The Complexity of an Adaptive Subdivision Method for Approximating Real Curves

Burr, Michael A.; Gao, Shuhong; Tsigaridas, Elias

doi:10.1145/3087604.3087654

Cited by 12 publications

(27 citation statements)

References 39 publications

(56 reference statements)

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“…For , w ∈ W , , w ≤ , x + x, w ≤ π /2, by the triangle inequality. e interval approximations in [6] are based on Taylor expansion at the midpoint, so they are different from ours. However, our complexity analysis also applies to the interval approximations considered in [6], see §6.2 below for the details.…”

Section: Lipschitz Propertiesmentioning

confidence: 92%

“…In Section 5, we introduce the condition number along with some of its main properties. In Section 6, we present the existing results of complexity of the subdivision method of the PV Algorithm based on local size bound functions from [6] and we relate them to the local condition number. In Section 7, we rely on the bounds for the condition number obtained in Section 5 to derive average and smoothed complexity bounds under (quite) general randomness assumptions.…”

Section: Outlinementioning

confidence: 99%

“…Moreover, these additional computations have been implemented only for n ≤ 3. So, proceeding as in [6], we will only analyze the complexity of the subdivision routine, keeping track of the dependency on n. Also as in [6], our complexity analysis will not deal with the precision needed for the algorithm.…”

Section: The Pv Algorithmmentioning

confidence: 99%

“…An article doing so was published in 2017 by Burr, Gao and Tsigaridas [6]. e functions this article deals with are polynomials with integer coefficients and smooth zero set.…”

Section: Introductionmentioning

confidence: 99%

“…To begin with, the fact that a condition number is a perfect fit for the notion of an "additional geometric and intrinsic parameter. " To which we may add the fact that the obvious set of ill-posed inputs, the set of polynomials having a non-smooth zero set, is precisely the set of data which are not allowed as inputs in [6]. Of course, such a condition-based analysis would drop the assumption of integer coefficients and replace it by that of real coefficients but this is a common practice for numerical algorithms and, as we will see, it pays off in our case as it yields small (i.e., polynomial) average complexity bounds for a large class of probability measures.…”

Section: Introductionmentioning

confidence: 99%

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Ergür

Tonelli-Cueto

2019

Proceedings of the 2019 International Symposium on Symbolic and Algebraic Computation

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We exhibit a condition-based analysis of the adaptive subdivision algorithm due to Plantinga and Vegter. e first complexity analysis of the PV Algorithm is due to Burr, Gao and Tsigaridas who proved a O 2 τ d 4 log d worst-case cost bound for degree d plane curves with maximum coefficient bit-size τ .is exponential bound, it was observed, is in stark contrast with the good performance of the algorithm in practice. More in line with this performance, we show that, with respect to a broad family of measures, the expected time complexity of the PV Algorithm is bounded by O(d 7 ) for real, degree d, plane curves. We also exhibit a smoothed analysis of the PV Algorithm that yields similar complexity estimates. To obtain these results we combine robust probabilistic techniques coming from geometric functional analysis with condition numbers and the continuous amortization paradigm introduced by Burr, Krahmer and Yap. We hope this will motivate a fruitful exchange of ideas between the different approaches to numerical computation. CCS CONCEPTS•Mathematics of computing → Computations on polynomials; Interval arithmetic; • eory of computation → Design and analysis of algorithms; Computational geometry; KEYWORDS computational algebraic geometry, numerical methods, adaptive subdivision methods, isotopy of curves, complexity ACKNOWLEDGMENTSWe thank Michael Burr for useful discussions.

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Section: Lipschitz Propertiesmentioning

confidence: 92%

Section: Outlinementioning

confidence: 99%

Section: The Pv Algorithmmentioning

confidence: 99%

“…An article doing so was published in 2017 by Burr, Gao and Tsigaridas [6]. e functions this article deals with are polynomials with integer coefficients and smooth zero set.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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