High-performance implementations of the Descartes method

  • Authors:

  • Jeremy R. Johnson;Werner Krandick;Kevin Lynch;David G. Richardson;Anatole D. Ruslanov

  • Affiliations:

  • Drexel University, Philadelphia, PA;Drexel University, Philadelphia, PA;Drexel University, Philadelphia, PA;Drexel University, Philadelphia, PA;Drexel University, Philadelphia, PA

  • Venue:
  • Proceedings of the 2006 international symposium on Symbolic and algebraic computation
  • Year:
  • 2006

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Abstract

The Descartes method for polynomial real root isolation can be performed with respect to monomial bases and with respect to Bernstein bases. The first variant uses Taylor shift by 1 as its main subalgorithm, the second uses de Casteljau's algorithm. When applied to integer polynomials, the two variants have co-dominant, almost tight computing time bounds. Implementations of either variant can obtain speed-ups over previous state-of-the-art implementations by more than an order of magnitude if they use features of the processor architecture. We present an implementation of the Bernstein-bases variant of the Descartes method that automatically generates architecture-aware high-level code and leaves further optimizations to the compiler. We compare the performance of our implementation, algorithmically tuned implementations of the monomial and Bernstein variants, and architecture-unaware implementations of both variants on four different processor architectures and for three classes of input polynomials.