Curves and surfaces for computer aided geometric design
Curves and surfaces for computer aided geometric design
Bezier and B-Spline Techniques
Bezier and B-Spline Techniques
Polynomial real root isolation using Descarte's rule of signs
SYMSAC '76 Proceedings of the third ACM symposium on Symbolic and algebraic computation
Efficient isolation of polynomial's real roots
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the international conference on linear algebra and arithmetic, Rabat, Morocco, 28-31 May 2001
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
New bounds for the Descartes method
Journal of Symbolic Computation
Fast and exact geometric analysis of real algebraic plane curves
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Exact and efficient 2D-arrangements of arbitrary algebraic curves
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
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If an open interval I contains a k-fold root @a of a real polynomial f, then, after transforming I to (0,~), Descartes' Rule of Signs counts exactly k roots of f in I, provided I is such that Descartes' Rule counts no roots of the kth derivative of f. We give a simple proof using the Bernstein basis. The above condition on I holds if its width does not exceed the minimum distance @s from @a to any complex root of the kth derivative. We relate @s to the minimum distance s from @a to any other complex root of f using Szego's composition theorem. For integer polynomials, log(1/@s) obeys the same asymptotic worst-case bound as log(1/s).