arXiv subjects

Elias P. Tsigaridas

Publications and source records attributed to Elias P. Tsigaridas.

7 recordsLinked to original sources

arXiv2016

Accelerated Approximation of the Complex Roots and Factors of a Univariate Polynomial

The algorithms of Pan (1995) and(2002) approximate the roots of a complex univariate polynomial in nearly optimal arithmetic and Boolean time but require precision of computing that exceeds the degree of the polynomial. This causes numerical stability problems when the degree is large. We observe, however, that such a difficulty disappears at the initial stage of the algorithms, and in our present paper we extend this stage to root-finding within a nearly optimal arithmetic and Boolean complexity bounds provided that some mild initial isolation of the roots of the input polynomial has been ensured. Furthermore our algorithm is nearly optimal for the approximation of the roots isolated in a fixed disc, square or another region on the complex plane rather than all complex roots of a polynomial. Moreover the algorithm can be applied to a polynomial given by a black box for its evaluation (even if its coefficients are not known); it promises to be of practical value for polynomial root-finding and factorization, the latter task being of interest on its own right. We also provide a new support for a winding number algorithm, which enables extension of our progress to obtaining mild initial approximations to the roots. We conclude with summarizing our algorithms and their extension to the approximation of isolated multiple roots and root clusters.

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arXiv2013

Chapter 10: Algebraic Algorithms

Our Chapter in the upcoming Volume I: Computer Science and Software Engineering of Computing Handbook (Third edition), Allen Tucker, Teo Gonzales and Jorge L. Diaz-Herrera, editors, covers Algebraic Algorithms, both symbolic and numerical, for matrix computations and root-finding for polynomials and systems of polynomials equations. We cover part of these large subjects and include basic bibliography for further study. To meet space limitation we cite books, surveys, and comprehensive articles with pointers to further references, rather than including all the original technical papers.

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arXiv2012

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 software

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arXiv2009

Algebraic methods for counting Euclidean embeddings of rigid graphs

The study of (minimally) rigid graphs is motivated by numerous applications, mostly in robotics and bioinformatics. A major open problem concerns the number of embeddings of such graphs, up to rigid motions, in Euclidean space. We capture embeddability by polynomial systems with suitable structure, so that their mixed volume, which bounds the number of common roots, to yield interesting upper bounds on the number of embeddings. We focus on $\RR^2$ and $\RR^3$, where Laman graphs and 1-skeleta of convex simplicial polyhedra, respectively, admit inductive Henneberg constructions. We establish the first lower bound in $\RR^3$ of about $2.52^n$, where $n$ denotes the number of vertices. Moreover, our implementation yields upper bounds for $n \le 10$ in $\RR^2$ and $\RR^3$, which reduce the existing gaps, and tight bounds up to $n=7$ in $\RR^3$.

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arXiv2008

Guarding curvilinear art galleries with vertex or point guards

One of the earliest and most well known problems in computational geometry is the so-called art gallery problem. The goal is to compute the minimum possible number guards placed on the vertices of a simple polygon in such a way that they cover the interior of the polygon. In this paper we consider the problem of guarding an art gallery which is modeled as a polygon with curvilinear walls. Our main focus is on polygons the edges of which are convex arcs pointing towards the exterior or interior of the polygon (but not both), named piecewise-convex and piecewise-concave polygons. We prove that, in the case of piecewise-convex polygons, if we only allow vertex guards, $\lfloor\frac{4n}{7}\rfloor-1$ guards are sometimes necessary, and $\lfloor\frac{2n}{3}\rfloor$ guards are always sufficient. Moreover, an $O(n\log{}n)$ time and O(n) space algorithm is described that produces a vertex guarding set of size at most $\lfloor\frac{2n}{3}\rfloor$. When we allow point guards the afore-mentioned lower bound drops down to $\lfloor\frac{n}{2}\rfloor$. In the special case of monotone piecewise-convex polygons we can show that $\lfloor\frac{n}{2}\rfloor$ vertex guards are always sufficient and sometimes necessary; these bounds remain valid even if we allow point guards. In the case of piecewise-concave polygons, we show that $2n-4$ point guards are always sufficient and sometimes necessary, whereas it might not be possible to guard such polygons by vertex guards. We conclude with bounds for other types of curvilinear polygons and future work.

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arXiv2006

Univariate polynomial real root isolation: Continued Fractions revisited

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 $d τ$, 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 $\sOB(d^4 τ^2)$ 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 \texttt{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.

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arXiv2005

Computations with one and two real algebraic numbers

We present algorithmic and complexity results concerning computations with one and two real algebraic numbers, as well as real solving of univariate polynomials and bivariate polynomial systems with integer coefficients using Sturm-Habicht sequences. Our main results, in the univariate case, concern the problems of real root isolation (Th. 19) and simultaneous inequalities (Cor.26) and in the bivariate, the problems of system real solving (Th.42), sign evaluation (Th. 37) and simultaneous inequalities (Cor. 43).

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