268 research outputs found
Near Optimal Subdivision Algorithms for Real Root Isolation
We describe a subroutine that improves the running time of any subdivision
algorithm for real root isolation. The subroutine first detects clusters of
roots using a result of Ostrowski, and then uses Newton iteration to converge
to them. Near a cluster, we switch to subdivision, and proceed recursively. The
subroutine has the advantage that it is independent of the predicates used to
terminate the subdivision. This gives us an alternative and simpler approach to
recent developments of Sagraloff (2012) and Sagraloff-Mehlhorn (2013), assuming
exact arithmetic.
The subdivision tree size of our algorithm using predicates based on
Descartes's rule of signs is bounded by , which is better by
compared to known results. Our analysis differs in two key
aspects. First, we use the general technique of continuous amortization from
Burr-Krahmer-Yap (2009), and second, we use the geometry of clusters of roots
instead of the Davenport-Mahler bound. The analysis naturally extends to other
predicates.Comment: 19 pages, 3 figure
SqFreeEVAL: An (almost) optimal real-root isolation algorithm
Let f be a univariate polynomial with real coefficients, f in 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
Real Root Isolation of Regular Chains
We present an algorithm RealRootIsolate for isolating the real roots of a system of multivariate polynomials given by a zerodimensional squarefree regular chain. The output of the algorithm is guaranteed in the sense that all real roots are obtained and are described by boxes of arbitrary precision. Real roots are encoded with a hybrid representation, combining a symbolic object, namely a regular chain, and a numerical approximation given by intervals. Our isolation algorithm is a generalization, for regular chains, of the algorithm proposed by Collins and Akritas. We have implemented RealRootIsolate as a command of the module SemiAlgebraicSetTools of the RegularChains library in Maple. Benchmarks are reported.
Model Checking Quantum Continuous-Time Markov Chains
Verifying quantum systems has attracted a lot of interests in the last decades. In this paper, we initialise the model checking of quantum continuous-time Markov chain (QCTMC). As a real-time system, we specify the temporal properties on QCTMC by signal temporal logic (STL). To effectively check the atomic propositions in STL, we develop a state-of-the-art real root isolation algorithm under Schanuel's conjecture; further, we check the general STL formula by interval operations with a bottom-up fashion, whose query complexity turns out to be linear in the size of the input formula by calling the real root isolation algorithm. A running example of an open quantum walk is provided to demonstrate our method
A Comparative Study of Two Real Root Isolation Methods
Recent progress in polynomial elimination has rendered the computation of the real roots of ill-conditioned polynomials of high degree (over 1000) with huge coefficients (several thousand digits) a critical operation in computer algebra.
To rise to the occasion, the only method-candidate that has been considered by various authors for modification and improvement has been the Collins-Akritas bisection method [1], which is a based on a variation of Vincentâs theorem [2].
The most recent example is the paper by Rouillier and Zimmermann [3], where the authors present â... a new algorithm, which is optimal in terms of memory usage and as fast as both Collins and Akritasâ algorithm and Krandick variant ...â [3]
In this paper we compare our own continued fractions method CF [4] (which is directly based on Vincentâs theorem) with the best bisection method REL described in [3]. Experimentation with the data presented in [3] showed that, with respect to time, our continued fractions method CF is by far superior to REL, whereas the two are about equal with respect to space
SLV: a software for real root isolation
International audienceSLV is a software package in C that provides routines for isolating (and subsequently refine) the real roots of univariate polynomials with integer or rational coefficients based on subdivision algorithms. Special attention is given so that the package can handle polynomials that have degree several thousands and size of coefficients hundreds of Megabytes. Currently the code consists of âŒ5 000 lines
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