The numerical stability of evaluation schemes for polynomials based on the lagrange interpolation form

Rack, Heinz-Joachim; Reimer, Manfred

doi:10.1007/bf01934399

Cited by 9 publications

(8 citation statements)

References 4 publications

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“…Ours is not the first numerical investigation of the modified Lagrange formula. Rack and Reimer [7] give a rounding error analysis that concludes with a weaker bound than (6) below and they do not identify the backward stability of the formula.…”

Section: Introductionmentioning

confidence: 96%

The numerical stability of barycentric Lagrange interpolation

Higham

2004

IMA Journal of Numerical Analysis

292

226

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The Lagrange representation of the interpolating polynomial can be rewritten in two more computationally attractive forms: a modified Lagrange form and a barycentric form. We give an error analysis of the evaluation of the interpolating polynomial using these two forms. The modified Lagrange formula is shown to be backward stable. The barycentric formula has a less favourable error analysis, but is forward stable for any set of interpolating points with a small Lebesgue constant. Therefore the barycentric formula can be significantly less accurate than the modified Lagrange formula only for a poor choice of interpolating points. This analysis provides further weight to the argument of Berrut and Trefethen that barycentric Lagrange interpolation should be the polynomial interpolation method of choice.

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Section: Introductionmentioning

confidence: 96%

The numerical stability of barycentric Lagrange interpolation

Higham

2004

IMA Journal of Numerical Analysis

292

226

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“…There is ongoing interest in exploring different strategies for fast summation of particle interactions, and the present work contributes a kernel-independent treecode (KITC) with operation count O(Nlog N) in which the far-field approximation uses barycentric Lagrange interpolation at Chebyshev points [6,56]. The barycentric Lagrange interpolant can be efficiently implemented and has good stability properties [28,43,47]; the 1D case is reviewed in [6,56] and here we apply it in 3D using a tensor product to compute well-separated particle-cluster approximations.…”

Section: Present Workmentioning

confidence: 99%

“…This means that the simple weights (3.3) can be used for any interval [a,b], along with the linearly mapped Chebyshev points; this is important in the present work because the treecode uses intervals of different sizes. Note also that barycentric Lagrange interpolation is stable in finite precision arithmetic [28,43,47], and the Chebfun software package uses this form of polynomial interpolation [14].…”

Section: Barycentric Lagrange Interpolationmentioning

confidence: 99%

A Kernel-Independent Treecode Based on Barycentric Lagrange Interpolation

Wang¹

2020

CICP

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A kernel-independent treecode (KITC) is presented for fast summation of particle interactions. The method employs barycentric Lagrange interpolation at Chebyshev points to approximate well-separated particle-cluster interactions. The KITC requires only kernel evaluations, is suitable for non-oscillatory kernels, and relies on the scale-invariance property of barycentric Lagrange interpolation. For a given level of accuracy, the treecode reduces the operation count for pairwise interactions from O(N 2 ) to O(Nlog N), where N is the number of particles in the system. The algorithm is demonstrated for systems of regularized Stokeslets and rotlets in 3D, and numerical results show the treecode performance in terms of error, CPU time, and memory consumption. The KITC is a relatively simple algorithm with low memory consumption, and this enables a straightforward OpenMP parallelization.

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“…Lebesgue constants as tools for detecting numerical problems. The numerical stability of (5) and (6) over [ −1, 1] was first looked at in a rigorous manner in [46] and later refined in [28]. This second reference shows how the modified Lagrange formula (5) is backward stable in general, whereas (6) is shown to be forward stable for vectors of points x ∈ R n+2 having a small Lebesgue constant 3 .…”

Section: 22mentioning

confidence: 99%

A Robust and Scalable Implementation of the Parks-McClellan Algorithm for Designing FIR Filters

Filip

2016

ACM Trans. Math. Softw.

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With a long history dating back to the beginning of the 1970s, the Parks-McClellan algorithm is probably the most well-known approach for designing finite impulse response filters. Despite being a standard routine in many signal processing packages, it is possible to find practical design specifications where existing codes fail to work. Our goal is twofold. We first examine and present solutions for the practical difficulties related to weighted minimax polynomial approximation problems on multi-interval domains (i.e., the general setting under which the Parks-McClellan algorithm operates). Using these ideas, we then describe a robust implementation of this algorithm. It routinely outperforms existing minimax filter design routines.

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The numerical stability of evaluation schemes for polynomials based on the lagrange interpolation form

Cited by 9 publications

References 4 publications

The numerical stability of barycentric Lagrange interpolation

The numerical stability of barycentric Lagrange interpolation

A Kernel-Independent Treecode Based on Barycentric Lagrange Interpolation

A Robust and Scalable Implementation of the Parks-McClellan Algorithm for Designing FIR Filters

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