Displaying similar documents to “On a Theorem by Van Vleck Regarding Sturm Sequences”

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...

Sturm Sequences and Modified Subresultant Polynomial Remainder Sequences

Akritas, Alkiviadis, Malaschonok, Gennadi, Vigklas, Panagiotis (2014)

Serdica Journal of Computing

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ACM Computing Classification System (1998): F.2.1, G.1.5, I.1.2. In 1971 using pseudo-divisions - that is, by working in Z[x] - Brown and Traub computed Euclid’s polynomial remainder sequences (prs’s) and (proper) subresultant prs’s using sylvester1, the most widely known form of Sylvester’s matrix, whose determinant defines the resultant of two polynomials. In this paper we use, for the first time in the literature, the Pell-Gordon Theorem of 1917, and sylvester2, a little...

Some new formulas for π .

Almkvist, Gert, Krattenthaler, Christian, Petersson, Joakim (2003)

Experimental Mathematics

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On stable polynomials

Miloslav Nekvinda (1989)

Aplikace matematiky

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The article is a survey on problem of the theorem of Hurwitz. The starting point of explanations is Schur's decomposition theorem for polynomials. It is showed how to obtain the well-known criteria on the distribution of roots of polynomials. The theorem on uniqueness of constants in Schur's decomposition seems to be new.

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...

Discrete-time symmetric polynomial equations with complex coefficients

Didier Henrion, Jan Ježek, Michael Šebek (2002)

Kybernetika

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Discrete-time symmetric polynomial equations with complex coefficients are studied in the scalar and matrix case. New theoretical results are derived and several algorithms are proposed and evaluated. Polynomial reduction algorithms are first described to study theoretical properties of the equations. Sylvester matrix algorithms are then developed to solve numerically the equations. The algorithms are implemented in the Polynomial Toolbox for Matlab.

Remarks on inverse of matrix polynomials

Fischer, Cyril, Náprstek, Jiří

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Analysis of a non-classically damped engineering structure, which is subjected to an external excitation, leads to the solution of a system of second order ordinary differential equations. Although there exists a large variety of powerful numerical methods to accomplish this task, in some cases it is convenient to formulate the explicit inversion of the respective quadratic fundamental system. The presented contribution uses and extends concepts in matrix polynomial theory and proposes...