[/macro definitions specific to octonions] [def __R ['[*R]]] [def __C ['[*C]]] [def __H ['[*H]]] [def __O ['[*O]]] [def __R3 ['[*'''R3''']]] [def __R4 ['[*'''R4''']]] [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]]] [mathpart octonions Octonions] [section:oct_overview Overview] Octonions, like [link quaternions 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:oct_overview Overview] [section:oct_header Header File] The interface and implementation are both supplied by the header file [@../../../../boost/math/octonion.hpp octonion.hpp]. [endsect] [section:oct_synopsis Synopsis] namespace boost{ namespace math{ template class ``[link math_toolkit.octonion octonion]``; template<> class ``[link math_toolkit.oct_specialization octonion]``; template<> class ``[link math_octonion_double octonion]``; template<> class ``[link math_octonion_long_double octonion]``; // operators template octonion ``[link math_toolkit.oct_non_mem.binary_addition_operators operator +]`` (T const & lhs, octonion const & rhs); template octonion ``[link math_toolkit.oct_non_mem.binary_addition_operators operator +]`` (octonion const & lhs, T const & rhs); template octonion ``[link math_toolkit.oct_non_mem.binary_addition_operators operator +]`` (::std::complex const & lhs, octonion const & rhs); template octonion ``[link math_toolkit.oct_non_mem.binary_addition_operators operator +]`` (octonion const & lhs, ::std::complex const & rhs); template octonion ``[link math_toolkit.oct_non_mem.binary_addition_operators operator +]`` (::boost::math::quaternion const & lhs, octonion const & rhs); template octonion ``[link math_toolkit.oct_non_mem.binary_addition_operators operator +]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); template octonion ``[link math_toolkit.oct_non_mem.binary_addition_operators operator +]`` (octonion const & lhs, octonion const & rhs); template octonion ``[link math_toolkit.oct_non_mem.binary_subtraction_operators operator -]`` (T const & lhs, octonion const & rhs); template octonion ``[link math_toolkit.oct_non_mem.binary_subtraction_operators operator -]`` (octonion const & lhs, T const & rhs); template octonion ``[link math_toolkit.oct_non_mem.binary_subtraction_operators operator -]`` (::std::complex const & lhs, octonion const & rhs); template octonion ``[link math_toolkit.oct_non_mem.binary_subtraction_operators operator -]`` (octonion const & lhs, ::std::complex const & rhs); template octonion ``[link math_toolkit.oct_non_mem.binary_subtraction_operators operator -]`` (::boost::math::quaternion const & lhs, octonion const & rhs); template octonion ``[link math_toolkit.oct_non_mem.binary_subtraction_operators operator -]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); template octonion ``[link math_toolkit.oct_non_mem.binary_subtraction_operators operator -]`` (octonion const & lhs, octonion const & rhs); template octonion ``[link math_toolkit.oct_non_mem.binary_multiplication_operators operator *]`` (T const & lhs, octonion const & rhs); template octonion ``[link math_toolkit.oct_non_mem.binary_multiplication_operators operator *]`` (octonion const & lhs, T const & rhs); template octonion ``[link math_toolkit.oct_non_mem.binary_multiplication_operators operator *]`` (::std::complex const & lhs, octonion const & rhs); template octonion ``[link math_toolkit.oct_non_mem.binary_multiplication_operators operator *]`` (octonion const & lhs, ::std::complex const & rhs); template octonion ``[link math_toolkit.oct_non_mem.binary_multiplication_operators operator *]`` (::boost::math::quaternion const & lhs, octonion const & rhs); template octonion ``[link math_toolkit.oct_non_mem.binary_multiplication_operators operator *]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); template octonion ``[link math_toolkit.oct_non_mem.binary_multiplication_operators operator *]`` (octonion const & lhs, octonion const & rhs); template octonion ``[link math_toolkit.oct_non_mem.binary_division_operators operator /]`` (T const & lhs, octonion const & rhs); template octonion ``[link math_toolkit.oct_non_mem.binary_division_operators operator /]`` (octonion const & lhs, T const & rhs); template octonion ``[link math_toolkit.oct_non_mem.binary_division_operators operator /]`` (::std::complex const & lhs, octonion const & rhs); template octonion ``[link math_toolkit.oct_non_mem.binary_division_operators operator /]`` (octonion const & lhs, ::std::complex const & rhs); template octonion ``[link math_toolkit.oct_non_mem.binary_division_operators operator /]`` (::boost::math::quaternion const & lhs, octonion const & rhs); template octonion ``[link math_toolkit.oct_non_mem.binary_division_operators operator /]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); template octonion ``[link math_toolkit.oct_non_mem.binary_division_operators operator /]`` (octonion const & lhs, octonion const & rhs); template octonion ``[link math_toolkit.oct_non_mem.unary_plus_and_minus_operators operator +]`` (octonion const & o); template octonion ``[link math_toolkit.oct_non_mem.unary_plus_and_minus_operators operator -]`` (octonion const & o); template bool ``[link math_toolkit.oct_non_mem.binary_equality_operators operator ==]`` (T const & lhs, octonion const & rhs); template bool ``[link math_toolkit.oct_non_mem.binary_equality_operators operator ==]`` (octonion const & lhs, T const & rhs); template bool ``[link math_toolkit.oct_non_mem.binary_equality_operators operator ==]`` (::std::complex const & lhs, octonion const & rhs); template bool ``[link math_toolkit.oct_non_mem.binary_equality_operators operator ==]`` (octonion const & lhs, ::std::complex const & rhs); template bool ``[link math_toolkit.oct_non_mem.binary_equality_operators operator ==]`` (::boost::math::quaternion const & lhs, octonion const & rhs); template bool ``[link math_toolkit.oct_non_mem.binary_equality_operators operator ==]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); template bool ``[link math_toolkit.oct_non_mem.binary_equality_operators operator ==]`` (octonion const & lhs, octonion const & rhs); template bool ``[link math_toolkit.oct_non_mem.binary_inequality_operators operator !=]`` (T const & lhs, octonion const & rhs); template bool ``[link math_toolkit.oct_non_mem.binary_inequality_operators operator !=]`` (octonion const & lhs, T const & rhs); template bool ``[link math_toolkit.oct_non_mem.binary_inequality_operators operator !=]`` (::std::complex const & lhs, octonion const & rhs); template bool ``[link math_toolkit.oct_non_mem.binary_inequality_operators operator !=]`` (octonion const & lhs, ::std::complex const & rhs); template bool ``[link math_toolkit.oct_non_mem.binary_inequality_operators operator !=]`` (::boost::math::quaternion const & lhs, octonion const & rhs); template bool ``[link math_toolkit.oct_non_mem.binary_inequality_operators operator !=]`` (octonion const & lhs, ::boost::math::quaternion const & rhs); template bool ``[link math_toolkit.oct_non_mem.binary_inequality_operators operator !=]`` (octonion const & lhs, octonion const & rhs); template ::std::basic_istream & ``[link math_toolkit.oct_non_mem.stream_extractor operator >>]`` (::std::basic_istream & is, octonion & o); template ::std::basic_ostream & ``[link math_toolkit.oct_non_mem.stream_inserter operator <<]`` (::std::basic_ostream & os, octonion const & o); // values template T ``[link math_toolkit.oct_value_ops.real_and_unreal real]``(octonion const & o); template octonion ``[link math_toolkit.oct_value_ops.real_and_unreal unreal]``(octonion const & o); template T ``[link math_toolkit.oct_value_ops.sup sup]``(octonion const & o); template T ``[link math_toolkit.oct_value_ops.l1 l1]``(octonionconst & o); template T ``[link math_toolkit.oct_value_ops.abs abs]``(octonion const & o); template T ``[link math_toolkit.oct_value_ops.norm norm]``(octonionconst & o); template octonion ``[link math_toolkit.oct_value_ops.conj conj]``(octonion const & o); template octonion ``[link math_toolkit.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 octonion ``[link math_toolkit.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 octonion ``[link math_toolkit.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 octonion ``[link math_toolkit.oct_trans.exp exp]``(octonion const & o); template octonion ``[link math_toolkit.oct_trans.cos cos]``(octonion const & o); template octonion ``[link math_toolkit.oct_trans.sin sin]``(octonion const & o); template octonion ``[link math_toolkit.oct_trans.tan tan]``(octonion const & o); template octonion ``[link math_toolkit.oct_trans.cosh cosh]``(octonion const & o); template octonion ``[link math_toolkit.oct_trans.sinh sinh]``(octonion const & o); template octonion ``[link math_toolkit.oct_trans.tanh tanh]``(octonion const & o); template octonion ``[link math_toolkit.oct_trans.pow pow]``(octonion const & o, int n); } } // namespaces [endsect] [/section:oct_header Header File] [section:octonion Template Class octonion] namespace boost{ namespace math { template class octonion { public: typedef T value_type; explicit ``[link math_toolkit.oct_mem_fun.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 math_toolkit.oct_mem_fun.constructors octonion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex(), ::std::complex const & z2 = ::std::complex(), ::std::complex const & z3 = ::std::complex()); explicit ``[link math_toolkit.oct_mem_fun.constructors octonion]``(::boost::math::quaternion const & q0, ::boost::math::quaternion const & q1 = ::boost::math::quaternion()); template explicit ``[link math_toolkit.oct_mem_fun.constructors octonion]``(octonion const & a_recopier); T ``[link math_toolkit.oct_mem_fun.real_and_unreal_parts real]``() const; octonion ``[link math_toolkit.oct_mem_fun.real_and_unreal_parts unreal]``() const; T ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_1]``() const; T ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_2]``() const; T ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_3]``() const; T ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_4]``() const; T ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_5]``() const; T ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_6]``() const; T ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_7]``() const; T ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_8]``() const; ::std::complex ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_1]``() const; ::std::complex ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_2]``() const; ::std::complex ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_3]``() const; ::std::complex ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_4]``() const; ::boost::math::quaternion ``[link math_toolkit.oct_mem_fun.individual_quaternion_components H_component_1]``() const; ::boost::math::quaternion ``[link math_toolkit.oct_mem_fun.individual_quaternion_components H_component_2]``() const; octonion & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (octonion const & a_affecter); template octonion & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (octonion const & a_affecter); octonion & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (T const & a_affecter); octonion & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (::std::complex const & a_affecter); octonion & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (::boost::math::quaternion const & a_affecter); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (T const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (::std::complex const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (::boost::math::quaternion const & rhs); template octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (octonion const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (T const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (::std::complex const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (::boost::math::quaternion const & rhs); template octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (octonion const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (T const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (::std::complex const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (::boost::math::quaternion const & rhs); template octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (octonion const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (T const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (::std::complex const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (::boost::math::quaternion const & rhs); template octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (octonion const & rhs); }; } } // namespaces [endsect] [/section:octonion Template Class octonion] [section:oct_specialization Octonion Specializations] namespace boost{ namespace math{ template<> class octonion { public: typedef float value_type; explicit ``[link math_toolkit.oct_mem_fun.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 math_toolkit.oct_mem_fun.constructors octonion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex(), ::std::complex const & z2 = ::std::complex(), ::std::complex const & z3 = ::std::complex()); explicit ``[link math_toolkit.oct_mem_fun.constructors octonion]``(::boost::math::quaternion const & q0, ::boost::math::quaternion const & q1 = ::boost::math::quaternion()); explicit ``[link math_toolkit.oct_mem_fun.constructors octonion]``(octonion const & a_recopier); explicit ``[link math_toolkit.oct_mem_fun.constructors octonion]``(octonion const & a_recopier); float ``[link math_toolkit.oct_mem_fun.real_and_unreal_parts real]``() const; octonion ``[link math_toolkit.oct_mem_fun.real_and_unreal_parts unreal]``() const; float ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_1]``() const; float ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_2]``() const; float ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_3]``() const; float ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_4]``() const; float ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_5]``() const; float ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_6]``() const; float ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_7]``() const; float ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_8]``() const; ::std::complex ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_1]``() const; ::std::complex ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_2]``() const; ::std::complex ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_3]``() const; ::std::complex ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_4]``() const; ::boost::math::quaternion ``[link math_toolkit.oct_mem_fun.individual_quaternion_components H_component_1]``() const; ::boost::math::quaternion ``[link math_toolkit.oct_mem_fun.individual_quaternion_components H_component_2]``() const; octonion & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (octonion const & a_affecter); template octonion & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (octonion const & a_affecter); octonion & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (float const & a_affecter); octonion & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (::std::complex const & a_affecter); octonion & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (::boost::math::quaternion const & a_affecter); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (float const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (::std::complex const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (::boost::math::quaternion const & rhs); template octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (octonion const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (float const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (::std::complex const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (::boost::math::quaternion const & rhs); template octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (octonion const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (float const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (::std::complex const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (::boost::math::quaternion const & rhs); template octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (octonion const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (float const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (::std::complex const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (::boost::math::quaternion const & rhs); template octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (octonion const & rhs); }; [#math_octonion_double] template<> class octonion { public: typedef double value_type; explicit ``[link math_toolkit.oct_mem_fun.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 math_toolkit.oct_mem_fun.constructors octonion]``(::std::complex const & z0, ::std::complex const & z1 = ::std::complex(), ::std::complex const & z2 = ::std::complex(), ::std::complex const & z3 = ::std::complex()); explicit ``[link math_toolkit.oct_mem_fun.constructors octonion]``(::boost::math::quaternion const & q0, ::boost::math::quaternion const & q1 = ::boost::math::quaternion()); explicit ``[link math_toolkit.oct_mem_fun.constructors octonion]``(octonion const & a_recopier); explicit ``[link math_toolkit.oct_mem_fun.constructors octonion]``(octonion const & a_recopier); double ``[link math_toolkit.oct_mem_fun.real_and_unreal_parts real]``() const; octonion ``[link math_toolkit.oct_mem_fun.real_and_unreal_parts unreal]``() const; double ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_1]``() const; double ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_2]``() const; double ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_3]``() const; double ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_4]``() const; double ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_5]``() const; double ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_6]``() const; double ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_7]``() const; double ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_8]``() const; ::std::complex ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_1]``() const; ::std::complex ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_2]``() const; ::std::complex ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_3]``() const; ::std::complex ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_4]``() const; ::boost::math::quaternion ``[link math_toolkit.oct_mem_fun.individual_quaternion_components H_component_1]``() const; ::boost::math::quaternion ``[link math_toolkit.oct_mem_fun.individual_quaternion_components H_component_2]``() const; octonion & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (octonion const & a_affecter); template octonion & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (octonion const & a_affecter); octonion & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (double const & a_affecter); octonion & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (::std::complex const & a_affecter); octonion & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (::boost::math::quaternion const & a_affecter); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (double const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (::std::complex const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (::boost::math::quaternion const & rhs); template octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (octonion const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (double const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (::std::complex const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (::boost::math::quaternion const & rhs); template octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (octonion const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (double const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (::std::complex const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (::boost::math::quaternion const & rhs); template octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (octonion const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (double const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (::std::complex const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (::boost::math::quaternion const & rhs); template octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (octonion const & rhs); }; [#math_octonion_long_double] template<> class octonion { public: typedef long double value_type; explicit ``[link math_toolkit.oct_mem_fun.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 math_toolkit.oct_mem_fun.constructors octonion]``( ::std::complex const & z0, ::std::complex const & z1 = ::std::complex(), ::std::complex const & z2 = ::std::complex(), ::std::complex const & z3 = ::std::complex()); explicit ``[link math_toolkit.oct_mem_fun.constructors octonion]``( ::boost::math::quaternion const & q0, ::boost::math::quaternion const & z1 = ::boost::math::quaternion()); explicit ``[link math_toolkit.oct_mem_fun.constructors octonion]``(octonion const & a_recopier); explicit ``[link math_toolkit.oct_mem_fun.constructors octonion]``(octonion const & a_recopier); long double ``[link math_toolkit.oct_mem_fun.real_and_unreal_parts real]``() const; octonion ``[link math_toolkit.oct_mem_fun.real_and_unreal_parts unreal]``() const; long double ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_1]``() const; long double ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_2]``() const; long double ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_3]``() const; long double ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_4]``() const; long double ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_5]``() const; long double ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_6]``() const; long double ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_7]``() const; long double ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_8]``() const; ::std::complex ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_1]``() const; ::std::complex ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_2]``() const; ::std::complex ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_3]``() const; ::std::complex ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_4]``() const; ::boost::math::quaternion ``[link math_toolkit.oct_mem_fun.individual_quaternion_components H_component_1]``() const; ::boost::math::quaternion ``[link math_toolkit.oct_mem_fun.individual_quaternion_components H_component_2]``() const; octonion & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (octonion const & a_affecter); template octonion & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (octonion const & a_affecter); octonion & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (long double const & a_affecter); octonion & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (::std::complex const & a_affecter); octonion & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (::boost::math::quaternion const & a_affecter); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (long double const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (::std::complex const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (::boost::math::quaternion const & rhs); template octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (octonion const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (long double const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (::std::complex const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (::boost::math::quaternion const & rhs); template octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (octonion const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (long double const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (::std::complex const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (::boost::math::quaternion const & rhs); template octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (octonion const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (long double const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (::std::complex const & rhs); octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (::boost::math::quaternion const & rhs); template octonion & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (octonion const & rhs); }; } } // namespaces [endsect] [/section:oct_specialization Octonion Specializations] [section:oct_typedefs 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 access to the type the template is built upon. [endsect] [/section:oct_typedefs Octonion Member Typedefs] [section:oct_mem_fun 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 const & z0, ::std::complex const & z1 = ::std::complex(), ::std::complex const & z2 = ::std::complex(), ::std::complex const & z3 = ::std::complex()); explicit octonion(::boost::math::quaternion const & q0, ::boost::math::quaternion const & q1 = ::boost::math::quaternion()); template explicit octonion(octonion 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 const & z0, ::std::complex const & z1 = ::std::complex(), ::std::complex const & z2 = ::std::complex(), ::std::complex const & z3 = ::std::complex()); explicit octonion(::boost::math::quaternion const & q0, ::boost::math::quaternion const & q1 = ::boost::math::quaternion()); explicit octonion(octonion const & a_recopier); explicit octonion(octonion 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 const & z0, ::std::complex const & z1 = ::std::complex(), ::std::complex const & z2 = ::std::complex(), ::std::complex const & z3 = ::std::complex()); explicit octonion(::boost::math::quaternion const & q0, ::boost::math::quaternion const & q1 = ::boost::math::quaternion()); explicit octonion(octonion const & a_recopier); explicit octonion(octonion 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 const & z0, ::std::complex const & z1 = ::std::complex(), ::std::complex const & z2 = ::std::complex(), ::std::complex const & z3 = ::std::complex()); explicit octonion(::boost::math::quaternion const & q0, ::boost::math::quaternion const & q1 = ::boost::math::quaternion()); explicit octonion(octonion const & a_recopier); explicit octonion(octonion 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 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 C_component_1() const; ::std::complex C_component_2() const; ::std::complex C_component_3() const; ::std::complex 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 H_component_1() const; ::boost::math::quaternion 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 & operator = (octonion const & a_affecter); template octonion & operator = (octonion const & a_affecter); octonion & operator = (T const & a_affecter); octonion & operator = (::std::complex const & a_affecter); octonion & operator = (::boost::math::quaternion 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 & operator += (T const & rhs) octonion & operator += (::std::complex const & rhs); octonion & operator += (::boost::math::quaternion const & rhs); template octonion & operator += (octonion 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 & operator -= (T const & rhs) octonion & operator -= (::std::complex const & rhs); octonion & operator -= (::boost::math::quaternion const & rhs); template octonion & operator -= (octonion 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 & operator *= (T const & rhs) octonion & operator *= (::std::complex const & rhs); octonion & operator *= (::boost::math::quaternion const & rhs); template octonion & operator *= (octonion 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 & operator /= (T const & rhs) octonion & operator /= (::std::complex const & rhs); octonion & operator /= (::boost::math::quaternion const & rhs); template octonion & operator /= (octonion 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:oct_mem_fun Octonion Member Functions] [section:oct_non_mem Octonion Non-Member Operators] [h4 Unary Plus and Minus Operators] template octonion operator + (octonion const & o); This unary operator simply returns o. template octonion operator - (octonion const & o); This unary operator returns the opposite of o. [h4 Binary Addition Operators] template octonion operator + (T const & lhs, octonion const & rhs); template octonion operator + (octonion const & lhs, T const & rhs); template octonion operator + (::std::complex const & lhs, octonion const & rhs); template octonion operator + (octonion const & lhs, ::std::complex const & rhs); template octonion operator + (::boost::math::quaternion const & lhs, octonion const & rhs); template octonion operator + (octonion const & lhs, ::boost::math::quaternion const & rhs); template octonion operator + (octonion const & lhs, octonion const & rhs); These operators return `octonion(lhs) += rhs`. [h4 Binary Subtraction Operators] template octonion operator - (T const & lhs, octonion const & rhs); template octonion operator - (octonion const & lhs, T const & rhs); template octonion operator - (::std::complex const & lhs, octonion const & rhs); template octonion operator - (octonion const & lhs, ::std::complex const & rhs); template octonion operator - (::boost::math::quaternion const & lhs, octonion const & rhs); template octonion operator - (octonion const & lhs, ::boost::math::quaternion const & rhs); template octonion operator - (octonion const & lhs, octonion const & rhs); These operators return `octonion(lhs) -= rhs`. [h4 Binary Multiplication Operators] template octonion operator * (T const & lhs, octonion const & rhs); template octonion operator * (octonion const & lhs, T const & rhs); template octonion operator * (::std::complex const & lhs, octonion const & rhs); template octonion operator * (octonion const & lhs, ::std::complex const & rhs); template octonion operator * (::boost::math::quaternion const & lhs, octonion const & rhs); template octonion operator * (octonion const & lhs, ::boost::math::quaternion const & rhs); template octonion operator * (octonion const & lhs, octonion const & rhs); These operators return `octonion(lhs) *= rhs`. [h4 Binary Division Operators] template octonion operator / (T const & lhs, octonion const & rhs); template octonion operator / (octonion const & lhs, T const & rhs); template octonion operator / (::std::complex const & lhs, octonion const & rhs); template octonion operator / (octonion const & lhs, ::std::complex const & rhs); template octonion operator / (::boost::math::quaternion const & lhs, octonion const & rhs); template octonion operator / (octonion const & lhs, ::boost::math::quaternion const & rhs); template octonion operator / (octonion const & lhs, octonion const & rhs); These operators return `octonion(lhs) /= rhs`. It is of course still an error to divide by zero... [h4 Binary Equality Operators] template bool operator == (T const & lhs, octonion const & rhs); template bool operator == (octonion const & lhs, T const & rhs); template bool operator == (::std::complex const & lhs, octonion const & rhs); template bool operator == (octonion const & lhs, ::std::complex const & rhs); template bool operator == (::boost::math::quaternion const & lhs, octonion const & rhs); template bool operator == (octonion const & lhs, ::boost::math::quaternion const & rhs); template bool operator == (octonion const & lhs, octonion const & rhs); These return true if and only if the four components of `octonion(lhs)` are equal to their counterparts in `octonion(rhs)`. As with any floating-type entity, this is essentially meaningless. [h4 Binary Inequality Operators] template bool operator != (T const & lhs, octonion const & rhs); template bool operator != (octonion const & lhs, T const & rhs); template bool operator != (::std::complex const & lhs, octonion const & rhs); template bool operator != (octonion const & lhs, ::std::complex const & rhs); template bool operator != (::boost::math::quaternion const & lhs, octonion const & rhs); template bool operator != (octonion const & lhs, ::boost::math::quaternion const & rhs); template bool operator != (octonion const & lhs, octonion const & rhs); These return true if and only if `octonion(lhs) == octonion(rhs)` is false. As with any floating-type entity, this is essentially meaningless. [h4 Stream Extractor] template ::std::basic_istream & operator >> (::std::basic_istream & is, octonion & 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 ::std::basic_ostream & operator << (::std::basic_ostream & os, octonion const & o); Inserts the octonion `o` onto the stream `os` as if it were implemented as follows: template ::std::basic_ostream & operator << ( ::std::basic_ostream & os, octonion const & o) { ::std::basic_ostringstream 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:oct_non_mem Octonion Non-Member Operators] [section:oct_value_ops Octonion Value Operations] [h4 Real and Unreal] template T real(octonion const & o); template octonion unreal(octonion const & o); These return `o.real()` and `o.unreal()` respectively. [h4 conj] template octonion conj(octonion const & o); This returns the conjugate of the octonion. [h4 sup] template T sup(octonion 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 T l1(octonion const & o); This return the l1 norm (`abs(o.R_component_1())+...+abs(o.R_component_8())`) of the octonion. [h4 abs] template T abs(octonion const & o); This return the magnitude (Euclidean norm) of the octonion. [h4 norm] template T norm(octonionconst & o); This return the (Cayley) norm of the octonion. The term "norm" might be confusing, as most people associate it with the Euclidean norm (and quadratic functionals). For this version of (the mathematical objects known as) octonions, the Euclidean norm (also known as magnitude) is the square root of the Cayley norm. [endsect] [/section:oct_value_ops Octonion Value Operations] [section:oct_create Octonion Creation Functions] template octonion 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 octonion 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 octonion 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:oct_create Octonion Creation Functions] [section:oct_trans 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 octonion exp(octonion const & o); Computes the exponential of the octonion. [h4 cos] template octonion cos(octonion const & o); Computes the cosine of the octonion [h4 sin] template octonion sin(octonion const & o); Computes the sine of the octonion. [h4 tan] template octonion tan(octonion const & o); Computes the tangent of the octonion. [h4 cosh] template octonion cosh(octonion const & o); Computes the hyperbolic cosine of the octonion. [h4 sinh] template octonion sinh(octonion const & o); Computes the hyperbolic sine of the octonion. [h4 tanh] template octonion tanh(octonion const & o); Computes the hyperbolic tangent of the octonion. [h4 pow] template octonion pow(octonion const & o, int n); Computes the n-th power of the octonion q. [endsect] [section:oct_tests Test Program] The [@../../test/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 helpful 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:oct_trans Octonions Transcendentals] [section:acknowledgements 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:acknowledgements Acknowledgements] [section:oct_history History] * 1.5.9 - 13/5/2013: Incorporated into Boost.Math. * 1.5.8 - 17/12/2005: Converted documentation to Quickbook Format. * 1.5.7 - 25/02/2003: transitioned to the unit test framework; now included by the library header (rather than the test files), via . * 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 and instead of ; 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:oct_history History] [section:oct_todo To Do] * Improve testing. * Rewrite input operators using Spirit (creates a dependency). * Put in place an Expression Template mechanism (perhaps borrowing from uBlas). [endsect] [/section:oct_todo To Do] [endmathpart] [/ Copyright 1999, 2005, 2013 Hubert Holin. 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). ]