Numerical Algorithms

, Volume 68, Issue 1, pp 167–183 | Cite as

An improved Talbot method for numerical Laplace transform inversion

  • Benedict Dingfelder
  • J. A. C. Weideman
Original Paper


The classical Talbot method for the computation of the inverse Laplace transform is improved for the case where the transform is analytic in the complex plane except for the negative real axis. First, by using a truncated Talbot contour rather than the classical contour that goes to infinity in the left half-plane, faster convergence is achieved. Second, a control mechanism for improving numerical stability is introduced.


Inverse Laplace transform Talbot’s method Trapezoidal rule Midpoint rule 

Mathematics Subject Classifications (2010)

65R10 65Z05 


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  1. 1.
    Abate, J., Valkó, P.P.: NumericalLaplaceInversion (2002).
  2. 2.
    Abate, J., Valkó, P.P.: Multi-precision Laplace transform inversion. Int. J. Numer. Methods Fluids 60 (5), 979–993 (2004)CrossRefzbMATHGoogle Scholar
  3. 3.
    Butcher, J.C.: On the numerical inversion of Laplace and Mellin transforms. Conference on Data Processing and Automatic Computing Machines,Salisbury,Australia (1957)Google Scholar
  4. 4.
    Dingfelder, B.: Raten- und stabilitätsoptimierte Algorithmen zur Berechnung eines Integrals mit Hankelintegrationsweg. Bachelor’s thesis, Technische Universität München,Germany (2012)Google Scholar
  5. 5.
    Dingfelder, B., Weideman, J.A.C.: ModifiedTalbot (2014).
  6. 6.
    Duffy, D.G.: On the numerical inversion of Laplace transforms: comparison of three new methods on characteristic problems from applications. ACM Trans. Math. Software 19 (3), 333–359 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Gavrilyuk, I.P., Makarov, V.L.: Exponentially convergent algorithms for the operator exponential with applications to inhomogeneous problems in Banach spaces. SIAM J. Numer. Ana.l 43 (5), 2144–2171 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Kress, R.: Numerical analysis. Springer-Verlag, New York (1998)CrossRefzbMATHGoogle Scholar
  9. 9.
    López-Fernández, M., Palencia, C.: On the numerical inversion of the Laplace transform of certain holomorphic mappings. Appl. Numer. Math. 51 (2-3), 289–303 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    López-Fernández, M., Palencia, C., Schädle, A.: A spectral order method for inverting sectorial Laplace transforms. SIAM J. Numer. Anal 44 (3), 1332–1350 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
  12. 12.
    Murli, A., Rizzardi, M.: Algorithm 682: Talbot’s method for the Laplace inversion problem. ACM Trans. Math. Software 16 (2), 158–168 (1990)CrossRefzbMATHGoogle Scholar
  13. 13.
    Rizzardi, M.: A modification of Talbot’s method for the simultaneous approximation of several values of the inverse Laplace transform. ACM Trans. Math. Software 21 (4), 347–371 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Sheen, D., Sloan, I.H., Thomée, V.: A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature. IMA J. Numer. Anal. 23 (2), 269–299 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Strang, G.: Computational science and engineering. Wellesley-Cambridge Press, Wellesley (2007)zbMATHGoogle Scholar
  16. 16.
    Talbot, A.: The accurate numerical inversion of Laplace transforms. J. Inst. Math. Appl 23 (1), 97–120 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Trefethen, L.N.: Private communication (2006)Google Scholar
  18. 18.
    Trefethen, L.N., Weideman, J.A.C., Schmelzer, T.: Talbot quadratures and rational approximations. BIT 46 (3), 653–670 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Weideman, J.A.C.: Optimizing Talbot’s contours for the inversion of the Laplace transform. SIAM J. Numer. Anal 44 (6), 2342–2362 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Weideman, J.A.C.: Improved contour integral methods for parabolic PDEs. IMA J. Numer. Anal. 30 (1), 334–350 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Weideman, J.A.C., Trefethen, L.N.: Parabolic and hyperbolic contours for computing the Bromwich integral. Math. Comp. 76 (259), 1341–1356 (2007)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Zentrum Mathematik, M3Technische Universität MünchenGarching bei MünchenGermany
  2. 2.Department of Mathematical SciencesStellenbosch UniversityStellenboschSouth Africa

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