#ifndef TCG_POLY_OPS_H

#define TCG_POLY_OPS_H

// tcg includes

#include "macros.h"

/*!

  \file     tcg_poly_ops.h

  \brief    This file contains useful functions for operating with polynomials.

*/

//*******************************************************************************

//    Polynomial Operations

//*******************************************************************************

namespace tcg

{

//! Contains useful functions for operating with polynomials.

namespace poly_ops

{

/*!

  \brief    Evaluates a polynomial using Horner's algorithm

  \return   The value of the input polynomial at the specified parameter

*/

template <typename scalar=""></typename>

Scalar evaluate(

	const Scalar poly[], //!< Coefficients of the input polynomial, indexed by degree

	int deg,			 //!< Degree of the polynomial function

	Scalar x)			 //!< Parameter the polynomial will be evaluated on

{

	// ((poly[deg] * x) + poly[deg-1]) * x + poly[deg - 2] + ...

	Scalar value = poly[deg];

	for (int d = deg - 1; d >= 0; --d)

		value = value * x + poly[d];

	return value;

}

//-------------------------------------------------------------------------------------------

/*!

  \brief  Reduces the degree of the input polynomial, discarding all leading coefficients

          whose absolute value is below the specified tolerance threshold.

*/

template <typename scalar=""></typename>

void reduceDegree(

	const Scalar poly[], //!< Input polynomial to be reduced.

	int °,			 //!< Input/output polynomial degree.

	Scalar tolerance	 //!< Coefficients threshold to reduce the polynomial with.

	)

{

	while ((deg > 0) && (std::abs(poly[deg]) < tolerance))

		--deg;

}

//-------------------------------------------------------------------------------------------

/*!

  \brief    Adds two polynomials and returns the sum.

  \remark   The supplied polynomials can actually be the same.

*/

template <typename a,="" b,="" c,="" deg="" int="" typename=""></typename>

void add(

	const A(&poly1)[deg], //!< First polynomial addendum.

	const B(&poly2)[deg], //!< Second polynomial addendum.

	C(&result)[deg])	  //!< Resulting sum.

{

	for (int d = 0; d != deg; ++d)

		result[d] = poly1[d] + poly2[d];

}

//-------------------------------------------------------------------------------------------

/*!

  \brief  Subtracts two polynomials /p poly1 and \p poly2 and returns the

          difference <tt>poly1 - poly2</tt>.

*/

template <typename a,="" b,="" c,="" deg="" int="" typename=""></typename>

void sub(

	const A(&poly1)[deg], //!< First polynomial addendum.

	const B(&poly2)[deg], //!< Second polynomial addendum.

	C(&result)[deg])	  //!< Resulting difference.

{

	for (int d = 0; d != deg; ++d)

		result[d] = poly1[d] - poly2[d];

}

//-------------------------------------------------------------------------------------------

/*!

  \brief    Multiplies two polynomials into a polynomial whose degree is the

            \a sum of the multiplicands' degrees.

  \warning  The resulting polynomial is currently not allowed to be one

            of the multiplicands.

*/

template <typename a,="" b,="" c,="" deg1,="" deg2,="" degr="" int="" typename=""></typename>

void mul(

	const A(&poly1)[deg1], //!< First multiplicand.

	const B(&poly2)[deg2], //!< Second multiplicand.

	C(&result)[degR])	  //!< Resulting polynomial.

{

	TCG_STATIC_ASSERT(degR == deg1 + deg2 - 1);

	for (int a = 0; a != deg1; ++a) {

		for (int b = 0; b != deg2; ++b)

			result[a + b] += poly1[a] * poly2[b];

	}

}

//-------------------------------------------------------------------------------------------

/*!

  \brief    Calculates the chaining <tt>poly1 o poly2</tt> of two given polynomial

            \p poly1 and \p poly2.

  \warning  The resulting polynomial is currently not allowed to be one

            of the multiplicands.

  \warning  This function is still \b untested.

*/

template <typename a,="" b,="" c,="" deg1,="" deg2,="" degr="" int="" typename=""></typename>

void chain(

	const A(&poly1)[deg1], //!< First polynomial.

	const B(&poly2)[deg2], //!< Second polynomial.

	C(&result)[degR])	  //!< Resulting polynomial.

{

	for (int a = 0; a != deg1; ++a) {

		B pow[degR][2] = {{}};

		// Build poly2 powers

		{

			std::copy(poly2, poly2 + deg2, pow[1]);

			for (int p = 1; p < a; ++p)

				poly_mul(pow[p % 2], poly2, pow[(p + 1) % 2]);

		}

		B(&pow_add)

		[degR] = pow[a % 2];

		// multiply by poly1[a]

		C addendum[degR];

		for (int c = 0; c != degR; ++c)

			addendum[c] = poly1[c] * pow_add[c];

		poly_add(addendum, result);

	}

}

//-------------------------------------------------------------------------------------------

template <typename a,="" b,="" deg,="" degr="" int="" typename=""></typename>

void derivative(

	const A(&poly)[deg], //!< Polynomial to be derived.

	B(&result)[degR])	//!< Resulting derivative polynomial.

{

	TCG_STATIC_ASSERT(degR == deg - 1);

	for (int c = 1; c != deg; ++c)

		result[c - 1] = c * poly[c];

}

//-------------------------------------------------------------------------------------------

/*!

  \brief    Solves the 1st degree equation: $c[1] t + c[0] = 0$

  \return   The number of solutions found under the specified divide-by tolerance

*/

template <typename scalar=""></typename>

inline unsigned int solve_1(

	Scalar c[2],	//!< Polynomial coefficients array

	Scalar s[1],	//!< Solutions array

	Scalar tol = 0) //!< Leading coefficient tolerance, the equation has no solution

					//!  if the leading coefficient is below this threshold

{

	if (std::abs(c[1]) <= tol)

		return 0;

	s[0] = -c[0] / c[1];

	return 1;

}

//-------------------------------------------------------------------------------------------

/*!

  \brief    Solves the 2nd degree equation: $c[2] t^2 + c[1] t + c[0] = 0$

  \return   The number of #real# solutions found under the specified divide-by tolerance

  \remark   The returned solutions are sorted, with $s[0] <= s[1]$

*/

template <typename scalar=""></typename>

unsigned int solve_2(

	Scalar c[3],	//!< Polynomial coefficients array

	Scalar s[2],	//!< Solutions array

	Scalar tol = 0) //!< Leading coefficient tolerance, the equation has no solution

					//!  if the leading coefficient is below this threshold

{

	if (std::abs(c[2]) <= tol)

		return solve_1(c, s, tol); // Reduces to first degree

	Scalar nc[2] = {c[0] / c[2], c[1] / (c[2] + c[2])}; // NOTE: nc[1] gets further divided by 2

	Scalar delta = nc[1] * nc[1] - nc[0];

	if (delta < 0)

		return 0;

	delta = sqrt(delta);

	s[0] = -delta - nc[1];

	s[1] = delta - nc[1];

	return 2;

}

//-------------------------------------------------------------------------------------------

/*!

  \brief    Solves the 3rd degree equation: $c[3]t^3 + c[2] t^2 + c[1] t + c[0] = 0$

  \return   The number of #real# solutions found under the specified divide-by tolerance

  \remark   The returned solutions are sorted, with $s[0] <= s[1] <= s[2]$

*/

template <typename scalar=""></typename>

unsigned int solve_3(

	Scalar c[4],	//!< Polynomial coefficients array

	Scalar s[3],	//!< Solutions array

	Scalar tol = 0) //!< Leading coefficient tolerance, the equation is reduced to 2nd degree

					//!  if the leading coefficient is below this threshold

{

	if (std::abs(c[3]) <= tol)

		return solve_2(c, s, tol); // Reduces to second degree

	Scalar nc[3] = {c[0] / c[3], c[1] / c[3], c[2] / c[3]};

	Scalar b2 = nc[2] * nc[2];

	Scalar p = nc[1] - b2 / 3;

	Scalar q = nc[2] * (b2 + b2 - 9 * nc[1]) / 27 + nc[0];

	Scalar p3 = p * p * p;

	Scalar d = q * q + 4 * p3 / 27;

	Scalar offset = -nc[2] / 3;

	if (d >= 0) {

		// Single solution

		Scalar z = sqrt(d);

		Scalar u = (-q + z) / 2;

		Scalar v = (-q - z) / 2;

		u = (u < 0) ? -pow(-u, 1 / Scalar(3)) : pow(u, 1 / Scalar(3));

		v = (v < 0) ? -pow(-v, 1 / Scalar(3)) : pow(v, 1 / Scalar(3));

		s[0] = offset + u + v;

		return 1;

	}

	assert(p3 < 0);

	Scalar u = sqrt(-p / Scalar(3));

	Scalar v = acos(-sqrt(-27 / p3) * q / Scalar(2)) / Scalar(3);

	Scalar m = cos(v), n = sin(v) * 1.7320508075688772935274463415059; // sqrt(3)

	s[0] = offset - u * (n + m);

	s[1] = offset + u * (n - m);

	s[2] = offset + u * (m + m);

	assert(s[0] <= s[1] && s[1] <= s[2]);

	return 3;

}

}

} // namespace tcg::poly_ops

#endif // TCG_POLY_OPS_H