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Numerical Algorithms

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21.07.2020 | Original Paper

Generalizing the trapezoidal rule in the complex plane

verfasst von: Bengt Fornberg

Erschienen in: Numerical Algorithms | Ausgabe 1/2021

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Abstract

In computational contexts, analytic functions are often best represented by grid-based function values in the complex plane. For integrating periodic functions, the spectrally accurate trapezoidal rule (TR) then becomes a natural choice, due to both accuracy and simplicity. The two key present observations are (i) the accuracy of TR in the periodic case can be greatly increased (doubling or tripling the number of correct digits) by using function values also along grid lines adjacent to the line of integration and (ii) a recently developed end correction strategy for finite interval integrations applies just as well when using these enhanced TR schemes.

Vorheriger Artikel High-dimensional sparse Fourier algorithms

Nächster Artikel The Bakhvalov mesh: a complete finite-difference analysis of two-dimensional singularly perturbed convection-diffusion problems

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If an analytic functions is somewhat costly to evaluate, it may be desirable to pay a one-time cost of evaluating over a grid and then re-use this data for multiple purposes, such as for graphical displays (e.g., [7, 12]) and, as focused on here, for highly accurate contour integrations (assuming some vicinity of the integration paths to be free of singularities). In some cases, the fastest evaluation method for the function is naturally grid based (e.g., the Painlevé functions [8]).

The present application is atypical in that Clenshaw-Curtis requires roughly twice as many evaluation points as Gaussian quadrature for matching accuracies; c.f., [10] and the discussion in Section 4.2 of [4].

First observed by Poisson in the 1820s [9]

Although many papers discuss and give error estimates for the periodic TR, there seem to be few—if any—previous references on utilizing this type of supplementary information for improving the accuracy. A somewhat related procedure was used in [3] for approximating derivatives/Taylor coefficients (c.f., its equations (5)–(9)).

Repeated application of Birkhoff-Young type formulas (first introduced in [1]) also produce multi-line TR-like formulas in the periodic case. However, as noted in [4], Section 4.3, their weights become such that they reduce rather than improve the accuracy compared with the regular TR.

c.f., [13], Example 5, and [7], page 353.

We recall the assumption that grid-based data is more economically available than “fresh” function evaluations. If not, it would be more cost effective to double and triple the number of nodes in the 1-line TR than to apply the 3-line and 5-line TR versions, respectively.

Appendix A in [5] sketches three derivations with references to still further ones.

As was done in [4]

The last parameter in the NSolve statement specifies the precision (number of decimal digits) to be used in this calculation of the coefficients.

Advanpix, Multiprecision Computing Toolbox for MATLAB, http://www.advanpix.com/, Advanpix LLC., Yokohama, Japan.

Figure 6 in [4] shows similarly the nodes used when integrating along the rectangle in the Cartesian node case.

One higher than the number of nodes in the correction stencil.

In Section 4.3 of [4], end corrected 1-line TR is compared against Gaussian and Clenshaw-Curtis quadratures for the test problem \({\int \limits }_{-1}^{1}f(z)dz\).

When expressed as derivatives of odd orders at the end point(s)

After the leading term, both expansions are geometric progressions that are readily summed in closed form, giving \(\pi \cot (\pi (x+iy))\) in their respective regions of convergence.

Birkhoff, G., Young, D.: Numerical quadrature of analytic and harmonic functions. J. Math. Physics 29, 217–221 (1950)MathSciNetCrossRef

Burne, C. R.: Derivative corrections to the trapezoidal rule. arXiv:1808.04743v1 [math.NA] (2018)

Fornberg, B.: Numerical differentiation of analytic functions. ACM Trans. Math. Software 7(4), 512–526 (1981)MathSciNetCrossRef

Fornberg, B.: Contour integrals of analytic functions given on a grid in the complex plane. IMA Journal of Num. Anal (to appear) (2020)

Fornberg, B.: Euler-Maclaurin expansions without analytic derivatives. submitted (2020)

Fornberg, B.: Improving the accuracy of the trapezoidal rule. SIAM Review (to appear) (2020)

Fornberg, B., Piret, C.: Complex Variables and Analytic Functions: an Illustrated Introduction. SIAM (2020)

Fornberg, B., Weideman, J.A.C.: A numerical method for the Painlevé equations. J. Comput. Phys. 230, 5957–5973 (2011)MathSciNetCrossRef

Poisson, S.D.: Sur le calcul numérique des intégrales définies. Mémoires de l’Académie Royale des Sciences de l’Institute de France 4, 571–602 (1827)

10.

Trefethen, L.N.: Is Gauss quadrature better than Clenshaw-Curtis? SIAM Rev. 50(1), 67–87 (2008)MathSciNetCrossRef

11.

Trefethen, L.N., Weideman, J.A.C.: The exponentially convergent trapezoidal rule. SIAM Rev. 56, 384–458 (2014)MathSciNetCrossRef

12.

Wegert, E.: Visual Complex Functions. Birkhäuser (2012)

13.

Weideman, J.A.C.: Numerical integration of periodic functions: a few examples. Amer. Math. Monthly 109, 21–36 (2002)MathSciNetCrossRef

Titel: Generalizing the trapezoidal rule in the complex plane
verfasst von: Bengt Fornberg
Publikationsdatum: 21.07.2020
Verlag: Springer US
Erschienen in: Numerical Algorithms / Ausgabe 1/2021
Print ISSN: 1017-1398
Elektronische ISSN: 1572-9265
DOI: https://doi.org/10.1007/s11075-020-00963-0

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