Orthogonal Laurent polynomials and strong moment theory: a survey

Jones, William B.; Njåstad, Olav

doi:10.1016/s0377-0427(99)00027-8

Cited by 49 publications

(47 citation statements)

References 63 publications

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“…A survey of this and related results is provided by Jones and Njåstad [22]. An extension of the recursion relation discussed by Njåstad and Thron [24] to the sequence of rational Krylov subspaces…”

Section: Introduction Let a ∈ Rmentioning

confidence: 95%

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The structure of matrices in rational Gauss quadrature

Jagels

Reichel

2013

Math. Comp.

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Abstract. This paper is concerned with the approximation of matrix functionals defined by a large, sparse or structured, symmetric definite matrix. These functionals are Stieltjes integrals with a measure supported on a compact real interval. Rational Gauss quadrature rules that are designed to exactly integrate Laurent polynomials with a fixed pole in the vicinity of the support of the measure may yield better approximations of these functionals than standard Gauss quadrature rules with the same number of nodes. It therefore can be attractive to approximate matrix functionals by these rational Gauss rules. We describe the structure of the matrices associated with these quadrature rules, derive remainder terms, and discuss computational aspects. Also, rational Gauss-Radau rules and the applicability of pairs of rational Gauss and Gauss-Radau rules to computing lower and upper bounds for certain matrix functionals are discussed.

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Section: Introduction Let a ∈ Rmentioning

confidence: 95%

“…A classical approach to rational Gauss quadrature rules for f ∈ L 2m−2,2m+1 that does not explicitly utilize the recursion matrix H 2m is found in [22], where they are referred to as "strong Gaussian quadrature rules". This section demonstrates that more general rational Gauss rules exist for f ∈ L 2m−2,2mi+1 .…”

mentioning

confidence: 99%

The structure of matrices in rational Gauss quadrature

Jagels

Reichel

2013

Math. Comp.

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“…From this theorem one of the most commonly used orderings in the literature on orthogonal L-polynomials appears (see, e.g., [20]). …”

Section: Orthogonal Laurent Polynomials Associated With a Certain Strmentioning

confidence: 99%

“…Such distributions were earlier introduced in 1980 by Jones et al in their celebrated paper A Strong Stieltjes Moment Problem [17]. After this, the theory of the so-called orthogonal Laurent polynomials made their appearance [18] (see also [22,16,5]) and connections were established with certain kinds of continued fractions, two-point Padé approximants and new quadrature rules (see [20] for a survey). On the other hand, both orthogonal Laurent polynomials and proper orthogonal polynomials can be considered as particular cases within a more general framework: orthogonal rational functions with prescribed poles [1].…”

Section: Introductionmentioning

confidence: 99%

Orthogonal Laurent polynomials corresponding to certain strong Stieltjes distributions with applications to numerical quadratures

et al. 2005

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Abstract. In this paper we shall be mainly concerned with sequences of orthogonal Laurent polynomials associated with a class of strong Stieltjes distributions introduced by A.S. Ranga. Algebraic properties of certain quadratures formulae exactly integrating Laurent polynomials along with an application to estimate weighted integrals on [−1, 1] with nearby singularities are given. Finally, numerical examples involving interpolatory rules whose nodes are zeros of orthogonal Laurent polynomials are also presented.

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“…It is a fact [10,11,15,17] (on R) with the given (real) moments. For the case of the SSMP, there are four (Hankel determinant) inequalities (in this latter case, the c k , k ∈ Z, which appear in Equations (1.1) should be replaced by c SSMP k , k ∈ Z) which guarantee the existence of a non-negative measure µ SS MP (on [0, +∞)) with the given moments, namely [8] (see, also, [10,11] The φ n 's are normalised so that they all have real coefficients; in particular, the leading coefficients, LC(φ 2n ) := ξ [12,17]; see, also, [25]): φ 2m (ζ)(zφ 2m−1 (z)−ζφ 2m−1 (ζ))−ζφ 2m−1 (ζ)(φ 2m (z)−φ 2m (ζ)) = (z−ζ) ξ moreover, it can be shown that (see, for example, [15,17]), for n ∈ Z For each m ∈ Z + 0 , let µ 2m := card{z; π π π 2m (z) = 0} and µ 2m+1 := card{z; π π π 2m+1 (z) = 0}.…”

Section: Introductionmentioning

confidence: 99%

Asymptotics of Laurent polynomials of even degree orthogonal with respect to varying exponential weights

McLaughlin

Vartanian

Zhou

2010

International Mathematics Research Papers

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Let Λ R denote the linear space over R spanned by z k , k ∈ Z. Define the real inner product (with varying exponential weights). . } with respect to · · ·, · · · L yields the even degree and odd degree orthonormal Laurent polynomialsn > 0, and φ 2n+1 (z) = ξ n , φ 2n (z) (in the entire complex plane), and Hankel determinant ratios associated with the real-valued, bi-infinite, strong moment sequenceare obtained by formulating the even degree monic orthogonal Laurent polynomial problem as a matrix Riemann-Hilbert problem on R, and then extracting the largen behaviour by applying the non-linear steepest-descent method introduced in [1] and further developed in [2,3].

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Orthogonal Laurent polynomials and strong moment theory: a survey

Cited by 49 publications

References 63 publications

The structure of matrices in rational Gauss quadrature

The structure of matrices in rational Gauss quadrature

Orthogonal Laurent polynomials corresponding to certain strong Stieltjes distributions with applications to numerical quadratures

Asymptotics of Laurent polynomials of even degree orthogonal with respect to varying exponential weights

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