Univariate Polynomial Real Root Isolation: Continued Fractions Revisited

Abstract

We present algorithmic, complexity and implementation results concerning real root isolation of integer univariate polynomials using the continued fraction expansion of real algebraic numbers. One motivation is to explain the method's good performance in practice. We improve the previously known bound by a factor of dτ , where d is the polynomial degree and τ bounds the coefficient bitsize, thus matching the current record complexity for real root isolation by exact methods. Namely, the complexity bound is OB(d 4 τ 2 ) using the standard bound on the expected bitsize of the integers in the continued fraction expansion. We show how to compute the multiplicities within the same complexity and extend the algorithm to non square-free polynomials. Finally, we present an efficient open-source C++ implementation in the algebraic library synaps, and illustrate its efficiency as compared to other available software. We use polynomials with coefficient bitsize up to 8000 and degree up to 1000.

Complexity of real root isolation using continued fractions

“…(12) This bound is also derived in [26,Thm. 8], and improves the bound by Akritas [3] by a factor of n.…”

“…But what is interesting about the algorithm is that, unlike the Descartes method, it utilizes the distribution of the real roots of the polynomial for isolating them; the advantage of this approach is evident [26,Tab. 1] when isolating the real roots of Mignotte's polynomials [19], where it is known [11,Thm.…”

“…In practice, Akritas' formulation of the continued fraction algorithm almost always performs better than the Descartes method [26,4]. But what is interesting about the algorithm is that, unlike the Descartes method, it utilizes the distribution of the real roots of the polynomial for isolating them; the advantage of this approach is evident [26,Tab.…”

“…His analysis, however, has two flaws: first, he assumes the ideal Positive Lower Bound (PLB) function, i.e., a function that can determine whether a polynomial has positive real roots, and if there are such roots then returns the floor of the smallest positive root of the polynomial; and second, his analysis does not account for the increased coefficient size of Ai+1(X) after performing Taylor shift by b on Ai(X). The latter drawback was partially resolved by Tsigaridas and Emiris [26] by using bounds by Khinchin [15] on the expected bit-size of the partial denominators appearing in the continued fraction expansion of a real number; they derived an e O(n 4 L 2 ) bound on the expected running time of the algorithm assuming asymptotically fast Taylor shifts in their analysis. For bounding the size of the recursion tree of the algorithm, however, their analysis also assumed the ideal PLB function; nevertheless, they do improve the bound on the tree size by a factor of n as compared to the e O(n 2 L) bound by Akritas.…”

Construction of a numerical method for finding the zeros of both smooth and nonsmooth functions

“…Article [2] presented the result of algorithmic complexity relating to the isolation of the roots of integer one-variant polynomials.…”