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.
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.
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.
Biorthogonal polynomials P-n((i, j)) include as particular cases vector orthogonal polynomials of dimension d and - d(d is an element of N). We pay special attention to the cases of dimension 1 and -1. We discuss the ...
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Biorthogonal polynomials P-n((i, j)) include as particular cases vector orthogonal polynomials of dimension d and - d(d is an element of N). We pay special attention to the cases of dimension 1 and -1. We discuss the problem of computing P-n((i, j)) using only one or several recurrence relations. Furthermore, we deduce all recurrence relations of a certain type that give P-n((i, j)) from two other biorthogonal polynomials. The coefficients that appear in any two independent relations satisfy some identities from which it is possible to establish qd-like algorithms. (C) 1999 Elsevier Science 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.
The convergence of columns in the univariate qd-algorithm to reciprocals of polar singularities of meromorphic functions has often proved to be very useful. A multivariate qd-algorithm was discovered in 1982 for the c...
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The convergence of columns in the univariate qd-algorithm to reciprocals of polar singularities of meromorphic functions has often proved to be very useful. A multivariate qd-algorithm was discovered in 1982 for the construction of the so-called homogeneous Pade approximants. In the first section we repeat the univariate convergence results. In the second section we summarize the ''homogeneous'' multivariate qd-algorithm. In the third section a multivariate convergence result is proved by combining results from the previous sections. This convergence result is compared with another theorem for the general order multivariate qdg-algorithm. The main difference lies in the fact that the homogeneous form detects the polar singularities ''pointwise'' while the general form detects them ''curvewise''.
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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The convergence of the vector qd-algorithm, associated to d meromorphic functions, is established. As a consequence a De Montessus–De Ballore theorem for vector Padé approximants is proved. A short numerical stu...
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The convergence of the vector qd-algorithm, associated to d meromorphic functions, is established. As a consequence a De Montessus–De Ballore theorem for vector Padé approximants is proved. A short numerical study is done in conclusion.
Vector-Padé approximants to a function F = (ƒ 1 ; … ƒ d ) from C to C d have been defined, uniquely, without any auxiliary choice than the degrees of the numerator and the denominator (the same for all the compo...
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Vector-Padé approximants to a function F = (ƒ 1 ; … ƒ d ) from C to C d have been defined, uniquely, without any auxiliary choice than the degrees of the numerator and the denominator (the same for all the components ƒ i ), as in the scalar case [1,5]. The denominators are associated to polynomials P s r , which are given by vector orthogonal properties (R) and which satisfy for each s , recurrence relations of order d + 1 (i.e. with d + 2 terms), called relations (D). We study here consequences of (R) and (D): first we prove an algorithm similar to the generalized MNA-algorithm; then we define a vector qd-algorithm which links two diagonals ( P s r ) r and ( P s + 1 r ) r . Conversely if a family ( P r ) r ⩾ 0 verifying (D) is given, it is possible to build ( P s r ) r ⩾ 0, s ⩾ 0 , and d linear functionals C α , α = 1,…, d , such that P 0 r = P r and ( P s r ) verify the orthogonal relations (R), with respect to the C α .
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