Barycentric-Remez algorithms for best polynomial approximation in the chebfun system

Pachon, Ricardo; Trefethen, Lloyd N.

doi:10.1007/s10543-009-0240-1

Cited by 59 publications

(42 citation statements)

References 37 publications

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“…Quantifying E n [ f ] A and obtaining the polynomial that achieves it turned out to be extremely challenging. Remez [159] in 1934 proposed an efficient algorithm for computing the best polynomial approximation, and it was recently implemented and highly optimized in Matlab by the Chebfun team [160], [161]. Regarding the theoretical understanding of E n [ f ] A , de la Vallée-Poussin, Bernstein, Ibragimov, Markov, Kolmogorov and others have made significant contributions, and it is still an active research area.…”

Section: Motivation Methodology and Related Workmentioning

confidence: 99%

“…The only overhead that deserves special mention is the computation of the best polynomial approximation, which is performed via the Remez algorithm [159] offline before obtaining any samples. The Chebfun team [160] provides a highly optimized implementation of the Remez algorithm in Matlab [161]. In numerical analysis, the convergence of an algorithm is called quadratic if the error e m after the m -th computation satisfies e m ≤ Cα 2 m for some C > 0 and 0 < α < 1.…”

Section: Methodsmentioning

confidence: 99%

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Minimax Estimation of Functionals of Discrete Distributions

Jiao

Venkat

Han

et al. 2015

IEEE Trans. Inform. Theory

144

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Abstract-We propose a general methodology for the construction and analysis of essentially minimax estimators for a wide class of functionals of finite dimensional parameters, and elaborate on the case of discrete distributions, where the support size S is unknown and may be comparable with or even much larger than the number of observations n. We treat the respective regions where the functional is "nonsmooth" and "smooth" separately. In the "nonsmooth" regime, we apply an unbiased estimator for the best polynomial approximation of the functional whereas, in the "smooth" regime, we apply a biascorrected version of the Maximum Likelihood Estimator (MLE).We illustrate the merit of this approach by thoroughly analyzing the performance of the resulting schemes for estimating two important information measures: the entropyWe obtain the minimax L2 rates for estimating these functionals. In particular, we demonstrate that our estimator achieves the optimal sample complexity n S/ ln S for entropy estimation. We also demonstrate that the sample complexity for estimating Fα(P ), 0 < α < 1, is n S 1/α / ln S, which can be achieved by our estimator but not the MLE. For 1 < α < 3/2, we show the minimax L2 rate for estimating Fα(P ) is (n ln n) −2(α−1) for infinite support size, while the maximum L2 rate for the MLE is n −2(α−1) . For all the above cases, the behavior of the minimax rate-optimal estimators with n samples is essentially that of the MLE (plug-in rule) with n ln n samples, which we term "effective sample size enlargement".We highlight the practical advantages of our schemes for the estimation of entropy and mutual information. We compare our performance with various existing approaches, and demonstrate that our approach reduces running time and boosts the accuracy. Moreover, we show that the minimax rate-optimal mutual information estimator yielded by our framework leads to significant performance boosts over the Chow-Liu algorithm in learning graphical models. The wide use of information measure estimation suggests that the insights and estimators obtained in this work could be broadly applicable.

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Section: Motivation Methodology and Related Workmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Minimax Estimation of Functionals of Discrete Distributions

Jiao

Venkat

Han

et al. 2015

IEEE Trans. Inform. Theory

144

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“…Their numerical work gave an enormous impact into the analytical investigation of approximation problems, not only restricted to the Bernstein constants. We would also like to mention the numerical work of Pachón and Trefethen ( [13], Figure 4.4) from 2008, when they recomputed {nE n (|x| , L ∞ [−1, 1]) : n = 1, . .…”

Section: The Bernstein Constantmentioning

confidence: 99%

Extremal Polynomials and Entire Functions of Exponential Type

Revers¹

2018

Results Math

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In this paper, we discuss asymptotic relations for the approximation of |x| α , α > 0 in L ∞ [−1, 1] by Lagrange interpolation polynomials based on the zeros of the Chebyshev polynomials of first kind. The limiting process reveals an entire function of exponential type for which we can present an explicit formula. As a consequence, we further deduce an asymptotic relation for the approximation error when α → ∞. Finally, we present connections of our results together with some recent work of Ganzburg [5] and Lubinsky [10], by presenting numerical results, indicating a possible constructive way towards a representation for the Bernstein constants.MSC Classification (2010): 41A05, 41A10, 41A60, 65D05

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“…• piecewise smooth functions 16 • periodic functions (Fourier not Chebyshev) 7 • fast edge detection for determining breakpoints 16 • infinite intervals [a, ∞ ), (− ∞ , b], (− ∞ , ∞ ) • functions with poles and other singularities • delta functions of arbitrary order • Padé, Remez, CF rational approximations 8,17,23 • fast Gauss and Gauss-Jacobi quadrature 9,11 • fast Chebyshev ↔ Legendre conversions 10 • continuous QR factorization, SVD, least-squares 1 We shall not attempt to describe these developments, but here are a few comments. For solving ODE boundary value problems, whether scalars or systems and smooth or just piecewise smooth, Chebfun and its interface Chebgui have emerged as the most convenient and flexible tool in existence, making it possible to solve all kinds of problems with minimal effort with accuracy close to machine precision (these developments are due especially to Ásgeir Birkisson, Toby Driscoll, and Nick Hale).…”

Section: Chebfun Software Projectmentioning

confidence: 99%

Computing numerically with functions instead of numbers

Trefethen

2015

Commun. ACM

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Science and engineering depend upon computation of functions such as flow fields, charge distributions, and quantum states. Ultimately, such computations require some kind of discretization, but in recent years, it has become possible in many cases to hide the discretizations from the user. We present the Chebfun system for numerical computation with functions, which is based on a key idea: an analogy of floating-point arithmetic for functions rather than numbers.

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Barycentric-Remez algorithms for best polynomial approximation in the chebfun system

Cited by 59 publications

References 37 publications

Minimax Estimation of Functionals of Discrete Distributions

Minimax Estimation of Functionals of Discrete Distributions

Extremal Polynomials and Entire Functions of Exponential Type

Computing numerically with functions instead of numbers

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