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ReLaTIve. An Ansi C90 software package for the Real Laplace Transform Inversion

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An Erratum to this article was published on 10 June 2013

Abstract

A software package for numerical inversion of Laplace transforms computable everywhere on the real axis is described. Besides the function to invert the user has only to provide the numerical value (even if it is an approximate value) of the abscissa of convergence and the accuracy required for the inverse function. The software provides a controlled accuracy, i.e. it dynamically computes the so-called maximum attainable accuracy such that numerical results are provided within the greatest value between the user’s required accuracy and the maximum attainable accuracy. This is done because the intrinsic ill posedness of the real inversion problem sometime may prevent to reach the desired accuracy. The method implemented is based on a Laguerre polynomial series expansion of the inverse function and belongs to the class of polynomial-type methods of inversion of the Laplace transform, formally characterized as Collocation methods (C-methods).

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References

  1. Abate J., Choudhury G., Whitt W.: On the Laguerre method for numerically inversting Laplace transforms. INFORMS J. Comput. 8(4), 413–427 (1996)

    Article  MATH  Google Scholar 

  2. Abate, J., Valko, P.: Uniform Resource Locators (URL). Wolfram Information Center. http://library.wolfram.com/infocenter/MathSource/4738/ (2002)

  3. Abate, J., Valko, P.: Uniform Resource Locators (URL). Wolfram Information Center. http://library.wolfram.com/infocenter/MathSource/5026/ (2003)

  4. Abate, J., Valko, P.: Numerical Laplace inversion in rheological characterization. J. Non-Newton. Fluid Mech. 116, 395–406 (2004)

    Article  MATH  Google Scholar 

  5. Bjorck, A., Pereira, V.: Solution of Vandermonde systems of equations. Math. Comput. 112(24), 893—903 (1970)

    Google Scholar 

  6. Binous, H.: Uniform Resource Locators (URL). Wolfram Information Center. http://library.wolfram.com/infocenter/MathSource/6557/ (2010)

  7. Borgia, G.C., Brown, R.J.S., Fantazzini, P.: Uniform penalty inversion of multiexponential decay data II. J. Magn. Reson. 147, 273–285 (2000)

    Article  Google Scholar 

  8. Cohen A.M.: Numerical Methods for Laplace Transform Inversion. Springer (2007)

  9. Cuomo S., D’Amore L., Murli A., Rizzardi, M.: Computation of the inverse Laplace transform based on a collocation method which uses only real values. J. Comput. Appl. Math. 198, 98–115 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. D’Amore, L., Laccetti, G., Murli, A.: An implementation of a Fourier series method for the numerical inversion of the Laplace transform. ACM Trans. Math. Softw. 25(30), 279–305 (1999)

    Article  MATH  Google Scholar 

  11. D’Amore L., Laccetti G., Murli, A.: Algorithm 796: a Fortran software package for the numerical inversion of the Laplace transform based on a Fourier series method. ACM Trans. Math. Softw. 25, 306–315 (1999)

    Article  MATH  Google Scholar 

  12. Davies, B., Martin, B.: Numerical inversion of the Laplace transform: a survey and comparison of methods. J. Comput. Phys. 33, 1–32 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  13. Duffy, D.G.: On the numerical inversion of Laplace transforms: comparison of three new methods on characteristic problems from applications. ACM Trans. Math. Softw. 19(3), 333–359 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fair, W. Jr.: Numerical Laplace transforms and inverse transforms in C#. http://www.codeproject.com/KB/recipes/LaplaceTransforms.aspx?msg=3150794 (2008)

  15. Garbow, S., Giunta, G., Lyness, N.J., Murli, A.: Algorithm 662: a Fortran software package for the numerical inversion of a Laplace transform based on Week’s method. ACM Trans. Math. Softw 54, 163–170 (1988)

    Article  MathSciNet  Google Scholar 

  16. Giunta, G., Laccetti, G., Rizzardi, M.: More on weeks method for the numerical inversion of the Laplace trasform. Numer. Math. 193, 193–200 (1988)

    MathSciNet  Google Scholar 

  17. Giunta, G., Murli, A., Schmid, G.: An analysis of bilinear transform-polynomial methods of inversion of Laplace transforms. Numer. Math. 69(1), 269–282 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  18. Giunta, G., Murli, A., Schmid, G.: Error analysis of Rjabov algorithm for inverting Laplace transforms. Ric. Mat. XLIV(1), 207–219 (1995)

    MathSciNet  Google Scholar 

  19. Halsted, D.J., Brown, D.E.: Zakian’s technique for inverting Laplace transforms. Chem. Eng. J. 3, 312–313 (1972)

    Article  Google Scholar 

  20. Henrici, P.: Applied and Computational Complex Analysis, vol. 2. John Wiley & Sons (1977)

  21. Higham, N.: Error analysis of the Björk–Pereira algorthm for solving Vandermonde systems. Numer. Math. 50, 613–632 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kryzhniy, V.V.: On regularization of numerical inversion of Laplace transforms. J. Inverse Ill-Posed Probl. 12(3), 279–296 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kryzhniy, V.V.: InvertLT. http://www-users.cs.umn.edu/~yelena/web/software.php. Accessed 2006

  24. Lyness, N.J., Giunta, G.: A modification of the weeks method for the inversion of the Laplace tranform. Math. Comput. 47, 313–322 (1987)

    MathSciNet  Google Scholar 

  25. Mallet, A.: Uniform Resource Locators (URL). Wolfram Information Center. http://library.wolfram.com/infocenter/MathSource/2691/ (2000)

  26. Membrez, J., Infelta, P.P., Renken, A.: Use of the Laplace transform technique for simple kinetic parameters evaluation. Application to the adsorption of a protein on porous beads. Chem. Eng. Sci. 51(19), 4489–4498 (1996)

    Article  Google Scholar 

  27. Murli, A., Rizzardi, M.: Algorithm 682: Talbot’s method for the Laplace inversion problem. ACM Trans. Math. Softw. 16, 347–371 (1990)

    Article  Google Scholar 

  28. NAg Documentation: Uniform Resource Locators (URL). Numerical Algorithms Group Ltd. http://www.num-alg-grp.co.uk/numeric/Fl/manual/html/examples/source/c06lafe.f (1989)

  29. Piessens, R., Branders, P.: Numerical inversion of the Laplace tranform usinge generalized Laguerre polynomials. Proc. IEEE 118, 1517–1522 (1971)

    MathSciNet  Google Scholar 

  30. Piessens, R.: A new numerical method for the inversion of the Laplace tranform. J. Inst. Math. Appl. 10, 185–192 (1972)

    Article  MATH  Google Scholar 

  31. Piessens, R., Huysmans, R.: Algorithm 619: automatic numerical inversion of the Laplace transform. ACM Trans. Math. Softw. 10(3), 348–353 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  32. Provencher, S.: CONTIN. http://s-provencher.com/pages/contin.shtml. Accessed 1982

  33. Rjabov, V.M.: On the numerical inversion of the Laplace tranform. Vestn. Leningr. Univ. 7, 177–185 (1974)

    MathSciNet  Google Scholar 

  34. Sykora, S.: Bortolotti, V., Fantazzini, P.: PERFIDI: parametrically enabled relaxation filters with double and multiple inversion. Magn. Reson. Imaging 25, 529–532 (2007)

    Article  Google Scholar 

  35. Spinelli, R.A.: Numerical inversion of a Laplace transform. SIAM J. Numer. Anal. 3(4), 636–649 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  36. Stehfest, H.: Algorithm 368: numerical inversion of Laplace transform. Commun. ACM 13, 47–49 (1970)

    Article  Google Scholar 

  37. Saravanathamizhana, R., Paranthamana, R., Balasubramaniana, N., Ahmed Bashab, C.: Residence time distribution in continuous stirred tank electrochemical reactor. Chem. Eng. J. 142, 209–216 (2008)

    Article  Google Scholar 

  38. Thornhill, N.F., Patwardhan, S.C., Shah, S.L.: A continuous stirred tank heater simulation model with applications. J. Process Control 18, 347–360 (2008)

    Article  Google Scholar 

  39. Valko, P.: Numerical Inversion of Laplace Transform. http://www.pe.tamu.edu/valko/Nil/ (2002)

  40. Valko, P., Vaida, S.: Inversion of noise-free Laplace transforms: towards a standardized set of test problems. Inverse Probl. Eng. 10, 467–483 (2002)

    Article  Google Scholar 

  41. Weeks, W.: Numerical inversion of the Laplace tranform using Laguerre functions. J. ACM 13, 419–429 (1966)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Applications, University of Naples Federico II, Naples, Italy

    Luisa D’Amore, Rosanna Campagna & Valeria Mele

  2. CMCC (Centro Euro-Mediterraneo sui Cambiamenti Climatici), Lecce, Italy

    Almerico Murli

Authors
  1. Luisa D’Amore
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  2. Rosanna Campagna
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  3. Valeria Mele
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  4. Almerico Murli
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Correspondence to Luisa D’Amore.

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D’Amore, L., Campagna, R., Mele, V. et al. ReLaTIve. An Ansi C90 software package for the Real Laplace Transform Inversion. Numer Algor 63, 187–211 (2013). https://doi.org/10.1007/s11075-012-9636-0

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  • DOI: https://doi.org/10.1007/s11075-012-9636-0

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