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Quadrature of integrands with a logarithmic singularity

Author: John A. Crow
Journal: Math. Comp. 60 (1993), 297-301
MSC: Primary 65D32; Secondary 65D30
MathSciNet review: 1155572
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Abstract: A quadrature rule is presented that is exact for integrands of the form $ \pi (\xi ) + \varpi (\xi )\log \xi $ on the interval (0, 1), where $ \pi $ and $ \varpi $ are polynomials. The computed weights and abscissae are given for one- through seven-point rules. In particular, the four-point rule is exact for integral operators with log-arithmically singular kernel on a cubic B-spline basis, and it is expected these results shall prove useful for numerical applications of weighted-residual finite element methods.

References [Enhancements On Off] (What's this?)

  • [1] M. Abramowitz and I. A. Stegun (eds.), Handbook of mathematical functions, ninth printing, Dover, New York, 1965.
  • [2] V. U. Aihie and G. A. Evans, Variable transformation methods and their use in general and singular quadratures, Internat. J. Comput. Math. 27 (1989), 91-101.
  • [3] H. Engels and U. Eckhardt, Algorithm 33: Wilf-quadrature, Computing 18 (1977), 271-279.
  • [4] G. A. Evans, R. C. Forbes, and J. Hyslop, The tanh transformation for singular integrals, Internat. J. Comput. Math. 5 (1984), 339-358. MR 754275 (86d:65034)
  • [5] C. G. Harris and W. A. B. Evans, Extension of numerical quadrature formulae to cater for end point singular behaviours over finite intervals, Internat. J. Comput. Math. (B) 6 (1977), 219-227. MR 0474708 (57:14342)
  • [6] M. Iri, S. Moriguti, and Y. Takasawa, On a certain quadrature formula, J. Comput. Math. 17 (1987), 3-20. MR 884257 (88j:65057)
  • [7] K. Murota and M. Iri, Parameter tuning and repeated application of the IMT-type transformation in numerical quadrature, Numer. Math. 38 (1982), 347-363. MR 654102 (83d:65061)
  • [8] A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and series, vol. 2, Special functions, Gordon and Breach, New York, 1986. MR 874987 (88f:00014)
  • [9] D. M. Smith, Algorithm 693: A FORTRAN package for floating-point multiple-precision arithmetic, ACM Trans. Math. Software 17 (1991), 273-283.
  • [10] A. H. Stroud and D. Secrest, Gaussian quadrature formulas, Prentice-Hall, Englewood Cliffs, NJ, 1966. MR 0202312 (34:2185)
  • [11] H. Takahasi and M. Mori, Quadrature formulas obtained by variable transformations, Numer. Math. 21 (1973), 206-219. MR 0331738 (48:10070)
  • [12] H. S. Wilf, Exactness conditions in numerical quadrature, Numer. Math. 6 (1964), 315-319. MR 0179941 (31:4178)

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Keywords: Numerical integration, quadrature formula, Galerkin method
Article copyright: © Copyright 1993 American Mathematical Society

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