#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>
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>
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, typename B, typename C, int deg>
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, typename B, typename C, int deg>
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 <B>not allowed</B> to be one
of the multiplicands.
*/
template <typename A, typename B, typename C, int deg1, int deg2, int degR>
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 <B>not allowed</B> to be one
of the multiplicands.
\warning This function is still \b untested.
*/
template <typename A, typename B, typename C, int deg1, int deg2, int degR>
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, typename B, int deg, int degR>
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>
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>
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>
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