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

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 14-01-0059814-01-00671à 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:

UDC: 519.95
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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• http://mi.mathnet.ru/eng/mz10176
• https://doi.org/10.4213/mzm10176
• http://mi.mathnet.ru/eng/mz/v97/i4/p529

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Citing articles on Google Scholar: Russian citations, English citations
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2. Mukhopadhyay S., Hossain S., Ghosal S.K., Sarkar R., “Secured Image Steganography Based on Catalan Transform”, Multimed. Tools Appl.
3. S. B. Gashkov, I. S. Sergeev, “On the Additive Complexity of GCD and LCM Matrices”, Math. Notes, 100:2 (2016), 199–212
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