Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Movable singularities and quadrature


Authors: R. F. Goodrich and F. Stenger
Journal: Math. Comp. 24 (1970), 283-300
MSC: Primary 65.55
DOI: https://doi.org/10.1090/S0025-5718-1970-0275669-4
MathSciNet review: 0275669
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A general procedure is described for treating a movable singularity in an integral. This enabĺes us to change the original integral $ {I_0}$ into $ G{I_1}$, where $ G$ depends only on the parameters of the singularity and $ {I_1}$, is a new integral which exists for all values of the parameters. The results are then applied to the particular problem of evaluating

$\displaystyle \int_{ - 1}^1 {\frac{{f(x)dx}} {{{{\{ (1 - {x^2})(1 - {k^2}{x^2})\} }^{1/2}}}}} ,$

where $ f$ is entire and $ k$ varies between 0 and $ 1$. Some new quadrature schemes and new effective methods of evaluating incomplete elliptic integrals are derived.

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

  • [1] P. J. Davis & P. Rabinowitz, Numerical Integration, Blaisdell, Waltham, Mass., 1967. MR 35 #2482. MR 0211604 (35:2482)
  • [2] H. Hönl, A. W. Maue & K. Westpfahl, Theorie der Beugung, Handbuch der Physik, Band 25/1, Springer, Berlin, 1961, pp. 218-544. MR 31 #1877. MR 0177615 (31:1877)
  • [3] F. Stenger, "Bounds on the error of Gauss-type quadratures," Numer. Math., v. 8, 1966, pp. 150- 160. MR 33 #5120. MR 0196936 (33:5120)
  • [4] C. Caratheodory, Conformal Representation, 2nd ed., Cambridge Tracts in Math. and Math. Physics, no. 28, Cambridge Univ. Press, New York, 1958. MR 13 #734.
  • [5] W. Gautschi, Algorithm $ \ldots $ Gaussian Quadrature Formulas (Submitted for publication.)
  • [6] J. McNamee, "Error-bounds for the evaluation of integrals by the Euler-Maclaurin formula and by Gauss-type formulae," Math. Comp., v. 18, 1964, pp. 368-381. MR 32 #3264. MR 0185804 (32:3264)
  • [7] P. F. Byrd & M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, Die Grundlehren der math. Wissenschaften, Band LXVII, Springer-Verlag, Berlin and New York, 1954. MR 15, 702. MR 0060642 (15:702a)
  • [8] H. Kober, Dictionary of Conformal Representations, Dover, New York, 1960. MR 0049326 (14:156d)
  • [9] E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, Clarendon Press, Oxford, 1950.
  • [10] W. G. Fair & Y. L. Luke, "Rational approximations to the incomplete elliptic integrals of the first and second kinds," Math. Comp., v. 21, 1967, pp. 418-422. MR 36 #5400. MR 0222348 (36:5400)
  • [11] M. Abramowitz & I. A. Stegun (Editors), Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Nat. Bur. Standards Appl. Math. Series, 55, Superintendent of Documents, U.S. Government Printing Office, Washington, D.C, 1964; 3rd printing, with corrections, 1965. MR 29 #4914; MR 31 #1400. MR 0167642 (29:4914)
  • [12] L. Fox, "Romberg integration for a class of singular integrands," Comput. J., v. 10, 1967, pp. 87- 93. MR 35 #3878. MR 0213013 (35:3878)
  • [13] E. Isaacson & H. B. Keller, Analysis of Numerical Methods, Wiley, New York, 1966. MR 34 #924. MR 0201039 (34:924)
  • [14] Y. L. Luke, "Approximations for elliptic integrals," Math. Comp., v. 22, 1968, pp. 627-634. MR 37 #2412. MR 0226825 (37:2412)
  • [15] P. Barrucand, "Quadratures numériques, fonctions elliptiques, et facteur de convergence," C. R. Acad. Sci. Paris, t. 258, 1964, pp. 2742-2744. MR 29 #5381. MR 0168117 (29:5381)
  • [16] F. W. J. Olver, "Error bounds for the Liouville-Green (or WKB) approximation," Proc. Cambridge Philos. Soc., v. 57, 1961, pp. 790-810. MR 24 #A313. MR 0130452 (24:A313)
  • [17] G. Meinardus, Approximation von Funktionen und ihre numerische Behandlung, Springer Tracts in Natural Philosophy, vol. 4, Springer-Verlag, Berlin and New York, 1964. MR 31 #547. MR 0176272 (31:547)
  • [18] A. F. Timan, Theory of Approximation of Functions of a Real Variable, Fizmatgiz, Moscow, 1960; English transl., Internat. Series of Monographs in Pure and Appl. Math., vol. 34, Macmillan, New York, 1963. MR 22 #8257; MR 33 #465. MR 0117478 (22:8257)
  • [19] R. E. Barnhill & T. A. Wixom, "Quadratures with remainders of minimum norm. I," Math. Comp., v. 21, 1967, pp. 66-75. MR 36 #6138. MR 0223089 (36:6138)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65.55

Retrieve articles in all journals with MSC: 65.55


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1970-0275669-4
Keywords: Quadrature, singularities, error, bound
Article copyright: © Copyright 1970 American Mathematical Society

American Mathematical Society