Computational benefit of smoothness: Parameterized bit-complexity of numerical operators on analytic functions and Gevrey’s hierarchy

Kawamura, Akitoshi; Müller, Norbert; Rösnick, Carsten; Ziegler, Martin

doi:10.1016/j.jco.2015.05.001

Cited by 19 publications

(23 citation statements)

References 55 publications

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“…This representation has been investigated by different authors for instance in [8,11,14]. For simplicity restrict to the case of analytic functions on the unit disk.…”

Section: The Set Of Analytic Functions Is Denoted By C ω (D) Each Anmentioning

confidence: 99%

Comparing Representations for Function Spaces in Computable Analysis

Pauly

Steinberg

2017

Theory Comput Syst

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This paper compares different representations (in the sense of computable analysis) of a number of function spaces that are of interest in analysis. In particular subspace representations inherited from a larger function space are compared to more natural representations for these spaces. The formal framework for the comparisons is provided by Weihrauch reducibility. The centrepiece of the paper considers several representations of the analytic functions on the unit disk and their mutual translations. All translations that are not already computable are shown to be Weihrauch equivalent to closed choice on the natural numbers. Subsequently some similar considerations are carried out for representations of polynomials. In this case in addition to closed choice the Weihrauch degree LPO * shows up as the difficulty of finding the degree or the zeros. As a final example, the smooth functions are contrasted with functions with bounded support and Schwartz functions. Here closed choice on the natural numbers and the lim degree appear.

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“…This representation has been investigated by different authors for instance in [8,11,14]. For simplicity restrict to the case of analytic functions on the unit disk.…”

Section: The Set Of Analytic Functions Is Denoted By C ω (D) Each Anmentioning

confidence: 99%

Comparing Representations for Function Spaces in Computable Analysis

Pauly

Steinberg

2017

Theory Comput Syst

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“…There is a more suitable representation for the analytic functions than the restriction of the representation of continuous functions. This representation has been investigated by different authors for instance in [4], [7], [9]. For simplicity we restrict to the case of analytic functions on the unit disk.…”

Section: Representations Of Analytic Functionsmentioning

confidence: 99%

“…This function turns up in the results of this paper. In the terminology of [4], one would say that C ω (D) arises from the restriction C(D)| C ω (D) by enriching with the discrete advice Adv C ω . The topology induced by the representation of C ω (D) is well known and used in analysis: It can be constructed as a direct limit topology and makes C ω (D) a so called Silva-Space.…”

Section: Representations Of Analytic Functionsmentioning

confidence: 99%

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Representations of Analytic Functions and Weihrauch Degrees

Pauly

Steinberg

2016

Computer Science – Theory and Applications

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This paper considers several representations of the analytic functions on the unit disk and their mutual translations. All translations that are not already computable are shown to be Weihrauch equivalent to closed choice on the natural numbers. Subsequently some similar considerations are carried out for representations of polynomials. In this case in addition to closed choice the Weihrauch degree LPO * shows up as the difficulty of finding the degree or the zeros.

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“…It is also shown that integral and maximum of a function are uniformly polytime computable from such a sequence. These results were strengthened and refined in various ways by Kawamura, Müller, Rösnick, and Ziegler [8] who studied the uniform complexity of maximisation and integration for analytic functions and functions in Gevrey's hierarchy in dependence on certain parameters which control the growth of the derivatives or the proximity of singularities in the complex plane.…”

Section: Introductionmentioning

confidence: 95%

Representations and evaluation strategies for feasibly approximable functions

Konečný

Neumann

2021

COM

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A famous result due to Ko and Friedman (1982) asserts that the problems of integration and maximisation of a univariate real function are computationally hard in a well-defined sense. Yet, both functionals are routinely computed at great speed in practice. We aim to resolve this apparent paradox by studying classes of functions which can be feasibly integrated and maximised, together with representations for these classes of functions which encode the information which is necessary to uniformly compute integral and maximum in polynomial time. The theoretical framework for this is the second-order complexity theory for operators in analysis which was introduced by Kawamura and Cook (2012). The representations we study are based on approximation by polynomials, piecewise polynomials, and rational functions. We compare these representations with respect to polytime reducibility. We show that the representation based on approximation by piecewise polynomials is polytime equivalent to the representation based on approximation by rational functions. With this representation, all terms in a certain language, which is expressive enough to contain the maximum and integral of most functions of practical interest, can be evaluated in polynomial time. By contrast, both the representation based on polynomial approximation and the standard representation based on function evaluation, which implicitly underlies the Ko-Friedman result, require exponential time to evaluate certain terms in this language. We confirm our theoretical results by an implementation in Haskell, which provides some evidence that second-order polynomial time computability is similarly closely tied with practical feasibility as its first-order counterpart. 1 This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 731143. 2 Most of this work was carried out when Eike Neumann was affiliated with Aston University.

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Computational benefit of smoothness: Parameterized bit-complexity of numerical operators on analytic functions and Gevrey’s hierarchy

Cited by 19 publications

References 55 publications

Comparing Representations for Function Spaces in Computable Analysis

Comparing Representations for Function Spaces in Computable Analysis

Representations of Analytic Functions and Weihrauch Degrees

Representations and evaluation strategies for feasibly approximable functions

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