Accurate recovery of recursion coefficients from Gaussian quadrature formulas

Laurie, Dirk

doi:10.1016/s0377-0427(99)00228-9

Cited by 20 publications

(20 citation statements)

References 18 publications

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“…2 consists in whether or not the number of points of increase (i.e., the 'size' of the support) is changed when ω is perturbed to formω. We will show that if there is no change in the number of points of increase, then a result by Laurie [36] explains the observed insensitivity of Gauss-quadrature for small enough perturbations.…”

Section: Application To Motivating Examples: When Larger Support Mattersmentioning

confidence: 80%

“…considered in [29, p. 325], does not play a role. In the matlab routine pftoqd.m we have also implemented the algorithm by Laurie which requires no subtractions; see [36]. We emphasize that the same sensitivity phenomenon can be observed, with differences which are here insignificant, using various computations of the recurrence coefficients from the spectral data.…”

Section: Motivating Examplesmentioning

confidence: 87%

“…The map H k is said to be generally well-conditioned in [23, p. 59], though numerically stable computation of the entries of the Jacobi matrix from the quadrature nodes and weights is not easy; see [ (See also the last two paragraphs of this section, which explain in detail history of that problem and the fact that the algorithmic construction of Laurie in [36] also gives the perturbation result.) Further results on the condition numbers of the map H k (and also of its inverse H −1 k ) can be found in [1, relation (7), Sect.…”

Section: T1mentioning

confidence: 99%

“…They concluded that the problem of reconstructing a Jacobi matrix from the weights and nodes is ill-conditioned [29, p. 330 and 332]. This conclusion has been examined by Laurie [36], who pointed out that the negative statement is linked to the use of the max-norm for vectors. He suggested instead measuring the perturbation of the weights in the componentwise relative sense [36, p. 179], [38,Sect.…”

Section: T1mentioning

confidence: 99%

“…6]. The main part of [36] is devoted to the constructive proof of the following statement [36, Theorem on p. 168]: given the weights and the N −1 positive differences between the consecutive nodes, the main diagonal entries of the corresponding Jacobi matrix (shifted by the smallest node) and the off-diagonal entries can be computed in 9 2 N 2 + O(N ) arithmetic operations, all of which can involve only addition, multiplication and division of positive numbers. Consequently, in finite precision arithmetic they can be computed to a relative accuracy no worse than 9 2 N 2 ε + O(N ε), where ε denotes machine precision.…”

Section: T1mentioning

confidence: 99%

See 4 more Smart Citations

On sensitivity of Gauss–Christoffel quadrature

2007

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In numerical computations the question how much does a function change under perturbations of its arguments is of central importance. In this work, we investigate sensitivity of Gauss-Christoffel quadrature with respect to small perturbations of the distribution function. In numerical quadrature, a definite integral is approximated by a finite sum of functional values evaluated at given quadrature nodes and multiplied by given weights. Consider a sufficiently smooth integrated function uncorrelated with the perturbation of the distribution function. Then it seems natural that given the same number of function evaluations, the difference between the quadrature approximations is of the same order as the difference between the (original and perturbed) approximated integrals. 123 148 D. P. O'Leary et al.several other sensitivity problems, motivated, in particular, by computation of the quadrature nodes and weights from moments, have been thoroughly studied by many authors. We survey existing particular results and show that even a small perturbation of a distribution function can cause large differences in Gauss-Christoffel quadrature estimates. We then discuss conditions under which the Gauss-Christoffel quadrature is insensitive under perturbation of the distribution function, present illustrative examples, and relate our observations to known conjectures on some sensitivity problems.

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Section: Application To Motivating Examples: When Larger Support Mattersmentioning

confidence: 80%

Section: Motivating Examplesmentioning

confidence: 87%

Section: T1mentioning

confidence: 99%

Section: T1mentioning

confidence: 99%

Section: T1mentioning

confidence: 99%

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On sensitivity of Gauss–Christoffel quadrature

2007

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Numerical Quadrature Rules

2020

The Nyström Method in Electromagnetics

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Direct and inverse computation of Jacobi matrices of infinite iterated function systems

Mantica

2013

Numer. Math.

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We introduce a new set of algorithms to compute the Jacobi matrices associated with invariant measures of infinite iterated function systems, composed of one–dimensional, homogeneous affine maps. We demonstrate their utility in the study of theoretical problems, like the conjectured almost periodicity of such Jacobi matrices, the singularity of the measures, and the logarithmic capacity of their support. Since our technique is based on a reversible transformation between pairs of Jacobi matrices, it can also be applied to solve an inverse/approximation problem. The proposed algorithms are tested in significant, highly sensitive cases: they perform in a stable fashion, and can reliably compute Jacobi matrices of large order

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Accurate recovery of recursion coefficients from Gaussian quadrature formulas

Cited by 20 publications

References 18 publications

On sensitivity of Gauss–Christoffel quadrature

On sensitivity of Gauss–Christoffel quadrature

Numerical Quadrature Rules

Direct and inverse computation of Jacobi matrices of infinite iterated function systems

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