On the Complexity of Real Root Isolation

We introduce a new approach to isolate the real roots of a square-free polynomial $F=\sum_{i=0}^n A_i x^i$ with real coefficients. It is assumed that each coefficient of $F$ can be approximated to any specified error bound. The presented method is exact, complete and deterministic. Due to its similarities to the Descartes method, we also consider it practical and easy to implement. Compared to previous approaches, our new method achieves a significantly better bit complexity. It is further shown that the hardness of isolating the real roots of $F$ is exclusively determined by the geometry of the roots and not by the complexity or the size of the coefficients. For the special case where $F$ has integer coefficients of maximal bitsize $\tau$, our bound on the bit complexity writes as $\tilde{O}(n^3\tau^2)$ which improves the best bounds known for existing practical algorithms by a factor of $n=deg F$. The crucial idea underlying the new approach is to run an approximate version of the Descartes method, where, in each subdivision step, we only consider approximations of the intermediate results to a certain precision. We give an upper bound on the maximal precision that is needed for isolating the roots of $F$. For integer polynomials, this bound is by a factor $n$ lower than that of the precision needed when using exact arithmetic explaining the improved bound on the bit complexity

On the asymptotic and practical complexity of solving bivariate systems over the reals

This paper is concerned with exact real solving of well-constrained, bivariate polynomial systems. The main problem is to isolate all common real roots in rational rectangles, and to determine their intersection multiplicities. We present three algorithms and analyze their asymptotic bit complexity, obtaining a bound of \sOB(N^{14}) for the purely projection-based method, and \sOB(N^{12}) for two subresultant-based methods: this notation ignores polylogarithmic factors, where $N$ bounds the degree and the bitsize of the polynomials. The previous record bound was \sOB(N^{14}). Our main tool is signed subresultant sequences. We exploit recent advances on the complexity of univariate root isolation, and extend them to sign evaluation of bivariate polynomials over two algebraic numbers, and real root counting for polynomials over an extension field. Our algorithms apply to the problem of simultaneous inequalities; they also compute the topology of real plane algebraic curves in \sOB(N^{12}), whereas the previous bound was \sOB(N^{14}). All algorithms have been implemented in MAPLE, in conjunction with numeric filtering. We compare them against FGB/RS, system solvers from SYNAPS, and MAPLE libraries INSULATE and TOP, which compute curve topology. Our software is among the most robust, and its runtimes are comparable, or within a small constant factor, with respect to the C/C++ libraries. Key words: real solving, polynomial systems, complexity, MAPLE softwareComment: 17 pages, 4 algorithms, 1 table, and 1 figure with 2 sub-figure

Efficiently Computing Real Roots of Sparse Polynomials

We propose an efficient algorithm to compute the real roots of a sparse polynomial $f\in\mathbb{R}[x]$ having $k$ non-zero real-valued coefficients. It is assumed that arbitrarily good approximations of the non-zero coefficients are given by means of a coefficient oracle. For a given positive integer $L$, our algorithm returns disjoint disks $\Delta_{1},\ldots,\Delta_{s}\subset\mathbb{C}$, with $s<2k$, centered at the real axis and of radius less than $2^{-L}$ together with positive integers $\mu_{1},\ldots,\mu_{s}$ such that each disk $\Delta_{i}$ contains exactly $\mu_{i}$ roots of $f$ counted with multiplicity. In addition, it is ensured that each real root of $f$ is contained in one of the disks. If $f$ has only simple real roots, our algorithm can also be used to isolate all real roots. The bit complexity of our algorithm is polynomial in $k$ and $\log n$, and near-linear in $L$ and $\tau$, where $2^{-\tau}$ and $2^{\tau}$ constitute lower and upper bounds on the absolute values of the non-zero coefficients of $f$, and $n$ is the degree of $f$. For root isolation, the bit complexity is polynomial in $k$ and $\log n$, and near-linear in $\tau$ and $\log\sigma^{-1}$, where $\sigma$ denotes the separation of the real roots

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 note on the complexity of univariate root isolation

This paper presents the average-case bit complexity of subdivision-based univariate solvers, namely those named after Sturm, Descartes, and Bernstein. By real solving we mean real root isolation. We prove bounds of \sOB(N^5) for all methods, where $N$ bounds the polynomial degree and the coefficient bitsize, whereas their worst-case complexity is in \sOB(N^6). In the case of the Sturm solver, our bound depends on the number of real roots. Our work is a step towards understanding the practical complexity of real root isolation. This enables a better juxtaposition against numerical solvers, the latter having worst-case complexity in \sOB(N^4). % Our approach extends to complex root isolation, where we offer a simple proof leading to bounds % for the number of steps that the subdivision algorithm performs on the worst and average-case complexities of \sOB(N^7 ) and \sOB(N^6) respectively, where the latter is output-sensitive

A Generic Position Based Method for Real Root Isolation of Zero-Dimensional Polynomial Systems

We improve the local generic position method for isolating the real roots of a zero-dimensional bivariate polynomial system with two polynomials and extend the method to general zero-dimensional polynomial systems. The method mainly involves resultant computation and real root isolation of univariate polynomial equations. The roots of the system have a linear univariate representation. The complexity of the method is $\tilde{O}_B(N^{10})$ for the bivariate case, where $N=\max(d,\tau)$, $d$ resp., $\tau$ is an upper bound on the degree, resp., the maximal coefficient bitsize of the input polynomials. The algorithm is certified with probability 1 in the multivariate case. The implementation shows that the method is efficient, especially for bivariate polynomial systems.Comment: 24 pages, 5 figure

On the complexity of real root isolation using Continued Fractions

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 $d \tau$, where $d$ is the polynomial degree and $\tau$ bounds the coefficient bit size, thus matching the current record complexity for real root isolation by exact methods (Sturm, Descartes and Bernstein subdivision). Namely our complexity bound is \sOB(d^4 \tau^2) using a standard bound on the expected bit size of the integers in the continued fraction expansion. Moreover, using a homothetic transformation we improve the expected complexity bound to \sOB( d^3 \tau) under the assumption that d = \OO( \tau). We compute the multiplicities within the same complexity and extend the algorithm to non square-free polynomials. Finally, we present an efficient open-source \texttt{C++} implementation and illustrate its completeness and efficiency as compared to other available software. For this we use polynomials with coefficient bit size up to 8000 bits and degree up to 1000

When Newton meets Descartes: A Simple and Fast Algorithm to Isolate the Real Roots of a Polynomial

We introduce a new algorithm denoted DSC2 to isolate the real roots of a univariate square-free polynomial f with integer coefficients. The algorithm iteratively subdivides an initial interval which is known to contain all real roots of f. The main novelty of our approach is that we combine Descartes' Rule of Signs and Newton iteration. More precisely, instead of using a fixed subdivision strategy such as bisection in each iteration, a Newton step based on the number of sign variations for an actual interval is considered, and, only if the Newton step fails, we fall back to bisection. Following this approach, our analysis shows that, for most iterations, we can achieve quadratic convergence towards the real roots. In terms of complexity, our method induces a recursion tree of almost optimal size O(nlog(n tau)), where n denotes the degree of the polynomial and tau the bitsize of its coefficients. The latter bound constitutes an improvement by a factor of tau upon all existing subdivision methods for the task of isolating the real roots. In addition, we provide a bit complexity analysis showing that DSC2 needs only \tilde{O}(n^3tau) bit operations to isolate all real roots of f. This matches the best bound known for this fundamental problem. However, in comparison to the much more involved algorithms by Pan and Sch\"onhage (for the task of isolating all complex roots) which achieve the same bit complexity, DSC2 focuses on real root isolation, is very easy to access and easy to implement