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///////////////////////////////////////////////////////////////////////////
//
// Copyright (c) 2002-2012, Industrial Light & Magic, a division of Lucas
// Digital Ltd. LLC
//
// All rights reserved.
//
// Redistribution and use in source and binary forms, with or without
// modification, 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 the above
// copyright notice, this list of conditions and the following disclaimer
// in the documentation and/or other materials provided with the
// distribution.
// * Neither the name of Industrial Light & Magic nor the names of
// its contributors may be used to endorse or promote products derived
// from this software without specific prior written permission.
//
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
// OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
// SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
// LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
// DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
// THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
// (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
// OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
//
///////////////////////////////////////////////////////////////////////////
#ifndef INCLUDED_IMATHROOTS_H
#define INCLUDED_IMATHROOTS_H
//---------------------------------------------------------------------
//
// Functions to solve linear, quadratic or cubic equations
//
//---------------------------------------------------------------------
#include "ImathMath.h"
#include "ImathNamespace.h"
#include <complex>
IMATH_INTERNAL_NAMESPACE_HEADER_ENTER
//--------------------------------------------------------------------------
// Find the real solutions of a linear, quadratic or cubic equation:
//
// function equation solved
//
// solveLinear (a, b, x) a * x + b == 0
// solveQuadratic (a, b, c, x) a * x*x + b * x + c == 0
// solveNormalizedCubic (r, s, t, x) x*x*x + r * x*x + s * x + t == 0
// solveCubic (a, b, c, d, x) a * x*x*x + b * x*x + c * x + d == 0
//
// Return value:
//
// 3 three real solutions, stored in x[0], x[1] and x[2]
// 2 two real solutions, stored in x[0] and x[1]
// 1 one real solution, stored in x[1]
// 0 no real solutions
// -1 all real numbers are solutions
//
// Notes:
//
// * It is possible that an equation has real solutions, but that the
// solutions (or some intermediate result) are not representable.
// In this case, either some of the solutions returned are invalid
// (nan or infinity), or, if floating-point exceptions have been
// enabled with Iex::mathExcOn(), an Iex::MathExc exception is
// thrown.
//
// * Cubic equations are solved using Cardano's Formula; even though
// only real solutions are produced, some intermediate results are
// complex (std::complex<T>).
//
//--------------------------------------------------------------------------
template <class T> int solveLinear (T a, T b, T &x);
template <class T> int solveQuadratic (T a, T b, T c, T x[2]);
template <class T> int solveNormalizedCubic (T r, T s, T t, T x[3]);
template <class T> int solveCubic (T a, T b, T c, T d, T x[3]);
//---------------
// Implementation
//---------------
template <class T>
int
solveLinear (T a, T b, T &x)
{
if (a != 0)
{
x = -b / a;
return 1;
}
else if (b != 0)
{
return 0;
}
else
{
return -1;
}
}
template <class T>
int
solveQuadratic (T a, T b, T c, T x[2])
{
if (a == 0)
{
return solveLinear (b, c, x[0]);
}
else
{
T D = b * b - 4 * a * c;
if (D > 0)
{
T s = Math<T>::sqrt (D);
T q = -(b + (b > 0 ? 1 : -1) * s) / T(2);
x[0] = q / a;
x[1] = c / q;
return 2;
}
if (D == 0)
{
x[0] = -b / (2 * a);
return 1;
}
else
{
return 0;
}
}
}
template <class T>
int
solveNormalizedCubic (T r, T s, T t, T x[3])
{
T p = (3 * s - r * r) / 3;
T q = 2 * r * r * r / 27 - r * s / 3 + t;
T p3 = p / 3;
T q2 = q / 2;
T D = p3 * p3 * p3 + q2 * q2;
if (D == 0 && p3 == 0)
{
x[0] = -r / 3;
x[1] = -r / 3;
x[2] = -r / 3;
return 1;
}
std::complex<T> u = std::pow (-q / 2 + std::sqrt (std::complex<T> (D)),
T (1) / T (3));
std::complex<T> v = -p / (T (3) * u);
const T sqrt3 = T (1.73205080756887729352744634150587); // enough digits
// for long double
std::complex<T> y0 (u + v);
std::complex<T> y1 (-(u + v) / T (2) +
(u - v) / T (2) * std::complex<T> (0, sqrt3));
std::complex<T> y2 (-(u + v) / T (2) -
(u - v) / T (2) * std::complex<T> (0, sqrt3));
if (D > 0)
{
x[0] = y0.real() - r / 3;
return 1;
}
else if (D == 0)
{
x[0] = y0.real() - r / 3;
x[1] = y1.real() - r / 3;
return 2;
}
else
{
x[0] = y0.real() - r / 3;
x[1] = y1.real() - r / 3;
x[2] = y2.real() - r / 3;
return 3;
}
}
template <class T>
int
solveCubic (T a, T b, T c, T d, T x[3])
{
if (a == 0)
{
return solveQuadratic (b, c, d, x);
}
else
{
return solveNormalizedCubic (b / a, c / a, d / a, x);
}
}
IMATH_INTERNAL_NAMESPACE_HEADER_EXIT
#endif // INCLUDED_IMATHROOTS_H