## Asymptotically accurate error estimates of exponential convergence for the trapezoidal rule

**Authors:**Belov A.A.^{1}^{,2}, Khokhlachev V.S.^{1}**Affiliations:**- M.V. Lomonosov Moscow State University
- Peoples’ Friendship University of Russia (RUDN University)

**Issue:**Vol 29, No 3 (2021)**Pages:**251-259**Section:**Articles**URL:**http://journals.rudn.ru/miph/article/view/27529**DOI:**https://doi.org/10.22363/2658-4670-2021-29-3-251-259- Cite item

# Abstract

In many applied problems, efficient calculation of quadratures with high accuracy is required. The examples are: calculation of special functions of mathematical physics, calculation of Fourier coefficients of a given function, Fourier and Laplace transformations, numerical solution of integral equations, solution of boundary value problems for partial differential equations in integral form, etc. For grid calculation of quadratures, the trapezoidal, the mean and the Simpson methods are usually used. Commonly, the error of these methods depends quadratically on the grid step, and a large number of steps are required to obtain good accuracy. However, there are some cases when the error of the trapezoidal method depends on the step value not quadratically, but exponentially. Such cases are integral of a periodic function over the full period and the integral over the entire real axis of a function that decreases rapidly enough at infinity. If the integrand has poles of the first order on the complex plane, then the Trefethen-Weidemann majorant accuracy estimates are valid for such quadratures. In the present paper, new error estimates of exponentially converging quadratures from periodic functions over the full period are constructed. The integrand function can have an arbitrary number of poles of an integer order on the complex plane. If the grid is sufficiently detailed, i.e., it resolves the profile of the integrand function, then the proposed estimates are not majorant, but asymptotically sharp. Extrapolating, i.e., excluding this error from the numerical quadrature, it is possible to calculate the integrals of these classes with the accuracy of rounding errors already on extremely coarse grids containing only ∼ 10 steps.

# Full Text

1. Introduction Applied tasks. In many physical problems it is needed to calculate integrals, that cannot be obtained in terms of elementary functions. Here are some examples: © Belov A.A., Khokhlachev V.S., 2021 This work is licensed under a Creative Commons Attribution 4.0 International License http://creativecommons.org/licenses/by/4.0/ 1) Calculation of special functions of mathematical physics: the Fermi-Dirac functions, which are equal to the moments of the Fermi distribution, the Gamma function, cylindrical functions and a number of others. 2) Calculation of the Fourier coefficients of a given function, Fourier and Laplace transform. 3) Numerical solution of integral equations, both well-posed and ill-posed. 4) Solving boundary value problems for partial differential equations (including eigenvalue problems) written in integral form, etc. Calculation of quadratures. Trapezoidal rule, rectangle rule and Simpson’s rule are commonly used for grid computation of quadratures. Usually the error of these methods quadratically depends on the grid step, and a large number of steps is needed to obtain good accuracy. However, there are a number of cases when the error of the trapezoidal rule depends on the grid step exponentially, i.e. when the step is reduced by half, the number of correct signs of the numerical result is approximately doubled. This rate of convergence is similar to that of Newton’s method. Two such cases are known. These are: the integral of the periodic function over the full period and the improper integral of a function that decreases rapidly enough at infinity. If the integrand has first order poles on the complex plane, then for such quadratures there are majorant error estimates of Trefethen and Weidemann [1], see also [2]-[10]. In [11], [12] the generalization of Trefethen and Weidemann estimates is built for the case when the nearest pole of an integrand function is multiple. In this paper, new error estimates of exponentially convergent quadratures of periodic functions over the full period are described. Integrand function can have an arbitrary number of poles of an integer order on the complex plane. If the mesh is detailed enough and the profile of the integrand resolved well, then the proposed estimates are not majorant, but asymptotically accurate. It is possible to calculate the integrals of the indicated classes with the accuracy of round-off errors even on extremely coarse grids containing only ∼ 10 steps by extrapolation (i.e., subtraction) of this error from the numerical value of the quadrature. 2. Exponentially convergent quadratures One of the classes of exponentially convergent quadratures are integrals of periodic functions over the full period. By replacing

# About the authors

### Aleksandr A. Belov

M.V. Lomonosov Moscow State University; Peoples’ Friendship University of Russia (RUDN University)
**Author for correspondence.**

Email: aa.belov@physics.msu.ru

ORCID iD: 0000-0002-0918-9263

1, bld. 2, Leninskie Gory, Moscow, 119991, Russian Federation; 6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation

Candidate of Physical and Mathematical Sciences, Assistant professor of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University); Researcher of Faculty of Physics, M.V. Lomonosov Moscow State University

### Valentin S. Khokhlachev

M.V. Lomonosov Moscow State University
Email: valentin.mycroft@yandex.ru

ORCID iD: 0000-0002-6590-5914

1, bld. 2, Leninskie Gory, Moscow, 119991, Russian Federation

Master’s degree student of Faculty of Physics

# References

- L. N. Trefethen and J. A. C. Weideman, “The exponentially convergent trapezoidal rule,” SIAM Review, vol. 56, no. 3, pp. 385-458, 2014. doi: 10.1137/130932132.
- J. Mohsin and L. N. Trefethen, “A trapezoidal rule error bound unifying the Euler-Maclaurin formula and geometric convergence for periodic functions,” in Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 470, 2014, p. 20130571. doi: 10.1098/rspa.2013.0571.
- J. A. C. Weideman, “Numerical integration of periodic functions: A few examples,” The American Mathematical Monthly, vol. 109, no. 1, pp. 21- 36, 2002. doi: 10.2307/2695765.
- N. Eggert and J. Lund, “The trapezoidal rule for analytic functions of rapid decrease,” Journal of Computational and Applied Mathematics, vol. 27, no. 3, pp. 389-406, 1989. doi: 10.1016/0377-0427(89)90024-1.
- H. Al Kafri, D. J. Jeffrey, and R. M. Corless, “Rapidly convergent integrals and function evaluation,” Lecture Notes in Computer Science, vol. 10693, pp. 270-274, 2017. doi: 10.1007/978-3-319-72453-9_20.
- J. Waldvogel, “Towards a general error theory of the trapezoidal rule,” in Springer Optimization and Its Applications. 2010, vol. 42, pp. 267- 282. doi: 10.1007/978-1-4419-6594-3_17.
- E. T. Goodwin, “The evaluation of integrals of the form f(x)e-x2dx,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 45, no. 2, pp. 241–245, 1949. DOI: 10.1017/ S0305004100024786.
- N. N. Kalitkin and S. A. Kolganov, “Quadrature formulas with exponential convergence and calculation of the Fermi–Dirac integrals,” Doklady Mathematics, vol. 95, no. 2, pp. 157–160, 2017. doi: 10.1134/S1064562417020156.
- N. N. Kalitkin and S. A. Kolganov, “Refinements of precision approximations of Fermi–Dirak functions of integer indices,” Mathematical Models and Computer Simulations, vol. 9, no. 5, pp. 554–560, 2017. doi: 10.1134/S2070048217050052.
- N. N. Kalitkin and S. A. Kolganov, “Computing the Fermi–Dirac functions by exponentially convergent quadratures,” Mathematical Models and Computer Simulations, vol. 10, no. 4, pp. 472–482, 2018. doi: 10.1134/S2070048218040063.
- A. A. Belov, N. N. Kalitkin, and V. S. Khokhlachev, “Improved error estimates for an exponentially convergent quadratures [Uluchshennyye otsenki pogreshnosti dlya eksponentsial’no skhodyashchikhsya kvadratur],” Preprints of IPM im. M.V. Keldysh, no. 75, 2020, in Russian. doi: 10.20948/prepr-2020-75.
- V. S. Khokhlachev, A. A. Belov, and N. N. Kalitkin, “Improvement of error estimates for exponentially convergent quadratures [Uluchsheniye otsenok pogreshnosti dlya eksponentsial’no skhodyashchikhsya kvadratur],” Izv. RAN. Ser. fiz., vol. 85, no. 2, pp. 282–288, 2021, in Russian. doi: 10.31857/S0367676521010166.