On the complexity of skew arithmetic

Hoeven, Joris van der

doi:10.1007/s00200-015-0269-0

Cited by 8 publications

(5 citation statements)

References 30 publications

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“…Remark 4.2. When 𝑠 = 𝑑, a more direct approach for step 6 is to write 𝑹 as a product of first-order factors with power series coefficients; see [38,Theorem 8] for a fast algorithm. Remark 4.3.…”

Section: Minimal Annihilatorsmentioning

confidence: 99%

Symbolic-Numeric Factorization of Differential Operators

Chyzak

Goyer

Mezzarobba

2022

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation

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We present a symbolic-numeric Las Vegas algorithm for factoring Fuchsian ordinary differential operators with rational function coefficients. The new algorithm combines ideas of van Hoeij's "local-toglobal" method and of the "analytic" approach proposed by van der Hoeven. It essentially reduces to the former in "easy" cases where the local-to-global method succeeds, and to an optimized variant of the latter in the "hardest" cases, while handling intermediate cases more efficiently than both. CCS CONCEPTS• Computing methodologies → Hybrid symbolic-numeric methods.

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Section: Minimal Annihilatorsmentioning

confidence: 99%

Symbolic-Numeric Factorization of Differential Operators

Chyzak

Goyer

Mezzarobba

2022

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation

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“…For instance, the inverse of an r × r matrix with entries in a field K can be computed in time O(r ω ) [115]. If SM(d, r ) stands for the complexity of multiplying two operators in K[x, ∂] of degree ⩽ d in x and order ⩽ r in ∂, then exact right division of two such operators can be done in time O(SM(d, r ) log d) [62].…”

Section: Basic Operationsmentioning

confidence: 99%

On the Complexity of Symbolic Computation

Hoeven

2022

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation

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“…By using simple interval arithmetic [15], Senn and Kumazawa [16] have shown how this idea can be leveraged to train robust echo state networks. We build on this approach and show how it could be applied to create robust physical reservoir computing systems in simulations.…”

Section: Introductionmentioning

confidence: 99%

Abstract Reservoir Computing

Senn

Kumazawa

2022

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Noise of any kind can be an issue when translating results from simulations to the real world. We suddenly have to deal with building tolerances, faulty sensors, or just noisy sensor readings. This is especially evident in systems with many free parameters, such as the ones used in physical reservoir computing. By abstracting away these kinds of noise sources using intervals, we derive a regularized training regime for reservoir computing using sets of possible reservoir states. Numerical simulations are used to show the effectiveness of our approach against different sources of errors that can appear in real-world scenarios and compare them with standard approaches. Our results support the application of interval arithmetics to improve the robustness of mass-spring networks trained in simulations.

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On the complexity of skew arithmetic

Cited by 8 publications

References 30 publications

Symbolic-Numeric Factorization of Differential Operators

Symbolic-Numeric Factorization of Differential Operators

On the Complexity of Symbolic Computation

Abstract Reservoir Computing

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