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cs440-acg/ext/openexr/IlmBase/Imath/ImathRoots.h

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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