Anti-Gaussian quadrature formulas

Laurie, Dirk

doi:10.1090/s0025-5718-96-00713-2

Cited by 98 publications

(68 citation statements)

References 15 publications

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“…(For stratified and related quadrature rules, see [12,19]). Note that when γ = 0, the modified anti-Gauss and average rules given by (9) and (11) agree with the anti-Gauss and average rules introduced by Laurie in [13]. Ehrich [6] investigated the properties of (9) and (11) for the Laguerre and classical Hermite weight functions, and he obtained the degree optimal average rules for these weights.…”

Section: Introductionsupporting

confidence: 56%

“…Lemma 2.1 can easily be proved using this property (see for example [13] or [3]). The following result is a consequence of Lemma 2.1.…”

Section: Some Properties Of Modified Anti-gauss Rulesmentioning

confidence: 99%

“…A common approach to estimate the error in practice is to consider a second rule A of degree greater than 2n − 1 and to use Af − G n f (see e.g. [5,13,15]). It can be shown that any such quadrature rule requires at least n + 1 additional nodes.…”

Section: Introductionmentioning

confidence: 99%

“…[1,17,18]). As a different approach, Laurie [13] introduced the so-called antiGauss rules and corresponding average rules to estimate the error of the Gauss quadrature rules. The anti-Gauss and average rules and their modified versions introduced by Ehrich [6], which will be defined below, exist for any nonnegative weight function.…”

Section: Introductionmentioning

confidence: 99%

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Degree optimal average quadrature rules for the generalized Hermite weight function

Hascelik¹

2006

imf

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For the practical estimation of the error of Gauss quadrature rules Gauss-Kronrod rules are widely used; but, it is well known that for the generalized Hermite weight function, ω α (x) = |x| 2α exp(−x 2 ) over [−∞, ∞], real positive Gauss-Kronrod rules do not exist. Among the alternatives which are available in the literature, the anti-Gauss and average rules introduced by Laurie, and their modified versions, are of particular interest. In this paper, we investigate the properties of the modified anti-Gauss and average quadrature rules for ω α , and we determine the degree optimal average rules by proving that for each n-point Gauss rule for ω α there exists a unique average rule with the precise degree of exactness 2n+3. We also give some numerical examples to test the performance of the average rules obtained in this paper. Mathematics Subject Classification: 65D30, 65D32

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Section: Introductionsupporting

confidence: 56%

“…Lemma 2.1 can easily be proved using this property (see for example [13] or [3]). The following result is a consequence of Lemma 2.1.…”

Section: Some Properties Of Modified Anti-gauss Rulesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Degree optimal average quadrature rules for the generalized Hermite weight function

Hascelik¹

2006

imf

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“…Let G n denote the n-point Gauss quadrature rule with respect to dμ. Laurie [37] introduced so-called anti-Gauss quadrature rules for the approximation of If . The (n + 1)-point anti-Gauss quadrature rule H n+1 associated with the Gauss rule G n is characterized by (1.3) (I − H n+1 )p = −(I − G n )p ∀p ∈ P 2n+1 , where P 2n+1 denotes the set of all polynomials of degree at most 2n + 1 (with scalar coefficients).…”

Section: Introductionmentioning

confidence: 99%

Block Gauss and Anti-Gauss Quadrature with Application to Networks

Fenu¹,

Martin²,

Reichel³

et al. 2013

SIAM J. Matrix Anal. & Appl.

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Abstract. Approximations of matrix-valued functions of the form W T f (A)W , where A ∈ R m×m is symmetric, W ∈ R m×k , with m large and k m, has orthonormal columns, and f is a function, can be computed by applying a few steps of the symmetric block Lanczos method to A with initial block-vector W ∈ R m×k . Golub and Meurant have shown that the approximants obtained in this manner may be considered block Gauss quadrature rules associated with a matrix-valued measure. This paper generalizes anti-Gauss quadrature rules, introduced by Laurie for real-valued measures, to matrix-valued measures, and shows that under suitable conditions pairs of block Gauss and block anti-Gauss rules provide upper and lower bounds for the entries of the desired matrix-valued function. Extensions to matrix-valued functions of the form W T f (A)V , where A ∈ R m×m may be nonsymmetric, and the matrices V, W ∈ R m×k satisfy V T W = I k are also discussed. Approximations of the latter functions are computed by applying a few steps of the nonsymmetric block Lanczos method to A with initial block-vectors V and W . We describe applications to the evaluation of functions of a symmetric or nonsymmetric adjacency matrix for a network. Numerical examples illustrate that a combination of block Gauss and anti-Gauss quadrature rules typically provides upper and lower bounds for such problems. We introduce some new quantities that describe properties of nodes in directed or undirected networks, and demonstrate how these and other quantities can be computed inexpensively with the quadrature rules of the present paper.

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The Remainder Term of Gauss–Turán Quadratures for Analytic Functions

Spalević¹,

Pranić²

2010

Approximation and Computation

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Anti-Gaussian quadrature formulas

Cited by 98 publications

References 15 publications

Degree optimal average quadrature rules for the generalized Hermite weight function

Degree optimal average quadrature rules for the generalized Hermite weight function

Block Gauss and Anti-Gauss Quadrature with Application to Networks

The Remainder Term of Gauss–Turán Quadratures for Analytic Functions

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