A discrete-time integrable system, called the DSF chain, is derived from the compatibility condition of spectral equations for biorthogonal polynomials associated with the discrete Schur flow. The DSF algorithm for co...
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A discrete-time integrable system, called the DSF chain, is derived from the compatibility condition of spectral equations for biorthogonal polynomials associated with the discrete Schur flow. The DSF algorithm for computing eigenvalues of a type of matrix eigenvalue problem is designed. A discrete analogue of the Miura transformation between the discrete Schur flow and the qd-algorithm is constructed. Two special solutions to the DSF chain related to q-special functions are presented. (C) 2018 Elsevier B.V. All rights reserved.
Given a, b ? R, m = min{a, b} and M = max{a, b}, we consider the orthogonal polynomials associated with nontrivial positive measures f for which supp(f) ? (-8, m] ? [M, 8) and (x - a)df (x) = -(x - b)df (-x + a + b). ...
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Given a, b ? R, m = min{a, b} and M = max{a, b}, we consider the orthogonal polynomials associated with nontrivial positive measures f for which supp(f) ? (-8, m] ? [M, 8) and (x - a)df (x) = -(x - b)df (-x + a + b). For this class of measures, formulas in order to compute the moments, as well as formulas for the weights and nodes in the associated Gaussian quadrature rules are provided. We also show that the qd-algorithm can be applied in order to generate new orthogonal polynomials in a simple way. Several examples are given to illustrate the results obtained.
To reconstruct a black box multivariate sparse polynomial from its floating point evaluations, the existing algorithms need to know upper bounds for both the number of terms in the polynomial and the partial degree in...
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To reconstruct a black box multivariate sparse polynomial from its floating point evaluations, the existing algorithms need to know upper bounds for both the number of terms in the polynomial and the partial degree in each of the variables. Here we present a new technique, based on Rutishauser's qd-algorithm, in which we overcome both drawbacks. (c) 2008 Elsevier B.V. All rights reserved.
The aim of this paper is to define and to study orthogonal polynomials with respect to a linear functional whose moments are vectors. We show how a Clifford algebra allows us to construct such polynomials in a natural...
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The aim of this paper is to define and to study orthogonal polynomials with respect to a linear functional whose moments are vectors. We show how a Clifford algebra allows us to construct such polynomials in a natural way. This new definition is motivated by the fact that there exist natural links between this theory of orthogonal polynomials and the theory of the vector valued Pade approximants in the sense of Graves-Morris and Roberts.
In this paper, a generalization of the G-transformation is proposed together with the corresponding auxiliary rs-algorithm for implementation. We show that the related generalization of the rs-algorithm is equivalent ...
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In this paper, a generalization of the G-transformation is proposed together with the corresponding auxiliary rs-algorithm for implementation. We show that the related generalization of the rs-algorithm is equivalent to a generalized qd-algorithm. Applications of the generalization of the G-transformation to the computation of infinite integrals are also given. (C) 2010 IMACS. Published by Elsevier B.V. All rights reserved.
In this paper a new presentation of orthogonal polynomials is given. It is based on the introduction of two auxiliary sequences of arbitrary monic polynomials and it leads to a very simple derivation of the usual dete...
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We present a method to extract factors of multivariate polynomials with complex coefficients in floating point arithmetic. We establish the connection between the reciprocal of a multivariate polynomial and its Taylor...
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ISBN:
(纸本)9781605586649
We present a method to extract factors of multivariate polynomials with complex coefficients in floating point arithmetic. We establish the connection between the reciprocal of a multivariate polynomial and its Taylor expansion. Since the multivariate Taylor coefficients are determined by the irreducible factors of the given polynomial, we reconstruct the factors from the Taylor expansion. As each irreducible factor, regardless of its multiplicity, can be separately extracted, our method can lead toward the complete numerical factorization of multivariate polynomials.
In this paper, a generalization of the G-transformation is proposed together with the corresponding auxiliary rs-algorithm for implementation. We show that the related generalization of the rs-algorithm is equivalent ...
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In this paper, a generalization of the G-transformation is proposed together with the corresponding auxiliary rs-algorithm for implementation. We show that the related generalization of the rs-algorithm is equivalent to a generalized qd-algorithm. Applications of the generalization of the G-transformation to the computation of infinite integrals are also given. (C) 2010 IMACS. Published by Elsevier B.V. All rights reserved.
The multivariate homogeneous two-point Pade approximants have been defined and studied recently. In the current work, we consider higher-order approximants and derive error formulas of these approximants using orthogo...
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The multivariate homogeneous two-point Pade approximants have been defined and studied recently. In the current work, we consider higher-order approximants and derive error formulas of these approximants using orthogonality conditions. Diverse three-term recurrence relations satisfied by the monic orthogonal polynomials are presented. Various continued fractions provided by these relations and the quotient-difference algorithm applied to a power series (positive or negative exponents) are described in terms of their relationships with the multivariate homogeneous two-point Pade table. Numerical examples are furnished to illustrate our results.
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