weak_upper_root_bound and sign_variation functions move to this file from Polynomial.h

.gitattributes

@@ -1724,6 +1724,7 @@ Polynomial/doc_tex/Polynomial_ref/Polynomial_1.tex -text
 Polynomial/include/CGAL/Polynomial/Coercion_traits.h -text
 Polynomial/include/CGAL/Polynomial/hgdelta_update.h -text
 Polynomial/include/CGAL/Polynomial/ipower.h -text
+Polynomial/include/CGAL/Polynomial/univariate_polynomial_utils.h -text
 Polytope_distance_d/test/Polytope_distance_d/create_test_PD_cin -text
 Polytope_distance_d/test/Polytope_distance_d/test_PD.cin -text
 Polytope_distance_d/test/Polytope_distance_d/test_PD_data/intersecting_segments.data -text

Polynomial/include/CGAL/Polynomial/univariate_polynomial_utils.h

@@ -0,0 +1,75 @@
+// TODO: Add licence
+//
+// This file is provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE
+// WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
+//
+// $URL:$
+// $Id: $
+// 
+//
+// Author(s)     :  
+//
+// ============================================================================
+
+// TODO: The comments are all original EXACUS comments and aren't adapted. So
+//         they may be wrong now.
+
+#ifndef CGAL_POLYNOMIAL_UNIVARIATE_POLYNOMIAL_UTILS_H
+#define CGAL_POLYNOMIAL_UNIVARIATE_POLYNOMIAL_UTILS_H
+
+#include <CGAL/Polynomial.h>
+
+CGAL_BEGIN_NAMESPACE
+
+//! return an upper bound on the absolute value of all real roots of \c P.
+/*! The upper bound is a power of two. Only works for univariate polynomials.
+ *  \pre \c NT must be \c RealComparable.
+ *  \relates NiX::Polynomial
+ */
+template <class NT>
+NT weak_upper_root_bound(const Polynomial<NT>& P) { 
+    // code comes from Kurt Mehlhorn
+    // see [Mignotte, 1992], p.144 for a proof
+    CGAL_precondition(Polynomial_traits_d<NT>::d == 0);
+    typename Real_embeddable_traits<NT>::Abs abs;
+    const int n = P.degree();
+    NT x(1); 
+    NT val;
+    for (;;) {
+        val = -abs(P[n]);
+        for (int i = n-1; i >= 0; i--) {
+            val = val*x + abs(P[i]);
+        }
+        if (val < NT(0)) return x;
+        x *= NT(2);
+    }
+}
+
+//! return the number of sign variations in the coefficient sequence of \c P.
+/*! This is the number of sign changes (+ to - or - to +) in the
+ *  coefficient sequence of the polynomial, ignoring zeroes.
+ *  Only meaningful for univariate polynomials.
+ *  \pre \c NT must be \c RealComparable.
+ *  \relates NiX::Polynomial
+ */
+template <class NT>
+int sign_variations(const Polynomial<NT>& P) { 
+    typename Real_embeddable_traits<NT>::Sign sign;
+    const int n = P.degree();
+    int variations = 0;
+    int old_sign = sign(P[n]); // never zero unless P is zero
+    for (int i = n-1; i >= 0; i--) {
+        int s = sign(P[i]);
+        if (s == 0) continue;
+        if (old_sign != s) {
+            old_sign = s;
+            variations++;
+        }
+    }
+    return variations;
+}
+
+CGAL_END_NAMESPACE
+
+
+#endif // CGAL_POLYNOMIAL_UNIVARIATE_POLYNOMIAL_UTILS_H