An algorithm for the numerical evaluation of certain Cauchy principal value integrals

Paget, D. F.; Elliott, David

doi:10.1007/bf01404920

Cited by 89 publications

(42 citation statements)

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“…The integral (1.2) has two practical difficulties-it is oscillatory and has a singularity of Cauchy type; for the latter see [3], [9], [11], [13], [16]. To deal with these pertinent problems, we present a method based on a modified Lagrangian interpolation formula and on properties of orthogonal polynomials.…”

Section: [-11]-mentioning

confidence: 99%

“…A Stable Algorithm. The quadrature rules given by (2.6), (4.3), (5.3), (5.15), and (5.17) are numerically unstable when t is close to one of the points x,; to avoid this, the following algorithm has been proposed [7], [16]. Let P"_x(g; •) be the polynomial of degree < n -I interpolating g at the zeros x, of Mn, written in the form (6.1) U«;^)=EaA(4…”

Section: [-11]-mentioning

confidence: 99%

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Quadrature Formulae for Cauchy Principal Value Integrals of Oscillatory Kind

Okecha¹

1987

Mathematics of Computation

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Abstract. The problem considered is that of evaluating numerically an integral of the form /-1 e"*xf(x) dx, where / has one simple pole in the interval [ -1,1). Modified forms of the Lagrangian interpolation formula, taking account of the simple pole are obtained, and form the bases for the numerical quadrature rules obtained. Further modification to deal with the case when an abscissa in the interpolation formula is coincident with the pole is also considered. An error bound is provided and some numerical examples are given to illustrate the formulae developed.

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Section: [-11]-mentioning

confidence: 99%

Section: [-11]-mentioning

confidence: 99%

Quadrature Formulae for Cauchy Principal Value Integrals of Oscillatory Kind

Okecha¹

1987

Mathematics of Computation

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“…A great many papers devoted to the problem of numerical evaluation of the integrals of the form (1.1) have been published so far. Some of them are [4,8,13,16,17,21]. A nice survey on the subject, along with a large number of references, is presented in [3, § 2.12.8].…”

Section: Introductionmentioning

confidence: 99%

Computing Cauchy principal value integrals using a standard adaptive quadrature

Keller

Wróbel

2016

Journal of Computational and Applied Mathematics

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We investigate the possibility of fast, accurate and reliable computation of the Cauchy principal value integrals − b a f (x)(x − τ ) −1 dx (a < τ < b) using standard adaptive quadratures. In order to properly control the error tolerance for the adaptive quadrature and to obtain a reliable estimation of the approximation error, we research the possible influence of roundoff errors on the computed result. As the numerical experiments confirm, the proposed method can successfully compete with other algorithms for computing such type integrals. Moreover, the presented method is very easy to implement on any system equipped with a reliable adaptive integration subroutine.

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“…It should be mentioned that the above Gaussian quadrature rules ( (7) to (10)) should not be confused with the interpolatory quadrature rules based on Gaussian nodes (and sometimes called Gaussian rules too) suggested in [14], [19] for p = 0. The convergence of the latter class of quadrature rules for p = 0 was studied in detail both for pointwise [3]- [5], [9], [16], [21] and for uniform [14], [20], [21] convergence.…”

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confidence: 99%

“…This is so because, obviously, h(np)x(t, x) is a polynomial of degree n -p -1 (if « > p + 1) or 0 (if n < p + 1) and for such a polynomial the Gaussian quadrature rule with n nodes is exact. Now we take into account that Finally, by applying the Gaussian quadrature rule to K(p)(x), (15), and taking into account the previous developments, as well as the fact that (19) f w(t) dt=Z /*,,"…”

mentioning

confidence: 99%