Merge branch 'feature/upstream' into develop

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diff --git a/numlib/roots.c b/numlib/roots.c
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+
+/*
+ *  Roots3And4.c
+ *
+ *  Utility functions to find cubic and quartic roots.
+ *
+ *  Coefficients are passed like this:
+ *
+ *	  c[0] + c[1]*x + c[2]*x^2 + c[3]*x^3 + c[4]*x^4 = 0
+ *
+ *  The functions return the number of non-complex roots and
+ *  put the values into the s array.
+ *
+ *  Author:		 Jochen Schwarze (schwarze@isa.de)
+ *
+ *  Jan 26, 1990	Version for Graphics Gems
+ *  Oct 11, 1990	Fixed sign problem for negative q's in SolveQuartic
+ *  						(reported by Mark Podlipec),
+ *  						Old-style function definitions,
+ *  						IsZero() as a macro
+ *  Nov 23, 1990	Some systems do not declare acos() and cbrt() in
+ *				  <math.h>, though the functions exist in the library.
+ *				  If large coefficients are used, EQN_EPS should be
+ *				  reduced considerably (e.g. to 1E-30), results will be
+ *				  correct but multiple roots might be reported more
+ *				  than once.
+ *  Apr 18, 2018  Reformat for inclusing in ArgyllCMS - GWG
+ *
+ *  The authors and the publisher hold no copyright restrictions 
+ *  on any of these files; this source code is public domain, and 
+ *  is freely available to the entire computer graphics community 
+ *  for study, use, and modification.  We do request that the 
+ *  comment at the top of each file, identifying the original 
+ *  author and its original publication in the book Graphics 
+ *  Gems, be retained in all programs that use these files.
+ *
+ */
+
+#include	<math.h>
+#ifndef M_PI
+#define M_PI		  3.14159265358979323846
+#endif
+
+/* epsilon surrounding for near zero values */
+#define	EQN_EPS 1e-9
+#define	IsZero(x) ((x) > -EQN_EPS && (x) < EQN_EPS)
+
+#define	CBRT(x) ((x) > 0.0 ? pow((double)(x), 1.0/3.0) : \
+                ((x) < 0.0 ? -pow((double)-(x), 1.0/3.0) : 0.0))
+
+int SolveQuadric(double c[3], double s[2]) {
+	double p, q, D;
+
+	/* normal form: x^2 + px + q = 0 */
+	p = c[1] / (2 * c[2]);
+	q = c[0] / c[2];
+
+	D = p * p - q;
+
+	if (IsZero(D)) {
+		s[ 0 ] = - p;
+		return 1;
+	} else if (D < 0) {
+		return 0;
+	} else /* if (D > 0) */ {
+		double sqrt_D = sqrt(D);
+
+		s[0] =   sqrt_D - p;
+		s[1] = - sqrt_D - p;
+		return 2;
+	}
+}
+
+
+int SolveCubic(double c[4], double s[3]) {
+	int i, num;
+	double sub;
+	double A, B, C;
+	double sq_A, p, q;
+	double cb_p, D;
+
+	/* normal form: x^3 + Ax^2 + Bx + C = 0 */
+	A = c[2] / c[3];
+	B = c[1] / c[3];
+	C = c[0] / c[3];
+
+	/* substitute x = y - A/3 to eliminate quadric term: */
+	/* x^3 +px + q = 0 */
+	sq_A = A * A;
+	p = 1.0/3 * (-1.0/3 * sq_A + B);
+	q = 1.0/2 * (2.0/27 * A * sq_A - 1.0/3 * A * B + C);
+
+	/* use Cardano's formula */
+	cb_p = p * p * p;
+	D = q * q + cb_p;
+
+	if (IsZero(D)) {
+		if (IsZero(q)) { /* one triple solution */
+			s[0] = 0.0;
+			num = 1;
+		} else { /* one single and one double solution */
+			double u = CBRT(-q);
+			s[0] = 2.0 * u;
+			s[1] = - u;
+			num = 2;
+		}
+	} else if (D < 0) { /* Casus irreducibilis: three real solutions */
+		double phi = 1.0/3 * acos(-q / sqrt(-cb_p));
+		double t = 2 * sqrt(-p);
+
+		s[0] =  t * cos(phi);
+		s[1] = -t * cos(phi + M_PI / 3.0);
+		s[2] = -t * cos(phi - M_PI / 3.0);
+		num = 3;
+	} else { /* one real solution */
+		double sqrt_D = sqrt(D);
+		double u = CBRT(sqrt_D - q);
+		double v = -CBRT(sqrt_D + q);
+
+		s[0] = u + v;
+		num = 1;
+	}
+
+	/* resubstitute */
+	sub = 1.0/3.0 * A;
+
+	for (i = 0; i < num; i++)
+		s[i] -= sub;
+
+	return num;
+}
+
+
+int SolveQuartic(double c[5], double s[4]) {
+	double coeffs[4];
+	double z, u, v, sub;
+	double A, B, C, D;
+	double sq_A, p, q, r;
+	int	i, num;
+
+	/* normal form: x^4 + Ax^3 + Bx^2 + Cx + D = 0 */
+	A = c[3] / c[4];
+	B = c[2] / c[4];
+	C = c[1] / c[4];
+	D = c[0] / c[4];
+
+	/*  substitute x = y - A/4 to eliminate cubic term:
+		x^4 + px^2 + qx + r = 0 */
+	sq_A = A * A;
+	p = - 3.0/8 * sq_A + B;
+	q = 1.0/8 * sq_A * A - 1.0/2 * A * B + C;
+	r = - 3.0/256*sq_A*sq_A + 1.0/16*sq_A*B - 1.0/4*A*C + D;
+
+	if (IsZero(r)) {
+		/* no absolute term: y(y^3 + py + q) = 0 */
+
+		coeffs[0] = q;
+		coeffs[1] = p;
+		coeffs[2] = 0;
+		coeffs[3] = 1.0;
+
+		num = SolveCubic(coeffs, s);
+
+		s[num++] = 0;
+	} else {
+		/* solve the resolvent cubic ... */
+		coeffs[0] = 1.0/2.0 * r * p - 1.0/8.0 * q * q;
+		coeffs[1] = - r;
+		coeffs[2] = - 1.0/2.0 * p;
+		coeffs[3] = 1.0;
+
+		(void) SolveCubic(coeffs, s);
+
+		/* ... and take the one real solution ... */
+		z = s[0];
+
+		/* ... to build two quadric equations */
+		u = z * z - r;
+		v = 2 * z - p;
+
+		if (IsZero(u))
+			u = 0;
+		else if (u > 0)
+			u = sqrt(u);
+		else
+			return 0;
+
+		if (IsZero(v))
+			v = 0;
+		else if (v > 0)
+			v = sqrt(v);
+		else
+			return 0;
+
+		coeffs[0] = z - u;
+		coeffs[1] = q < 0 ? -v : v;
+		coeffs[2] = 1.0;
+
+		num = SolveQuadric(coeffs, s);
+
+		coeffs[0]= z + u;
+		coeffs[1] = q < 0 ? v : -v;
+		coeffs[2] = 1.0;
+
+		num += SolveQuadric(coeffs, s + num);
+	}
+
+	/* resubstitute */
+	sub = 1.0/4.0 * A;
+
+	for (i = 0; i < num; i++)
+		s[i] -= sub;
+
+	return num;
+}
+
+