula" is a result 
of the author (the existence and definition of the exponential, on the 
octonions among others, on the other hand, is a few centuries old). 
Basically, any converging power series with real coefficients which 
allows for a closed formula in __C can be transposed to __O. More 
transcendentals of this type could be added in a further revision upon 
request. It should be noted that it is these functions which force the 
dependency upon the 
[@../../../../../boost/math/special_functions/sinc.hpp boost/math/special_functions/sinc.hpp]
and the 
[@../../../../../boost/math/special_functions/sinhc.hpp boost/math/special_functions/sinhc.hpp] 
headers.

[h4 exp]

   template<typename T> 
   octonion<T> exp(octonion<T> const & o);

Computes the exponential of the octonion.

[h4 cos]

   template<typename T> 
   octonion<T> cos(octonion<T> const & o);

Computes the cosine of the octonion

[h4 sin]

   template<typename T> 
   octonion<T> sin(octonion<T> const & o);

Computes the sine of the octonion.

[h4 tan]

   template<typename T> 
   octonion<T> tan(octonion<T> const & o);

Computes the tangent of the octonion.

[h4 cosh]

   template<typename T> 
   octonion<T> cosh(octonion<T> const & o);

Computes the hyperbolic cosine of the octonion.

[h4 sinh]

   template<typename T> 
   octonion<T> sinh(octonion<T> const & o);

Computes the hyperbolic sine of the octonion.

[h4 tanh]

   template<typename T> 
   octonion<T> tanh(octonion<T> const & o);

Computes the hyperbolic tangent of the octonion.

[h4 pow]

   template<typename T> 
   octonion<T>  pow(octonion<T> const & o, int n);

Computes the n-th power of the octonion q.

[endsect]

[section Test Program]

The [@../../../octonion/octonion_test.cpp octonion_test.cpp] 
test program tests octonions specialisations for float, double and long double 
([@../../../octonion/output.txt sample output]).

If you define the symbol BOOST_OCTONION_TEST_VERBOSE, you will get additional 
output ([@../../../octonion/output_more.txt verbose output]); this will 
only be helpfull if you enable message output at the same time, of course 
(by uncommenting the relevant line in the test or by adding --log_level=messages 
to your command line,...). In that case, and if you are running interactively, 
you may in addition define the symbol BOOST_INTERACTIVE_TEST_INPUT_ITERATOR to 
interactively test the input operator with input of your choice from the 
standard input (instead of hard-coding it in the test).

[endsect]

[section Acknowledgements]

The mathematical text has been typeset with 
[@http://www.nisus-soft.com/ Nisus Writer]. 
Jens Maurer has helped with portability and standard adherence, and was the 
Review Manager for this library. More acknowledgements in the 
History section. Thank you to all who contributed to the discussion about this library.

[endsect]

[section History]

* 1.5.8 - 17/12/2005: Converted documentation to Quickbook Format.
* 1.5.7 - 25/02/2003: transitionned to the unit test framework; <boost/config.hpp> now included by the library header (rather than the test files), via <boost/math/quaternion.hpp>.
* 1.5.6 - 15/10/2002: Gcc2.95.x and stlport on linux compatibility by Alkis Evlogimenos (alkis@routescience.com).
* 1.5.5 - 27/09/2002: Microsoft VCPP 7 compatibility, by Michael Stevens (michael@acfr.usyd.edu.au); requires the /Za compiler option.
* 1.5.4 - 19/09/2002: fixed problem with multiple inclusion (in different translation units); attempt at an improved compatibility with Microsoft compilers, by Michael Stevens (michael@acfr.usyd.edu.au) and Fredrik Blomqvist; other compatibility fixes.
* 1.5.3 - 01/02/2002: bugfix and Gcc 2.95.3 compatibility by Douglas Gregor (gregod@cs.rpi.edu).
* 1.5.2 - 07/07/2001: introduced namespace math.
* 1.5.1 - 07/06/2001: (end of Boost review) now includes <boost/math/special_functions/sinc.hpp> and <boost/math/special_functions/sinhc.hpp> instead of <boost/special_functions.hpp>; corrected bug in sin (Daryle Walker); removed check for self-assignment (Gary Powel); made converting functions explicit (Gary Powel); added overflow guards for division operators and abs (Peter Schmitteckert); added sup and l1; used Vesa Karvonen's CPP metaprograming technique to simplify code.
* 1.5.0 - 23/03/2001: boostification, inlining of all operators except input, output and pow, fixed exception safety of some members (template version).
* 1.4.0 - 09/01/2001: added tan and tanh.
* 1.3.1 - 08/01/2001: cosmetic fixes.
* 1.3.0 - 12/07/2000: pow now uses Maarten Hilferink's (mhilferink@tip.nl) algorithm.
* 1.2.0 - 25/05/2000: fixed the division operators and output; changed many signatures.
* 1.1.0 - 23/05/2000: changed sinc into sinc_pi; added sin, cos, sinh, cosh.
* 1.0.0 - 10/08/1999: first public version. 

[endsect]

[section To Do]

* Improve testing.
* Rewrite input operatore using Spirit (creates a dependency).
* Put in place an Expression Template mechanism (perhaps borrowing from uBlas). 

[endsect]

[endsect]
