boost/math/octonion.hpp
// boost octonion.hpp header file // (C) Copyright Hubert Holin 2001. // 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) // See http://www.boost.org for updates, documentation, and revision history. #ifndef BOOST_OCTONION_HPP #define BOOST_OCTONION_HPP #include <boost/math/quaternion.hpp> #include <valarray> namespace boost { namespace math { #define BOOST_OCTONION_ACCESSOR_GENERATOR(type) \ type real() const \ { \ return(a); \ } \ \ octonion<type> unreal() const \ { \ return( octonion<type>(static_cast<type>(0),b,c,d,e,f,g,h)); \ } \ \ type R_component_1() const \ { \ return(a); \ } \ \ type R_component_2() const \ { \ return(b); \ } \ \ type R_component_3() const \ { \ return(c); \ } \ \ type R_component_4() const \ { \ return(d); \ } \ \ type R_component_5() const \ { \ return(e); \ } \ \ type R_component_6() const \ { \ return(f); \ } \ \ type R_component_7() const \ { \ return(g); \ } \ \ type R_component_8() const \ { \ return(h); \ } \ \ ::std::complex<type> C_component_1() const \ { \ return(::std::complex<type>(a,b)); \ } \ \ ::std::complex<type> C_component_2() const \ { \ return(::std::complex<type>(c,d)); \ } \ \ ::std::complex<type> C_component_3() const \ { \ return(::std::complex<type>(e,f)); \ } \ \ ::std::complex<type> C_component_4() const \ { \ return(::std::complex<type>(g,h)); \ } \ \ ::boost::math::quaternion<type> H_component_1() const \ { \ return(::boost::math::quaternion<type>(a,b,c,d)); \ } \ \ ::boost::math::quaternion<type> H_component_2() const \ { \ return(::boost::math::quaternion<type>(e,f,g,h)); \ } #define BOOST_OCTONION_MEMBER_ASSIGNMENT_GENERATOR(type) \ template<typename X> \ octonion<type> & operator = (octonion<X> const & a_affecter) \ { \ a = static_cast<type>(a_affecter.R_component_1()); \ b = static_cast<type>(a_affecter.R_component_2()); \ c = static_cast<type>(a_affecter.R_component_3()); \ d = static_cast<type>(a_affecter.R_component_4()); \ e = static_cast<type>(a_affecter.R_component_5()); \ f = static_cast<type>(a_affecter.R_component_6()); \ g = static_cast<type>(a_affecter.R_component_7()); \ h = static_cast<type>(a_affecter.R_component_8()); \ \ return(*this); \ } \ \ octonion<type> & operator = (octonion<type> const & a_affecter) \ { \ a = a_affecter.a; \ b = a_affecter.b; \ c = a_affecter.c; \ d = a_affecter.d; \ e = a_affecter.e; \ f = a_affecter.f; \ g = a_affecter.g; \ h = a_affecter.h; \ \ return(*this); \ } \ \ octonion<type> & operator = (type const & a_affecter) \ { \ a = a_affecter; \ \ b = c = d = e = f= g = h = static_cast<type>(0); \ \ return(*this); \ } \ \ octonion<type> & operator = (::std::complex<type> const & a_affecter) \ { \ a = a_affecter.real(); \ b = a_affecter.imag(); \ \ c = d = e = f = g = h = static_cast<type>(0); \ \ return(*this); \ } \ \ octonion<type> & operator = (::boost::math::quaternion<type> const & a_affecter) \ { \ a = a_affecter.R_component_1(); \ b = a_affecter.R_component_2(); \ c = a_affecter.R_component_3(); \ d = a_affecter.R_component_4(); \ \ e = f = g = h = static_cast<type>(0); \ \ return(*this); \ } #define BOOST_OCTONION_MEMBER_DATA_GENERATOR(type) \ type a; \ type b; \ type c; \ type d; \ type e; \ type f; \ type g; \ type h; \ template<typename T> class octonion { public: typedef T value_type; // constructor for O seen as R^8 // (also default constructor) 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()) : a(requested_a), b(requested_b), c(requested_c), d(requested_d), e(requested_e), f(requested_f), g(requested_g), h(requested_h) { // nothing to do! } // constructor for H seen as C^4 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>()) : a(z0.real()), b(z0.imag()), c(z1.real()), d(z1.imag()), e(z2.real()), f(z2.imag()), g(z3.real()), h(z3.imag()) { // nothing to do! } // constructor for O seen as H^2 explicit octonion( ::boost::math::quaternion<T> const & q0, ::boost::math::quaternion<T> const & q1 = ::boost::math::quaternion<T>()) : a(q0.R_component_1()), b(q0.R_component_2()), c(q0.R_component_3()), d(q0.R_component_4()), e(q1.R_component_1()), f(q1.R_component_2()), g(q1.R_component_3()), h(q1.R_component_4()) { // nothing to do! } // UNtemplated copy constructor // (this is taken care of by the compiler itself) // templated copy constructor template<typename X> explicit octonion(octonion<X> const & a_recopier) : a(static_cast<T>(a_recopier.R_component_1())), b(static_cast<T>(a_recopier.R_component_2())), c(static_cast<T>(a_recopier.R_component_3())), d(static_cast<T>(a_recopier.R_component_4())), e(static_cast<T>(a_recopier.R_component_5())), f(static_cast<T>(a_recopier.R_component_6())), g(static_cast<T>(a_recopier.R_component_7())), h(static_cast<T>(a_recopier.R_component_8())) { // nothing to do! } // destructor // (this is taken care of by the compiler itself) // accessors // // Note: 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 an octonion, and usually // nothing simpler (as opposed to the complex number case). // However, for practicality, there are accessors for the other components // (these are necessary for the templated copy constructor, for instance). BOOST_OCTONION_ACCESSOR_GENERATOR(T) // assignment operators BOOST_OCTONION_MEMBER_ASSIGNMENT_GENERATOR(T) // other assignment-related operators // // NOTE: Octonion multiplication is *NOT* commutative; // symbolically, "q *= rhs;" means "q = q * rhs;" // and "q /= rhs;" means "q = q * inverse_of(rhs);"; // octonion multiplication is also *NOT* associative octonion<T> & operator += (T const & rhs) { T at = a + rhs; // exception guard a = at; return(*this); } octonion<T> & operator += (::std::complex<T> const & rhs) { T at = a + rhs.real(); // exception guard T bt = b + rhs.imag(); // exception guard a = at; b = bt; return(*this); } octonion<T> & operator += (::boost::math::quaternion<T> const & rhs) { T at = a + rhs.R_component_1(); // exception guard T bt = b + rhs.R_component_2(); // exception guard T ct = c + rhs.R_component_3(); // exception guard T dt = d + rhs.R_component_4(); // exception guard a = at; b = bt; c = ct; d = dt; return(*this); } template<typename X> octonion<T> & operator += (octonion<X> const & rhs) { T at = a + static_cast<T>(rhs.R_component_1()); // exception guard T bt = b + static_cast<T>(rhs.R_component_2()); // exception guard T ct = c + static_cast<T>(rhs.R_component_3()); // exception guard T dt = d + static_cast<T>(rhs.R_component_4()); // exception guard T et = e + static_cast<T>(rhs.R_component_5()); // exception guard T ft = f + static_cast<T>(rhs.R_component_6()); // exception guard T gt = g + static_cast<T>(rhs.R_component_7()); // exception guard T ht = h + static_cast<T>(rhs.R_component_8()); // exception guard a = at; b = bt; c = ct; d = dt; e = et; f = ft; g = gt; h = ht; return(*this); } octonion<T> & operator -= (T const & rhs) { T at = a - rhs; // exception guard a = at; return(*this); } octonion<T> & operator -= (::std::complex<T> const & rhs) { T at = a - rhs.real(); // exception guard T bt = b - rhs.imag(); // exception guard a = at; b = bt; return(*this); } octonion<T> & operator -= (::boost::math::quaternion<T> const & rhs) { T at = a - rhs.R_component_1(); // exception guard T bt = b - rhs.R_component_2(); // exception guard T ct = c - rhs.R_component_3(); // exception guard T dt = d - rhs.R_component_4(); // exception guard a = at; b = bt; c = ct; d = dt; return(*this); } template<typename X> octonion<T> & operator -= (octonion<X> const & rhs) { T at = a - static_cast<T>(rhs.R_component_1()); // exception guard T bt = b - static_cast<T>(rhs.R_component_2()); // exception guard T ct = c - static_cast<T>(rhs.R_component_3()); // exception guard T dt = d - static_cast<T>(rhs.R_component_4()); // exception guard T et = e - static_cast<T>(rhs.R_component_5()); // exception guard T ft = f - static_cast<T>(rhs.R_component_6()); // exception guard T gt = g - static_cast<T>(rhs.R_component_7()); // exception guard T ht = h - static_cast<T>(rhs.R_component_8()); // exception guard a = at; b = bt; c = ct; d = dt; e = et; f = ft; g = gt; h = ht; return(*this); } octonion<T> & operator *= (T const & rhs) { T at = a * rhs; // exception guard T bt = b * rhs; // exception guard T ct = c * rhs; // exception guard T dt = d * rhs; // exception guard T et = e * rhs; // exception guard T ft = f * rhs; // exception guard T gt = g * rhs; // exception guard T ht = h * rhs; // exception guard a = at; b = bt; c = ct; d = dt; e = et; f = ft; g = gt; h = ht; return(*this); } octonion<T> & operator *= (::std::complex<T> const & rhs) { T ar = rhs.real(); T br = rhs.imag(); T at = +a*ar-b*br; T bt = +a*br+b*ar; T ct = +c*ar+d*br; T dt = -c*br+d*ar; T et = +e*ar+f*br; T ft = -e*br+f*ar; T gt = +g*ar-h*br; T ht = +g*br+h*ar; a = at; b = bt; c = ct; d = dt; e = et; f = ft; g = gt; h = ht; return(*this); } octonion<T> & operator *= (::boost::math::quaternion<T> const & rhs) { T ar = rhs.R_component_1(); T br = rhs.R_component_2(); T cr = rhs.R_component_2(); T dr = rhs.R_component_2(); T at = +a*ar-b*br-c*cr-d*dr; T bt = +a*br+b*ar+c*dr-d*cr; T ct = +a*cr-b*dr+c*ar+d*br; T dt = +a*dr+b*cr-c*br+d*ar; T et = +e*ar+f*br+g*cr+h*dr; T ft = -e*br+f*ar-g*dr+h*cr; T gt = -e*cr+f*dr+g*ar-h*br; T ht = -e*dr-f*cr+g*br+h*ar; a = at; b = bt; c = ct; d = dt; e = et; f = ft; g = gt; h = ht; return(*this); } template<typename X> octonion<T> & operator *= (octonion<X> const & rhs) { T ar = static_cast<T>(rhs.R_component_1()); T br = static_cast<T>(rhs.R_component_2()); T cr = static_cast<T>(rhs.R_component_3()); T dr = static_cast<T>(rhs.R_component_4()); T er = static_cast<T>(rhs.R_component_5()); T fr = static_cast<T>(rhs.R_component_6()); T gr = static_cast<T>(rhs.R_component_7()); T hr = static_cast<T>(rhs.R_component_8()); T at = +a*ar-b*br-c*cr-d*dr-e*er-f*fr-g*gr-h*hr; T bt = +a*br+b*ar+c*dr-d*cr+e*fr-f*er-g*hr+h*gr; T ct = +a*cr-b*dr+c*ar+d*br+e*gr+f*hr-g*er-h*fr; T dt = +a*dr+b*cr-c*br+d*ar+e*hr-f*gr+g*fr-h*er; T et = +a*er-b*fr-c*gr-d*hr+e*ar+f*br+g*cr+h*dr; T ft = +a*fr+b*er-c*hr+d*gr-e*br+f*ar-g*dr+h*cr; T gt = +a*gr+b*hr+c*er-d*fr-e*cr+f*dr+g*ar-h*br; T ht = +a*hr-b*gr+c*fr+d*er-e*dr-f*cr+g*br+h*ar; a = at; b = bt; c = ct; d = dt; e = et; f = ft; g = gt; h = ht; return(*this); } octonion<T> & operator /= (T const & rhs) { T at = a / rhs; // exception guard T bt = b / rhs; // exception guard T ct = c / rhs; // exception guard T dt = d / rhs; // exception guard T et = e / rhs; // exception guard T ft = f / rhs; // exception guard T gt = g / rhs; // exception guard T ht = h / rhs; // exception guard a = at; b = bt; c = ct; d = dt; e = et; f = ft; g = gt; h = ht; return(*this); } octonion<T> & operator /= (::std::complex<T> const & rhs) { T ar = rhs.real(); T br = rhs.imag(); T denominator = ar*ar+br*br; T at = (+a*ar-b*br)/denominator; T bt = (-a*br+b*ar)/denominator; T ct = (+c*ar-d*br)/denominator; T dt = (+c*br+d*ar)/denominator; T et = (+e*ar-f*br)/denominator; T ft = (+e*br+f*ar)/denominator; T gt = (+g*ar+h*br)/denominator; T ht = (+g*br+h*ar)/denominator; a = at; b = bt; c = ct; d = dt; e = et; f = ft; g = gt; h = ht; return(*this); } octonion<T> & operator /= (::boost::math::quaternion<T> const & rhs) { T ar = rhs.R_component_1(); T br = rhs.R_component_2(); T cr = rhs.R_component_2(); T dr = rhs.R_component_2(); T denominator = ar*ar+br*br+cr*cr+dr*dr; T at = (+a*ar+b*br+c*cr+d*dr)/denominator; T bt = (-a*br+b*ar-c*dr+d*cr)/denominator; T ct = (-a*cr+b*dr+c*ar-d*br)/denominator; T dt = (-a*dr-b*cr+c*br+d*ar)/denominator; T et = (+e*ar-f*br-g*cr-h*dr)/denominator; T ft = (+e*br+f*ar+g*dr-h*cr)/denominator; T gt = (+e*cr-f*dr+g*ar+h*br)/denominator; T ht = (+e*dr+f*cr-g*br+h*ar)/denominator; a = at; b = bt; c = ct; d = dt; e = et; f = ft; g = gt; h = ht; return(*this); } template<typename X> octonion<T> & operator /= (octonion<X> const & rhs) { T ar = static_cast<T>(rhs.R_component_1()); T br = static_cast<T>(rhs.R_component_2()); T cr = static_cast<T>(rhs.R_component_3()); T dr = static_cast<T>(rhs.R_component_4()); T er = static_cast<T>(rhs.R_component_5()); T fr = static_cast<T>(rhs.R_component_6()); T gr = static_cast<T>(rhs.R_component_7()); T hr = static_cast<T>(rhs.R_component_8()); T denominator = ar*ar+br*br+cr*cr+dr*dr+er*er+fr*fr+gr*gr+hr*hr; T at = (+a*ar+b*br+c*cr+d*dr+e*er+f*fr+g*gr+h*hr)/denominator; T bt = (-a*br+b*ar-c*dr+d*cr-e*fr+f*er+g*hr-h*gr)/denominator; T ct = (-a*cr+b*dr+c*ar-d*br-e*gr-f*hr+g*er+h*fr)/denominator; T dt = (-a*dr-b*cr+c*br+d*ar-e*hr+f*gr-g*fr+h*er)/denominator; T et = (-a*er+b*fr+c*gr+d*hr+e*ar-f*br-g*cr-h*dr)/denominator; T ft = (-a*fr-b*er+c*hr-d*gr+e*br+f*ar+g*dr-h*cr)/denominator; T gt = (-a*gr-b*hr-c*er+d*fr+e*cr-f*dr+g*ar+h*br)/denominator; T ht = (-a*hr+b*gr-c*fr-d*er+e*dr+f*cr-g*br+h*ar)/denominator; a = at; b = bt; c = ct; d = dt; e = et; f = ft; g = gt; h = ht; return(*this); } protected: BOOST_OCTONION_MEMBER_DATA_GENERATOR(T) private: }; // declaration of octonion specialization template<> class octonion<float>; template<> class octonion<double>; template<> class octonion<long double>; // helper templates for converting copy constructors (declaration) namespace detail { template< typename T, typename U > octonion<T> octonion_type_converter(octonion<U> const & rhs); } // implementation of octonion specialization #define BOOST_OCTONION_CONSTRUCTOR_GENERATOR(type) \ explicit octonion( type const & requested_a = static_cast<type>(0), \ type const & requested_b = static_cast<type>(0), \ type const & requested_c = static_cast<type>(0), \ type const & requested_d = static_cast<type>(0), \ type const & requested_e = static_cast<type>(0), \ type const & requested_f = static_cast<type>(0), \ type const & requested_g = static_cast<type>(0), \ type const & requested_h = static_cast<type>(0)) \ : a(requested_a), \ b(requested_b), \ c(requested_c), \ d(requested_d), \ e(requested_e), \ f(requested_f), \ g(requested_g), \ h(requested_h) \ { \ } \ \ explicit octonion( ::std::complex<type> const & z0, \ ::std::complex<type> const & z1 = ::std::complex<type>(), \ ::std::complex<type> const & z2 = ::std::complex<type>(), \ ::std::complex<type> const & z3 = ::std::complex<type>()) \ : a(z0.real()), \ b(z0.imag()), \ c(z1.real()), \ d(z1.imag()), \ e(z2.real()), \ f(z2.imag()), \ g(z3.real()), \ h(z3.imag()) \ { \ } \ \ explicit octonion( ::boost::math::quaternion<type> const & q0, \ ::boost::math::quaternion<type> const & q1 = ::boost::math::quaternion<type>()) \ : a(q0.R_component_1()), \ b(q0.R_component_2()), \ c(q0.R_component_3()), \ d(q0.R_component_4()), \ e(q1.R_component_1()), \ f(q1.R_component_2()), \ g(q1.R_component_3()), \ h(q1.R_component_4()) \ { \ } #define BOOST_OCTONION_MEMBER_ADD_GENERATOR_1(type) \ octonion<type> & operator += (type const & rhs) \ { \ a += rhs; \ \ return(*this); \ } #define BOOST_OCTONION_MEMBER_ADD_GENERATOR_2(type) \ octonion<type> & operator += (::std::complex<type> const & rhs) \ { \ a += rhs.real(); \ b += rhs.imag(); \ \ return(*this); \ } #define BOOST_OCTONION_MEMBER_ADD_GENERATOR_3(type) \ octonion<type> & operator += (::boost::math::quaternion<type> const & rhs) \ { \ a += rhs.R_component_1(); \ b += rhs.R_component_2(); \ c += rhs.R_component_3(); \ d += rhs.R_component_4(); \ \ return(*this); \ } #define BOOST_OCTONION_MEMBER_ADD_GENERATOR_4(type) \ template<typename X> \ octonion<type> & operator += (octonion<X> const & rhs) \ { \ a += static_cast<type>(rhs.R_component_1()); \ b += static_cast<type>(rhs.R_component_2()); \ c += static_cast<type>(rhs.R_component_3()); \ d += static_cast<type>(rhs.R_component_4()); \ e += static_cast<type>(rhs.R_component_5()); \ f += static_cast<type>(rhs.R_component_6()); \ g += static_cast<type>(rhs.R_component_7()); \ h += static_cast<type>(rhs.R_component_8()); \ \ return(*this); \ } #define BOOST_OCTONION_MEMBER_SUB_GENERATOR_1(type) \ octonion<type> & operator -= (type const & rhs) \ { \ a -= rhs; \ \ return(*this); \ } #define BOOST_OCTONION_MEMBER_SUB_GENERATOR_2(type) \ octonion<type> & operator -= (::std::complex<type> const & rhs) \ { \ a -= rhs.real(); \ b -= rhs.imag(); \ \ return(*this); \ } #define BOOST_OCTONION_MEMBER_SUB_GENERATOR_3(type) \ octonion<type> & operator -= (::boost::math::quaternion<type> const & rhs) \ { \ a -= rhs.R_component_1(); \ b -= rhs.R_component_2(); \ c -= rhs.R_component_3(); \ d -= rhs.R_component_4(); \ \ return(*this); \ } #define BOOST_OCTONION_MEMBER_SUB_GENERATOR_4(type) \ template<typename X> \ octonion<type> & operator -= (octonion<X> const & rhs) \ { \ a -= static_cast<type>(rhs.R_component_1()); \ b -= static_cast<type>(rhs.R_component_2()); \ c -= static_cast<type>(rhs.R_component_3()); \ d -= static_cast<type>(rhs.R_component_4()); \ e -= static_cast<type>(rhs.R_component_5()); \ f -= static_cast<type>(rhs.R_component_6()); \ g -= static_cast<type>(rhs.R_component_7()); \ h -= static_cast<type>(rhs.R_component_8()); \ \ return(*this); \ } #define BOOST_OCTONION_MEMBER_MUL_GENERATOR_1(type) \ octonion<type> & operator *= (type const & rhs) \ { \ a *= rhs; \ b *= rhs; \ c *= rhs; \ d *= rhs; \ e *= rhs; \ f *= rhs; \ g *= rhs; \ h *= rhs; \ \ return(*this); \ } #define BOOST_OCTONION_MEMBER_MUL_GENERATOR_2(type) \ octonion<type> & operator *= (::std::complex<type> const & rhs) \ { \ type ar = rhs.real(); \ type br = rhs.imag(); \ \ type at = +a*ar-b*br; \ type bt = +a*br+b*ar; \ type ct = +c*ar+d*br; \ type dt = -c*br+d*ar; \ type et = +e*ar+f*br; \ type ft = -e*br+f*ar; \ type gt = +g*ar-h*br; \ type ht = +g*br+h*ar; \ \ a = at; \ b = bt; \ c = ct; \ d = dt; \ e = et; \ f = ft; \ g = gt; \ h = ht; \ \ return(*this); \ } #define BOOST_OCTONION_MEMBER_MUL_GENERATOR_3(type) \ octonion<type> & operator *= (::boost::math::quaternion<type> const & rhs) \ { \ type ar = rhs.R_component_1(); \ type br = rhs.R_component_2(); \ type cr = rhs.R_component_2(); \ type dr = rhs.R_component_2(); \ \ type at = +a*ar-b*br-c*cr-d*dr; \ type bt = +a*br+b*ar+c*dr-d*cr; \ type ct = +a*cr-b*dr+c*ar+d*br; \ type dt = +a*dr+b*cr-c*br+d*ar; \ type et = +e*ar+f*br+g*cr+h*dr; \ type ft = -e*br+f*ar-g*dr+h*cr; \ type gt = -e*cr+f*dr+g*ar-h*br; \ type ht = -e*dr-f*cr+g*br+h*ar; \ \ a = at; \ b = bt; \ c = ct; \ d = dt; \ e = et; \ f = ft; \ g = gt; \ h = ht; \ \ return(*this); \ } #define BOOST_OCTONION_MEMBER_MUL_GENERATOR_4(type) \ template<typename X> \ octonion<type> & operator *= (octonion<X> const & rhs) \ { \ type ar = static_cast<type>(rhs.R_component_1()); \ type br = static_cast<type>(rhs.R_component_2()); \ type cr = static_cast<type>(rhs.R_component_3()); \ type dr = static_cast<type>(rhs.R_component_4()); \ type er = static_cast<type>(rhs.R_component_5()); \ type fr = static_cast<type>(rhs.R_component_6()); \ type gr = static_cast<type>(rhs.R_component_7()); \ type hr = static_cast<type>(rhs.R_component_8()); \ \ type at = +a*ar-b*br-c*cr-d*dr-e*er-f*fr-g*gr-h*hr; \ type bt = +a*br+b*ar+c*dr-d*cr+e*fr-f*er-g*hr+h*gr; \ type ct = +a*cr-b*dr+c*ar+d*br+e*gr+f*hr-g*er-h*fr; \ type dt = +a*dr+b*cr-c*br+d*ar+e*hr-f*gr+g*fr-h*er; \ type et = +a*er-b*fr-c*gr-d*hr+e*ar+f*br+g*cr+h*dr; \ type ft = +a*fr+b*er-c*hr+d*gr-e*br+f*ar-g*dr+h*cr; \ type gt = +a*gr+b*hr+c*er-d*fr-e*cr+f*dr+g*ar-h*br; \ type ht = +a*hr-b*gr+c*fr+d*er-e*dr-f*cr+g*br+h*ar; \ \ a = at; \ b = bt; \ c = ct; \ d = dt; \ e = et; \ f = ft; \ g = gt; \ h = ht; \ \ return(*this); \ } // There is quite a lot of repetition in the code below. This is intentional. // The last conditional block is the normal form, and the others merely // consist of workarounds for various compiler deficiencies. Hopefully, when // more compilers are conformant and we can retire support for those that are // not, we will be able to remove the clutter. This is makes the situation // (painfully) explicit. #define BOOST_OCTONION_MEMBER_DIV_GENERATOR_1(type) \ octonion<type> & operator /= (type const & rhs) \ { \ a /= rhs; \ b /= rhs; \ c /= rhs; \ d /= rhs; \ \ return(*this); \ } #if defined(BOOST_NO_ARGUMENT_DEPENDENT_LOOKUP) #define BOOST_OCTONION_MEMBER_DIV_GENERATOR_2(type) \ octonion<type> & operator /= (::std::complex<type> const & rhs) \ { \ using ::std::valarray; \ using ::std::abs; \ \ valarray<type> tr(2); \ \ tr[0] = rhs.real(); \ tr[1] = rhs.imag(); \ \ type mixam = static_cast<type>(1)/(abs(tr).max)(); \ \ tr *= mixam; \ \ valarray<type> tt(8); \ \ tt[0] = +a*tr[0]-b*tr[1]; \ tt[1] = -a*tr[1]+b*tr[0]; \ tt[2] = +c*tr[0]-d*tr[1]; \ tt[3] = +c*tr[1]+d*tr[0]; \ tt[4] = +e*tr[0]-f*tr[1]; \ tt[5] = +e*tr[1]+f*tr[0]; \ tt[6] = +g*tr[0]+h*tr[1]; \ tt[7] = +g*tr[1]+h*tr[0]; \ \ tr *= tr; \ \ tt *= (mixam/tr.sum()); \ \ a = tt[0]; \ b = tt[1]; \ c = tt[2]; \ d = tt[3]; \ e = tt[4]; \ f = tt[5]; \ g = tt[6]; \ h = tt[7]; \ \ return(*this); \ } #else #define BOOST_OCTONION_MEMBER_DIV_GENERATOR_2(type) \ octonion<type> & operator /= (::std::complex<type> const & rhs) \ { \ using ::std::valarray; \ \ valarray<type> tr(2); \ \ tr[0] = rhs.real(); \ tr[1] = rhs.imag(); \ \ type mixam = static_cast<type>(1)/(abs(tr).max)(); \ \ tr *= mixam; \ \ valarray<type> tt(8); \ \ tt[0] = +a*tr[0]-b*tr[1]; \ tt[1] = -a*tr[1]+b*tr[0]; \ tt[2] = +c*tr[0]-d*tr[1]; \ tt[3] = +c*tr[1]+d*tr[0]; \ tt[4] = +e*tr[0]-f*tr[1]; \ tt[5] = +e*tr[1]+f*tr[0]; \ tt[6] = +g*tr[0]+h*tr[1]; \ tt[7] = +g*tr[1]+h*tr[0]; \ \ tr *= tr; \ \ tt *= (mixam/tr.sum()); \ \ a = tt[0]; \ b = tt[1]; \ c = tt[2]; \ d = tt[3]; \ e = tt[4]; \ f = tt[5]; \ g = tt[6]; \ h = tt[7]; \ \ return(*this); \ } #endif /* BOOST_NO_ARGUMENT_DEPENDENT_LOOKUP */ #if defined(BOOST_NO_ARGUMENT_DEPENDENT_LOOKUP) #define BOOST_OCTONION_MEMBER_DIV_GENERATOR_3(type) \ octonion<type> & operator /= (::boost::math::quaternion<type> const & rhs) \ { \ using ::std::valarray; \ using ::std::abs; \ \ valarray<type> tr(4); \ \ tr[0] = static_cast<type>(rhs.R_component_1()); \ tr[1] = static_cast<type>(rhs.R_component_2()); \ tr[2] = static_cast<type>(rhs.R_component_3()); \ tr[3] = static_cast<type>(rhs.R_component_4()); \ \ type mixam = static_cast<type>(1)/(abs(tr).max)(); \ \ tr *= mixam; \ \ valarray<type> tt(8); \ \ tt[0] = +a*tr[0]+b*tr[1]+c*tr[2]+d*tr[3]; \ tt[1] = -a*tr[1]+b*tr[0]-c*tr[3]+d*tr[2]; \ tt[2] = -a*tr[2]+b*tr[3]+c*tr[0]-d*tr[1]; \ tt[3] = -a*tr[3]-b*tr[2]+c*tr[1]+d*tr[0]; \ tt[4] = +e*tr[0]-f*tr[1]-g*tr[2]-h*tr[3]; \ tt[5] = +e*tr[1]+f*tr[0]+g*tr[3]-h*tr[2]; \ tt[6] = +e*tr[2]-f*tr[3]+g*tr[0]+h*tr[1]; \ tt[7] = +e*tr[3]+f*tr[2]-g*tr[1]+h*tr[0]; \ \ tr *= tr; \ \ tt *= (mixam/tr.sum()); \ \ a = tt[0]; \ b = tt[1]; \ c = tt[2]; \ d = tt[3]; \ e = tt[4]; \ f = tt[5]; \ g = tt[6]; \ h = tt[7]; \ \ return(*this); \ } #else #define BOOST_OCTONION_MEMBER_DIV_GENERATOR_3(type) \ octonion<type> & operator /= (::boost::math::quaternion<type> const & rhs) \ { \ using ::std::valarray; \ \ valarray<type> tr(4); \ \ tr[0] = static_cast<type>(rhs.R_component_1()); \ tr[1] = static_cast<type>(rhs.R_component_2()); \ tr[2] = static_cast<type>(rhs.R_component_3()); \ tr[3] = static_cast<type>(rhs.R_component_4()); \ \ type mixam = static_cast<type>(1)/(abs(tr).max)(); \ \ tr *= mixam; \ \ valarray<type> tt(8); \ \ tt[0] = +a*tr[0]+b*tr[1]+c*tr[2]+d*tr[3]; \ tt[1] = -a*tr[1]+b*tr[0]-c*tr[3]+d*tr[2]; \ tt[2] = -a*tr[2]+b*tr[3]+c*tr[0]-d*tr[1]; \ tt[3] = -a*tr[3]-b*tr[2]+c*tr[1]+d*tr[0]; \ tt[4] = +e*tr[0]-f*tr[1]-g*tr[2]-h*tr[3]; \ tt[5] = +e*tr[1]+f*tr[0]+g*tr[3]-h*tr[2]; \ tt[6] = +e*tr[2]-f*tr[3]+g*tr[0]+h*tr[1]; \ tt[7] = +e*tr[3]+f*tr[2]-g*tr[1]+h*tr[0]; \ \ tr *= tr; \ \ tt *= (mixam/tr.sum()); \ \ a = tt[0]; \ b = tt[1]; \ c = tt[2]; \ d = tt[3]; \ e = tt[4]; \ f = tt[5]; \ g = tt[6]; \ h = tt[7]; \ \ return(*this); \ } #endif /* BOOST_NO_ARGUMENT_DEPENDENT_LOOKUP */ #if defined(BOOST_NO_ARGUMENT_DEPENDENT_LOOKUP) #define BOOST_OCTONION_MEMBER_DIV_GENERATOR_4(type) \ template<typename X> \ octonion<type> & operator /= (octonion<X> const & rhs) \ { \ using ::std::valarray; \ using ::std::abs; \ \ valarray<type> tr(8); \ \ tr[0] = static_cast<type>(rhs.R_component_1()); \ tr[1] = static_cast<type>(rhs.R_component_2()); \ tr[2] = static_cast<type>(rhs.R_component_3()); \ tr[3] = static_cast<type>(rhs.R_component_4()); \ tr[4] = static_cast<type>(rhs.R_component_5()); \ tr[5] = static_cast<type>(rhs.R_component_6()); \ tr[6] = static_cast<type>(rhs.R_component_7()); \ tr[7] = static_cast<type>(rhs.R_component_8()); \ \ type mixam = static_cast<type>(1)/(abs(tr).max)(); \ \ tr *= mixam; \ \ valarray<type> tt(8); \ \ tt[0] = +a*tr[0]+b*tr[1]+c*tr[2]+d*tr[3]+e*tr[4]+f*tr[5]+g*tr[6]+h*tr[7]; \ tt[1] = -a*tr[1]+b*tr[0]-c*tr[3]+d*tr[2]-e*tr[5]+f*tr[4]+g*tr[7]-h*tr[6]; \ tt[2] = -a*tr[2]+b*tr[3]+c*tr[0]-d*tr[1]-e*tr[6]-f*tr[7]+g*tr[4]+h*tr[5]; \ tt[3] = -a*tr[3]-b*tr[2]+c*tr[1]+d*tr[0]-e*tr[7]+f*tr[6]-g*tr[5]+h*tr[4]; \ tt[4] = -a*tr[4]+b*tr[5]+c*tr[6]+d*tr[7]+e*tr[0]-f*tr[1]-g*tr[2]-h*tr[3]; \ tt[5] = -a*tr[5]-b*tr[4]+c*tr[7]-d*tr[6]+e*tr[1]+f*tr[0]+g*tr[3]-h*tr[2]; \ tt[6] = -a*tr[6]-b*tr[7]-c*tr[4]+d*tr[5]+e*tr[2]-f*tr[3]+g*tr[0]+h*tr[1]; \ tt[7] = -a*tr[7]+b*tr[6]-c*tr[5]-d*tr[4]+e*tr[3]+f*tr[2]-g*tr[1]+h*tr[0]; \ \ tr *= tr; \ \ tt *= (mixam/tr.sum()); \ \ a = tt[0]; \ b = tt[1]; \ c = tt[2]; \ d = tt[3]; \ e = tt[4]; \ f = tt[5]; \ g = tt[6]; \ h = tt[7]; \ \ return(*this); \ } #else #define BOOST_OCTONION_MEMBER_DIV_GENERATOR_4(type) \ template<typename X> \ octonion<type> & operator /= (octonion<X> const & rhs) \ { \ using ::std::valarray; \ \ valarray<type> tr(8); \ \ tr[0] = static_cast<type>(rhs.R_component_1()); \ tr[1] = static_cast<type>(rhs.R_component_2()); \ tr[2] = static_cast<type>(rhs.R_component_3()); \ tr[3] = static_cast<type>(rhs.R_component_4()); \ tr[4] = static_cast<type>(rhs.R_component_5()); \ tr[5] = static_cast<type>(rhs.R_component_6()); \ tr[6] = static_cast<type>(rhs.R_component_7()); \ tr[7] = static_cast<type>(rhs.R_component_8()); \ \ type mixam = static_cast<type>(1)/(abs(tr).max)(); \ \ tr *= mixam; \ \ valarray<type> tt(8); \ \ tt[0] = +a*tr[0]+b*tr[1]+c*tr[2]+d*tr[3]+e*tr[4]+f*tr[5]+g*tr[6]+h*tr[7]; \ tt[1] = -a*tr[1]+b*tr[0]-c*tr[3]+d*tr[2]-e*tr[5]+f*tr[4]+g*tr[7]-h*tr[6]; \ tt[2] = -a*tr[2]+b*tr[3]+c*tr[0]-d*tr[1]-e*tr[6]-f*tr[7]+g*tr[4]+h*tr[5]; \ tt[3] = -a*tr[3]-b*tr[2]+c*tr[1]+d*tr[0]-e*tr[7]+f*tr[6]-g*tr[5]+h*tr[4]; \ tt[4] = -a*tr[4]+b*tr[5]+c*tr[6]+d*tr[7]+e*tr[0]-f*tr[1]-g*tr[2]-h*tr[3]; \ tt[5] = -a*tr[5]-b*tr[4]+c*tr[7]-d*tr[6]+e*tr[1]+f*tr[0]+g*tr[3]-h*tr[2]; \ tt[6] = -a*tr[6]-b*tr[7]-c*tr[4]+d*tr[5]+e*tr[2]-f*tr[3]+g*tr[0]+h*tr[1]; \ tt[7] = -a*tr[7]+b*tr[6]-c*tr[5]-d*tr[4]+e*tr[3]+f*tr[2]-g*tr[1]+h*tr[0]; \ \ tr *= tr; \ \ tt *= (mixam/tr.sum()); \ \ a = tt[0]; \ b = tt[1]; \ c = tt[2]; \ d = tt[3]; \ e = tt[4]; \ f = tt[5]; \ g = tt[6]; \ h = tt[7]; \ \ return(*this); \ } #endif /* BOOST_NO_ARGUMENT_DEPENDENT_LOOKUP */ #define BOOST_OCTONION_MEMBER_ADD_GENERATOR(type) \ BOOST_OCTONION_MEMBER_ADD_GENERATOR_1(type) \ BOOST_OCTONION_MEMBER_ADD_GENERATOR_2(type) \ BOOST_OCTONION_MEMBER_ADD_GENERATOR_3(type) \ BOOST_OCTONION_MEMBER_ADD_GENERATOR_4(type) #define BOOST_OCTONION_MEMBER_SUB_GENERATOR(type) \ BOOST_OCTONION_MEMBER_SUB_GENERATOR_1(type) \ BOOST_OCTONION_MEMBER_SUB_GENERATOR_2(type) \ BOOST_OCTONION_MEMBER_SUB_GENERATOR_3(type) \ BOOST_OCTONION_MEMBER_SUB_GENERATOR_4(type) #define BOOST_OCTONION_MEMBER_MUL_GENERATOR(type) \ BOOST_OCTONION_MEMBER_MUL_GENERATOR_1(type) \ BOOST_OCTONION_MEMBER_MUL_GENERATOR_2(type) \ BOOST_OCTONION_MEMBER_MUL_GENERATOR_3(type) \ BOOST_OCTONION_MEMBER_MUL_GENERATOR_4(type) #define BOOST_OCTONION_MEMBER_DIV_GENERATOR(type) \ BOOST_OCTONION_MEMBER_DIV_GENERATOR_1(type) \ BOOST_OCTONION_MEMBER_DIV_GENERATOR_2(type) \ BOOST_OCTONION_MEMBER_DIV_GENERATOR_3(type) \ BOOST_OCTONION_MEMBER_DIV_GENERATOR_4(type) #define BOOST_OCTONION_MEMBER_ALGEBRAIC_GENERATOR(type) \ BOOST_OCTONION_MEMBER_ADD_GENERATOR(type) \ BOOST_OCTONION_MEMBER_SUB_GENERATOR(type) \ BOOST_OCTONION_MEMBER_MUL_GENERATOR(type) \ BOOST_OCTONION_MEMBER_DIV_GENERATOR(type) template<> class octonion<float> { public: typedef float value_type; BOOST_OCTONION_CONSTRUCTOR_GENERATOR(float) // UNtemplated copy constructor // (this is taken care of by the compiler itself) // explicit copy constructors (precision-losing converters) explicit octonion(octonion<double> const & a_recopier) { *this = detail::octonion_type_converter<float, double>(a_recopier); } explicit octonion(octonion<long double> const & a_recopier) { *this = detail::octonion_type_converter<float, long double>(a_recopier); } // destructor // (this is taken care of by the compiler itself) // accessors // // Note: 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 an octonion, and usually // nothing simpler (as opposed to the complex number case). // However, for practicality, there are accessors for the other components // (these are necessary for the templated copy constructor, for instance). BOOST_OCTONION_ACCESSOR_GENERATOR(float) // assignment operators BOOST_OCTONION_MEMBER_ASSIGNMENT_GENERATOR(float) // other assignment-related operators // // NOTE: Octonion multiplication is *NOT* commutative; // symbolically, "q *= rhs;" means "q = q * rhs;" // and "q /= rhs;" means "q = q * inverse_of(rhs);"; // octonion multiplication is also *NOT* associative BOOST_OCTONION_MEMBER_ALGEBRAIC_GENERATOR(float) protected: BOOST_OCTONION_MEMBER_DATA_GENERATOR(float) private: }; template<> class octonion<double> { public: typedef double value_type; BOOST_OCTONION_CONSTRUCTOR_GENERATOR(double) // UNtemplated copy constructor // (this is taken care of by the compiler itself) // converting copy constructor explicit octonion(octonion<float> const & a_recopier) { *this = detail::octonion_type_converter<double, float>(a_recopier); } // explicit copy constructors (precision-losing converters) explicit octonion(octonion<long double> const & a_recopier) { *this = detail::octonion_type_converter<double, long double>(a_recopier); } // destructor // (this is taken care of by the compiler itself) // accessors // // Note: 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 an octonion, and usually // nothing simpler (as opposed to the complex number case). // However, for practicality, there are accessors for the other components // (these are necessary for the templated copy constructor, for instance). BOOST_OCTONION_ACCESSOR_GENERATOR(double) // assignment operators BOOST_OCTONION_MEMBER_ASSIGNMENT_GENERATOR(double) // other assignment-related operators // // NOTE: Octonion multiplication is *NOT* commutative; // symbolically, "q *= rhs;" means "q = q * rhs;" // and "q /= rhs;" means "q = q * inverse_of(rhs);"; // octonion multiplication is also *NOT* associative BOOST_OCTONION_MEMBER_ALGEBRAIC_GENERATOR(double) protected: BOOST_OCTONION_MEMBER_DATA_GENERATOR(double) private: }; template<> class octonion<long double> { public: typedef long double value_type; BOOST_OCTONION_CONSTRUCTOR_GENERATOR(long double) // UNtemplated copy constructor // (this is taken care of by the compiler itself) // converting copy constructor explicit octonion(octonion<float> const & a_recopier) { *this = detail::octonion_type_converter<long double, float>(a_recopier); } explicit octonion(octonion<double> const & a_recopier) { *this = detail::octonion_type_converter<long double, double>(a_recopier); } // destructor // (this is taken care of by the compiler itself) // accessors // // Note: 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 an octonion, and usually // nothing simpler (as opposed to the complex number case). // However, for practicality, there are accessors for the other components // (these are necessary for the templated copy constructor, for instance). BOOST_OCTONION_ACCESSOR_GENERATOR(long double) // assignment operators BOOST_OCTONION_MEMBER_ASSIGNMENT_GENERATOR(long double) // other assignment-related operators // // NOTE: Octonion multiplication is *NOT* commutative; // symbolically, "q *= rhs;" means "q = q * rhs;" // and "q /= rhs;" means "q = q * inverse_of(rhs);"; // octonion multiplication is also *NOT* associative BOOST_OCTONION_MEMBER_ALGEBRAIC_GENERATOR(long double) protected: BOOST_OCTONION_MEMBER_DATA_GENERATOR(long double) private: }; #undef BOOST_OCTONION_CONSTRUCTOR_GENERATOR #undef BOOST_OCTONION_MEMBER_ALGEBRAIC_GENERATOR #undef BOOST_OCTONION_MEMBER_ADD_GENERATOR #undef BOOST_OCTONION_MEMBER_SUB_GENERATOR #undef BOOST_OCTONION_MEMBER_MUL_GENERATOR #undef BOOST_OCTONION_MEMBER_DIV_GENERATOR #undef BOOST_OCTONION_MEMBER_ADD_GENERATOR_1 #undef BOOST_OCTONION_MEMBER_ADD_GENERATOR_2 #undef BOOST_OCTONION_MEMBER_ADD_GENERATOR_3 #undef BOOST_OCTONION_MEMBER_ADD_GENERATOR_4 #undef BOOST_OCTONION_MEMBER_SUB_GENERATOR_1 #undef BOOST_OCTONION_MEMBER_SUB_GENERATOR_2 #undef BOOST_OCTONION_MEMBER_SUB_GENERATOR_3 #undef BOOST_OCTONION_MEMBER_SUB_GENERATOR_4 #undef BOOST_OCTONION_MEMBER_MUL_GENERATOR_1 #undef BOOST_OCTONION_MEMBER_MUL_GENERATOR_2 #undef BOOST_OCTONION_MEMBER_MUL_GENERATOR_3 #undef BOOST_OCTONION_MEMBER_MUL_GENERATOR_4 #undef BOOST_OCTONION_MEMBER_DIV_GENERATOR_1 #undef BOOST_OCTONION_MEMBER_DIV_GENERATOR_2 #undef BOOST_OCTONION_MEMBER_DIV_GENERATOR_3 #undef BOOST_OCTONION_MEMBER_DIV_GENERATOR_4 #undef BOOST_OCTONION_MEMBER_DATA_GENERATOR #undef BOOST_OCTONION_MEMBER_ASSIGNMENT_GENERATOR #undef BOOST_OCTONION_ACCESSOR_GENERATOR // operators #define BOOST_OCTONION_OPERATOR_GENERATOR_BODY(op) \ { \ octonion<T> res(lhs); \ res op##= rhs; \ return(res); \ } #define BOOST_OCTONION_OPERATOR_GENERATOR_1_L(op) \ template<typename T> \ inline octonion<T> operator op (T const & lhs, octonion<T> const & rhs) \ BOOST_OCTONION_OPERATOR_GENERATOR_BODY(op) #define BOOST_OCTONION_OPERATOR_GENERATOR_1_R(op) \ template<typename T> \ inline octonion<T> operator op (octonion<T> const & lhs, T const & rhs) \ BOOST_OCTONION_OPERATOR_GENERATOR_BODY(op) #define BOOST_OCTONION_OPERATOR_GENERATOR_2_L(op) \ template<typename T> \ inline octonion<T> operator op (::std::complex<T> const & lhs, octonion<T> const & rhs) \ BOOST_OCTONION_OPERATOR_GENERATOR_BODY(op) #define BOOST_OCTONION_OPERATOR_GENERATOR_2_R(op) \ template<typename T> \ inline octonion<T> operator op (octonion<T> const & lhs, ::std::complex<T> const & rhs) \ BOOST_OCTONION_OPERATOR_GENERATOR_BODY(op) #define BOOST_OCTONION_OPERATOR_GENERATOR_3_L(op) \ template<typename T> \ inline octonion<T> operator op (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs) \ BOOST_OCTONION_OPERATOR_GENERATOR_BODY(op) #define BOOST_OCTONION_OPERATOR_GENERATOR_3_R(op) \ template<typename T> \ inline octonion<T> operator op (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs) \ BOOST_OCTONION_OPERATOR_GENERATOR_BODY(op) #define BOOST_OCTONION_OPERATOR_GENERATOR_4(op) \ template<typename T> \ inline octonion<T> operator op (octonion<T> const & lhs, octonion<T> const & rhs) \ BOOST_OCTONION_OPERATOR_GENERATOR_BODY(op) #define BOOST_OCTONION_OPERATOR_GENERATOR(op) \ BOOST_OCTONION_OPERATOR_GENERATOR_1_L(op) \ BOOST_OCTONION_OPERATOR_GENERATOR_1_R(op) \ BOOST_OCTONION_OPERATOR_GENERATOR_2_L(op) \ BOOST_OCTONION_OPERATOR_GENERATOR_2_R(op) \ BOOST_OCTONION_OPERATOR_GENERATOR_3_L(op) \ BOOST_OCTONION_OPERATOR_GENERATOR_3_R(op) \ BOOST_OCTONION_OPERATOR_GENERATOR_4(op) BOOST_OCTONION_OPERATOR_GENERATOR(+) BOOST_OCTONION_OPERATOR_GENERATOR(-) BOOST_OCTONION_OPERATOR_GENERATOR(*) BOOST_OCTONION_OPERATOR_GENERATOR(/) #undef BOOST_OCTONION_OPERATOR_GENERATOR #undef BOOST_OCTONION_OPERATOR_GENERATOR_1_L #undef BOOST_OCTONION_OPERATOR_GENERATOR_1_R #undef BOOST_OCTONION_OPERATOR_GENERATOR_2_L #undef BOOST_OCTONION_OPERATOR_GENERATOR_2_R #undef BOOST_OCTONION_OPERATOR_GENERATOR_3_L #undef BOOST_OCTONION_OPERATOR_GENERATOR_3_R #undef BOOST_OCTONION_OPERATOR_GENERATOR_4 #undef BOOST_OCTONION_OPERATOR_GENERATOR_BODY template<typename T> inline octonion<T> operator + (octonion<T> const & o) { return(o); } template<typename T> inline octonion<T> operator - (octonion<T> const & o) { return(octonion<T>(-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())); } template<typename T> inline bool operator == (T const & lhs, octonion<T> const & rhs) { return( (rhs.R_component_1() == lhs)&& (rhs.R_component_2() == static_cast<T>(0))&& (rhs.R_component_3() == static_cast<T>(0))&& (rhs.R_component_4() == static_cast<T>(0))&& (rhs.R_component_5() == static_cast<T>(0))&& (rhs.R_component_6() == static_cast<T>(0))&& (rhs.R_component_7() == static_cast<T>(0))&& (rhs.R_component_8() == static_cast<T>(0)) ); } template<typename T> inline bool operator == (octonion<T> const & lhs, T const & rhs) { return( (lhs.R_component_1() == rhs)&& (lhs.R_component_2() == static_cast<T>(0))&& (lhs.R_component_3() == static_cast<T>(0))&& (lhs.R_component_4() == static_cast<T>(0))&& (lhs.R_component_5() == static_cast<T>(0))&& (lhs.R_component_6() == static_cast<T>(0))&& (lhs.R_component_7() == static_cast<T>(0))&& (lhs.R_component_8() == static_cast<T>(0)) ); } template<typename T> inline bool operator == (::std::complex<T> const & lhs, octonion<T> const & rhs) { return( (rhs.R_component_1() == lhs.real())&& (rhs.R_component_2() == lhs.imag())&& (rhs.R_component_3() == static_cast<T>(0))&& (rhs.R_component_4() == static_cast<T>(0))&& (rhs.R_component_5() == static_cast<T>(0))&& (rhs.R_component_6() == static_cast<T>(0))&& (rhs.R_component_7() == static_cast<T>(0))&& (rhs.R_component_8() == static_cast<T>(0)) ); } template<typename T> inline bool operator == (octonion<T> const & lhs, ::std::complex<T> const & rhs) { return( (lhs.R_component_1() == rhs.real())&& (lhs.R_component_2() == rhs.imag())&& (lhs.R_component_3() == static_cast<T>(0))&& (lhs.R_component_4() == static_cast<T>(0))&& (lhs.R_component_5() == static_cast<T>(0))&& (lhs.R_component_6() == static_cast<T>(0))&& (lhs.R_component_7() == static_cast<T>(0))&& (lhs.R_component_8() == static_cast<T>(0)) ); } template<typename T> inline bool operator == (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs) { return( (rhs.R_component_1() == lhs.R_component_1())&& (rhs.R_component_2() == lhs.R_component_2())&& (rhs.R_component_3() == lhs.R_component_3())&& (rhs.R_component_4() == lhs.R_component_4())&& (rhs.R_component_5() == static_cast<T>(0))&& (rhs.R_component_6() == static_cast<T>(0))&& (rhs.R_component_7() == static_cast<T>(0))&& (rhs.R_component_8() == static_cast<T>(0)) ); } template<typename T> inline bool operator == (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs) { return( (lhs.R_component_1() == rhs.R_component_1())&& (lhs.R_component_2() == rhs.R_component_2())&& (lhs.R_component_3() == rhs.R_component_3())&& (lhs.R_component_4() == rhs.R_component_4())&& (lhs.R_component_5() == static_cast<T>(0))&& (lhs.R_component_6() == static_cast<T>(0))&& (lhs.R_component_7() == static_cast<T>(0))&& (lhs.R_component_8() == static_cast<T>(0)) ); } template<typename T> inline bool operator == (octonion<T> const & lhs, octonion<T> const & rhs) { return( (rhs.R_component_1() == lhs.R_component_1())&& (rhs.R_component_2() == lhs.R_component_2())&& (rhs.R_component_3() == lhs.R_component_3())&& (rhs.R_component_4() == lhs.R_component_4())&& (rhs.R_component_5() == lhs.R_component_5())&& (rhs.R_component_6() == lhs.R_component_6())&& (rhs.R_component_7() == lhs.R_component_7())&& (rhs.R_component_8() == lhs.R_component_8()) ); } #define BOOST_OCTONION_NOT_EQUAL_GENERATOR \ { \ return(!(lhs == rhs)); \ } template<typename T> inline bool operator != (T const & lhs, octonion<T> const & rhs) BOOST_OCTONION_NOT_EQUAL_GENERATOR template<typename T> inline bool operator != (octonion<T> const & lhs, T const & rhs) BOOST_OCTONION_NOT_EQUAL_GENERATOR template<typename T> inline bool operator != (::std::complex<T> const & lhs, octonion<T> const & rhs) BOOST_OCTONION_NOT_EQUAL_GENERATOR template<typename T> inline bool operator != (octonion<T> const & lhs, ::std::complex<T> const & rhs) BOOST_OCTONION_NOT_EQUAL_GENERATOR template<typename T> inline bool operator != (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs) BOOST_OCTONION_NOT_EQUAL_GENERATOR template<typename T> inline bool operator != (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs) BOOST_OCTONION_NOT_EQUAL_GENERATOR template<typename T> inline bool operator != (octonion<T> const & lhs, octonion<T> const & rhs) BOOST_OCTONION_NOT_EQUAL_GENERATOR #undef BOOST_OCTONION_NOT_EQUAL_GENERATOR // Note: the default values in the constructors of the complex and quaternions make for // a very complex and ambiguous situation; we have made choices to disambiguate. template<typename T, typename charT, class traits> ::std::basic_istream<charT,traits> & operator >> ( ::std::basic_istream<charT,traits> & is, octonion<T> & o) { #ifdef BOOST_NO_STD_LOCALE #else const ::std::ctype<charT> & ct = ::std::use_facet< ::std::ctype<charT> >(is.getloc()); #endif /* BOOST_NO_STD_LOCALE */ T a = T(); T b = T(); T c = T(); T d = T(); T e = T(); T f = T(); T g = T(); T h = T(); ::std::complex<T> u = ::std::complex<T>(); ::std::complex<T> v = ::std::complex<T>(); ::std::complex<T> x = ::std::complex<T>(); ::std::complex<T> y = ::std::complex<T>(); ::boost::math::quaternion<T> p = ::boost::math::quaternion<T>(); ::boost::math::quaternion<T> q = ::boost::math::quaternion<T>(); charT ch = charT(); char cc; is >> ch; // get the first lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == '(') // read "(" { is >> ch; // get the second lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == '(') // read "((" { is >> ch; // get the third lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == '(') // read "(((" { is.putback(ch); is >> u; // read "((u" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((u)" { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // format: (((a))), (((a,b))) { o = octonion<T>(u); } else if (cc == ',') // read "((u)," { p = ::boost::math::quaternion<T>(u); is >> q; // read "((u),q" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // format: (((a)),q), (((a,b)),q) { o = octonion<T>(p,q); } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else if (cc ==',') // read "((u," { is >> v; // read "((u,v" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((u,v)" { p = ::boost::math::quaternion<T>(u,v); is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // format: (((a),v)), (((a,b),v)) { o = octonion<T>(p); } else if (cc == ',') // read "((u,v)," { is >> q; // read "(p,q" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // format: (((a),v),q), (((a,b),v),q) { o = octonion<T>(p,q); } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else // read "((a" { is.putback(ch); is >> a; // we extract the first component if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a)" { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a))" { o = octonion<T>(a); } else if (cc == ',') // read "((a)," { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == '(') // read "((a),(" { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == '(') // read "((a),((" { is.putback(ch); is.putback(ch); // we backtrack twice, with the same value! is >> q; // read "((a),q" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a),q)" { p = ::boost::math::quaternion<T>(a); o = octonion<T>(p,q); } else // error { is.setstate(::std::ios_base::failbit); } } else // read "((a),(c" or "((a),(e" { is.putback(ch); is >> c; if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a),(c)" (ambiguity resolution) { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a),(c))" { o = octonion<T>(a,b,c); } else if (cc == ',') // read "((a),(c)," { u = ::std::complex<T>(a); v = ::std::complex<T>(c); is >> x; // read "((a),(c),x" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a),(c),x)" { o = octonion<T>(u,v,x); } else if (cc == ',') // read "((a),(c),x," { is >> y; // read "((a),(c),x,y" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a),(c),x,y)" { o = octonion<T>(u,v,x,y); } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else if (cc == ',') // read "((a),(c," or "((a),(e," { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == '(') // read "((a),(e,(" (ambiguity resolution) { p = ::boost::math::quaternion<T>(a); x = ::std::complex<T>(c); // "c" was actually "e" is.putback(ch); // we can only backtrace once is >> y; // read "((a),(e,y" if (!is.good()) goto finish; is >> ch; // get the next lexeme #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a),(e,y)" { q = ::boost::math::quaternion<T>(x,y); is >> ch; // get the next lexeme #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a),(e,y))" { o = octonion<T>(p,q); } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else // read "((a),(c,d" or "((a),(e,f" { is.putback(ch); is >> d; if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a),(c,d)" (ambiguity resolution) { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a),(c,d))" { o = octonion<T>(a,b,c,d); } else if (cc == ',') // read "((a),(c,d)," { u = ::std::complex<T>(a); v = ::std::complex<T>(c,d); is >> x; // read "((a),(c,d),x" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a),(c,d),x)" { o = octonion<T>(u,v,x); } else if (cc == ',') // read "((a),(c,d),x," { is >> y; // read "((a),(c,d),x,y" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a),(c,d),x,y)" { o = octonion<T>(u,v,x,y); } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else if (cc == ',') // read "((a),(e,f," (ambiguity resolution) { p = ::boost::math::quaternion<T>(a); is >> g; // read "((a),(e,f,g" (too late to backtrack) if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a),(e,f,g)" { q = ::boost::math::quaternion<T>(c,d,g); // "c" was actually "e", and "d" was actually "f" is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a),(e,f,g))" { o = octonion<T>(p,q); } else // error { is.setstate(::std::ios_base::failbit); } } else if (cc == ',') // read "((a),(e,f,g," { is >> h; // read "((a),(e,f,g,h" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a),(e,f,g,h)" { q = ::boost::math::quaternion<T>(c,d,g,h); // "c" was actually "e", and "d" was actually "f" is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a),(e,f,g,h))" { o = octonion<T>(p,q); } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } } else // error { is.setstate(::std::ios_base::failbit); } } } else // read "((a),c" (ambiguity resolution) { is.putback(ch); is >> c; // we extract the third component if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a),c)" { o = octonion<T>(a,b,c); } else if (cc == ',') // read "((a),c," { is >> x; // read "((a),c,x" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a),c,x)" { o = octonion<T>(a,b,c,d,x.real(),x.imag()); } else if (cc == ',') // read "((a),c,x," { is >> y;if (!is.good()) goto finish; // read "((a),c,x,y" is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a),c,x,y)" { o = octonion<T>(a,b,c,d,x.real(),x.imag(),y.real(),y.imag()); } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } } else // error { is.setstate(::std::ios_base::failbit); } } else if (cc ==',') // read "((a," { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == '(') // read "((a,(" { u = ::std::complex<T>(a); is.putback(ch); // can only backtrack so much is >> v; // read "((a,v" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a,v)" { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a,v))" { o = octonion<T>(u,v); } else if (cc == ',') // read "((a,v)," { p = ::boost::math::quaternion<T>(u,v); is >> q; // read "((a,v),q" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a,v),q)" { o = octonion<T>(p,q); } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else { is.putback(ch); is >> b; // read "((a,b" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a,b)" { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a,b))" { o = octonion<T>(a,b); } else if (cc == ',') // read "((a,b)," { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == '(') // read "((a,b),(" { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == '(') // read "((a,b),((" { p = ::boost::math::quaternion<T>(a,b); is.putback(ch); is.putback(ch); // we backtrack twice, with the same value is >> q; // read "((a,b),q" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a,b),q)" { o = octonion<T>(p,q); } else // error { is.setstate(::std::ios_base::failbit); } } else // read "((a,b),(c" or "((a,b),(e" { is.putback(ch); is >> c; if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a,b),(c)" (ambiguity resolution) { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a,b),(c))" { o = octonion<T>(a,b,c); } else if (cc == ',') // read "((a,b),(c)," { u = ::std::complex<T>(a,b); v = ::std::complex<T>(c); is >> x; // read "((a,b),(c),x" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a,b),(c),x)" { o = octonion<T>(u,v,x); } else if (cc == ',') // read "((a,b),(c),x," { is >> y; // read "((a,b),(c),x,y" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a,b),(c),x,y)" { o = octonion<T>(u,v,x,y); } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else if (cc == ',') // read "((a,b),(c," or "((a,b),(e," { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == '(') // read "((a,b),(e,(" (ambiguity resolution) { u = ::std::complex<T>(a,b); x = ::std::complex<T>(c); // "c" is actually "e" is.putback(ch); is >> y; // read "((a,b),(e,y" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a,b),(e,y)" { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a,b),(e,y))" { o = octonion<T>(u,v,x,y); } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else // read "((a,b),(c,d" or "((a,b),(e,f" { is.putback(ch); is >> d; if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a,b),(c,d)" (ambiguity resolution) { u = ::std::complex<T>(a,b); v = ::std::complex<T>(c,d); is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a,b),(c,d))" { o = octonion<T>(u,v); } else if (cc == ',') // read "((a,b),(c,d)," { is >> x; // read "((a,b),(c,d),x if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a,b),(c,d),x)" { o = octonion<T>(u,v,x); } else if (cc == ',') // read "((a,b),(c,d),x," { is >> y; // read "((a,b),(c,d),x,y" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a,b),(c,d),x,y)" { o = octonion<T>(u,v,x,y); } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else if (cc == ',') // read "((a,b),(e,f," (ambiguity resolution) { p = ::boost::math::quaternion<T>(a,b); // too late to backtrack is >> g; // read "((a,b),(e,f,g" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a,b),(e,f,g)" { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a,b),(e,f,g))" { q = ::boost::math::quaternion<T>(c,d,g); // "c" is actually "e" and "d" is actually "f" o = octonion<T>(p,q); } else // error { is.setstate(::std::ios_base::failbit); } } else if (cc == ',') // read "((a,b),(e,f,g," { is >> h; // read "((a,b),(e,f,g,h" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a,b),(e,f,g,h)" { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read ((a,b),(e,f,g,h))" { q = ::boost::math::quaternion<T>(c,d,g,h); // "c" is actually "e" and "d" is actually "f" o = octonion<T>(p,q); } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } } else // error { is.setstate(::std::ios_base::failbit); } } } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else if (cc == ',') // read "((a,b," { is >> c; // read "((a,b,c" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a,b,c)" { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a,b,c))" { o = octonion<T>(a,b,c); } else if (cc == ',') // read "((a,b,c)," { p = ::boost::math::quaternion<T>(a,b,c); is >> q; // read "((a,b,c),q" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a,b,c),q)" { o = octonion<T>(p,q); } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else if (cc == ',') // read "((a,b,c," { is >> d; // read "((a,b,c,d" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a,b,c,d)" { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a,b,c,d))" { o = octonion<T>(a,b,c,d); } else if (cc == ',') // read "((a,b,c,d)," { p = ::boost::math::quaternion<T>(a,b,c,d); is >> q; // read "((a,b,c,d),q" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "((a,b,c,d),q)" { o = octonion<T>(p,q); } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } } else // error { is.setstate(::std::ios_base::failbit); } } } else // read "(a" { is.putback(ch); is >> a; // we extract the first component if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "(a)" { o = octonion<T>(a); } else if (cc == ',') // read "(a," { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == '(') // read "(a,(" { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == '(') // read "(a,((" { p = ::boost::math::quaternion<T>(a); is.putback(ch); is.putback(ch); // we backtrack twice, with the same value is >> q; // read "(a,q" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "(a,q)" { o = octonion<T>(p,q); } else // error { is.setstate(::std::ios_base::failbit); } } else // read "(a,(c" or "(a,(e" { is.putback(ch); is >> c; if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "(a,(c)" (ambiguity resolution) { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "(a,(c))" { o = octonion<T>(a,b,c); } else if (cc == ',') // read "(a,(c)," { u = ::std::complex<T>(a); v = ::std::complex<T>(c); is >> x; // read "(a,(c),x" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "(a,(c),x)" { o = octonion<T>(u,v,x); } else if (cc == ',') // read "(a,(c),x," { is >> y; // read "(a,(c),x,y" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "(a,(c),x,y)" { o = octonion<T>(u,v,x,y); } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else if (cc == ',') // read "(a,(c," or "(a,(e," { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == '(') // read "(a,(e,(" (ambiguity resolution) { u = ::std::complex<T>(a); x = ::std::complex<T>(c); // "c" is actually "e" is.putback(ch); // we backtrack is >> y; // read "(a,(e,y" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "(a,(e,y)" { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "(a,(e,y))" { o = octonion<T>(u,v,x,y); } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else // read "(a,(c,d" or "(a,(e,f" { is.putback(ch); is >> d; if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "(a,(c,d)" (ambiguity resolution) { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "(a,(c,d))" { o = octonion<T>(a,b,c,d); } else if (cc == ',') // read "(a,(c,d)," { u = ::std::complex<T>(a); v = ::std::complex<T>(c,d); is >> x; // read "(a,(c,d),x" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "(a,(c,d),x)" { o = octonion<T>(u,v,x); } else if (cc == ',') // read "(a,(c,d),x," { is >> y; // read "(a,(c,d),x,y" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "(a,(c,d),x,y)" { o = octonion<T>(u,v,x,y); } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else if (cc == ',') // read "(a,(e,f," (ambiguity resolution) { p = ::boost::math::quaternion<T>(a); is >> g; // read "(a,(e,f,g" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "(a,(e,f,g)" { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "(a,(e,f,g))" { q = ::boost::math::quaternion<T>(c,d,g); // "c" is actually "e" and "d" is actually "f" o = octonion<T>(p,q); } else // error { is.setstate(::std::ios_base::failbit); } } else if (cc == ',') // read "(a,(e,f,g," { is >> h; // read "(a,(e,f,g,h" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "(a,(e,f,g,h)" { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "(a,(e,f,g,h))" { q = ::boost::math::quaternion<T>(c,d,g,h); // "c" is actually "e" and "d" is actually "f" o = octonion<T>(p,q); } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } } else // error { is.setstate(::std::ios_base::failbit); } } } else // read "(a,b" or "(a,c" (ambiguity resolution) { is.putback(ch); is >> b; if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "(a,b)" (ambiguity resolution) { o = octonion<T>(a,b); } else if (cc == ',') // read "(a,b," or "(a,c," { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == '(') // read "(a,c,(" (ambiguity resolution) { u = ::std::complex<T>(a); v = ::std::complex<T>(b); // "b" is actually "c" is.putback(ch); // we backtrack is >> x; // read "(a,c,x" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "(a,c,x)" { o = octonion<T>(u,v,x); } else if (cc == ',') // read "(a,c,x," { is >> y; // read "(a,c,x,y" // read "(a,c,x" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "(a,c,x,y)" { o = octonion<T>(u,v,x,y); } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else // read "(a,b,c" or "(a,c,e" { is.putback(ch); is >> c; if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "(a,b,c)" (ambiguity resolution) { o = octonion<T>(a,b,c); } else if (cc == ',') // read "(a,b,c," or "(a,c,e," { is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == '(') // read "(a,c,e,(") (ambiguity resolution) { u = ::std::complex<T>(a); v = ::std::complex<T>(b); // "b" is actually "c" x = ::std::complex<T>(c); // "c" is actually "e" is.putback(ch); // we backtrack is >> y; // read "(a,c,e,y" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "(a,c,e,y)" { o = octonion<T>(u,v,x,y); } else // error { is.setstate(::std::ios_base::failbit); } } else // read "(a,b,c,d" (ambiguity resolution) { is.putback(ch); // we backtrack is >> d; if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "(a,b,c,d)" { o = octonion<T>(a,b,c,d); } else if (cc == ',') // read "(a,b,c,d," { is >> e; // read "(a,b,c,d,e" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "(a,b,c,d,e)" { o = octonion<T>(a,b,c,d,e); } else if (cc == ',') // read "(a,b,c,d,e," { is >> f; // read "(a,b,c,d,e,f" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "(a,b,c,d,e,f)" { o = octonion<T>(a,b,c,d,e,f); } else if (cc == ',') // read "(a,b,c,d,e,f," { is >> g; // read "(a,b,c,d,e,f,g" // read "(a,b,c,d,e,f" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "(a,b,c,d,e,f,g)" { o = octonion<T>(a,b,c,d,e,f,g); } else if (cc == ',') // read "(a,b,c,d,e,f,g," { is >> h; // read "(a,b,c,d,e,f,g,h" // read "(a,b,c,d,e,f,g" // read "(a,b,c,d,e,f" if (!is.good()) goto finish; is >> ch; // get the next lexeme if (!is.good()) goto finish; #ifdef BOOST_NO_STD_LOCALE cc = ch; #else cc = ct.narrow(ch, char()); #endif /* BOOST_NO_STD_LOCALE */ if (cc == ')') // read "(a,b,c,d,e,f,g,h)" { o = octonion<T>(a,b,c,d,e,f,g,h); } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } else // error { is.setstate(::std::ios_base::failbit); } } } else // error { is.setstate(::std::ios_base::failbit); } } } else // error { is.setstate(::std::ios_base::failbit); } } } else // error { is.setstate(::std::ios_base::failbit); } } } else // format: a { is.putback(ch); is >> a; // we extract the first component if (!is.good()) goto finish; o = octonion<T>(a); } finish: return(is); } 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()); #ifdef BOOST_NO_STD_LOCALE #else s.imbue(os.getloc()); #endif /* BOOST_NO_STD_LOCALE */ 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(); } // values template<typename T> inline T real(octonion<T> const & o) { return(o.real()); } template<typename T> inline octonion<T> unreal(octonion<T> const & o) { return(o.unreal()); } #define BOOST_OCTONION_VALARRAY_LOADER \ using ::std::valarray; \ \ valarray<T> temp(8); \ \ temp[0] = o.R_component_1(); \ temp[1] = o.R_component_2(); \ temp[2] = o.R_component_3(); \ temp[3] = o.R_component_4(); \ temp[4] = o.R_component_5(); \ temp[5] = o.R_component_6(); \ temp[6] = o.R_component_7(); \ temp[7] = o.R_component_8(); template<typename T> inline T sup(octonion<T> const & o) { #ifdef BOOST_NO_ARGUMENT_DEPENDENT_LOOKUP using ::std::abs; #endif /* BOOST_NO_ARGUMENT_DEPENDENT_LOOKUP */ BOOST_OCTONION_VALARRAY_LOADER return((abs(temp).max)()); } template<typename T> inline T l1(octonion<T> const & o) { #ifdef BOOST_NO_ARGUMENT_DEPENDENT_LOOKUP using ::std::abs; #endif /* BOOST_NO_ARGUMENT_DEPENDENT_LOOKUP */ BOOST_OCTONION_VALARRAY_LOADER return(abs(temp).sum()); } template<typename T> inline T abs(const octonion<T> & o) { #ifdef BOOST_NO_ARGUMENT_DEPENDENT_LOOKUP using ::std::abs; #endif /* BOOST_NO_ARGUMENT_DEPENDENT_LOOKUP */ using ::std::sqrt; BOOST_OCTONION_VALARRAY_LOADER T maxim = (abs(temp).max)(); // overflow protection if (maxim == static_cast<T>(0)) { return(maxim); } else { T mixam = static_cast<T>(1)/maxim; // prefer multiplications over divisions temp *= mixam; temp *= temp; return(maxim*sqrt(temp.sum())); } //return(::std::sqrt(norm(o))); } #undef BOOST_OCTONION_VALARRAY_LOADER // Note: This is the Cayley norm, not the Euclidean norm... template<typename T> inline T norm(octonion<T> const & o) { return(real(o*conj(o))); } template<typename T> inline octonion<T> conj(octonion<T> const & o) { return(octonion<T>( +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())); } // Note: There is little point, for the octonions, to introduce the equivalents // to the complex "arg" and the quaternionic "cylindropolar". template<typename T> inline 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) { using ::std::cos; using ::std::sin; //T a = cos(theta)*cos(phi1)*cos(phi2)*cos(phi3)*cos(phi4)*cos(phi5)*cos(phi6); //T b = sin(theta)*cos(phi1)*cos(phi2)*cos(phi3)*cos(phi4)*cos(phi5)*cos(phi6); //T c = sin(phi1)*cos(phi2)*cos(phi3)*cos(phi4)*cos(phi5)*cos(phi6); //T d = sin(phi2)*cos(phi3)*cos(phi4)*cos(phi5)*cos(phi6); //T e = sin(phi3)*cos(phi4)*cos(phi5)*cos(phi6); //T f = sin(phi4)*cos(phi5)*cos(phi6); //T g = sin(phi5)*cos(phi6); //T h = sin(phi6); T courrant = static_cast<T>(1); T h = sin(phi6); courrant *= cos(phi6); T g = sin(phi5)*courrant; courrant *= cos(phi5); T f = sin(phi4)*courrant; courrant *= cos(phi4); T e = sin(phi3)*courrant; courrant *= cos(phi3); T d = sin(phi2)*courrant; courrant *= cos(phi2); T c = sin(phi1)*courrant; courrant *= cos(phi1); T b = sin(theta)*courrant; T a = cos(theta)*courrant; return(rho*octonion<T>(a,b,c,d,e,f,g,h)); } template<typename T> inline 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) { using ::std::cos; using ::std::sin; T a = rho1*cos(theta1); T b = rho1*sin(theta1); T c = rho2*cos(theta2); T d = rho2*sin(theta2); T e = rho3*cos(theta3); T f = rho3*sin(theta3); T g = rho4*cos(theta4); T h = rho4*sin(theta4); return(octonion<T>(a,b,c,d,e,f,g,h)); } template<typename T> inline 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) { using ::std::cos; using ::std::sin; T a = r*cos(angle); T b = r*sin(angle); return(octonion<T>(a,b,h1,h2,h3,h4,h5,h6)); } template<typename T> inline octonion<T> exp(octonion<T> const & o) { using ::std::exp; using ::std::cos; using ::boost::math::sinc_pi; T u = exp(real(o)); T z = abs(unreal(o)); T w = sinc_pi(z); return(u*octonion<T>(cos(z), w*o.R_component_2(), w*o.R_component_3(), w*o.R_component_4(), w*o.R_component_5(), w*o.R_component_6(), w*o.R_component_7(), w*o.R_component_8())); } template<typename T> inline octonion<T> cos(octonion<T> const & o) { using ::std::sin; using ::std::cos; using ::std::cosh; using ::boost::math::sinhc_pi; T z = abs(unreal(o)); T w = -sin(o.real())*sinhc_pi(z); return(octonion<T>(cos(o.real())*cosh(z), w*o.R_component_2(), w*o.R_component_3(), w*o.R_component_4(), w*o.R_component_5(), w*o.R_component_6(), w*o.R_component_7(), w*o.R_component_8())); } template<typename T> inline octonion<T> sin(octonion<T> const & o) { using ::std::sin; using ::std::cos; using ::std::cosh; using ::boost::math::sinhc_pi; T z = abs(unreal(o)); T w = +cos(o.real())*sinhc_pi(z); return(octonion<T>(sin(o.real())*cosh(z), w*o.R_component_2(), w*o.R_component_3(), w*o.R_component_4(), w*o.R_component_5(), w*o.R_component_6(), w*o.R_component_7(), w*o.R_component_8())); } template<typename T> inline octonion<T> tan(octonion<T> const & o) { return(sin(o)/cos(o)); } template<typename T> inline octonion<T> cosh(octonion<T> const & o) { return((exp(+o)+exp(-o))/static_cast<T>(2)); } template<typename T> inline octonion<T> sinh(octonion<T> const & o) { return((exp(+o)-exp(-o))/static_cast<T>(2)); } template<typename T> inline octonion<T> tanh(octonion<T> const & o) { return(sinh(o)/cosh(o)); } template<typename T> octonion<T> pow(octonion<T> const & o, int n) { if (n > 1) { int m = n>>1; octonion<T> result = pow(o, m); result *= result; if (n != (m<<1)) { result *= o; // n odd } return(result); } else if (n == 1) { return(o); } else if (n == 0) { return(octonion<T>(static_cast<T>(1))); } else /* n < 0 */ { return(pow(octonion<T>(static_cast<T>(1))/o,-n)); } } // helper templates for converting copy constructors (definition) namespace detail { template< typename T, typename U > octonion<T> octonion_type_converter(octonion<U> const & rhs) { return(octonion<T>( static_cast<T>(rhs.R_component_1()), static_cast<T>(rhs.R_component_2()), static_cast<T>(rhs.R_component_3()), static_cast<T>(rhs.R_component_4()), static_cast<T>(rhs.R_component_5()), static_cast<T>(rhs.R_component_6()), static_cast<T>(rhs.R_component_7()), static_cast<T>(rhs.R_component_8()))); } } } } #endif /* BOOST_OCTONION_HPP */