Polynomials with respect to a general basis. I. Theory

Maroulas, John; Barnett, Stephen M.

doi:10.1016/0022-247x(79)90282-8

Cited by 37 publications

(15 citation statements)

References 6 publications

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“…For the Chebyshev basis, a colleague linearization can be used. In fact, both of these are confederate linearizations [32,33], which are defined by the polynomial basis' recurrence relations. To construct such a linearization, consider as per example the recurrence relations…”

Section: Notation and Preliminariesmentioning

confidence: 99%

Numerical Solution of Bivariate and Polyanalytic Polynomial Systems

Sorber¹,

Barel²,

Lathauwer³

2014

SIAM J. Numer. Anal.

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Abstract. Finding the real solutions of a bivariate polynomial system is a central problem in robotics, computer modeling and graphics, computational geometry and numerical optimization. We propose an efficient and numerically robust algorithm for solving bivariate and polyanalytic polynomial systems using a single generalized eigenvalue decomposition. In contrast to existing eigen-based solvers, the proposed algorithm does not depend on Gröbner bases or normal sets, nor does it require computing eigenvectors or solving additional eigenproblems to recover the solution. The method transforms bivariate systems into polyanalytic systems and then uses resultants in a novel way to project the variables onto the real plane associated with the two variables. Solutions are returned counting multiplicity and their accuracy is maximized by means of numerical balancing and Newton-Raphson refinement. Numerical experiments show that the proposed algorithm consistently recovers a higher percentage of solutions and is at the same time significantly faster and more accurate than competing double precision solvers.

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Section: Notation and Preliminariesmentioning

confidence: 99%

Numerical Solution of Bivariate and Polyanalytic Polynomial Systems

Sorber¹,

Barel²,

Lathauwer³

2014

SIAM J. Numer. Anal.

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“…On the other hand, Bezoutians have many applications in the theory of equations, system and control theory, etc., we refer the reader to the survey article of Helmke and Fuhrmann [10] and the book of Barnett [1] and the references therein. Recently the (classical) Bezoutian has been generalized to some other forms, in which the polynomial Bezoutian is an important direction of the research (e.g., see [3,4,7,12,13,14,18,19]). At the same time we have observed that in the recent work of Helmke and Fuhrmann [10], Fuhrmann and Datta [6], Mani and Hartwig [13], and Yang [18], etc., some properties of Bezoutians and their relation to system theoretic problems were derived by using operator approach and viewing the Bezoutian as a matrix representation of a certain operator in the dual bases.…”

Section: Elamentioning

confidence: 99%

“…The companion (confederate) matrix has intertwining relations with the classical (polynomial) Bezoutian and Hankel (generalized Hankel) matrix (see [5], [14]). We note that the matrix in (1.8) is in Hessenberg form.…”

Section: Elamentioning

confidence: 99%

More on polynomial Bezoutians with respect to a general basis

Wu¹,

Wu²,

Yang³

2010

ELA

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Abstract. We use unified algebraic methods to investigate the properties of polynomial Bezoutians with respect to a general basis. Not only can three known results be easily verified, but also some new properties of polynomial Bezoutians are obtained. Nonsymmetric Lyapunov-type equations of polynomial Bezoutians are also discussed. It turns out that most properties of classical Bezoutians can be analogously generalized to the case of polynomial Bezoutians in the framework of algebraic methods.

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“…[Many useful results on connection of Hessenberg matrices H Q to polynomials Q can be found in [MB79] where the name confederate matrix was associated with H Q in (7.20)]. 7.7.2.…”

Section: 7mentioning

confidence: 99%

Pivoting for structured matrices and rational tangential interpolation

Olshevsky

2003

Fast Algorithms for Structured Matrices: Theory and Applications

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Gaussian elimination is a standard tool for computing triangular factorizations for general matrices, and thereby solving associated linear systems of equations. As is well-known, when this classical method is implemented in finite-precision-arithmetic, it often fails to compute the solution accurately because of the accumulation of small roundoffs accompanying each elementary floating point operation. This problem motivated a number of interesting and important studies in modern numerical linear algebra; for our purposes in this paper we only mention that starting with the breakthrough work of Wilkinson, several pivoting techniques have been proposed to stabilize the numerical behavior of Gaussian elimination. Interestingly, matrix interpretations of many known and new algorithms for various applied problems can be seen as a way of computing triangular fac-torizations for the associated structured matrices, where different patterns of structure arise in the context of different physical problems. The special structure of such matrices [e.g., Toeplitz, Hankel, Cauchy, Vandermonde, etc.] often allows one to speed-up the computation of its triangular factorization, i.e., to efficiently obtain fast implementations of the Gaussian elimination procedure. There is a vast literature about such methods which are known under different names, e.g., fast Cholesky, fast Gaussian elimination, generalized Schur, or Schur-type algorithms. However, without further improvements they are efficient fast implementations of a numerically inaccurate [for indefinite matrices] method. In this paper we survey recent results on the fast implementation of various pivoting techniques which allowed us to improve numerical accuracy for a variety of fast algorithms. This approach led us to formulate new more accurate numerical methods for factorization of general and of J-unitary rational matrix functions, for solving various tangential interpolation problems, new Toeplitz-like and Toeplitz-plus-Hankel-like solvers, and new divided differences schemes. We beleive that similar methods can be used to design accurate fast algorithm for the other applied problems and a recent work of our colleagues supports this anticipation.

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Polynomials with respect to a general basis. I. Theory

Cited by 37 publications

References 6 publications

Numerical Solution of Bivariate and Polyanalytic Polynomial Systems

Numerical Solution of Bivariate and Polyanalytic Polynomial Systems

More on polynomial Bezoutians with respect to a general basis

Pivoting for structured matrices and rational tangential interpolation

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