On the condition number of Lagrangian numerical differentiation

Miel, George; Mooney, Rose

doi:10.1016/0096-3003(85)90031-1

Cited by 10 publications

(8 citation statements)

References 12 publications

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“…While the series representation method provides a tractable alternate for handling general fading, it often leads to situations where it is difficult to derive closed form expressions. Numerically evaluating a higher order derivative is also complex and prone to floating-point rounding errors [22].…”

Section: Introductionmentioning

confidence: 99%

A Comprehensive Analysis of 5G Heterogeneous Cellular Systems Operating Over $\kappa$ – $\mu$ Shadowed Fading Channels

Chun

Cotton

Dhillon

et al. 2017

IEEE Trans. Wireless Commun.

109

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Emerging cellular technologies such as those proposed for use in 5G communications will accommodate a wide range of usage scenarios with diverse link requirements. This will include the necessity to operate over a versatile set of wireless channels ranging from indoor to outdoor, from line-of-sight (LOS) to non-LOS, and from circularly symmetric scattering to environments which promote the clustering of scattered multipath waves. Unfortunately, many of the conventional fading models lack the flexibility to account for such disparate signal propagation mechanisms. To bridge the gap between theory and practical channels, we consider κ-µ shadowed fading, which contains every linear fading models proposed in the open literature as special cases. In particular, we propose an analytic framework to evaluate the average of an arbitrary function of the SINR over κ-µ shadowed fading channels by using a simplified orthogonal expression with tools from stochastic geometry. Using the proposed method, we evaluate the spectral efficiency, moments of the SINR, and outage probability of a K-tier HetNet with K classes of BSs, differing in terms of the transmit power, BS density, shadowing and fading. Building upon these results, we provide important new insights into the network performance of these emerging wireless applications while considering a diverse range of fading conditions and link qualities.

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Section: Introductionmentioning

confidence: 99%

A Comprehensive Analysis of 5G Heterogeneous Cellular Systems Operating Over $\kappa$ – $\mu$ Shadowed Fading Channels

Chun

Cotton

Dhillon

et al. 2017

IEEE Trans. Wireless Commun.

109

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show abstract

“…Much has been written on this topic [1,2,4,6,8,9,10,11,13,15,16,18,20,21,22,23,28,29,33,34,35,36], and a number of techniques have been developed. Most fall into one of three categories: difference methods, interpolation methods, and regularization methods.…”

Section: Introductionmentioning

confidence: 99%

A variational method for numerical differentiation

Knowles

Wallace

1995

Numerische Mathematik

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“…Computing derivatives of a function numerically is a notoriously ill-conditioned problem, especially for high-order derivatives [21]. It was shown by Bornemann in [4] that computation of high-order derivatives through Cauchy integrals in the complex plane is, in fact, stable.…”

Section: Accurate Computation Of High-order Derivativesmentioning

confidence: 99%

Fast and accurate computation of Chebyshev coefficients in the complex plane

Wang

Huybrechs

2016

IMA J Numer Anal

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Chebyshev expansion coefficients can be computed efficiently by using the FFT, and for smooth functions the resulting approximation is close to optimal, with computations that are numerically stable. Given sufficiently accurate function samples, the Chebyshev expansion coefficients can be computed to machine precision accuracy. However, the accuracy is only with respect to absolute error, and this implies that very small expansion coefficients typically have very large relative error. Upon differentiating a Chebyshev expansion, this relative error in the small coefficients is magnified and accuracy may be lost, especially after repeated differentiation. At first sight, this seems unavoidable. Yet, in this paper, we focus on an alternative computation of Chebyshev expansion coefficients using contour integrals in the complex plane. The main result is that the coefficients can be computed with machine precision relative error, rather than absolute error. This implies that even very small coefficients can be computed with full floating point accuracy, even when they are themselves much smaller than machine precision. As a result, no accuracy is lost after differentiating the expansion, and even the 100th derivative of an analytic function can be computed with near machine precision accuracy using standard floating point arithmetic. In some cases, the contour integrals can be evaluated using the FFT, making the approach both highly accurate and fast. Accurate computation of high-order derivativesComputing derivatives of a function numerically is a notoriously ill-conditioned problem, especially for high-order derivatives [21]. It was shown by Bornemann in [4] that computation of high-order derivatives through Cauchy integrals in the complex plane is, in fact, stable. To be precise, consider the power series of a function analytic at the

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On the condition number of Lagrangian numerical differentiation

Cited by 10 publications

References 12 publications

A Comprehensive Analysis of 5G Heterogeneous Cellular Systems Operating Over $\kappa$ – $\mu$ Shadowed Fading Channels

A Comprehensive Analysis of 5G Heterogeneous Cellular Systems Operating Over $\kappa$ – $\mu$ Shadowed Fading Channels

A variational method for numerical differentiation

Fast and accurate computation of Chebyshev coefficients in the complex plane

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