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An improvement of Euclid's algorithm

Zítko, Jan, Kuřátko, Jan (2010)

Programs and Algorithms of Numerical Mathematics

The paper introduces the calculation of a greatest common divisor of two univariate polynomials. Euclid’s algorithm can be easily simulated by the reduction of the Sylvester matrix to an upper triangular form. This is performed by using c - s transformation and Q R -factorization methods. Both procedures are described and numerically compared. Computations are performed in the floating point environment.

Approximate polynomial GCD

Eliaš, Ján, Zítko, Jan (2013)

Programs and Algorithms of Numerical Mathematics

The computation of polynomial greatest common divisor (GCD) ranks among basic algebraic problems with many applications, for example, in image processing and control theory. The problem of the GCD computing of two exact polynomials is well defined and can be solved symbolically, for example, by the oldest and commonly used Euclid’s algorithm. However, this is an ill-posed problem, particularly when some unknown noise is applied to the polynomial coefficients. Hence, new methods for the GCD computation...

Calculation of the greatest common divisor of perturbed polynomials

Zítko, Jan, Eliaš, Ján (2013)

Programs and Algorithms of Numerical Mathematics

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 j 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 greatest...

Comparison of algorithms for calculation of the greatest common divisor of several polynomials

Eckstein, Jiří, Zítko, Jan (2015)

Programs and Algorithms of Numerical Mathematics

The computation of the greatest common divisor (GCD) has many applications in several disciplines including computer graphics, image deblurring problem or computing multiple roots of inexact polynomials. In this paper, Sylvester and Bézout matrices are considered for this purpose. The computation is divided into three stages. A rank revealing method is shortly mentioned in the first one and then the algorithms for calculation of an approximation of GCD are formulated. In the final stage the coefficients...

Convergence of series of dilated functions and spectral norms of GCD matrices

Christoph Aistleitner, István Berkes, Kristian Seip, Michel Weber (2015)

Acta Arithmetica

We establish a connection between the L² norm of sums of dilated functions whose jth Fourier coefficients are ( j - α ) for some α ∈ (1/2,1), and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in L² and for the almost everywhere convergence of series of dilated functions.

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