/*
Copyright (c) 2019, Michael Kazhdan
All rights reserved.

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are permitted provided that the following conditions are met:

Redistributions of source code must retain the above copyright notice, this list of
conditions and the following disclaimer. Redistributions in binary form must reproduce
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in the documentation and/or other materials provided with the distribution. 

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may be used to endorse or promote products derived from this software without specific
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EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO THE IMPLIED WARRANTIES 
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*/

#ifndef POLYNOMIAL_INCLUDED
#define POLYNOMIAL_INCLUDED
#include <iostream>
#include "Geometry.h"
#include "Algebra.h"
#include "Poly34.h"
#include "Exceptions.h"

namespace MishaK
{
	namespace Polynomial
	{
		/** Helper functionality for computing the minimum of two integers.*/
		template< unsigned int D1 , unsigned int D2 > struct Min{ static const unsigned int Value = D1<D2 ? D1 : D2; };
		/** Helper functionality for computing the maximum of two integers.*/
		template< unsigned int D1 , unsigned int D2 > struct Max{ static const unsigned int Value = D1>D2 ? D1 : D2; };

		/** The generic, recursively defined, Polynomial class of total degree Degree. */
		template< unsigned int Dim , unsigned int Degree , typename T , typename Real=T >
		class Polynomial : public VectorSpace< Real , Polynomial< Dim , Degree , T , Real > >
		{
			template< unsigned int _Dim , unsigned int _Degree , typename _T , typename _Real > friend class Polynomial;
			template< unsigned int _Dim , unsigned int Degree1 , unsigned int Degree2 , typename _T , typename _Real > friend Polynomial< _Dim , Degree1 + Degree2 , _T , _Real > operator * ( const Polynomial< _Dim , Degree1 , _T , _Real > & , const Polynomial< _Dim , Degree2 , _T , _Real > & );
			template< unsigned int _Dim , unsigned int Degree1 , unsigned int Degree2 , typename _T , typename _Real > friend Polynomial< _Dim , Max< Degree1 , Degree2 >::Value , _Real > operator + ( const Polynomial< _Dim , Degree1 , _T , _Real > & , const Polynomial< _Dim , Degree2 , _T , _Real > & );
			template< unsigned int _Dim , unsigned int Degree1 , unsigned int Degree2 , typename _T , typename _Real > friend Polynomial< _Dim , Max< Degree1 , Degree2 >::Value , _Real > operator - ( const Polynomial< _Dim , Degree1 , _T , _Real > & , const Polynomial< _Dim , Degree2 , _T , _Real > & );
			template< unsigned int _Dim , unsigned int _Degree , typename _T , typename _Real > friend std::ostream &operator << ( std::ostream & , const Polynomial< _Dim , _Degree , _T , _Real > & );

			/** The polynomials in Dim-1 dimensions.
			*** The total polynomial is assumed to be _polynomials[0] + _polynomials[1] * (x_Dim) + _polynomials[2] * (x_Dim)^2 + ... */
			Polynomial< Dim-1 , Degree , T , Real > _polynomials[Degree+1];

			/** This method returns the specified coefficient of the polynomial.*/
			const T &_coefficient( const unsigned int indices[] , unsigned int maxDegree ) const;

			/** This method returns the specified coefficient of the polynomial.*/
			T &_coefficient( const unsigned int indices[] , unsigned int maxDegree );

			/** This method evaluates the polynomial at the specified set of coordinates.*/
			T _evaluate( const Real coordinates[] , unsigned int maxDegree ) const;

			/** This method computes the pull-back of the polynomial given the map from a _Dim-dimensional space to the Dim-dimensional space given by x -> A({x,1}). */	
			template< unsigned int _Dim >
			Polynomial< _Dim , Degree , T , Real > _pullBack( const Matrix< Real , _Dim+1 , Dim > &A , unsigned int maxDegree ) const;

			/** This method returns true if the polynomial is zero. */
			bool _isZero( unsigned int maxDegree ) const;

			/** This method returns true if the polynomial is a constant. */
			bool _isConstant( unsigned int maxDegree ) const;

			bool _print( std::ostream &ostream , const std::string varNames[] , bool first ) const;
			bool _print( std::ostream &ostream , const std::string varNames[] , std::string suffix , bool first ) const;

			unsigned int _setCoefficients( const T *coefficients , unsigned int maxDegree );
			unsigned int _getCoefficients(       T *coefficients , unsigned int maxDegree ) const;

			template< unsigned int _D=0 >
			static void _SetDegrees( unsigned int coefficientIndex , unsigned int degrees[Dim] );

		public:
			/** The number of coefficients in the polynomial. */
			static const unsigned int NumCoefficients = Polynomial< Dim , Degree-1 , Real >::NumCoefficients + Polynomial< Dim-1 , Degree , T , Real >::NumCoefficients;

			/** The default constructor initializes the coefficients to zero.*/
			Polynomial( void );

			/** This constructor creates a constant polynomial */
			Polynomial( T c );

			/** This constructor initializes the coefficients. */
			Polynomial( const T coefficients[NumCoefficients] );

			/** The constructor copies over as much of the polynomial as will fit.*/
			template< unsigned int _Degree >
			Polynomial( const Polynomial< Dim , _Degree , T , Real > &p );

			/** The equality operator copies over as much of the polynomial as will fit.*/
			template< unsigned int _Degree >
			Polynomial &operator = ( const Polynomial< Dim , _Degree , T , Real > &p );

			/** This method returns the associated coefficient of the polynomial */
			template< typename ... UnsignedInts >
			const T &coefficient( unsigned int idx , UnsignedInts ... indices ) const;

			/** This method returns the associated coefficient of the polynomial */
			template< typename ... UnsignedInts >
			T &coefficient( unsigned int idx , UnsignedInts ... indices );

			/** This method returns the associated coefficient of the polynomial */
			template< typename ... Ints >
			const T &coefficient(  int idx , Ints ... indices ) const;

			/** This method returns the associated coefficient of the polynomial */
			template< typename ... Ints >
			T &coefficient( int idx , Ints ... indices );

			/** This method returns the associated coefficient of the polynomial */
			T &coefficient( const unsigned int indices[ Dim ] );

			/** This method returns the associated coefficient of the polynomial */
			const T &coefficient( const unsigned int indices[ Dim ] ) const;

			/** This method returns the associated coefficient of the polynomial */
			T &operator[]( unsigned int idx );

			/** This method returns the associated coefficient of the polynomial */
			const T &operator[]( unsigned int idx ) const;

			/** This static method sets the degrees of the associated coefficient */
			static void SetDegrees( unsigned int coefficientIndex , unsigned int degrees[Dim] );

			/** This method returns a vector of the coefficients of the polynomial */
			Point< T , Polynomial< Dim , Degree , T , Real >::NumCoefficients , Real > coefficients( void ) const;

			/** This method evaluates the polynomial at the prescribed point.*/
			template< typename ... Reals >
			T operator()( Real coord , Reals ... coordinates ) const;

			/** This method evaluates the polynomial at the prescribed point.*/
			T operator()( Point< Real , Dim > p ) const;

			/** This method evaluates the gradient of the polynomial at the prescribed point.*/
			template< typename ... Reals >
			Point< T , Dim , Real > gradient( Reals ... coordinates ) const;

			/** This method evaluates the gradient of the  polynomial at the prescribed point.*/
			Point< T , Dim , Real > gradient( Point< Real , Dim > p ) const;

			/** This method evaluates the Hessian of the polynomial at the prescribed point.*/
			template< typename ... Reals >
			SquareMatrix< T , Dim > hessian( Reals ... coordinates ) const;

			/** This method evaluates the Hessian of the  polynomial at the prescribed point.*/
			SquareMatrix< T , Dim > hessian( Point< Real , Dim > p ) const;

			/** This method returns the partial derivative with respect to the prescribed dimension.*/
			Polynomial< Dim , (Degree>1) ? Degree-1 : 0 , T , Real > d( int dim ) const;

			/** This method computes the pull-back of the polynomial given the map from a (_Dim-1)-dimensional space to the Dim-dimensional space given by x -> A({x,1}). */	
			template< unsigned int _Dim >
			Polynomial< _Dim-1 , Degree , T , Real > operator()( Matrix< Real , _Dim , Dim > A ) const;
			template< unsigned int _Dim >
			Polynomial< _Dim-1 , Degree , T , Real > pullBack( Matrix< Real , _Dim , Dim > A ) const;
			Polynomial< 1 , Degree , T , Real > operator()( const Ray< Real , Dim > &ray ) const;

			/** Integrate the polynomial over a unit cube */
			T integrateUnitCube( void ) const;

			/** Integrate the polynomial over a unit right simplex */
			T integrateUnitRightSimplex( void ) const;

			/** Returns the matrix taking in the ccoefficients of the polynomial and returning the values at the prescribed points */
			static Matrix< Real , Polynomial< Dim , Degree , T , Real >::NumCoefficients , Polynomial< Dim , Degree , T , Real >::NumCoefficients > EvaluationMatrix( const Point< Real , Dim > positions[NumCoefficients] );

			/////////////////////////
			// VectorSpace methods //
			/////////////////////////
			/** This method scales the polynomial */
			void Scale( Real s );

			/** This method adds in the polynomial */
			void Add( const Polynomial & p );
		};

		/** This function returns the product of two polynomials.*/
		template< unsigned int Dim , unsigned int Degree1 , unsigned int Degree2 , typename T , typename Real >
		Polynomial< Dim , Degree1 + Degree2 , T , Real > operator * ( const Polynomial< Dim , Degree1 , T , Real > &p1 , const Polynomial< Dim , Degree2 , T , Real > &p2 );

		/** This function returns the sum of two polynomials. */
		template< unsigned int Dim , unsigned int Degree1 , unsigned int Degree2 , typename T , typename Real >
		Polynomial< Dim , Max< Degree1 , Degree2 >::Value , T , Real > operator + ( const Polynomial< Dim , Degree1 , T , Real > &p1 , const Polynomial< Dim , Degree2 , T , Real > &p2 );

		/** This function returns the difference of two polynomials. */
		template< unsigned int Dim , unsigned int Degree1 , unsigned int Degree2 , typename T , typename Real >
		Polynomial< Dim , Max< Degree1 , Degree2 >::Value , T , Real > operator - ( const Polynomial< Dim , Degree1 , T , Real > &p1 , const Polynomial< Dim , Degree2 , T , Real > &p2 );

		/** This function prints out the polynomial.*/
		template< unsigned int Dim , unsigned int Degree , typename T , typename Real >
		std::ostream &operator << ( std::ostream &stream , const Polynomial< Dim , Degree , T , Real > &poly );


		/** A specialized instance of the Polynomial class in one variable */
		template< unsigned int Degree , typename T , typename Real >
		class Polynomial< 0 , Degree , T , Real > : public VectorSpace< Real , Polynomial< 0 , Degree , T , Real > >
		{
			template< unsigned int _Dim , unsigned int _Degree , typename _T , typename _Real > friend class Polynomial;
			template< unsigned int Degree1 , unsigned int Degree2 , typename _T , typename _Real > friend Polynomial< 0 , Degree1 + Degree2 , _T , _Real > operator * ( const Polynomial< 0 , Degree1 , _T , _Real > & , const Polynomial< 0 , Degree2 , _T , _Real > & );
			template< unsigned int Degree1 , unsigned int Degree2 , typename _T , typename _Real > friend Polynomial< 0 , Max< Degree1 , Degree2 >::Value , _T , _Real > operator + ( const Polynomial< 0 , Degree1 , _T , _Real > & , const Polynomial< 0 , Degree2 , _T , _Real > & );
			template< unsigned int Degree1 , unsigned int Degree2 , typename _T , typename _Real > friend Polynomial< 0 , Max< Degree1 , Degree2 >::Value , _T , _Real > operator - ( const Polynomial< 0 , Degree1 , _T , _Real > & , const Polynomial< 0 , Degree2 , _T , _Real > & );
			template< unsigned int _Degree , typename _T , typename _Real > friend std::ostream &operator << ( std::ostream & , const Polynomial< 0 , _Degree , _Real > & );
			template< unsigned int Dim , unsigned int Degree1 , unsigned int Degree2 , typename _T , typename _Real > friend Polynomial< Dim , Degree1 + Degree2 , _T , _Real > operator * ( const Polynomial< Dim , Degree1 , _T , _Real > & , const Polynomial< Dim , Degree2 , _T , _Real > & );
			template< unsigned int Dim , unsigned int Degree1 , unsigned int Degree2 , typename _T , typename _Real > friend Polynomial< Dim , Max< Degree1 , Degree2 >::Value , _T , _Real > operator + ( const Polynomial< Dim , Degree1 , _T , _Real > & , const Polynomial< Dim , Degree2 , _T , _Real > & );
			template< unsigned int Dim , unsigned int Degree1 , unsigned int Degree2 , typename _T , typename _Real > friend Polynomial< Dim , Max< Degree1 , Degree2 >::Value , _T , _Real > operator - ( const Polynomial< Dim , Degree1 , _T , _Real > & , const Polynomial< Dim , Degree2 , _T , _Real > & );
			template< unsigned int Dim , unsigned int _Degree , typename _T , typename _Real > friend std::ostream &operator << ( std::ostream & , const Polynomial< Dim , _Degree , _T , _Real > & );
		protected:
			/** The coefficients of the polynomial. */
			T _coefficients[1];

			/** This method returns the specified coefficient of the polynomial.*/
			const T &_coefficient( const unsigned int indices[] , unsigned int maxDegree ) const;

			/** This method returns the specified coefficient of the polynomial.*/
			T &_coefficient( const unsigned int indices[] , unsigned int maxDegree );

			/** This method evaluates the polynomial at the specified set of coordinates.*/
			T _evaluate( const Real coordinates[] , unsigned int maxDegree ) const;

			/** This method computes the pull-back of the polynomial given the map from a _Dim-dimensional space to the 1-dimensional space given by x -> A({x,1}). */	
			template< unsigned int _Dim >
			Polynomial< _Dim , Degree , T , Real > _pullBack( const Matrix< Real , _Dim+1 , 0 > &A , unsigned int maxDegree ) const;

			/** This method returns true if the polynomial is zero. */
			bool _isZero( unsigned int maxDegree ) const;

			/** This method returns true if the polynomial is a constant. */
			bool _isConstant( unsigned int maxDegree ) const;

			bool _print( std::ostream &ostream , const std::string varNames[] , bool first ) const;
			bool _print( std::ostream &ostream , const std::string varNames[] , std::string suffix , bool first ) const;

			unsigned int _setCoefficients( const T *coefficients , unsigned int maxDegree );
			unsigned int _getCoefficients(       T *coefficients , unsigned int maxDegree ) const;
		public:
			/** The number of coefficients in the polynomial. */
			static const unsigned int NumCoefficients = 1;

			/** The default constructor initializes the coefficients to zero.*/
			Polynomial( void );

			/** This constructor creates a constant polynomial */
			Polynomial( T c );

			/** This constructor initializes the coefficients (starting with lower degrees). */
			Polynomial( const T coefficients[NumCoefficients] );

			/** The constructor copies over as much of the polynomial as will fit.*/
			template< unsigned int _Degree >
			Polynomial( const Polynomial< 0 , _Degree , T , Real > &p );

			/** The equality operator copies over as much of the polynomial as will fit.*/
			template< unsigned int _Degree >
			Polynomial &operator= ( const Polynomial< 0 , _Degree , T , Real > &p );

			/** This method returns the coefficient.*/
			const T &coefficient( void ) const;

			/** This method returns the coefficient.*/
			T &coefficient( void );

			/** This method returns the associated coefficient of the polynomial */
			const T &coefficient( const unsigned int indices[0] ) const;

			/** This method returns the associated coefficient of the polynomial */
			T &coefficient( const unsigned int indices[0] );

			/** This method returns the associated coefficient of the polynomial */
			T &operator[]( unsigned int idx );

			/** This method returns the associated coefficient of the polynomial */
			const T &operator[]( unsigned int idx ) const;

			/** This static method sets the degrees of the associated coefficient */
			static void SetDegrees( unsigned int coefficientIndex , unsigned int degrees[0] );

			/** This method returns a vector of the coefficients of the polynomial */
			Point< T , Polynomial< 0 , Degree , T , Real >::NumCoefficients , Real > coefficients( void ) const;

			/** This method evaluates the polynomial at a given value.*/
			T operator()( void ) const;

			/** This method evaluates the polynomial at the prescribed point.*/
			T operator()( Point< Real , 0 > p ) const;

			/** This method returns the derivative of the polynomial.*/
			Polynomial< 0 , (Degree>1) ? Degree-1 : 0 , T , Real > d( unsigned int d=0 ) const;

			/** This method computes the pull-back of the polynomial given the map from a (_Dim-1)-dimensional space to the 1-dimensional space given by x -> A({x,1}). */	
			template< unsigned int _Dim >
			Polynomial< _Dim-1 , Degree , T , Real > operator()( const Matrix< Real , _Dim , 0 > &A ) const;

			/** Integrate the polynomial over a unit interval */
			T integrateUnitCube( void ) const;

			/** Integrate the polynomial over a unit interval */
			T integrateUnitRightSimplex( void ) const;

			/** Returns the matrix taking in the ccoefficients of the polynomial and returning the values at the prescribed points */
			static Matrix< Real , Polynomial< 0 , Degree , T , Real >::NumCoefficients  , Polynomial< 0 , Degree , T , Real >::NumCoefficients> EvaluationMatrix( const Point< Real , 0 > positions[NumCoefficients] );

			/////////////////////////
			// VectorSpace methods //
			/////////////////////////
			/** This method scales the polynomial*/
			void Scale( Real s );

			/** This method adds in the polynomial */
			void Add( const Polynomial & p );
		};

		/** This function returns the product of two polynomials.*/
		template< unsigned int Degree1 , unsigned int Degree2 , typename T , typename Real >
		Polynomial< 0 , Degree1 + Degree2 , T , Real > operator * ( const Polynomial< 0 , Degree1 , T , Real > &p1 , const Polynomial< 0 , Degree2 , T , Real > &p2 );

		/** This function returns the sum of two polynomials. */
		template< unsigned int Degree1 , unsigned int Degree2 , typename T , typename Real >
		Polynomial< 0 , Max< Degree1 , Degree2 >::Value , T , Real > operator + ( const Polynomial< 0 , Degree1 , T , Real > &p1 , const Polynomial< 0 , Degree2 , T , Real > &p2 );

		/** This function returns the difference of two polynomials. */
		template< unsigned int Degree1 , unsigned int Degree2 , typename T , typename Real >
		Polynomial< 0 , Max< Degree1 , Degree2 >::Value , T , Real > operator - ( const Polynomial< 0 , Degree1 , T , Real > &p1 , const Polynomial< 0 , Degree2 , T , Real > &p2 );

		/** A specialization that allows us for the recursive computation of the number of coefficints, that allows us to avoid special-casing when the the Degree is zero. */
		template< unsigned int Dim , typename T , typename Real >
		class Polynomial< Dim , (unsigned int)-1 , T , Real >
		{
		public:
			static const unsigned int NumCoefficients = 0;
		};


		////////////////////////////////////////////////
		// Classes specialized for 1D, 2D, 3D, and 4D //
		////////////////////////////////////////////////
		/** A polynomial in one variable of degree Degree */
		template< unsigned int Degree >
		using Polynomial1D = Polynomial< 1 , Degree , double >;

		/** A polynomial in two variables of degree Degree */
		template< unsigned int Degree >
		using Polynomial2D = Polynomial< 2 , Degree , double >;

		/** A polynomial in three variable of degree Degree */
		template< unsigned int Degree >
		using Polynomial3D = Polynomial< 3 , Degree , double >;

		/** A polynomial in four variable of degree Degree */
		template< unsigned int Degree >
		using Polynomial4D = Polynomial< 4 , Degree , double >;


		/** Sets the roots of the 1D polynomial and returns the number of roots set.
		*** The method is only specialized for degrees 1, 2, 3, and 4. */
		template< unsigned int Degree , typename Real >
		unsigned int Roots( const Polynomial< 1 , Degree , Real > &p , Real *r , double eps=1e-14 );

#include "Polynomial.inl"
	}
}
#endif // POLYNOMIAL_INCLUDED
