Bernstein–Bezoutian matrices

Бини, Дарио Андреа; Gemignani, Luca

doi:10.1016/j.tcs.2004.01.016

Cited by 37 publications

(49 citation statements)

References 28 publications

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“…Explicit forms for the entries of the Bézout resultant matrix [3], the companion resultant matrix [16] and the Sylvester resultant matrix [18] (1) have been developed but there has been significantly less investigation into their numerical properties. These properties are worthy of consideration because these resultant matrices contain combinatorial terms, and thus even if the magnitude of the coefficientsâ i andb j is of order one, the entries of these matrices may span several orders of magnitude, which may cause numerical problems.…”

Section: Introductionmentioning

confidence: 99%

A non-linear structure-preserving matrix method for the computation of the coefficients of an approximate greatest common divisor of two Bernstein polynomials

Bourne

Winkler

2017

Journal of Computational and Applied Mathematics

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This paper describes a non-linear structure-preserving matrix method for the computation of the coefficients of an approximate greatest common divisor (AGCD) of degree t of two Bernstein polynomials f (y) and g(y). This method is applied to a modified form S t (f, g)Q t of the tth subresultant matrix S t (f, g) of the Sylvester resultant matrix S(f, g) of f (y) and g(y), where Q t is a diagonal matrix of combinatorial terms. This modified subresultant matrix has significant computational advantages with respect to the standard subresultant matrix S t (f, g), and it yields better results for AGCD computations. It is shown that f (y) and g(y) must be processed by three operations before S t (f, g)Q t is formed, and the consequence of these operations is the introduction of two parameters, α and θ, such that the entries of S t (f, g)Q t are non-linear functions of α, θ and the coefficients of f (y) and g(y). The values of α and θ are optimised, and it is shown that these optimal values allow an AGCD that has a small error, and a structured low rank approximation of S(f, g), to be computed.

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Section: Introductionmentioning

confidence: 99%

A non-linear structure-preserving matrix method for the computation of the coefficients of an approximate greatest common divisor of two Bernstein polynomials

Bourne

Winkler

2017

Journal of Computational and Applied Mathematics

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“…Note that, although we have expressed (2.1) in the monomial basis, any polynomial basis can be used, see for example [8]. Monomials are the standard basis in the literature [16,18], due to their simplicity and flexibility for algebraic manipulations.…”

Section: Resultant Methodsmentioning

confidence: 99%

“…Usually, resultant matrices are constructed from polynomials expressed in the monomial basis [9,32], but they can be derived when using any other bases, see for example [8].…”

Section: Resultant Methodsmentioning

confidence: 99%

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Computing the common zeros of two bivariate functions via Bézout resultants

2014

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Abstract. The common zeros of two bivariate functions can be computed by finding the common zeros of their polynomial interpolants expressed in a tensor Chebyshev basis. From here we develop a bivariate rootfinding algorithm based on the hidden variable resultant method and Bézout matrices with polynomial entries. Using techniques including domain subdivision, Bézoutian regularization and local refinement we are able to reliably and accurately compute the simple common zeros of two smooth functions with polynomial interpolants of very high degree (≥ 1000). We analyze the resultant method and its conditioning by noting that the Bézout matrices are matrix polynomials. Our robust algorithm is implemented in the roots command in Chebfun2, a software package written in object-oriented Matlab for computing with bivariate functions.

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“…One thread, exemplified by the papers [5,10,11,22,24] investigates polynomial computation via the Bernstein basis, which is well-suited to applications in computer-aided geometric design. The paper [23] compares the average-case behaviour of monomial and Bernstein bases and shows that Bernstein bases are better.…”

Section: Introductionmentioning

confidence: 99%

On a Generalized Companion Matrix Pencil for Matrix Polynomials Expressed in the Lagrange Basis

Corless

Trends in Mathematics

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Abstract. Experimental observations of univariate rootfinding by generalized companion matrix pencils expressed in the Lagrange basis show that the method can sometimes be numerically stable. It has recently been proved that a new condition number, defined for points on a set containing the interpolation points, is never larger than the rootfinding condition number for the Bernstein polynomial (which is itself optimal in a certain sense); and computation shows that sometimes it can be much smaller. These results also hold for the matrix polynomial case, when we are not looking for polynomial roots but rather for eigenvalues where the matrix polynomial is singular. This current paper explores the influence of the geometry of the interpolation nodes on the conditioning of the rootfinding and eigenvalue problems.

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Bernstein–Bezoutian matrices

Cited by 37 publications

References 28 publications

A non-linear structure-preserving matrix method for the computation of the coefficients of an approximate greatest common divisor of two Bernstein polynomials

A non-linear structure-preserving matrix method for the computation of the coefficients of an approximate greatest common divisor of two Bernstein polynomials

Computing the common zeros of two bivariate functions via Bézout resultants

On a Generalized Companion Matrix Pencil for Matrix Polynomials Expressed in the Lagrange Basis

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