Computing numerically with functions instead of numbers

Trefethen, Lloyd N.

doi:10.1145/2814847

Cited by 21 publications

(20 citation statements)

References 24 publications

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“…Since (4.1) is an asymptotic expansion, it can only be used for sufficiently large k. For this paper, j 0,k is tabulated for k ∈ [1,20] and computed by means of (4.1) if k > 20.…”

Section: Lagrange Inversion Theoremmentioning

confidence: 99%

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Iteration-Free Computation of Gauss--Legendre Quadrature Nodes and Weights

Bogaert¹

2014

SIAM J. Sci. Comput.

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Abstract. Gauss-Legendre quadrature rules are of considerable theoretical and practical interest because of their role in numerical integration and interpolation. In this paper, a series expansion for the zeros of the Legendre polynomials is constructed. In addition, a series expansion useful for the computation of the Gauss-Legendre weights is derived. Together, these two expansions provide a practical and fast iteration-free method to compute individual Gauss-Legendre node-weight pairs in O (1) complexity and with double precision accuracy. An expansion for the barycentric interpolation weights for the Gauss-Legendre nodes is also derived. A C++ implementation is available on line.

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“…Since (4.1) is an asymptotic expansion, it can only be used for sufficiently large k. For this paper, j 0,k is tabulated for k ∈ [1,20] and computed by means of (4.1) if k > 20.…”

Section: Lagrange Inversion Theoremmentioning

confidence: 99%

“…Because these functions need only be evaluated in the range φ ∈ [0, π 2 ], they are good candidates to be approximated by means of Chebyshev interpolants [19,20]. However, the presence of a singularity at φ = rπ, ∀r ∈ {±1, ±2, ±3, ...} slows down the convergence of the Chebyshev series.…”

Section: 2mentioning

confidence: 99%

Iteration-Free Computation of Gauss--Legendre Quadrature Nodes and Weights

Bogaert¹

2014

SIAM J. Sci. Comput.

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“…A comparison of univariate Taylor forms and Chebyshev forms in [27] showed that expansions in Chebyshev series may be orders of magnitude more accurate than expansions in Taylor series. The use of Chebyshev series expansion as an arithmetic is also the philosophy behind the package Chebfun by Trefethen et al [6,57], which has recently been extended to functions in two variables as well [56]. It should be noted that Chebfun relies on computations with functions to 15-digit accuracy, thus removing the need for propagating a remainder term, but the results are not validated per se.…”

Section: Introductionmentioning

confidence: 99%

Chebyshev model arithmetic for factorable functions

et al. 2016

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This article presents an arithmetic for the computation of Chebyshev models for factorable functions and an analysis of their convergence properties. Similar to Taylor models, Chebyshev models consist of a pair of a multivariate polynomial approximating the factorable function and an interval remainder term bounding the actual gap with this polynomial approximant. Propagation rules and local convergence bounds are established for the addition, multiplication and composition operations with Chebyshev models. The global convergence of this arithmetic as the polynomial expansion order increases is also discussed. A generic implementation of Chebyshev model arithmetic is available in the library MC++. It is shown through several numerical case studies that Chebyshev models provide tighter bounds than their Taylor model counterparts, but this comes at the price of extra computational burden.

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“…It was implemented in a 2004 paper [3] for the smooth functions on the interval [−1, 1] with the aim of building some links between the discrete and continuous linear algebra in particular to extend the MATLAB operations made on vectors and matrix in the functions and the operators. The basis of Chebfun is Chebyshev polynomials [33,34]. Chebfun computes Chebyshev coefficients by examining them in the machine precision of a function g and represents it to the Chebyshev points by using barycentric interpolation [10].…”

Section: Chebfun Systemmentioning

confidence: 99%

On Rational and Adaptive Spectral Methods

Ndeuzoumbet¹,

Mampassi²

2015

Int. J. of Appl. Math.

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In this work, we give a general overview on pseudo-spectral methods. Three types of its variants which are: the adaptive spectral methods conceived for functions with rapid variation in certain domains, the rational adaptive spectral method using rational interpolants, and the adaptive grid Chebyshev points thanks to conformal maps, another type of approximation by the Chebyshev spectral method constituted by a collection of algorithms coded in an objet-oriented MATLAB software environment are considered.

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Computing numerically with functions instead of numbers

Cited by 21 publications

References 24 publications

Iteration-Free Computation of Gauss--Legendre Quadrature Nodes and Weights

Iteration-Free Computation of Gauss--Legendre Quadrature Nodes and Weights

Chebyshev model arithmetic for factorable functions

On Rational and Adaptive Spectral Methods

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