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992 lines
62 KiB
Plaintext
992 lines
62 KiB
Plaintext
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[def __R ['[*R]]]
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[def __C ['[*C]]]
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[def __H ['[*H]]]
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[def __O ['[*O]]]
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[def __R3 ['[*'''R<superscript>3</superscript>''']]]
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[def __R4 ['[*'''R<superscript>4</superscript>''']]]
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[def __octulple ('''α,β,γ,δ,ε,ζ,η,θ''')]
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[def __oct_formula ['[^o = '''α + βi + γj + δk + εe' + ζi' + ηj' + θk' ''']]]
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[def __oct_complex_formula ['[^o = ('''α + βi) + (γ + δi)j + (ε + ζi)e' + (η - θi)j' ''']]]
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[def __oct_quat_formula ['[^o = ('''α + βi + γj + δk) + (ε + ζi + ηj - θj)e' ''']]]
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[def __oct_not_equal ['[^x(yz) '''≠''' (xy)z]]]
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[mathpart octonions Octonions]
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[section:oct_overview Overview]
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Octonions, like [link quaternions quaternions], are a relative of complex numbers.
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Octonions see some use in theoretical physics.
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In practical terms, an octonion is simply an octuple of real numbers __octulple,
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which we can write in the form __oct_formula, where ['[^i]], ['[^j]] and ['[^k]]
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are the same objects as for quaternions, and ['[^e']], ['[^i']], ['[^j']] and ['[^k']]
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are distinct objects which play essentially the same kind of role as ['[^i]] (or ['[^j]] or ['[^k]]).
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Addition and a multiplication is defined on the set of octonions,
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which generalize their quaternionic counterparts. The main novelty this time
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is that [*the multiplication is not only not commutative, is now not even
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associative] (i.e. there are octonions ['[^x]], ['[^y]] and ['[^z]] such that __oct_not_equal).
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A way of remembering things is by using the following multiplication table:
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[$../octonion/graphics/octonion_blurb17.jpeg]
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Octonions (and their kin) are described in far more details in this other
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[@../quaternion/TQE.pdf document]
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(with [@../quaternion/TQE_EA.pdf errata and addenda]).
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Some traditional constructs, such as the exponential, carry over without too
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much change into the realms of octonions, but other, such as taking a square root,
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do not (the fact that the exponential has a closed form is a result of the
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author, but the fact that the exponential exists at all for octonions is known
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since quite a long time ago).
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[endsect]
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[section:oct_header Header File]
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The interface and implementation are both supplied by the header file
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[@../../../../boost/math/octonion.hpp octonion.hpp].
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[endsect]
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[section:oct_synopsis Synopsis]
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namespace boost{ namespace math{
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template<typename T> class ``[link math_toolkit.octonion octonion]``;
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template<> class ``[link math_toolkit.oct_specialization octonion<float>]``;
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template<> class ``[link math_octonion_double octonion<double>]``;
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template<> class ``[link math_octonion_long_double octonion<long double>]``;
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// operators
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_addition_operators operator +]`` (T const & lhs, octonion<T> const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_addition_operators operator +]`` (octonion<T> const & lhs, T const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_addition_operators operator +]`` (::std::complex<T> const & lhs, octonion<T> const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_addition_operators operator +]`` (octonion<T> const & lhs, ::std::complex<T> const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_addition_operators operator +]`` (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_addition_operators operator +]`` (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_addition_operators operator +]`` (octonion<T> const & lhs, octonion<T> const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_subtraction_operators operator -]`` (T const & lhs, octonion<T> const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_subtraction_operators operator -]`` (octonion<T> const & lhs, T const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_subtraction_operators operator -]`` (::std::complex<T> const & lhs, octonion<T> const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_subtraction_operators operator -]`` (octonion<T> const & lhs, ::std::complex<T> const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_subtraction_operators operator -]`` (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_subtraction_operators operator -]`` (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_subtraction_operators operator -]`` (octonion<T> const & lhs, octonion<T> const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_multiplication_operators operator *]`` (T const & lhs, octonion<T> const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_multiplication_operators operator *]`` (octonion<T> const & lhs, T const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_multiplication_operators operator *]`` (::std::complex<T> const & lhs, octonion<T> const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_multiplication_operators operator *]`` (octonion<T> const & lhs, ::std::complex<T> const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_multiplication_operators operator *]`` (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_multiplication_operators operator *]`` (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_multiplication_operators operator *]`` (octonion<T> const & lhs, octonion<T> const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_division_operators operator /]`` (T const & lhs, octonion<T> const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_division_operators operator /]`` (octonion<T> const & lhs, T const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_division_operators operator /]`` (::std::complex<T> const & lhs, octonion<T> const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_division_operators operator /]`` (octonion<T> const & lhs, ::std::complex<T> const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_division_operators operator /]`` (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_division_operators operator /]`` (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.binary_division_operators operator /]`` (octonion<T> const & lhs, octonion<T> const & rhs);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.unary_plus_and_minus_operators operator +]`` (octonion<T> const & o);
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template<typename T> octonion<T> ``[link math_toolkit.oct_non_mem.unary_plus_and_minus_operators operator -]`` (octonion<T> const & o);
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template<typename T> bool ``[link math_toolkit.oct_non_mem.binary_equality_operators operator ==]`` (T const & lhs, octonion<T> const & rhs);
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template<typename T> bool ``[link math_toolkit.oct_non_mem.binary_equality_operators operator ==]`` (octonion<T> const & lhs, T const & rhs);
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template<typename T> bool ``[link math_toolkit.oct_non_mem.binary_equality_operators operator ==]`` (::std::complex<T> const & lhs, octonion<T> const & rhs);
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template<typename T> bool ``[link math_toolkit.oct_non_mem.binary_equality_operators operator ==]`` (octonion<T> const & lhs, ::std::complex<T> const & rhs);
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template<typename T> bool ``[link math_toolkit.oct_non_mem.binary_equality_operators operator ==]`` (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
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template<typename T> bool ``[link math_toolkit.oct_non_mem.binary_equality_operators operator ==]`` (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
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template<typename T> bool ``[link math_toolkit.oct_non_mem.binary_equality_operators operator ==]`` (octonion<T> const & lhs, octonion<T> const & rhs);
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template<typename T> bool ``[link math_toolkit.oct_non_mem.binary_inequality_operators operator !=]`` (T const & lhs, octonion<T> const & rhs);
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template<typename T> bool ``[link math_toolkit.oct_non_mem.binary_inequality_operators operator !=]`` (octonion<T> const & lhs, T const & rhs);
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template<typename T> bool ``[link math_toolkit.oct_non_mem.binary_inequality_operators operator !=]`` (::std::complex<T> const & lhs, octonion<T> const & rhs);
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template<typename T> bool ``[link math_toolkit.oct_non_mem.binary_inequality_operators operator !=]`` (octonion<T> const & lhs, ::std::complex<T> const & rhs);
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template<typename T> bool ``[link math_toolkit.oct_non_mem.binary_inequality_operators operator !=]`` (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
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template<typename T> bool ``[link math_toolkit.oct_non_mem.binary_inequality_operators operator !=]`` (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
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template<typename T> bool ``[link math_toolkit.oct_non_mem.binary_inequality_operators operator !=]`` (octonion<T> const & lhs, octonion<T> const & rhs);
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template<typename T, typename charT, class traits>
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::std::basic_istream<charT,traits> & ``[link math_toolkit.oct_non_mem.stream_extractor operator >>]`` (::std::basic_istream<charT,traits> & is, octonion<T> & o);
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template<typename T, typename charT, class traits>
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::std::basic_ostream<charT,traits> & ``[link math_toolkit.oct_non_mem.stream_inserter operator <<]`` (::std::basic_ostream<charT,traits> & os, octonion<T> const & o);
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// values
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template<typename T> T ``[link math_toolkit.oct_value_ops.real_and_unreal real]``(octonion<T> const & o);
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template<typename T> octonion<T> ``[link math_toolkit.oct_value_ops.real_and_unreal unreal]``(octonion<T> const & o);
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template<typename T> T ``[link math_toolkit.oct_value_ops.sup sup]``(octonion<T> const & o);
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template<typename T> T ``[link math_toolkit.oct_value_ops.l1 l1]``(octonion<T>const & o);
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template<typename T> T ``[link math_toolkit.oct_value_ops.abs abs]``(octonion<T> const & o);
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template<typename T> T ``[link math_toolkit.oct_value_ops.norm norm]``(octonion<T>const & o);
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template<typename T> octonion<T> ``[link math_toolkit.oct_value_ops.conj conj]``(octonion<T> const & o);
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template<typename T> octonion<T> ``[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);
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template<typename T> octonion<T> ``[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);
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template<typename T> octonion<T> ``[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);
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// transcendentals
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template<typename T> octonion<T> ``[link math_toolkit.oct_trans.exp exp]``(octonion<T> const & o);
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template<typename T> octonion<T> ``[link math_toolkit.oct_trans.cos cos]``(octonion<T> const & o);
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template<typename T> octonion<T> ``[link math_toolkit.oct_trans.sin sin]``(octonion<T> const & o);
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template<typename T> octonion<T> ``[link math_toolkit.oct_trans.tan tan]``(octonion<T> const & o);
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template<typename T> octonion<T> ``[link math_toolkit.oct_trans.cosh cosh]``(octonion<T> const & o);
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template<typename T> octonion<T> ``[link math_toolkit.oct_trans.sinh sinh]``(octonion<T> const & o);
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template<typename T> octonion<T> ``[link math_toolkit.oct_trans.tanh tanh]``(octonion<T> const & o);
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template<typename T> octonion<T> ``[link math_toolkit.oct_trans.pow pow]``(octonion<T> const & o, int n);
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} } // namespaces
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[endsect]
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[section:octonion Template Class octonion]
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namespace boost{ namespace math {
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template<typename T>
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class octonion
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{
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public:
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typedef T value_type;
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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());
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explicit ``[link math_toolkit.oct_mem_fun.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>());
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explicit ``[link math_toolkit.oct_mem_fun.constructors octonion]``(::boost::math::quaternion<T> const & q0, ::boost::math::quaternion<T> const & q1 = ::boost::math::quaternion<T>());
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template<typename X>
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explicit ``[link math_toolkit.oct_mem_fun.constructors octonion]``(octonion<X> const & a_recopier);
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T ``[link math_toolkit.oct_mem_fun.real_and_unreal_parts real]``() const;
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octonion<T> ``[link math_toolkit.oct_mem_fun.real_and_unreal_parts unreal]``() const;
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T ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_1]``() const;
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T ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_2]``() const;
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T ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_3]``() const;
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T ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_4]``() const;
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T ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_5]``() const;
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T ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_6]``() const;
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T ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_7]``() const;
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T ``[link math_toolkit.oct_mem_fun.individual_real_components R_component_8]``() const;
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::std::complex<T> ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_1]``() const;
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::std::complex<T> ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_2]``() const;
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::std::complex<T> ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_3]``() const;
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::std::complex<T> ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_4]``() const;
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::boost::math::quaternion<T> ``[link math_toolkit.oct_mem_fun.individual_quaternion_components H_component_1]``() const;
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::boost::math::quaternion<T> ``[link math_toolkit.oct_mem_fun.individual_quaternion_components H_component_2]``() const;
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octonion<T> & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (octonion<T> const & a_affecter);
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template<typename X>
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octonion<T> & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (octonion<X> const & a_affecter);
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octonion<T> & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (T const & a_affecter);
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octonion<T> & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (::std::complex<T> const & a_affecter);
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octonion<T> & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (::boost::math::quaternion<T> const & a_affecter);
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octonion<T> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (T const & rhs);
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octonion<T> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (::std::complex<T> const & rhs);
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octonion<T> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (::boost::math::quaternion<T> const & rhs);
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template<typename X>
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octonion<T> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (octonion<X> const & rhs);
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octonion<T> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (T const & rhs);
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octonion<T> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (::std::complex<T> const & rhs);
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octonion<T> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (::boost::math::quaternion<T> const & rhs);
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template<typename X>
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octonion<T> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (octonion<X> const & rhs);
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octonion<T> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (T const & rhs);
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octonion<T> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (::std::complex<T> const & rhs);
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octonion<T> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (::boost::math::quaternion<T> const & rhs);
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template<typename X>
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octonion<T> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (octonion<X> const & rhs);
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octonion<T> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (T const & rhs);
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octonion<T> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (::std::complex<T> const & rhs);
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octonion<T> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (::boost::math::quaternion<T> const & rhs);
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template<typename X>
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octonion<T> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (octonion<X> const & rhs);
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};
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} } // namespaces
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[endsect]
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[section:oct_specialization Octonion Specializations]
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namespace boost{ namespace math{
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template<>
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class octonion<float>
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{
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public:
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typedef float value_type;
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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);
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explicit ``[link math_toolkit.oct_mem_fun.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>());
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explicit ``[link math_toolkit.oct_mem_fun.constructors octonion]``(::boost::math::quaternion<float> const & q0, ::boost::math::quaternion<float> const & q1 = ::boost::math::quaternion<float>());
|
|
explicit ``[link math_toolkit.oct_mem_fun.constructors octonion]``(octonion<double> const & a_recopier);
|
|
explicit ``[link math_toolkit.oct_mem_fun.constructors octonion]``(octonion<long double> const & a_recopier);
|
|
|
|
float ``[link math_toolkit.oct_mem_fun.real_and_unreal_parts real]``() const;
|
|
octonion<float> ``[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<float> ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_1]``() const;
|
|
::std::complex<float> ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_2]``() const;
|
|
::std::complex<float> ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_3]``() const;
|
|
::std::complex<float> ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_4]``() const;
|
|
|
|
::boost::math::quaternion<float> ``[link math_toolkit.oct_mem_fun.individual_quaternion_components H_component_1]``() const;
|
|
::boost::math::quaternion<float> ``[link math_toolkit.oct_mem_fun.individual_quaternion_components H_component_2]``() const;
|
|
|
|
octonion<float> & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (octonion<float> const & a_affecter);
|
|
template<typename X>
|
|
octonion<float> & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (octonion<X> const & a_affecter);
|
|
octonion<float> & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (float const & a_affecter);
|
|
octonion<float> & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (::std::complex<float> const & a_affecter);
|
|
octonion<float> & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (::boost::math::quaternion<float> const & a_affecter);
|
|
|
|
octonion<float> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (float const & rhs);
|
|
octonion<float> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (::std::complex<float> const & rhs);
|
|
octonion<float> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (::boost::math::quaternion<float> const & rhs);
|
|
template<typename X>
|
|
octonion<float> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (octonion<X> const & rhs);
|
|
|
|
octonion<float> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (float const & rhs);
|
|
octonion<float> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (::std::complex<float> const & rhs);
|
|
octonion<float> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (::boost::math::quaternion<float> const & rhs);
|
|
template<typename X>
|
|
octonion<float> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (octonion<X> const & rhs);
|
|
|
|
octonion<float> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (float const & rhs);
|
|
octonion<float> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (::std::complex<float> const & rhs);
|
|
octonion<float> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (::boost::math::quaternion<float> const & rhs);
|
|
template<typename X>
|
|
octonion<float> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (octonion<X> const & rhs);
|
|
|
|
octonion<float> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (float const & rhs);
|
|
octonion<float> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (::std::complex<float> const & rhs);
|
|
octonion<float> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (::boost::math::quaternion<float> const & rhs);
|
|
template<typename X>
|
|
octonion<float> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (octonion<X> const & rhs);
|
|
};
|
|
|
|
[#math_octonion_double]
|
|
|
|
template<>
|
|
class octonion<double>
|
|
{
|
|
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<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 math_toolkit.oct_mem_fun.constructors octonion]``(::boost::math::quaternion<double> const & q0, ::boost::math::quaternion<double> const & q1 = ::boost::math::quaternion<double>());
|
|
explicit ``[link math_toolkit.oct_mem_fun.constructors octonion]``(octonion<float> const & a_recopier);
|
|
explicit ``[link math_toolkit.oct_mem_fun.constructors octonion]``(octonion<long double> const & a_recopier);
|
|
|
|
double ``[link math_toolkit.oct_mem_fun.real_and_unreal_parts real]``() const;
|
|
octonion<double> ``[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<double> ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_1]``() const;
|
|
::std::complex<double> ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_2]``() const;
|
|
::std::complex<double> ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_3]``() const;
|
|
::std::complex<double> ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_4]``() const;
|
|
|
|
::boost::math::quaternion<double> ``[link math_toolkit.oct_mem_fun.individual_quaternion_components H_component_1]``() const;
|
|
::boost::math::quaternion<double> ``[link math_toolkit.oct_mem_fun.individual_quaternion_components H_component_2]``() const;
|
|
|
|
octonion<double> & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (octonion<double> const & a_affecter);
|
|
template<typename X>
|
|
octonion<double> & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (octonion<X> const & a_affecter);
|
|
octonion<double> & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (double const & a_affecter);
|
|
octonion<double> & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (::std::complex<double> const & a_affecter);
|
|
octonion<double> & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (::boost::math::quaternion<double> const & a_affecter);
|
|
|
|
octonion<double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (double const & rhs);
|
|
octonion<double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (::std::complex<double> const & rhs);
|
|
octonion<double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (::boost::math::quaternion<double> const & rhs);
|
|
template<typename X>
|
|
octonion<double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (octonion<X> const & rhs);
|
|
|
|
octonion<double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (double const & rhs);
|
|
octonion<double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (::std::complex<double> const & rhs);
|
|
octonion<double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (::boost::math::quaternion<double> const & rhs);
|
|
template<typename X>
|
|
octonion<double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (octonion<X> const & rhs);
|
|
|
|
octonion<double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (double const & rhs);
|
|
octonion<double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (::std::complex<double> const & rhs);
|
|
octonion<double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (::boost::math::quaternion<double> const & rhs);
|
|
template<typename X>
|
|
octonion<double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (octonion<X> const & rhs);
|
|
|
|
octonion<double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (double const & rhs);
|
|
octonion<double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (::std::complex<double> const & rhs);
|
|
octonion<double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (::boost::math::quaternion<double> const & rhs);
|
|
template<typename X>
|
|
octonion<double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (octonion<X> const & rhs);
|
|
};
|
|
|
|
[#math_octonion_long_double]
|
|
|
|
template<>
|
|
class octonion<long double>
|
|
{
|
|
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<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 math_toolkit.oct_mem_fun.constructors octonion]``( ::boost::math::quaternion<long double> const & q0, ::boost::math::quaternion<long double> const & z1 = ::boost::math::quaternion<long double>());
|
|
explicit ``[link math_toolkit.oct_mem_fun.constructors octonion]``(octonion<float> const & a_recopier);
|
|
explicit ``[link math_toolkit.oct_mem_fun.constructors octonion]``(octonion<double> const & a_recopier);
|
|
|
|
long double ``[link math_toolkit.oct_mem_fun.real_and_unreal_parts real]``() const;
|
|
octonion<long double> ``[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<long double> ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_1]``() const;
|
|
::std::complex<long double> ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_2]``() const;
|
|
::std::complex<long double> ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_3]``() const;
|
|
::std::complex<long double> ``[link math_toolkit.oct_mem_fun.individual_complex_components C_component_4]``() const;
|
|
|
|
::boost::math::quaternion<long double> ``[link math_toolkit.oct_mem_fun.individual_quaternion_components H_component_1]``() const;
|
|
::boost::math::quaternion<long double> ``[link math_toolkit.oct_mem_fun.individual_quaternion_components H_component_2]``() const;
|
|
|
|
octonion<long double> & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (octonion<long double> const & a_affecter);
|
|
template<typename X>
|
|
octonion<long double> & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (octonion<X> const & a_affecter);
|
|
octonion<long double> & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (long double const & a_affecter);
|
|
octonion<long double> & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (::std::complex<long double> const & a_affecter);
|
|
octonion<long double> & ``[link math_toolkit.oct_mem_fun.assignment_operators operator =]`` (::boost::math::quaternion<long double> const & a_affecter);
|
|
|
|
octonion<long double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (long double const & rhs);
|
|
octonion<long double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (::std::complex<long double> const & rhs);
|
|
octonion<long double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (::boost::math::quaternion<long double> const & rhs);
|
|
template<typename X>
|
|
octonion<long double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator +=]`` (octonion<X> const & rhs);
|
|
|
|
octonion<long double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (long double const & rhs);
|
|
octonion<long double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (::std::complex<long double> const & rhs);
|
|
octonion<long double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (::boost::math::quaternion<long double> const & rhs);
|
|
template<typename X>
|
|
octonion<long double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator -=]`` (octonion<X> const & rhs);
|
|
|
|
octonion<long double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (long double const & rhs);
|
|
octonion<long double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (::std::complex<long double> const & rhs);
|
|
octonion<long double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (::boost::math::quaternion<long double> const & rhs);
|
|
template<typename X>
|
|
octonion<long double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator *=]`` (octonion<X> const & rhs);
|
|
|
|
octonion<long double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (long double const & rhs);
|
|
octonion<long double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (::std::complex<long double> const & rhs);
|
|
octonion<long double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (::boost::math::quaternion<long double> const & rhs);
|
|
template<typename X>
|
|
octonion<long double> & ``[link math_toolkit.oct_mem_fun.other_member_operators operator /=]`` (octonion<X> const & rhs);
|
|
};
|
|
|
|
} } // namespaces
|
|
|
|
[endsect]
|
|
|
|
[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 acces to the type the template is built upon.
|
|
|
|
[endsect]
|
|
|
|
[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<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:oct_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:oct_value_ops 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: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.
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[h4 exp]
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template<typename T>
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octonion<T> exp(octonion<T> const & o);
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Computes the exponential of the octonion.
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[h4 cos]
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template<typename T>
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octonion<T> cos(octonion<T> const & o);
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Computes the cosine of the octonion
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[h4 sin]
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template<typename T>
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octonion<T> sin(octonion<T> const & o);
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Computes the sine of the octonion.
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[h4 tan]
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template<typename T>
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octonion<T> tan(octonion<T> const & o);
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Computes the tangent of the octonion.
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[h4 cosh]
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template<typename T>
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octonion<T> cosh(octonion<T> const & o);
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Computes the hyperbolic cosine of the octonion.
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[h4 sinh]
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template<typename T>
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octonion<T> sinh(octonion<T> const & o);
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Computes the hyperbolic sine of the octonion.
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[h4 tanh]
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template<typename T>
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octonion<T> tanh(octonion<T> const & o);
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Computes the hyperbolic tangent of the octonion.
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[h4 pow]
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template<typename T>
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octonion<T> pow(octonion<T> const & o, int n);
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Computes the n-th power of the octonion q.
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[endsect]
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[section:oct_tests Test Program]
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The [@../../test/octonion_test.cpp octonion_test.cpp]
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test program tests octonions specialisations for float, double and long double
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([@../octonion/output.txt sample output]).
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|
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).
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|
[endsect]
|
|
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|
[section:acknowledgements Acknowledgements]
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|
|
|
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: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: 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:oct_todo To Do]
|
|
|
|
* Improve testing.
|
|
* Rewrite input operatore using Spirit (creates a dependency).
|
|
* Put in place an Expression Template mechanism (perhaps borrowing from uBlas).
|
|
|
|
[endsect]
|
|
|
|
[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).
|
|
]
|