Gauss quadrature rules for a generalized Hermite weight function

Hascelik, A. Ihsan

doi:10.1016/j.amc.2005.11.152

Cited by 6 publications

(3 citation statements)

References 15 publications

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“…But as far as we know these coefficients are not available analytically; hence one has to use a suitable method to compute them approximately to sufficient accuracy. There are mainly two approaches for this purpose: the first approach is based on the discretization methods (Stieltjes procedure, Lanczos-type algorithm) and the second one is based on the moments (moment based methods); see [12,Chapter 2] and the references therein, and see also [15] for an application of the Lanczos algorithm. Although the first approach is numerically more stable than the second one, our numerical experiments show that neither the Stieltjes procedure nor the Lanczos algorithm is suitable for the weight functions considered here.…”

Section: Computation Of the Recurrence Coefficientsmentioning

confidence: 99%

On numerical computation of integrals with integrands of the form f(x)sin(w/xr) on [0, 1]

Hascelik

2009

Journal of Computational and Applied Mathematics

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With existing numerical integration methods and algorithms it is difficult in general to obtain accurate approximations to integrals of the formwhere f is a sufficiently smooth function on [0, 1]. Gautschi has developed software (as scripts in Matlab) for computing these integrals for the special case r = ω = 1. In this paper, an algorithm (as a Mathematica program) is developed for computing these integrals to arbitrary precision for any given values of the parameters in a certain range. Numerical examples are given of testing the performance of the algorithm/program.

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Section: Computation Of the Recurrence Coefficientsmentioning

confidence: 99%

On numerical computation of integrals with integrands of the form f(x)sin(w/xr) on [0, 1]

Hascelik

2009

Journal of Computational and Applied Mathematics

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“…As a first attempt to solve this problem, one could consider using numerical integration techniques. In the scalar case (n = n y = 1) and if the function f is well approximated by a polynomial of order 2N 1, the Gaussian quadrature methods give approximate solutions for (14) of the form (see [5], [51]- [54])…”

Section: G Conclusion Of the Literature Reviewmentioning

confidence: 99%

A Systematization of the Unscented Kalman Filter Theory

Menegaz

Ishihara

Borges

et al. 2015

IEEE Trans. Automat. Contr.

220

154

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Abstract-In this paper, we propose a systematization of the (discrete-time) Unscented Kalman Filter (UKF) theory. We gather all available UKF variants in the literature, present corrections to theoretical inconsistencies, and provide a tool for the construction of new UKF's in a consistent way. This systematization is done, mainly, by revisiting the concepts of Sigma-Representation, Unscented Transformation (UT), Scaled Unscented Transformation (SUT), UKF, and Square-Root Unscented Kalman Filter (SRUKF). Inconsistencies are related to 1) matching the order of the transformed covariance and cross-covariance matrices of both the UT and the SUT; 2) multiple UKF definitions; 3) issue with some reduced sets of sigma points described in the literature; 4) the conservativeness of the SUT; 5) the scaling effect of the SUT on both its transformed covariance and crosscovariance matrices; and 6) possibly ill-conditioned results in SRUKF's. With the proposed systematization, the symmetric sets of sigma points in the literature are formally justified, and we are able to provide new consistent variations for UKF's, such as the Scaled SRUKF's and the UKF's composed by the minimum number of sigma points. Furthermore, our proposed SRUKF has improved computational properties when compared to state-ofthe-art methods.

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“…As a first attempt to solve this problem, we could consider using numerical integration techniques. In the scalar case (n = n y = 1) and if the function f is well approximated by a polynomial of order 2N − 1 for a N ∈ N, Gaussian quadrature methods give approximate solutions for (3.1) of the form (see [78,[99][100][101][102])…”

Section: Estimating a Posterior Expected Valuementioning

confidence: 99%

Unscented kalman filtering on euclidean and riemannian manifolds

Menegaz¹

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Gauss quadrature rules for a generalized Hermite weight function

Cited by 6 publications

References 15 publications

On numerical computation of integrals with integrands of the form f(x)sin(w/xr) on [0, 1]

On numerical computation of integrals with integrands of the form f(x)sin(w/xr) on [0, 1]

A Systematization of the Unscented Kalman Filter Theory

Unscented kalman filtering on euclidean and riemannian manifolds

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