Experimental evaluation and cross-benchmarking of univariate real solvers

Abstract

Real solving of univariate polynomials is a fundamental problem with several important applications. This paper is focused on the comparison of black-box implementations of state-of-the-art algorithms for isolating real roots of univariate polynomials over the integers. We have tested 9 different implementations based on symbolic-numeric methods, Sturm sequences, Continued Fractions and Descartes' rule of sign. The methods under consideration were developed at the GALAAD group at INRIA, the VEGAS group at LO-RIA and the MPI-Saarbrücken. We compared their sensitivity with respect to various aspects such as degree, bitsize or root separation of the input polynomials. Our datasets consist of 5 000 polynomials from many different settings, which have maximum coefficient bitsize up to bits 8 000, and the total running time of the experiments was about 50 hours. Thereby, all implementations of the theoretically exact methods always provided correct results throughout this extensive study. For each scenario we identify the currently most adequate method, and we point to weaknesses in each approach, which should lead to further improvements. Our results indicate that there is no "best method" overall, but one can say that for most instances the solvers based on Continued Fractions are among the best methods. To the best of our knowledge, this is the largest number of tests for univariate real solving up to date.

A General Approach to Isolating Roots of a Bitstream Polynomial

“…For very large coefficients, the Bitstream approaches sometimes outperform the methods for integer polynomials as the full precision of the coefficients might not be needed. This was already observed [16,28] for the previous Bitstream solvers and should even more be true for the new method, in particular, for polynomials of larger degree. Our results are crucially based on the usage of an adaptive precision management in comparison to the usage of worst case perturbation bounds as proposed in [12] or [23].…”

“…They both start on an initial interval and perform a recursive binary search. In practice, methods based on Descartes' Rule of Signs have proven to be more efficient [16,17,28] than Sturm's approach, but both approaches behave equally in terms of worst case complexity [8,13,20]. More precisely, for F a polynomial of degree n with integer coefficients of bitsize L, the induced subdivision tree has size O(n(log n + L)) and isolating all real roots demands forÕ(n 4 L 2 ) bit operations.…”

“…There exist several variants of subdivision methods dating back to Vincent's work [36], either using a Bernstein representation instead of the monomial basis or subdividing an interval not at the midpoint but at an arbitrary point by the use of root bounds (continued fraction methods [2,32,35,36]). Although continued fraction methods yield the same theoretical worst case bounds [22,32,35], experiments [1,16,35] have shown that they outperform other methods for hard instances such as Mignotte polynomials. For the problem of isolating all complex roots, there is a considerable discrepancy between asymptotically fast algorithms and algorithms which are fast in practice.…”

GRASS GIS: A multi-purpose open source GIS

“…For an empirical comparison of these methods, see Hemmer et al (2009). In this paper, we show that the subdivision tree for the SqFreeEVAL algorithm also achieves this bound; therefore, the SqFreeEVAL algorithm should also be considered on equal footing with the other more well-known root finding algorithms via the Sturm or Descartes methods.…”

3D Kinetic Alpha Complexes and Their Implementation

“…Both the internal kds and the ak d kernels use subdivision methods for root isolation, but they differ in the strategy for detecting empty intervals and isolating intervals. The kds kernel uses Sturm theory [17, §7], while the ak d kernel is based on Descartes' rule of sign [7], which leads to a better performance in practice; see [9] for a comparison of various root solvers. The difference between the third and fourth columns in Table 4 shows that exchanging the kernel yields another improvement of roughly a factor of two.…”