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21
Numerical Computation Of A Polynomial GCD And Extensions
, 1996
"... In the first part of this paper, we dene approximate polynomial gcds (greatest common divisors) and extended gcds provided that approximations to the zeros of the input polynomials are available. We relate our novel definition to the older and weaker ones, based on perturbation of the coefficients o ..."
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Cited by 19 (6 self)
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In the first part of this paper, we dene approximate polynomial gcds (greatest common divisors) and extended gcds provided that approximations to the zeros of the input polynomials are available. We relate our novel definition to the older and weaker ones, based on perturbation of the coefficients of the input polynomials, we demonstrate some deficiency of the latter definitions (which our denition avoids), and we propose new effective sequential and parallel (RNC and NC) algorithms for computing approximate gcds and extended gcds. Our stronger results are obtained with no increase of the asymptotic bounds on the computational cost. This is partly due to application of our recent nearly optimal algorithms for approximating polynomial zeros. In the second part of our paper, working under the older and more customary definition of approximate gcds, we modify and develop an alternative approach, which was previously based on the computation of the Singular Value Decomposition (SVD) of the associat...
Exact simulation of integrateandfire models with synaptic conductances
 Neural Comp
, 2006
"... Computational neuroscience relies heavily on the simulation of large networks of neuron models. There are essentially two simulation strategies: 1) using an approximation method (e.g. RungeKutta) with spike times binned to the time step; 2) calculating spike times exactly in an eventdriven fashion ..."
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Cited by 15 (1 self)
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Computational neuroscience relies heavily on the simulation of large networks of neuron models. There are essentially two simulation strategies: 1) using an approximation method (e.g. RungeKutta) with spike times binned to the time step; 2) calculating spike times exactly in an eventdriven fashion. In large networks, the computation time of the best algorithm for either strategy scales linearly with the number of synapses, but each strategy has its own assets and constraints: approximation methods can be applied to any model but are inexact; exact simulation avoids numerical artefacts but is limited to simple models. Previous work has focused on improving the accuracy of approximation methods. In this paper we extend the range of models that can be simulated exactly to a more realistic model, namely an integrateandfire model with exponential synaptic conductances.
On Approximating Complex Polynomial Zeros: Modified Quadtree (Weyl's) Construction and Improved Newton's Iteration
, 1996
"... The known record complexity estimates for approximating polynomial zeros rely on geometric constructions on the complex plane, which achieve initial approximation to the zeros and/or their clusters as well as their isolation from each other, and on the subsequent fast analytic refinement of the init ..."
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Cited by 14 (7 self)
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The known record complexity estimates for approximating polynomial zeros rely on geometric constructions on the complex plane, which achieve initial approximation to the zeros and/or their clusters as well as their isolation from each other, and on the subsequent fast analytic refinement of the initial approximations. We modify Weyl's classical geometric construction for approximating all the n polynomial zeros in order to more rapidly achieve their strong isolation. For approximating the isolated zeros or clusters of zeros, we propose a new extension of Newton's iteration to yield quadratic global convergence (right from the start), under substantially weaker requirements to their initial isolation than one needs in the known algorithms.
Shapes based Trajectory Queries for Moving Objects
 Proceedings of ACM GIS
, 2005
"... An interesting issue in moving objects databases is to find similar trajectories of moving objects. Previous work on this topic focuses on movement patterns (trajectories with time dimension) of moving objects, rather than spatial shapes (trajectories without time dimension) of their trajectories. I ..."
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Cited by 11 (0 self)
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An interesting issue in moving objects databases is to find similar trajectories of moving objects. Previous work on this topic focuses on movement patterns (trajectories with time dimension) of moving objects, rather than spatial shapes (trajectories without time dimension) of their trajectories. In this paper we propose a simple and effective way to compare spatial shapes of moving object trajectories. We introduce a new distance function based on “one way distance” (OWD). Algorithms for evaluating OWD in both continuous (piece wise linear) and discrete (grid representation) cases are developed. An index structure for OWD in grid representation, which guarantees no false dismissals, is also given to improve the efficiency of similarity search. Empirical studies show that OWD outperforms existent methods not only in precision, but also in efficiency. And the results of OWD in continuous case can be approximated by discrete case efficiently.
Towards Practical Constraint Databases (Extended Abstract)
, 1996
"... ) St ephane Grumbach I.N.R.I.A. Rocquencourt BP 105 78153 Le Chesnay, France stephane.grumbach@inria.fr Jianwen Su Computer Science Department University of California Santa Barbara, California 93106, USA su@cs.ucsb.edu Abstract We develop a framework for (real) constraint databases based on ..."
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Cited by 8 (3 self)
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) St ephane Grumbach I.N.R.I.A. Rocquencourt BP 105 78153 Le Chesnay, France stephane.grumbach@inria.fr Jianwen Su Computer Science Department University of California Santa Barbara, California 93106, USA su@cs.ucsb.edu Abstract We develop a framework for (real) constraint databases based on finite precision arithmetic which fulfills the main requirements of practical constraint databases. First, it allows the manipulation of approximate values, standard in scientific applications. More importantly, it permits the extension of the relational calculus with aggregate functions, while preserving the fundamental property of closed form evaluation with PTIME data complexity. This is an important step since the initial model of [KKR90] cannot be extended to aggregate functions. Moreover, finite precision computation plays a central role in efficient query processing. We introduce the finite precision semantics of queries and prove expressive power results concerning it. We then prese...
Some Recent Algebraic/Numerical Algorithms
, 1998
"... Combination of algebraic and numerical techniques for improving the computations in algebra and geometry is a popular research topic of growing interest. We survey some recent progress that we made in this area, in particular, regarding polynomial rootfinding, the solution of a polynomial system of ..."
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Cited by 4 (3 self)
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Combination of algebraic and numerical techniques for improving the computations in algebra and geometry is a popular research topic of growing interest. We survey some recent progress that we made in this area, in particular, regarding polynomial rootfinding, the solution of a polynomial system of equations, the computation of an approximate greatest common divisor of two polynomials as well as various computations with dense structured matrices and their further applications to polynomial and rational interpolation and multipoint polynomial evaluation. In some cases our algorithms reach nearly optimal time bounds and/or improve the previously known methods by order of magnitude, in other cases we yield other gains, such as improved numerical stability. Key words: algebraic/numerical algorithms, polynomial rootfinding, solution of a polynomial system of equations, approximate gcd, dense structured matrices, Toeplitz matrices, Cauchy matrices, polynomial interpolation, rational interpo...
Fast Solvers and Domain Decomposition Preconditioners for Spectral Element Discretizations of Problems in H(curl
, 2001
"... ..."
Schur Aggregation for Linear Systems and Determinants
 Graduate Center, City University of New York
, 2008
"... According to our previous theoretical and experimental study, additive preconditioners can be readily computed for ill conditioned matrices, but it is not straightforward how such preconditioners can help us to facilitate the solution of linear systems of equations, computation of determinants, and ..."
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Cited by 2 (2 self)
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According to our previous theoretical and experimental study, additive preconditioners can be readily computed for ill conditioned matrices, but it is not straightforward how such preconditioners can help us to facilitate the solution of linear systems of equations, computation of determinants, and other fundamental matrix computations. We develop some nontrivial techniques for this task. By applying the Sherman–Morrison–Woodbury formula and its new variations, we confine the original numerical problems to the computation of the Schur aggregates of smaller sizes. We overcome these problems by extending the classical algorithm for iterative refinement and applying advanced doubleprecision algorithms for high precision computation of sums and products.
The Similarities (and Differences) between Polynomials and Integers
, 1994
"... The purpose of this paper is to examine the two domains of the integers and the polynomials, in an attempt to understand the nature of complexity in these very basic situations. Can we formalize the integer algorithms which shed light on the polynomial domain, and vice versa? When will the casti ..."
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Cited by 2 (0 self)
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The purpose of this paper is to examine the two domains of the integers and the polynomials, in an attempt to understand the nature of complexity in these very basic situations. Can we formalize the integer algorithms which shed light on the polynomial domain, and vice versa? When will the casting of one in the other speed up an existing algorithm? Why do some problems not lend themselves to this kind of speedup? We give several simple and natural theorems that show how problems in one domain can be embedded in the other, and we examine the complexitytheoretic consequences of these embeddings. We also prove several results on the impossibility of solving integer problems by mimicking their polynomial counterparts. 1 Introduction It is a fact frequently remarked upon that polynomials and integers share a number of characteristics. Usually the Fast Fourier Transform is then Supported by NSF grants DMS8807202 and CCR9204630. y Supported by NSF grant CCR9207797. 1 giv...