Hermite–Padé approximation and simultaneous quadrature formulas

Prieto, U. Fidalgo; Illán, Juan Carlos Sánchez; Lagomasino, G. López

doi:10.1016/j.jat.2004.01.004

Cited by 23 publications

(4 citation statements)

References 14 publications

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“…with j p 0, , 1 = ¼ -, and of type I [13,25,28]. In particular, in [25] the convergence properties of simultaneous quadrature rules of a given function with respect to different weights is studied.From our developments we can derive such multiple Gauss quadrature formulas easily. First, we define and we are handling a Jacobi matrix, and we are asking for which entries of T are involved in ( )…”

Section: Favard Theoremmentioning

confidence: 99%

“…A remarkable achievement here is to determine the degree of precision and not just lower bounds for them. The determination of the numbers ν a , { } a p 1, , Î ¼ is missing in the works [13,25] .…”

Section: Favard Theoremmentioning

confidence: 99%

“…, Remark 1. Multiple quadrature formulas have been studied in [13,25]. In particular, in [25] the convergence properties of simultaneous quadrature rules of a given function with respect to different weights is studied.…”

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confidence: 99%

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Oscillatory banded Hessenberg matrices, multiple orthogonal polynomials and Markov chains

2023

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A spectral Favard theorem for bounded banded lower Hessenberg matrices that admit a positive bidiagonal factorization is found. The large knowledge on the spectral and factorization properties of oscillatory matrices leads to this spectral Favard theorem in terms of sequences of multiple orthogonal polynomials of types I and II with respect to a set of positive Lebesgue--Stieltjes measures. Also a multiple Gauss quadrature is proven and corresponding degrees of precision are found. This spectral Favard theorem is applied to Markov chains with (p+2)-diagonal transition matrices, i.e. beyond birth and death, that admit a positive stochastic bidiagonal factorization. In the finite case, the Karlin--McGregor spectral representation is given. It is shown that the Markov chains are recurrent and explicit expressions in terms of the orthogonal polynomials for the stationary distributions are given.Similar results are obtained for the countable infinite Markov chain. Now the Markov chain is not necessarily recurrent, and it is characterized in terms of the first measure. Ergodicity of the Markov chain is discussed in terms of the existence of a mass at 1, which is an eigenvalue corresponding to the right and left eigenvectors.

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Section: Favard Theoremmentioning

confidence: 99%

Section: Favard Theoremmentioning

confidence: 99%

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Oscillatory banded Hessenberg matrices, multiple orthogonal polynomials and Markov chains

2023

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“…Hermite-Padé approximation and its relatives have applications in various areas, for example, in number theory (see [1,[5][6][7][8][9]), numerical analysis (see [10][11][12][13][14][15][16][17][18]), multiple orthogonal polynomials (see [18][19][20][21]), linear algebraic equations (see [22]), nonlinear dynamical systems (see [23]), Brownian motion (see [24]), in random matrices (see [19,25]), Gibbs phenomenon (see [26]), and Lie algebra solution of differential equations (see [27]). In addition to the proof of the transcendence of e, Hermite-Padé approximation was used in various irrationality and transcendence proofs of important numbers (see, e.g., [1,[5][6][7][8]).…”

Section: Introductionmentioning

confidence: 99%

On Row Sequences of Hermite–Padé Approximation and Its Generalizations

Bosuwan

2020

Mathematics

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Hermite–Padé approximation has been a mainstay of approximation theory since the concept was introduced by Charles Hermite in his proof of the transcendence of e in 1873. This subject occupies a large place in the literature and it has applications in different subjects. Most of the studies of Hermite–Padé approximation have mainly concentrated on diagonal sequences. Recently, there were some significant contributions in the direction of row sequences of Type II Hermite–Padé approximation. Moreover, various generalizations of Type II Hermite–Padé approximation were introduced and studied on row sequences. The purpose of this paper is to reflect the current state of the study of Type II Hermite–Padé approximation and its generalizations on row sequences. In particular, we focus on the relationship between the convergence of zeros of the common denominators of such approximants and singularities of the vector of approximated functions. Some conjectures concerning these studies are posed.

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Convergence and computation of simultaneous rational quadrature formulas

Prieto¹,

González

Lagomasino

2007

Numer. Math.

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Abstract. We discuss the theoretical convergence and numerical evaluation of simultaneous interpolation quadrature formulas which are exact for rational functions. Basically, the problem consists in integrating a single function with respect to different measures by using a common set of quadrature nodes. Given a multi-index n, the nodes of the integration rule are the zeros of the multiorthogonal Hermite-Padé polynomial with respect to (S, α, n), where S is a collection of measures, and α is a polynomial which modifies S. The theory is based on the connection between Gausslike simultaneous quadrature formulas of rational type and multipoint Hermite-Padé approximation. As for the numerical treatment we present a procedure based on the technique of modifying the integrand by means of a change of variable when the integrand has real poles close to the integration interval. The output of some tests show the power of this approach when compared to other ones.Key words. simultaneous integration rule, Gauss-like rational quadrature formula, meromorphic integrand AMS subject classifications. Primary 41A55. Secondary 41A28, 65D321. Introduction. Simultaneous integration is a problem which arises in computer graphics to determinate the color of light which emanates from a given point on a surface toward the viewer (see Borges [4] and the bibliography therein). In an abstract setting, this problem was earlier studied by Nikishin [19] in connection with Hermite-Padé approximation. Simultaneous integration means that we are integrating a single function f : I → R, with respect to m distinct measures ds 1 , ..., ds m on I, respectively.A reference index Φ = (Ov + 1)/N u (performance ratio) is considered in [4] to state the efficiency of the procedure. Here Ov is the overall degree of exactness and N u is the number of integrand evaluations. Borges [4] remarked that when m ≥ 3 the use of the m corresponding Gaussian rules of polynomial type yields Φ < 1, which indicates a low performance. Instead he suggests quadrature rules whose nodes are the zeros of multi-orthogonal polynomials for which Φ > 1 holds.Geometric rate of convergence has been proved by the authors in [8] for polynomials methods when the integrand is analytic in a neighborhood of I. Nevertheless, instability shows up when the corresponding numerical method is applied to meromorphic integrands with poles close to I. Several methods to integrate functions with singularities on or near the integration interval have been developed in the past few years.An interesting approach, whose starting point seems to be [13], is that based on the use of rational Gaussian integration rules connected with multipoint Padé

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Hermite–Padé approximation and simultaneous quadrature formulas

Cited by 23 publications

References 14 publications

Oscillatory banded Hessenberg matrices, multiple orthogonal polynomials and Markov chains

Oscillatory banded Hessenberg matrices, multiple orthogonal polynomials and Markov chains

On Row Sequences of Hermite–Padé Approximation and Its Generalizations

Convergence and computation of simultaneous rational quadrature formulas

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