Efficient isolation of polynomial’s real roots. This paper revisits an algorithm isolating the real roots of a univariate polynomial using Descartes’ rule of signs. It follows work of Vincent, Uspensky, Collins and Akritas, Johnson, Krandick. Our first contribution is a generic algorithm which enables one to describe all the known algorithms based on Descartes’ rule of sign and the bisection strategy in a unified framework. Using that framework, a new algorithm is presented, which is optimal in terms of memory usage and as fast as both Collins and Akritas’ algorithm and Krandick’s variant, independently of the input polynomial. From this new algorithm, we derive an adaptive semi-numerical version, using multi-precision interval arithmetic. We finally show that these critical optimizations have important consequences since our new algorithm still works with huge polynomials, including orthogonal polynomials of degree 1000 and more, which were out of reach of previous methods.

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  1. Alcázar, Juan Gerardo; Hermoso, Carlos: Involutions of polynomially parametrized surfaces (2016)
  2. Burr, Michael A.: Continuous amortization and extensions: with applications to bisection-based root isolation (2016)
  3. Hu, Shenglong; Ye, Ke: Multiplicities of tensor eigenvalues (2016)
  4. Massri, César: Solving a sparse system using linear algebra (2016)
  5. Mehrabi, Esmaeil; Schost, Éric: A softly optimal Monte Carlo algorithm for solving bivariate polynomial systems over the integers (2016)
  6. Sagraloff, Michael; Mehlhorn, Kurt: Computing real roots of real polynomials (2016)
  7. Bouzidi, Yacine; Lazard, Sylvain; Pouget, Marc; Rouillier, Fabrice: Separating linear forms and rational univariate representations of bivariate systems (2015)
  8. Cheng, Jin-San; Jin, Kai: A generic position based method for real root isolation of zero-dimensional polynomial systems (2015)
  9. Falbel, E.; Koseleff, P.-V.; Rouillier, F.: Representations of fundamental groups of 3-manifolds into $\mathrmPGL(3,\mathbb C)$: exact computations in low complexity (2015)
  10. Kerber, Michael; Sagraloff, Michael: Root refinement for real polynomials using quadratic interval refinement (2015)
  11. Lebreton, Romain: Relaxed Hensel lifting of triangular sets (2015)
  12. Boulier, François; Chen, Changbo; Lemaire, François; Maza, Marc Moreno: Real root isolation of regular chains (2014)
  13. Galeani, Sergio; Henrion, Didier; Jacquemard, Alain; Zaccarian, Luca: Design of Marx generators as a structured eigenvalue assignment (2014)
  14. Sagraloff, Michael: On the complexity of the Descartes method when using approximate arithmetic (2014)
  15. Shen, Fei; Wu, Wenyuan; Xia, Bican: Real root isolation of polynomial equations based on hybrid computation (2014)
  16. Xiao, Shuijing; Zeng, Guangxing: Determination of the limits for multivariate rational functions (2014)
  17. Berberich, Eric; Emeliyanenko, Pavel; Kobel, Alexander; Sagraloff, Michael: Exact symbolic-numeric computation of planar algebraic curves (2013)
  18. Gao, Xiao-Shan; Li, Wei; Yuan, Chun-Ming: Intersection theory in differential algebraic geometry: generic intersections and the differential Chow form (2013)
  19. Herrero, María Isabel; Jeronimo, Gabriela; Sabia, Juan: Affine solution sets of sparse polynomial systems (2013)
  20. Lasserre, Jean-Bernard; Laurent, Monique; Mourrain, Bernard; Rostalski, Philipp; Trébuchet, Philippe: Moment matrices, border bases and real radical computation (2013)

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