On the Extended Hensel Construction and its application to the computation of real limit points

Alvandi, Parisa; Ataei, Masoud; Kazemi, Mahsa; Maza, Marc Moreno

doi:10.1016/j.jsc.2019.07.009

Cited by 5 publications

(4 citation statements)

References 17 publications

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“…Our method computes the first k terms of all factors of f within O(d 3 y k + d 2 y k 2 ) operations in K. Moreover, we conjecture in Section 4 that our method can achieve O(d 3 y k+d 2 y M (k) log k) operations in K through relaxed algorithms [15]. The Hensel-Sasaki Construction of [1] computes all factors in O(d 3 M (d)+k 2 dM (d)). Kung and Traub show that, over the complex numbers C, the Newton-Puiseux method can be performed in O(d 2 kM (k)) (resp.…”

Section: Introductionmentioning

confidence: 89%

“…where f has total degree d [1]. The method of Kung and Traub [10], requires O(d 2 kM (k)) (using linear lifting) or O(d 2 M (k)) (using quadratic lifting).…”

Section: Corollary 3 (Hensel Factorization Complexity)mentioning

confidence: 99%

“…In the same paper, an extension of the Hensel-Sasaki construction was proposed for multivariate coefficients. In [1], EHC was improved in terms of complexity estimates and practical implementation.…”

Section: Introductionmentioning

confidence: 99%

“…Nonetheless, the formulation of EHC in [1] is shown to be practically much more efficient than the method of Kung and Traub. Further, our serial implementation of lazy Hensel factorization has already been shown in [5] to be orders of magnitude faster than the implementation of EHC in [1]. We now extend our work by considering parallel processing techniques to further improve the performance of our lazy Hensel factorization.…”

Section: Introductionmentioning

confidence: 99%

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On the Complexity and Parallel Implementation of Hensel's Lemma and Weierstrass Preparation

Brandt¹,

Maza²

2021

Preprint

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Hensel's lemma, combined with repeated applications of Weierstrass preparation theorem, allows for the factorization of polynomials with multivariate power series coefficients. We present a complexity analysis for this method and leverage those results to guide the load-balancing of a parallel implementation to concurrently update all factors. In particular, the factorization creates a pipeline where the terms of degree k of the first factor are computed simultaneously with the terms of degree k − 1 of the second factor, etc. An implementation challenge is the inherent irregularity of computational work between factors, as our complexity analysis reveals. Additional resource utilization and load-balancing is achieved through the parallelization of Weierstrass preparation. Experimental results show the efficacy of this mixed parallel scheme, achieving up to 9× speedup on 12 cores.

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Section: Introductionmentioning

confidence: 89%

“…where f has total degree d [1]. The method of Kung and Traub [10], requires O(d 2 kM (k)) (using linear lifting) or O(d 2 M (k)) (using quadratic lifting).…”

Section: Corollary 3 (Hensel Factorization Complexity)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%