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Mat. Zametki, 2016, Volume 100, Issue 2, Pages 196–211 (Mi mz10626)  

On the Additive Complexity of GCD and LCM Matrices

S. B. Gashkova, I. S. Sergeevb

a Lomonosov Moscow State University
b Research Institute "Kvant"

Abstract: In the paper, the additive complexity of matrices formed by positive integer powers of greatest common divisors and least common multiples of the indices of the rows and columns is considered. It is proved that the complexity of the $n\times n$ matrix formed by the numbers $GCD^r(i,k)$ over the basis $\{x+y\}$ is asymptotically equal to $rn \log_2 n$ as $n\to\infty$, and the complexity of the $n\times n$ matrix formed by the numbers $LCM^r(i,k)$ over the basis $\{x+y,-x\}$ is asymptotically equal to $2rn \log_2 n$ as $n\to \infty$.

Keywords: greatest common divisor, least common multiple, additive complexity of matrices, Smith determinant, circuit complexity.

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-00671а
This work was financially supported by the Russian Foundation for Basic Research under grant 14-01-00671a.


DOI: https://doi.org/10.4213/mzm10626

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English version:
Mathematical Notes, 2016, 100:2, 199–212

Bibliographic databases:

Document Type: Article
UDC: 519.714+511.17
Received: 24.09.2014

Citation: S. B. Gashkov, I. S. Sergeev, “On the Additive Complexity of GCD and LCM Matrices”, Mat. Zametki, 100:2 (2016), 196–211; Math. Notes, 100:2 (2016), 199–212

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