/******************************************
 * C++ Octonions
 * Version: 1.0.9
 * Author:  Douglas Wilhelm Harder
 * Date:    2008/03/03
 *
 * Copyright (c) 2006-2008 by Douglas Wilhelm Harder.
 * All rights reserved.
 ******************************************/

#ifndef CA_UWATERLOO_ALUMNI_DWHARDER_OCTONION
#define CA_UWATERLOO_ALUMNI_DWHARDER_OCTONION

#include <iostream>
#include <cmath>
#include <string>

template <typename T> class Octonion;

template <typename T> std::ostream & operator << ( std::ostream &, const Octonion<T> & );

template <typename T> Octonion<T> operator + ( T, const Octonion<T> & );
template <typename T> Octonion<T> operator + ( long, const Octonion<T> & );
template <typename T> Octonion<T> operator - ( T, const Octonion<T> & );
template <typename T> Octonion<T> operator - ( long, const Octonion<T> & );
template <typename T> Octonion<T> operator * ( T, const Octonion<T> & );
template <typename T> Octonion<T> operator * ( long, const Octonion<T> & );
template <typename T> Octonion<T> operator / ( T, const Octonion<T> & );
template <typename T> Octonion<T> operator / ( long, const Octonion<T> & );

template <typename T> bool operator == ( T, const Octonion<T> & );
template <typename T> bool operator == ( long, const Octonion<T> & );
template <typename T> bool operator != ( T, const Octonion<T> & );
template <typename T> bool operator != ( long, const Octonion<T> & );

template <typename T> T real( const Octonion<T> & );
template <typename T> T imag_i( const Octonion<T> & );
template <typename T> T imag_j( const Octonion<T> & );
template <typename T> T imag_k( const Octonion<T> & );
template <typename T> T imag_u1( const Octonion<T> & );
template <typename T> T imag_i1( const Octonion<T> & );
template <typename T> T imag_j1( const Octonion<T> & );
template <typename T> T imag_k1( const Octonion<T> & );
template <typename T> T csgn( const Octonion<T> & );
template <typename T> T abs( const Octonion<T> & );
template <typename T> T norm( const Octonion<T> & );
template <typename T> T abs_imag( const Octonion<T> & );
template <typename T> T norm_imag( const Octonion<T> & );
template <typename T> T arg( const Octonion<T> & );
template <typename T> Octonion<T> imag( const Octonion<T> & );
template <typename T> Octonion<T> conj( const Octonion<T> & );
template <typename T> Octonion<T> signum( const Octonion<T> & );
template <typename T> Octonion<T> sqr( const Octonion<T> & );
template <typename T> Octonion<T> sqrt( const Octonion<T> & );
template <typename T> Octonion<T> rotate( const Octonion<T> &, const Octonion<T> & );
template <typename T> Octonion<T> exp( const Octonion<T> & );
template <typename T> Octonion<T> log( const Octonion<T> & );
template <typename T> Octonion<T> log10( const Octonion<T> & );
template <typename T> Octonion<T> pow( const Octonion<T> &, const Octonion<T> & );
template <typename T> Octonion<T> pow( const Octonion<T> &, T );
template <typename T> Octonion<T> inverse( const Octonion<T> & );
template <typename T> Octonion<T> sin( const Octonion<T> & );
template <typename T> Octonion<T> cos( const Octonion<T> & );
template <typename T> Octonion<T> tan( const Octonion<T> & );
template <typename T> Octonion<T> sec( const Octonion<T> & );
template <typename T> Octonion<T> csc( const Octonion<T> & );
template <typename T> Octonion<T> cot( const Octonion<T> & );
template <typename T> Octonion<T> sinh( const Octonion<T> & );
template <typename T> Octonion<T> cosh( const Octonion<T> & );
template <typename T> Octonion<T> tanh( const Octonion<T> & );
template <typename T> Octonion<T> sech( const Octonion<T> & );
template <typename T> Octonion<T> csch( const Octonion<T> & );
template <typename T> Octonion<T> coth( const Octonion<T> & );
template <typename T> Octonion<T> asin( const Octonion<T> &, const Octonion<T> & = Octonion<T>::I );
template <typename T> Octonion<T> acos( const Octonion<T> &, const Octonion<T> & = Octonion<T>::I );
template <typename T> Octonion<T> atan( const Octonion<T> & );
template <typename T> Octonion<T> asec( const Octonion<T> &, const Octonion<T> & = Octonion<T>::I );
template <typename T> Octonion<T> acsc( const Octonion<T> &, const Octonion<T> & = Octonion<T>::I );
template <typename T> Octonion<T> acot( const Octonion<T> & );
template <typename T> Octonion<T> asinh( const Octonion<T> & );
template <typename T> Octonion<T> acosh( const Octonion<T> &, const Octonion<T> & = Octonion<T>::I );
template <typename T> Octonion<T> atanh( const Octonion<T> &, const Octonion<T> & = Octonion<T>::I );
template <typename T> Octonion<T> asech( const Octonion<T> &, const Octonion<T> & = Octonion<T>::I );
template <typename T> Octonion<T> acsch( const Octonion<T> & );
template <typename T> Octonion<T> acoth( const Octonion<T> &, const Octonion<T> & = Octonion<T>::I );
template <typename T> Octonion<T> bessel_J( int, const Octonion<T> & );
template <typename T> Octonion<T> floor( const Octonion<T> & );
template <typename T> Octonion<T> ceil( const Octonion<T> & );
template <typename T> Octonion<T> horner( const Octonion<T> &, T *, unsigned int );
template <typename T> Octonion<T> horner( const Octonion<T> &, T *, T *, unsigned int );

template <typename T = double> class Octonion {
	private:
		T r, i, j, k, u1, i1, j1, k1;

		inline static Octonion multiplier( T, T, const Octonion & );
		inline static Octonion make_inf( T, T );
		inline static Octonion make_i( T, T );

	public:
		const static Octonion ZERO;
		const static Octonion ONE;
		const static Octonion I;
		const static Octonion J;
		const static Octonion K;
		const static Octonion U1;
		const static Octonion I1;
		const static Octonion J1;
		const static Octonion K1;

		const static Octonion UNITS[8];

		/******************************************
		 * Constructor and Copy Constructor
		 ******************************************/

		Octonion( T, T, T, T, T, T, T, T );
		Octonion( T = 0.0 );

		/******************************************
		 * Assignment Operator
		 ******************************************/

		const Octonion & operator = ( const T & );

		/******************************************
		 * Mutating Arithmetic Operators
		 ******************************************/

		Octonion & operator += ( const Octonion & );
		Octonion & operator -= ( const Octonion & );
		Octonion & operator *= ( const Octonion & );
		Octonion & operator /= ( const Octonion & );

		Octonion & operator += ( T );
		Octonion & operator -= ( T );
		Octonion & operator *= ( T );
		Octonion & operator /= ( T );

		Octonion & operator ++();
		Octonion operator ++( int );
		Octonion & operator --();
		Octonion operator --( int );

		/******************************************
		 * Real-valued Functions
		 ******************************************/

		T real() const;
		T operator []( int ) const;
		T& operator []( int );
		T imag_i() const;
		T imag_j() const;
		T imag_k() const;
		T imag_u1() const;
		T imag_i1() const;
		T imag_j1() const;
		T imag_k1() const;
		T csgn() const;
		T abs() const;
		T norm() const;
		T abs_imag() const;
		T norm_imag() const;
		T arg() const;

		/******************************************
		 * Octonion-valued Functions
		 ******************************************/

		Octonion imag() const;
		Octonion conj() const;
		Octonion operator * () const;
		Octonion signum() const;
		Octonion sqr() const;
		Octonion sqrt() const;
		Octonion rotate( const Octonion & ) const;

		/******************************************
		 * Boolean-valued Functions
		 ******************************************/

		bool is_imaginary() const;
		bool is_inf() const;
		bool is_nan() const;
		bool is_neg_inf() const;
		bool is_pos_inf() const;
		bool is_real() const;
		bool is_real_inf() const;
		bool is_zero() const;

		/******************************************
		 * Exponential and Logarithmic Functions
		 ******************************************/

		Octonion exp() const;
		Octonion log() const;
		Octonion log10() const;
		Octonion pow( const Octonion & ) const;
		Octonion pow( T ) const;
		Octonion inverse() const;

		/******************************************
		 * Trigonometric and Hyperbolic Functions
		 ******************************************/

		Octonion   sin() const;   Octonion   cos() const;   Octonion   tan() const;
		Octonion   sec() const;   Octonion   csc() const;   Octonion   cot() const;
		Octonion  sinh() const;   Octonion  cosh() const;   Octonion  tanh() const;
		Octonion  sech() const;   Octonion  csch() const;   Octonion  coth() const;
		Octonion  asin( const Octonion & = I ) const;
		Octonion  acos( const Octonion & = I ) const;       Octonion  atan() const;
		Octonion  asec( const Octonion & = I ) const;
		Octonion  acsc( const Octonion & = I ) const;       Octonion  acot() const;

		Octonion asinh() const;
		Octonion acosh( const Octonion & = I ) const;
		Octonion atanh( const Octonion & = I ) const;

		Octonion asech( const Octonion & = I ) const;
		Octonion acsch() const;
		Octonion acoth( const Octonion & = I ) const;

		Octonion bessel_J( int ) const;

		/******************************************
		 * Integer Functions
		 ******************************************/

		Octonion ceil() const;
		Octonion floor() const;

		/******************************************
		 * Horner's Rule
		 ******************************************/

		Octonion horner( T *, T *, unsigned int ) const;
		Octonion horner( T *, unsigned int ) const;

		/******************************************
		 * Random Factories
		 ******************************************/

		static Octonion random();
		static Octonion random_imag();
		static Octonion random_real();

		/******************************************
		 * Binary Arithmetic Operators
		 ******************************************/

		Octonion operator + ( const Octonion & ) const;
		Octonion operator + ( T ) const;

		Octonion operator - ( const Octonion & ) const;
		Octonion operator - ( T ) const;

		Octonion operator * ( const Octonion & ) const;
		Octonion operator * ( T ) const;

		Octonion operator / ( const Octonion & ) const;
		Octonion operator / ( T ) const;

		/******************************************
		 * Unary Arithmetic Operators
		 ******************************************/

		Octonion operator - () const;

		/******************************************
		 * Binary Boolean Operators
		 ******************************************/

		bool operator == ( const Octonion & ) const;
		bool operator == ( T ) const;

		bool operator != ( const Octonion & ) const;
		bool operator != ( T ) const;
};

#endif