An improvement of Euclid's algorithm

Zítko, Jan; Kuřátko, Jan

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Displaying similar documents to “An improvement of Euclid's algorithm”

The general methods of finding the sum $S_{n}$ for all kinds of series

K. Orlov (1981)

Matematički Vesnik

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Uniform convergence of the greedy algorithm with respect to the Walsh system

Martin Grigoryan (2010)

Studia Mathematica

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For any 0 < ϵ < 1, p ≥ 1 and each function $f \in L^{p} [0, 1]$ one can find a function $g \in L^{\infty} [0, 1)$ with mesx ∈ [0,1): g ≠ f < ϵ such that its greedy algorithm with respect to the Walsh system converges uniformly on [0,1) and the sequence $| c_{k} (g) | : k \in s p e c (g)$ is decreasing, where $c_{k} (g)$ is the sequence of Fourier coefficients of g with respect to the Walsh system.

Calculation of the greatest common divisor of perturbed polynomials

Zítko, Jan, Eliaš, Ján

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The coefficients of the greatest common divisor of two polynomials $f$ and $g$ (GCD $(f, g)$ ) can be obtained from the Sylvester subresultant matrix $S_{j} (f, g)$ transformed to lower triangular form, where $1 \leq j \leq d$ and $d =$ deg(GCD $(f, g)$ ) needs to be computed. Firstly, it is supposed that the coefficients of polynomials are given exactly. Transformations of $S_{j} (f, g)$ for an arbitrary allowable $j$ are in details described and an algorithm for the calculation of the GCD $(f, g)$ is formulated. If inexact polynomials are given, then an approximate...

Lanczos-like algorithm for the time-ordered exponential: The $*$ -inverse problem

Pierre-Louis Giscard, Stefano Pozza (2020)

Applications of Mathematics

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The time-ordered exponential of a time-dependent matrix $𝖠 (t)$ is defined as the function of $𝖠 (t)$ that solves the first-order system of coupled linear differential equations with non-constant coefficients encoded in $𝖠 (t)$ . The authors have recently proposed the first Lanczos-like algorithm capable of evaluating this function. This algorithm relies on inverses of time-dependent functions with respect to a non-commutative convolution-like product, denoted by $*$ . Yet, the existence of such inverses,...

The adaptation of the $k$ -means algorithm to solving the multiple ellipses detection problem by using an initial approximation obtained by the DIRECT global optimization algorithm

Rudolf Scitovski, Kristian Sabo (2019)

Applications of Mathematics

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We consider the multiple ellipses detection problem on the basis of a data points set coming from a number of ellipses in the plane not known in advance, whereby an ellipse $E$ is viewed as a Mahalanobis circle with center $S$ , radius $r$ , and some positive definite matrix $Σ$ . A very efficient method for solving this problem is proposed. The method uses a modification of the $k$ -means algorithm for Mahalanobis-circle centers. The initial approximation consists of the set of circles whose centers...

Seasonal time-series imputation of gap missing algorithm (STIGMA)

Eduardo Rangel-Heras, Pavel Zuniga, Alma Y. Alanis, Esteban A. Hernandez-Vargas, Oscar D. Sanchez (2023)

Kybernetika

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This work presents a new approach for the imputation of missing data in weather time-series from a seasonal pattern; the seasonal time-series imputation of gap missing algorithm (STIGMA). The algorithm takes advantage from a seasonal pattern for the imputation of unknown data by averaging available data. We test the algorithm using data measured every $10$ minutes over a period of $365$ days during the year 2010; the variables include global irradiance, diffuse irradiance, ultraviolet irradiance,...

Algorithm for the complement of orthogonal operations

Iryna V. Fryz (2018)

Commentationes Mathematicae Universitatis Carolinae

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G. B. Belyavskaya and G. L. Mullen showed the existence of a complement for a $k$ -tuple of orthogonal $n$ -ary operations, where $k < n$ , to an $n$ -tuple of orthogonal $n$ -ary operations. But they proposed no method for complementing. In this article, we give an algorithm for complementing a $k$ -tuple of orthogonal $n$ -ary operations to an $n$ -tuple of orthogonal $n$ -ary operations and an algorithm for complementing a $k$ -tuple of orthogonal $k$ -ary operations to an $n$ -tuple of orthogonal $n$ -ary operations. Also...

Factorization of CP-rank- $3$ completely positive matrices

Jan Brandts, Michal Křížek (2016)

Czechoslovak Mathematical Journal

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A symmetric positive semi-definite matrix $A$ is called completely positive if there exists a matrix $B$ with nonnegative entries such that $A = B B^{⊤}$ . If $B$ is such a matrix with a minimal number $p$ of columns, then $p$ is called the cp-rank of $A$ . In this paper we develop a finite and exact algorithm to factorize any matrix $A$ of cp-rank $3$ . Failure of this algorithm implies that $A$ does not have cp-rank $3$ . Our motivation stems from the question if there exist three nonnegative polynomials of degree at...