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A Divergent Sequence of Romberg Integrals

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Abstract

Romberg integrals are built in order to accelerate the convergence of sequences of trapezoidal rules for approximating the definite integral of a continuous function f. While every sequence of trapezoidal rules with decreasing step length converges whenever f is continuous, this does not always hold for Romberg integrals. In this note we present a concrete example for which the sequence formed by the diagonal elements of the Romberg table diverges when the number of points used to compute the trapezoidal rules grows too slowly.

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References

  1. Bauer, F.L., Rutishauser, H., H., Stiefel, E.: New aspects in numerical quadrature, In: Proceedings of Symposia in Applied Mathematics, vol. 15, American mathematical Society, Providence, pp. 199–218 (1963)

  2. Brass, H., Fischer, J.W.: Error bounds in Romberg quadrature. Numer. Math. 82, 389–408 (1999)

    Article  MathSciNet  Google Scholar 

  3. Brezinski, C.: Convergence acceleration during the 20th century. J. Comp. Appl. Math. 122, 1–21 (2000)

    Article  MathSciNet  Google Scholar 

  4. Bulirsch, R.: Bemerkungen zur Romberg-integration. Numer. Math. 6, 6–16 (1964)

    Article  MathSciNet  Google Scholar 

  5. Fischer, J.W.: Romberg quadrature using the Bulirsch sequence. Numer. Math. 90, 509–519 (2002)

    Article  MathSciNet  Google Scholar 

  6. Joyce, D.C.: Survey of extrapolation processes in numerical analysis. SIAM Rev. 13(4), 435–490 (1971)

    Article  MathSciNet  Google Scholar 

  7. Laurie, D.P.: Propagation of initial rounding error in Romberg-like quadrature. BIT 15, 277–282 (1975)

    Article  MathSciNet  Google Scholar 

  8. Shanks, J.A.: On forming the Romberg table. J. Comput. Appl. Math. 11, 343–351 (1984)

    Article  MathSciNet  Google Scholar 

  9. Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis, 2nd edn. Springer, New York (1993)

    Book  Google Scholar 

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  1. Centro de Matemática, Computação e Cognição, Universidade Federal do ABC - UFABC, Rua Santa Adélia, 166, bairro Bangu, Santo André, SP, CEP 09210-170, Brazil

    André Pierro de Camargo

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  1. André Pierro de Camargo
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Correspondence to André Pierro de Camargo.

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de Camargo, A.P. A Divergent Sequence of Romberg Integrals. Results Math 75, 11 (2020). https://doi.org/10.1007/s00025-019-1140-6

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  • DOI: https://doi.org/10.1007/s00025-019-1140-6

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