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Mat. Zametki, 2015, Volume 97, Issue 4, Pages 529–555 (Mi mz10176)  

This article is cited in 1 scientific paper (total in 1 paper)

Arithmetic Complexity of Certain Linear Transformations

S. B. Gashkov

M. V. Lomonosov Moscow State University

Abstract: We obtain order-sharp quadratic and slightly higher estimates of the computational complexity of certain linear transformations (binomial, Stirling, Lah, Gauss, Serpiński, Sylvester) in the basis $\{x+y \}\cup \{ax: \vert a \vert \leq C \}$ consisting of the operations of addition and inner multiplication by a bounded constant as well as upper bounds $O(n\log n)$ for the computational complexity in the basis $\{ax+by: a,b \in {\mathbb R}\}$ consisting of all linear functions. Lower bounds of the form $\Omega(n\log n)$ are obtained for the basis consisting of all monotone linear functions $\{ax+by: a, b > 0\}$. For the finite arithmetic basis $B_{+,*,/} = \{x\pm y,xy,1/x,1\}$ and for the bases $\{x\pm y\} \cup \{nx: n \in {\mathbb N}\}\cup \{x/n: n \in {\mathbb N}\}$ and $B_{+,*}=\{x+y,xy,-1\}$, estimates $O(n \log n \log \log n)$ are obtained in certain cases. In the case of a field of characteristic $p$, computations in the basis $\{x+y \operatorname{mod} p\}$ are also considered.

Keywords: linear transformation (binomial, Stirling, Lah, Gauss, Serpiński, Sylvester), arithmetic computational complexity of linear transformations, finite arithmetic basis, field of characteristic $p$,

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-00598
14-01-00671а
This work was supported by the Russian Foundation for Basic Research (grants no. 14-01-00598 and no. 14-01-00671a).


DOI: https://doi.org/10.4213/mzm10176

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English version:
Mathematical Notes, 2015, 97:4, 531–555

Bibliographic databases:

Document Type: Article
UDC: 519.95
Received: 01.09.2012
Revised: 05.09.2014

Citation: S. B. Gashkov, “Arithmetic Complexity of Certain Linear Transformations”, Mat. Zametki, 97:4 (2015), 529–555; Math. Notes, 97:4 (2015), 531–555

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. S. B. Gashkov, I. S. Sergeev, “On the Additive Complexity of GCD and LCM Matrices”, Math. Notes, 100:2 (2016), 199–212  mathnet  crossref  crossref  mathscinet  isi  elib
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