\(\mathrm{PVTSI}^{(m)}\): a novel approach to computation of Hadamard finite parts of nonperiodic singular integrals. (English) Zbl 07462048

Summary: We consider the numerical computation of \(I[f]=\int \!\!\!\!\!\!=^b_a f(x)\,dx\), the Hadamard finite part of the finite-range singular integral \(\int^b_a f(x)\,dx, f(x)=g(x)/(x-t)^m\) with \(a<t<b\) and \(m\in \{ 1,2,\ldots\}\), assuming that (i) \(g\in C^{\infty} (a,b)\) and (ii) \(g(x)\) is allowed to have arbitrary integrable singularities at the endpoints \(x=a\) and \(x=b\). We first prove that \(\int \!\!\!\!\!\!=^b_a f(x)\,dx\) is invariant under any legitimate variable transformation \(x=\psi (\xi), \psi :[\alpha,\beta]\rightarrow [a,b]\), hence there holds \(\int \!\!\!\!\!\!=^{\beta}_{\alpha} F(\xi)\,d\xi =\int \!\!\!\!\!\!=^b_a f(x)\,dx\), where \(F(\xi)=f(\psi (\xi))\,\psi^{\prime} (\xi)\). Based on this result, we next choose \(\psi (\xi)\) such that \(\mathcal{F}(\xi)\), the \(\mathcal{T}\)-periodic extension of \(F(\xi),\, \mathcal{T}=\beta -\alpha\), is sufficiently smooth, and prove, with the help of some recent extension/generalization of the Euler-Maclaurin expansion, that we can apply to \(\int \!\!\!\!\!\!=^{\beta}_{\alpha} F(\xi)\,d\xi\) the quadrature formulas derived for periodic singular integrals developed in an earlier work of the author: [ibid. 58, No. 2, Paper No. 22, 24 p. (2021; Zbl 1472.65031)]. We give a whole family of numerical quadrature formulas for \(\int \!\!\!\!\!\!=^{\beta}_{\alpha} F(\xi)\,d\xi\) for each \(m\), which we denote \(\widehat{T}^{(s)}_{m,n} [\mathcal{F}]\). Letting \(G(\xi)=(\xi -\tau)^m F(\xi)\), with \(\tau \in (\alpha,\beta)\) determined from \(t=\psi (\tau)\), and letting \(h=\mathcal{T}/n\), for \(m=3\), for example, we have the three formulas \[ \begin{gathered} \widehat{T}^{(0)}_{3,n}[\mathcal{F}]=h\sum^{n-1}_{j=1}\mathcal{F}(\tau +jh)-\frac{\pi^2}{3}\,G^{\prime}(\tau)\,h^{-1} +\frac{1}{6}\,G^{\prime\prime\prime}(\tau)\,h, \\ \widehat{T}^{(1)}_{3,n}[\mathcal{F}] =h\sum^n_{j=1}\mathcal{F}(\tau +jh-h/2)-\pi^2\,G^{\prime} (\tau)\,h^{-1}, \\ \widehat{T}^{(2)}_{3,n}[\mathcal{F}]=2h\sum^n_{j=1}\mathcal{F}(\tau +jh-h/2)- \frac{h}{2}\sum^{2n}_{j=1}\mathcal{F}(\tau +jh/2-h/4). \end{gathered} \] We show that all of the formulas \(\widehat{T}^{(s)}_{m,n} [\mathcal{F}]\) converge to \(I[f]\) as \(n\rightarrow \infty\); indeed, if \(\psi (\xi)\) is chosen such that \(\mathcal{F}^{(i)}(\alpha)=\mathcal{F}^{(i)}(\beta)=0, i=0,1,\ldots ,q-1\), and \(\mathcal{F}^{(q)}(\xi)\) is absolutely integrable in every closed interval not containing \(\xi =\tau\), then \[ \widehat{T}^{(s)}_{m,n}[\mathcal{F}]-I[f]=O(n^{-q})\quad \text{as }n\rightarrow \infty, \] where \(q\) is a positive integer determined by the behavior of \(g(x)\) at \(x=a\) and \(x=b\) and also by \(\psi (\xi)\). As such, \(q\) can be increased arbitrarily (even to \(q=\infty\), thus inducing spectral convergence) by choosing \(\psi (\xi)\) suitably. We provide several numerical examples involving nonperiodic integrands and confirm our theoretical results.


41A55 Approximate quadratures
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
45B05 Fredholm integral equations
45E05 Integral equations with kernels of Cauchy type
65B05 Extrapolation to the limit, deferred corrections
65B15 Euler-Maclaurin formula in numerical analysis
65D30 Numerical integration
65D32 Numerical quadrature and cubature formulas


Zbl 1472.65031
Full Text: DOI


[1] Beckers, M., Haegemans, A.: Transformations of integrands for lattice rules. In: Espelid, T.O., Genz, A. (eds.) Numerical Integration: Recent Developments. Software and Applications, NATO ASI, pp. 329-340. Kluwer Academic Publishers, Boston (1992) · Zbl 0747.65014
[2] Choi, UJ; Kim, SW; Yun, BI, Improvement of the asymptotic behavior of the Euler-Maclaurin formula for Cauchy principal value and Hadamard finite-part integrals, Int. J. Numer. Methods Eng., 61, 496-513 (2004) · Zbl 1078.65002
[3] Davis, PJ; Rabinowitz, P., Methods of Numerical Integration (1984), New York: Academic Press, New York · Zbl 0537.65020
[4] Elliott, D., Sigmoidal transformations and the trapezoidal rule, J. Austral. Math. Soc. B(E), 40, E, E77-E137 (1998) · Zbl 0928.65033
[5] Elliott, D., Sigmoidal-trapezoidal quadrature for ordinary and Cauchy principal value integrals, ANZIAM J., 46, E, E1-E69 (2004) · Zbl 1063.65531
[6] Elliott, D.; Venturino, E., Sigmoidal transformations and the Euler-Maclaurin expansion for evaluating certain Hadamard finite-part integrals, Numer. Math., 77, 453-465 (1997) · Zbl 0886.65021
[7] Evans, G., Practical Numerical Integration (1993), New York: Wiley, New York · Zbl 0811.65015
[8] Gakhov, FD, Boundary Value Problems (1966), Oxford: Pergamon Press, Oxford
[9] Kaya, AC; Erdogan, F., On the solution of integral equations with strongly singular kernels, Quart. Appl. Math., 45, 105-122 (1987) · Zbl 0631.65139
[10] Korobov, NM, Number-Theoretic Methods of Approximate Analysis (1963), Moscow: GIFL, Moscow · Zbl 0115.11703
[11] Krommer, AR; Ueberhuber, CW, Computational Integration (1998), Philadelphia: SIAM, Philadelphia
[12] Kythe, PK; Schäferkotter, MR, Handbook of Computational Methods for Integration (2005), New York: Chapman & Hall/CRC Press, New York · Zbl 1083.65027
[13] Lyness, JN, The Euler-Maclaurin expansion for the Cauchy principal value integral, Numer. Math., 46, 611-622 (1985) · Zbl 0551.65010
[14] Lyness, JN; Ninham, BW, Numerical quadrature and asymptotic expansions, Math. Comp., 21, 162-178 (1967) · Zbl 0178.18402
[15] Monegato, G., Definitions, properties and applications of finite-part integrals, J. Comp. Appl. Math., 229, 425-439 (2009) · Zbl 1166.65061
[16] Monegato, G.; Scuderi, L., Numerical integration of functions with boundary singularities, J. Comp. Appl. Math., 112, 201-214 (1999) · Zbl 0940.65027
[17] Navot, I., An extension of the Euler-Maclaurin summation formula to functions with a branch singularity, J. Math. Phys., 40, 271-276 (1961) · Zbl 0103.28804
[18] Prössdorf, S.; Rathsfeld, A.; Dym, H., Quadrature methods for strongly elliptic Cauchy singular integral equations on an interval, Topics in Analysis and Operator Theory, 435-471 (1991), Basel: Birkhäuser, Basel
[19] Ralston, A.; Rabinowitz, P., A First Course in Numerical Analysis (1978), New York: McGraw-Hill, New York · Zbl 0408.65001
[20] Sag, TW; Szekeres, G., Numerical evaluation of high-dimensional integrals, Math. Comp., 18, 245-253 (1964) · Zbl 0141.13903
[21] Sidi, A.: A new variable transformation for numerical integration. In: Brass, H., Hämmerlin, G. (eds.) Numerical Integration IV. ISNM, vol. number 112, pp. 359-373. Birkhäuser, Basel (1993) · Zbl 0791.41027
[22] Sidi, A., Practical Extrapolation Methods: Theory and Applications. Number 10 in Cambridge Monographs on Applied and Computational Mathematics (2003), Cambridge: Cambridge University Press, Cambridge · Zbl 1041.65001
[23] Sidi, A., Extension of a class of periodizing variable transformations for numerical integration, Math. Comp., 75, 327-343 (2006) · Zbl 1103.65029
[24] Sidi, A., A novel class of symmetric and nonsymmetric periodizing variable transformations for numerical integration, J. Sci. Comput., 31, 391-417 (2007) · Zbl 1133.65013
[25] Sidi, A., Further extension of a class of periodizing variable transformations for numerical integration, J. Comp. Appl. Math., 221, 132-149 (2008) · Zbl 1161.65019
[26] Sidi, A., Euler-Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities, Math. Comp., 81, 2159-2173 (2012) · Zbl 1271.30011
[27] Sidi, A., Euler-Maclaurin expansions for integrals with arbitrary algebraic-logarithmic endpoint singularities, Constr. Approx., 36, 331-352 (2012) · Zbl 1329.41041
[28] Sidi, A., Compact numerical quadrature formulas for hypersingular integrals and integral equations, J. Sci. Comput., 54, 145-176 (2013) · Zbl 1264.65033
[29] Sidi, A., Analysis of errors in some recent numerical quadrature formulas for periodic singular and hypersingular integrals via regularization, Appl. Numer. Math., 81, 30-39 (2014) · Zbl 1291.65078
[30] Sidi, A., Richardson extrapolation on some recent numerical quadrature formulas for singular and hypersingular integrals and its study of stability, J. Sci. Comput., 60, 141-159 (2014) · Zbl 1300.41018
[31] Sidi, A.: Unified compact numerical quadrature formulas for Hadamard finite parts of singular integrals of periodic functions. Calcolo, 58, (2021). Article number 22 · Zbl 1472.65031
[32] Sidi, A.: Exactness and convergence properties of some recent numerical quadrature formulas for supersingular integrals of periodic functions. Calcolo, 58, 2021. Article number 36 · Zbl 07380431
[33] Sidi, A.: Exponential convergence of some recent numerical quadrature methods for Hadamard finite parts of singular integrals of periodic analytic functions. Computer Science Department, Technion-Israel Institute of Technology, Technical report (2021) · Zbl 1472.65031
[34] Sidi, A., Israeli, M.: Quadrature methods for periodic singular and weakly singular Fredholm integral equations. J. Sci. Comput. 3, 201-231 (1988). (Originally appeared as Technical Report No. 384, Computer Science Dept., Technion-Israel Institute of Technology, (1985), and also as ICASE Report No. 86-50 (1986)) · Zbl 0662.65122
[35] Steffensen, JF, Interpolation (1950), New York: Chelsea, New York · Zbl 0041.02603
[36] Stoer, J.; Bulirsch, R., Introduction to Numerical Analysis (2002), New York: Springer, New York · Zbl 1004.65001
[37] Yun, BI, An efficient transformation with Gauss quadrature rule for weakly singular integrals, Comm. Numer. Methods Eng., 17, 881-891 (2001) · Zbl 0994.65024
[38] Yun, BI; Kim, P., A new sigmoidal transformation for weakly singular integrals in the boundary integral method, SIAM J. Sci. Comput., 24, 1203-1217 (2003) · Zbl 1036.65031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.