Approximate integration using iterated Levin transformations

  • Ricolindo Cariño
  • Elise de Doncker
  • Ian Robinson
Track 8: Numerical Analysis
Part of the Lecture Notes in Computer Science book series (LNCS, volume 507)


The efficiency of a quadrature scheme based on iterated Levin U transformations and composite rule approximations for a harmonic sequence of mesh ratios is demonstrated for typical problem classes. Numerical results indicate a favourable comparison with the well known nonlinear extrapolation procedures applied to a sequence of composite quadrature rule sums for a geometric progression of the mesh ratios.


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  1. [1]
    R.L. Cariño, I. Robinson, and E. de Doncker (1989), Approximate integration by the Levin transformation, In preparation.Google Scholar
  2. [2]
    J.S.R. Chisholm, A. Genz and G.E. Rowlands (1973), Accelerated convergence of quadrature approximations, J. Comp. Phys., v. 10, pp. 284–307.MathSciNetCrossRefGoogle Scholar
  3. [3]
    E. de Doncker (1978), An adaptive extrapolation algorithm for automatic integration, SIGNUM Newsl., v.13, pp. 12–18.Google Scholar
  4. [4]
    L. Fox (1967), Romberg integration for a class of singular integrands, Comput. J., v. 10, pp. 87–93.zbMATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    L. Fox and L. Hayes (1970), On the definite integration of singular integrals, SIAM Review, v. 12, pp. 449–457.zbMATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    D. Kahaner (1972), Numerical quadrature by the ε algorithm, Math. Comp., v. 26, pp. 689–693.zbMATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    D. Levin (1973), Development of non-linear transformations for improving convergence of sequences, Intern. J. Computer Math., v. B3, pp. 371–388.Google Scholar
  8. [8]
    A. Sidi (1979), Convergence properties of some nonlinear sequence transformations, Math. Comp., v. 33, pp. 315–326.zbMATHMathSciNetCrossRefGoogle Scholar
  9. [9]
    P. Wynn (1956), On a device for computing the e m(S n) transformation, Math. Comp., v. 10, pp. 91–96.zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Ricolindo Cariño
    • 1
  • Elise de Doncker
    • 2
  • Ian Robinson
    • 1
  1. 1.La Trobe UniversityBundooraAustralia
  2. 2.Dept. of Comp. ScienceWestern Mich. Univ.Kalamazoo

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