Displaying similar documents to “Structured Matrix Methods Computing the Greatest Common Divisor of Polynomials”

Some relations on Humbert matrix polynomials

Ayman Shehata (2016)

Mathematica Bohemica

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The Humbert matrix polynomials were first studied by Khammash and Shehata (2012). Our goal is to derive some of their basic relations involving the Humbert matrix polynomials and then study several generating matrix functions, hypergeometric matrix representations, matrix differential equation and expansions in series of some relatively more familiar matrix polynomials of Legendre, Gegenbauer, Hermite, Laguerre and modified Laguerre. Finally, some definitions of generalized Humbert matrix...

A fixed point method to compute solvents of matrix polynomials

Fernando Marcos, Edgar Pereira (2010)

Mathematica Bohemica

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Matrix polynomials play an important role in the theory of matrix differential equations. We develop a fixed point method to compute solutions of matrix polynomials equations, where the matricial elements of the matrix polynomial are considered separately as complex polynomials. Numerical examples illustrate the method presented.

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

Eckstein, Jiří, Zítko, Jan

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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...

Matrix quadratic equations column/row reduced factorizations and an inertia theorem for matrix polynomials

Irina Karelin, Leonid Lerer (2001)

International Journal of Applied Mathematics and Computer Science

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It is shown that a certain Bezout operator provides a bijective correspondence between the solutions of the matrix quadratic equation and factorizatons of a certain matrix polynomial (which is a specification of a Popov-type function) into a product of row and column reduced polynomials. Special attention is paid to the symmetric case, i.e. to the Algebraic Riccati Equation. In particular, it is shown that extremal solutions of such equations correspond to spectral factorizations of...

Factorization makes fast Walsh, PONS and other Hadamard-like transforms easy

Kautsky, Jaroslav

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A simple device, based on the factorization of invertible matrix polynomials, enabling to identify the possibility of fast implementation of linear transforms is presented. Its applicability is demonstrated in the case of Hadamard matrices and their generalization, Hadamard matrix polynomials.

Relations between the orthogonal matrix polynomials on [a, b], Dyukarev-Stieltjes parameters, and Schur complements

A.E. Choque-Rivero (2017)

Special Matrices

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We obtain explicit interrelations between new Dyukarev-Stieltjes matrix parameters and orthogonal matrix polynomials on a finite interval [a, b], as well as the Schur complements of the block Hankel matrices constructed through the moments of the truncated Hausdorff matrix moment (THMM) problem in the nondegenerate case. Extremal solutions of the THMM problem are described with the help of matrix continued fractions.

On the Remainders Obtained in Finding the Greatest Common Divisor of Two Polynomials

Akritas, Alkiviadis, Malaschonok, Gennadi, Vigklas, Panagiotis (2015)

Serdica Journal of Computing

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In 1917 Pell (1) and Gordon used sylvester2, Sylvester’s little known and hardly ever used matrix of 1853, to compute(2) the coefficients of a Sturmian remainder — obtained in applying in Q[x], Sturm’s algorithm on two polynomials f, g ∈ Z[x] of degree n — in terms of the determinants (3) of the corresponding submatrices of sylvester2. Thus, they solved a problem that had eluded both J. J. Sylvester, in 1853, and E. B. Van Vleck, in 1900. (4) In this paper we extend the work by Pell...

On characteristic and permanent polynomials of a matrix

Ranveer Singh, R. B. Bapat (2017)

Special Matrices

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There is a digraph corresponding to every square matrix over ℂ. We generate a recurrence relation using the Laplace expansion to calculate the characteristic and the permanent polynomials of a square matrix. Solving this recurrence relation, we found that the characteristic and the permanent polynomials can be calculated in terms of the characteristic and the permanent polynomials of some specific induced subdigraphs of blocks in the digraph, respectively. Interestingly, these induced...

Some relations satisfied by Hermite-Hermite matrix polynomials

Ayman Shehata, Lalit Mohan Upadhyaya (2017)

Mathematica Bohemica

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The classical Hermite-Hermite matrix polynomials for commutative matrices were first studied by Metwally et al. (2008). Our goal is to derive their basic properties including the orthogonality properties and Rodrigues formula. Furthermore, we define a new polynomial associated with the Hermite-Hermite matrix polynomials and establish the matrix differential equation associated with these polynomials. We give the addition theorems, multiplication theorems and summation formula for the...