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Equivalence of Nyström's method and Fourier methods for the numerical solution of Fredholm integral equations


Authors: Jean-Paul Berrut and Manfred R. Trummer
Journal: Math. Comp. 48 (1987), 617-623
MSC: Primary 45L10; Secondary 42A10, 65R20
DOI: https://doi.org/10.1090/S0025-5718-1987-0878694-8
MathSciNet review: 878694
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Abstract: Nyström's method with the trapezoidal rule, and the Fourier method, produce the same approximation to the solution of an integral equation at the collocation points for Nyström's method. This equivalence allows the derivation of error estimates for Nyström's method, and gives an intuitive explanation for its good performance in the periodic case. The equivalence holds for Fourier methods with arbitrary orthogonal basis functions. The quadrature rule for numerical integration must have the collocation points as abscissae, and must yield the exact entries of the Gramian matrix of the orthogonal basis.


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

  • [1] M. Abramowitz & I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965.
  • [2] K. E. Atkinson, A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind, SIAM, Philadelphia, Pa., 1976. MR 0483585 (58:3577)
  • [3] J.-P. Berrut, Integralgleichungen und Fourier-Methoden zur numerischen konformen Abbildung, Ph. D. Thesis, ETH Zürich, 1985.
  • [4] J.-P. Berrut, "A Fredholm integral equation of the second kind for conformal mapping," J. Comput. Appl. Math., v. 14, 1986, pp. 99-110. MR 829032 (87d:30011)
  • [5] J.-P. Berrut, "Baryzentrische Formeln zur trigonometrischen Interpolation. I," J. Appl. Math. Phys. (ZAMP), v. 35, 1984, pp. 91-105. MR 753088 (85m:65007)
  • [6] L. M. Delves & J. L. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, Cambridge, 1985. MR 837187 (87j:65159)
  • [7] P. Henrici, "Fast Fourier methods in computational complex analysis," SIAM Rev., v. 21, 1979, pp. 481-527. MR 545882 (80i:65031)
  • [8] P. Henrici, Essentials of Numerical Analysis, Wiley, New York, 1982. MR 655251 (83h:65002)
  • [9] P. Henrici, "Barycentric formulas for interpolating trigonometric polynomials and their conjugates," Numer. Math., v. 33, 1979, pp. 225-234. MR 549451 (80m:65009)
  • [10] N. Kerzman & M. R. Trummer, "Numerical conformal mapping via the Szegö kernel," J. Comput. Appl. Math., v. 14, 1986, pp. 111-124. MR 829033 (87f:30017)
  • [11] P. M. Prenter, "A collocation method for the numerical solution of integral equations," SIAM J. Numer. Anal., v. 10, 1973, pp. 570-581. MR 0327064 (48:5406)
  • [12] L. Reichel, "A fast method for solving certain integral equations of the first kind with application to conformal mapping," J. Comput. Appl. Math., v. 14, 1986, pp. 125-142. MR 829034 (87h:65223)
  • [13] M. Schleiff, "Über Näherungsverfahren zur Lösung einer singulären linearen Integrodifferentialgleichung," Z. Angew. Math. Mech., v. 48, 1968, pp. 477-483. MR 0242397 (39:3728)
  • [14] I. Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Grundlehren der mathematischen Wissenschaften, vol. 171, Springer, Berlin and New York, 1970. MR 0270044 (42:4937)
  • [15] G. T. Symm, "An integral equation method in conformal mapping," Numer. Math., v. 9, 1966, pp. 250-258. MR 0207240 (34:7056)
  • [16] M. R. Trummer, "An efficient implementation of a conformal mapping method based on the Szegö kernel," SIAM J. Numer. Anal., v. 23, 1986, pp. 853-872. MR 849287 (87k:30013)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1987-0878694-8
Keywords: Integral equations, Nyström's method, Fourier method, collocation, trigonometric approximation
Article copyright: © Copyright 1987 American Mathematical Society

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