×

ReLaTIve. An Ansi C90 software package for the Real Laplace Transform Inversion. (English) Zbl 1267.65202

Summary: 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 sometimes 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).

MSC:

65R10 Numerical methods for integral transforms
44A10 Laplace transform
65R30 Numerical methods for ill-posed problems for integral equations
65Y15 Packaged methods for numerical algorithms
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Abate, J; Choudhury, G; Whitt, W, On the Laguerre method for numerically inversting Laplace transforms, INFORMS J. Comput., 8, 413-427, (1996) · Zbl 0887.60100
[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) · Zbl 1106.76429
[5] Bjorck, A; Pereira, V, Solution of Vandermonde systems of equations, Math. Comput., 112, 893-903, (1970) · Zbl 0221.65054
[6] Binous, H.: Uniform Resource Locators (URL). Wolfram Information Center. http://library.wolfram.com/infocenter/MathSource/6557/ (2010) · Zbl 0166.13302
[7] Borgia, GC; Brown, RJS; Fantazzini, P, Uniform penalty inversion of multiexponential decay data II, J. Magn. Reson., 147, 273-285, (2000)
[8] Cohen A.M.: Numerical Methods for Laplace Transform Inversion. Springer (2007) · Zbl 1127.65094
[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) · Zbl 1105.65120
[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, 279-305, (1999) · Zbl 0962.65109
[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) · Zbl 0962.65110
[12] Davies, B; Martin, B, Numerical inversion of the Laplace transform: a survey and comparison of methods, J. Comput. Phys., 33, 1-32, (1979) · Zbl 0416.65077
[13] Duffy, DG, On the numerical inversion of Laplace transforms: comparison of three new methods on characteristic problems from applications, ACM Trans. Math. Softw., 19, 333-359, (1993) · Zbl 0892.65079
[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, NJ; 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) · Zbl 0642.65086
[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) · Zbl 0659.65138
[17] Giunta, G; Murli, A; Schmid, G, An analysis of bilinear transform-polynomial methods of inversion of Laplace transforms, Numer. Math., 69, 269-282, (1995) · Zbl 0840.65133
[18] Giunta, G; Murli, A; Schmid, G, Error analysis of rjabov algorithm for inverting Laplace transforms, Ric. Mat., XLIV, 207-219, (1995) · Zbl 0915.65131
[19] Halsted, DJ; Brown, DE, Zakian’s technique for inverting Laplace transforms, Chem. Eng. J., 3, 312-313, (1972)
[20] Henrici, P.: Applied and Computational Complex Analysis, vol. 2. John Wiley & Sons (1977) · Zbl 0363.30001
[21] Higham, N, Error analysis of the Björk-pereira algorthm for solving Vandermonde systems, Numer. Math., 50, 613-632, (1987) · Zbl 0595.65029
[22] Kryzhniy, VV, On regularization of numerical inversion of Laplace transforms, J. Inverse Ill-Posed Probl., 12, 279-296, (2004) · Zbl 1059.65119
[23] Kryzhniy, V.V.: InvertLT. http://www-users.cs.umn.edu/ yelena/web/software.php. Accessed 2006 · Zbl 0915.65131
[24] Lyness, NJ; Giunta, G, A modification of the weeks method for the inversion of the Laplace tranform, Math. Comput., 47, 313-322, (1987) · Zbl 0611.65088
[25] Mallet, A.: Uniform Resource Locators (URL). Wolfram Information Center. http://library.wolfram.com/infocenter/MathSource/2691/ (2000) · Zbl 0892.65079
[26] Membrez, J; Infelta, PP; 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, 4489-4498, (1996)
[27] Murli, A; Rizzardi, M, Algorithm 682: talbot’s method for the Laplace inversion problem, ACM Trans. Math. Softw., 16, 347-371, (1990) · Zbl 0900.65374
[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) · Zbl 1105.65120
[29] Piessens, R; Branders, P, Numerical inversion of the Laplace tranform usinge generalized Laguerre polynomials, Proc. IEEE, 118, 1517-1522, (1971)
[30] Piessens, R, A new numerical method for the inversion of the Laplace tranform, J. Inst. Math. Appl., 10, 185-192, (1972) · Zbl 0246.65035
[31] Piessens, R; Huysmans, R, Algorithm 619: automatic numerical inversion of the Laplace transform, ACM Trans. Math. Softw., 10, 348-353, (1984) · Zbl 0546.65087
[32] Provencher, S.: CONTIN. http://s-provencher.com/pages/contin.shtml. Accessed 1982
[33] Rjabov, VM, On the numerical inversion of the Laplace tranform, Vestn. Leningr. Univ., 7, 177-185, (1974)
[34] Sykora, S, Bortolotti, V., fantazzini, P.: PERFIDI: parametricallyenabled relaxation filters with double and multiple inversion, Magn. Reson. Imaging, 25, 529-532, (2007)
[35] Spinelli, RA, Numerical inversion of a Laplace transform, SIAM J. Numer. Anal., 3, 636-649, (1966) · Zbl 0166.13302
[36] Stehfest, H, Algorithm 368: numerical inversion of Laplace transform, Commun. ACM, 13, 47-49, (1970)
[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)
[38] Thornhill, NF; Patwardhan, SC; Shah, SL, A continuous stirred tank heater simulation model with applications, J. Process Control, 18, 347-360, (2008)
[39] Valko, P.: Numerical Inversion of Laplace Transform. http://www.pe.tamu.edu/valko/Nil/ (2002) · Zbl 0990.11054
[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)
[41] Weeks, W, Numerical inversion of the Laplace tranform using Laguerre functions, J. ACM, 13, 419-429, (1966) · Zbl 0141.33401
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.