SqFreeEVAL: An (almost) optimal real-root isolation algorithm

Abstract

Let f be a univariate polynomial with real coefficients, f ∈ R [X]. Subdivision algorithms based on algebraic techniques (e.g., Sturm or Descartes methods) are widely used for isolating the real roots of f in a given interval. In this paper, we consider a simple subdivision algorithm whose primitives are purely numerical (e.g., function evaluation). The complexity of this algorithm is adaptive because the algorithm makes decisions based on local data. The complexity analysis of adaptive algorithms (and this algorithm in particular) is a new challenge for computer science. In this paper, we compute the size of the subdivision tree for the SqFreeEVAL algorithm.The SqFreeEVAL algorithm is an evaluation-based numerical algorithm which is well-known in several communities. The algorithm itself is simple, but prior attempts to compute its complexity have proven to be quite technical and have yielded sub-optimal results. Our main result is a simple O(d(L + ln d)) bound on the size of the subdivision tree for the SqFreeEVAL algorithm on the benchmark problem of isolating all real roots of an integer polynomial f of degree d and whose coefficients can be written with at most L bits.Our proof uses two amortization-based techniques: First, we use the algebraic amortization technique of the standard Mahler-Davenport root bounds to interpret the integral in terms of d and L. Second, we use a continuous amortization technique based on an integral to bound the size of the subdivision tree. This paper is the first to use the novel analysis technique of continuous amortization to derive state of the art complexity bounds.

On soft predicates in subdivision motion planning

“…Excluding the cases of "Expanding Box T (α) fails" and "Expanding Box T (β) fails" (trivial cases of 'NO PATH'), the converse is true only for the BFS or Random search strategy. 6 We explicitly mark the entries in the last column with an asterisk (*) to indicate 'NO PATH'.…”

The Complexity of an Adaptive Subdivision Method for Approximating Real Curves

“…Recently, there is an extension to the case of analytic functions [17,38]. For the problem of isolating the roots of polynomials we refer the reader to [23,20,19,12,36,24,7,27,9] and the references therein. There are also approaches [24] that achieve locally quadratic convergence towards the simple roots of polynomial systems and there very efficient in practice.…”

“…As in [7], Σ fv (p) links the Taylor coefficients of f to the geometry of the zero set of f . This relationship is explicitly explored in the following two results.…”

“…In practice, this test is performed by comparing the sizes of the Taylor coefficients and the size of the input region to the value of the polynomial at a given point, see Remark 2.2. In particular, if the inequality appearing in the conclusion of the following corollary holds, then 0 is not included in the interval approximation, for additional details, see [7].…”

On soft predicates in subdivision motion planning

“…This only shows that adaptivity in subdivision is critical. Some recent examples [28,6,35] suggest that adaptive approaches can guarantee optimal tree sizes, even in the worst case sense. Also the exploitation of Newton-type techniques in subdivision is very promising (e.g., [27]).…”