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Computing Numerically with Functions Instead of Numbers
, 2007
"... Symbolic computation with functions of a real variable suffers from combinatorial explosion of memory and computation time. The alternative chebfun system for such computations is described, based on Chebyshev expansions and barycentric interpolation. ..."
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Cited by 16 (6 self)
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Symbolic computation with functions of a real variable suffers from combinatorial explosion of memory and computation time. The alternative chebfun system for such computations is described, based on Chebyshev expansions and barycentric interpolation.
An efficient and exact subdivision algorithm for isolating complex roots of a polynomial and its complexity analysis
, 2009
"... We introduce an exact subdivision algorithm CEVAL for isolating complex roots of a squarefree polynomial. The subdivision predicates are based on evaluating the original polynomial or its derivatives, and hence is easy to implement. It can be seen as a generalization of a previous real root isolati ..."
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Cited by 7 (4 self)
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We introduce an exact subdivision algorithm CEVAL for isolating complex roots of a squarefree polynomial. The subdivision predicates are based on evaluating the original polynomial or its derivatives, and hence is easy to implement. It can be seen as a generalization of a previous real root isolation algorithm called EVAL. Under suitable conditions, the algorithm is applicable for general analytic functions. We provide a complexity analysis of our algorithm on the benchmark problem of isolating all complex roots of a squarefree polynomial with Gaussian integer coefficients. The analysis is based on a novel technique called δclusters. This analysis shows, somewhat surprisingly, that the simple EVAL algorithm matches (up to logarithmic factors) the bit complexity bounds of current practical exact algorithms such as those based on Descartes, Continued Fraction or Sturm methods. Furthermore, the more general CEVAL also achieves the same complexity.
In Praise of Numerical Computation
, 2009
"... Theoretical Computer Science has developed an almost exclusively discrete/algebraic persona. We have effectively shut ourselves off from half of the world of computing: a host of problems in Computational Science & Engineering (CS&E) are defined on the continuum, and, for them, the discrete ..."
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Cited by 3 (1 self)
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Theoretical Computer Science has developed an almost exclusively discrete/algebraic persona. We have effectively shut ourselves off from half of the world of computing: a host of problems in Computational Science & Engineering (CS&E) are defined on the continuum, and, for them, the discrete viewpoint is inadequate. The computational techniques in such problems are wellknown to numerical analysis and applied mathematics, but are rarely discussed in theoretical algorithms: iteration, subdivision and approximation. By various case studies, I will indicate how our discrete/algebraic view of computing has many shortcomings in CS&E. We want embrace the continuous/analytic view, but in a new synthesis with the discrete/algebraic view. I will suggest a pathway, by way of an exact numerical model of computation, that allows us to incorporate iteration and approximation into our algorithms’ design. Some recent results give a peek into how this view of algorithmic development might look like, and its distinctive form suggests the name “numerical computational geometry ” for such activities.
Foundations of exact rounding
 Proc. WALCOM 2009
"... Abstract. Exact rounding of numbers and functions is a fundamental computational problem. This paper introduces the mathematical and computational foundations for exact rounding. We show that all the elementary functions in ISO standard (ISO/IEC 10967) for Language Independent Arithmetic can be exac ..."
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Abstract. Exact rounding of numbers and functions is a fundamental computational problem. This paper introduces the mathematical and computational foundations for exact rounding. We show that all the elementary functions in ISO standard (ISO/IEC 10967) for Language Independent Arithmetic can be exactly rounded, in any format, and to any precision. Moreover, a priori complexity bounds can be given for these rounding problems. Our conclusions are derived from results in transcendental number theory. 1
Tight bounds for dynamic convex hull queries (again)
 SCG'07
, 2007
"... The dynamic convex hull problem was recently solved in O(lg n) time per operation, and this result is best possible in models of computation with bounded branching (e.g., algebraic computation trees). From a data structures point of view, however, such models are considered unrealistic because they ..."
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Cited by 1 (0 self)
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The dynamic convex hull problem was recently solved in O(lg n) time per operation, and this result is best possible in models of computation with bounded branching (e.g., algebraic computation trees). From a data structures point of view, however, such models are considered unrealistic because they hide intrinsic notions of information in the input. In the standard wordRAM and cellprobe models of computation, we prove that the optimal query time for dynamic convex hulls is, in fact, Θ ` ´ lg n, for polylogarithmic uplg lg n date time (and word size). Our lower bound is based on a reduction from the markedancestor problem, and is one of the first data structural lower bounds for a nonorthogonal geometric problem. Our upper bounds follow a recent trend of attacking nonorthogonal geometric problems from an informationtheoretic perspective that has proved central to advanced data structures. Interestingly, our upper bounds are the first to successfully apply this perspective to dynamic geometric data structures, and require substantially different ideas from previous work.
ANALYSIS OF LONGESTEDGE ALGORITHMS FOR 2DIMENSIONAL MESH REFINEMENT
"... Las técnicas de generación y refinamiento de mallas no estructuradas son usadas para la descomposición de objetos geométricos. Estas técnicas son muy utilizadas en áreas como modelamiento geométrico, computación gráfica, computación cient́ıfica y aplicaciones de ingenieŕıa, entre otras, ..."
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Las técnicas de generación y refinamiento de mallas no estructuradas son usadas para la descomposición de objetos geométricos. Estas técnicas son muy utilizadas en áreas como modelamiento geométrico, computación gráfica, computación cient́ıfica y aplicaciones de ingenieŕıa, entre otras, lo que les da un interés interdisciplinario. Trabajando con triangulaciones (mallas compuestas por triángulos), el reto es generar una descomposición precisa del objeto geométrico o dominio, y al mismo tiempo satisfacer las restricciones adicionales impuestas por la aplicación, como restricciones en la forma de los elementos, el número de elementos, o la transición entre elementos de diferentes tamaños. Los algoritmos que ofrecen garant́ıas teóricas sobre estos temas son preferidos. Los algoritmos de arista más larga fueron diseñados para el refinamiento iterativo de triangulaciones en aplicaciones de método de elementos finitos adaptativo. Estos algoritmos esta ́ basados en la estrategia de propagación por la arista más larga. Comparados a otros algoritmos de refinamiento, los algoritmos de arista más larga rápidamente producen una descomposición del dominio (o de regiones de interés) a través de operaciones locales simples. Las triangulaciones obtenidas presentan buena densidad y la calidad de los triángulos
On the Use of Adaptive, Exact Decisions Number Types Based on ExpressionDags in Geometric Computing
"... We discuss how (not) to use number types based on expression dags and adaptive precision in geometric computing. Such number types provide exact decisions for (a subset of the) real algebraic numbers. 1 ..."
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We discuss how (not) to use number types based on expression dags and adaptive precision in geometric computing. Such number types provide exact decisions for (a subset of the) real algebraic numbers. 1