Fourier, Laplace and other Transforms
by Helmut Weber

Numerical Inversion/Computation of the Fourier Transform

Algorithms

• Weber's Algorithm

Numerical Inversion/Computation of the Hartley Transform

Algorithms

• Weber's Algorithm

Numerical Inversion/Computation of the Bilateral Laplace Transform

1. Inversion

• Algorithm of Weber and Weeks

2. Computation

• Gauss-Legendre Quadrature

Numerical Inversion/Computation of the Laplace Transform

1. Algorithms which evaluate the transform only at real arguments

• The Gaver/Stehfest-Algorithm
• Spinelli's Algorithm
• Piessens' Algorithm

2. Algorithms which evaluate the transform at complex arguments

• Weeks' Algorithm
• The Salzer/Piessens Algorithm
• Koizumi's Algorithm
• The Dubner&Abate Algorithm
• Veillon's Algorithm
• Crump's Algorithm
• The Durbin/Piessens&Huysmans Algorithm
• The Algorithm of Honig&Hirdes
• de Hoog's Algorithm
• Sidi's Algorithm
• Talbot's Algorithm

3. Numerical Computation of the Laplace Transform by Quadrature

• Gauss-Legendre Quadrature

Numerical Inversion of the Mellin Transform

• Algorithm of Weber and Weeks

4. References (Books, Book extracts, Surveys, Reviews, Comparisons, Web pages)

• Alan M. Cohen, Numerical Methods for Laplace Transform Inversion, Springer, New York (2007)
• Appendix of Alan. M. Cohen's Books with programs (2007)
• Prem K. Kythe, Michael R. Schäferkotter, Handbook of Computational Methods for Integration, Chapman & Hall/CRC, Boca Raton (2005), Chapter 7: Numerical Inversion of the Laplace Transform, 323-294
• Urs. E Graf, Applied Laplace Transforms and z-Transforms for Scientists and Engineers: A Computational Approach using a Mathematica Package, Birkhäuser, Basel (2004)
• Brian Davies, Integral Transforms and Their Applications, 3rd. ed., Springer-Verlag New York (2002), Chapter 19: Numerical Inversion of the Laplace Transform, 327-253
• Mark Craddock, David Heath, Eckhard Platen, Numerical inversion of Laplace Transforms: a survey of techniques with applications to derivative pricing, J. Comp. Finance 4 (2000), 57-81
• Joseph Abate, Gagan L. Choudhury, Ward Whitt, An Introduction to Numerical Transform Inversion and its Application to Probability Models, in: W. Grassman (ed.), Computational Probability, Kluwer, Boston (1999), 257-323
• Arnold R. Krommmer, Christoph W. Ueberhuber, Computational Integration, SIAM, Philadelphia (1998), 261-265
• Dean G. Duffy, On the Numerical Inversion of Laplace Transforms: Comparison of Three Methods on Characteristic Problems from Applications, ACM Transactions on Mathematical Software 19, No.3, 334-359 (1993)
• Philip J. Davis, Philip Rabinowitz, Methods of Numerical Integration, 2nd Edition, Academic Press, San Diego (1984), 264-270
• G.V. Narayanan, D.E. Beskos, Numerical Operational Methods for Time-Dependent Linear Problems, International Journal for Numerical Methods in Engineering 18, 1829-1854 (1982)
• Brian Davies, Brian Martin, Numerical Inversion of the Laplace Transform: a Survey and Comparison of Methods, Journal of Computational Physics 33, No.1, 2-32 (1979)
• Robert Piessens, N.D.P. Dang, A Bibliography on numerical inversion of the Laplace transform and applications: A Supplement, Journal of Applied and Computational Mathematics 2, No.3, 225-228 (1976)
• Robert Piessens, A Bibliography on numerical inversion of the Laplace transform and applications, Journal of Applied and Computational Mathematics 1, No.2, 115-128 (1975)
• Richard Bellman, Robert E. Kalaba, Jo Ann Lockett, Numerical Inversion of the Laplace transform, American Elsevier, New York (1966)
• Gustav Doetsch, Introduction to the Theory and Application of the Laplace Transformation, Springer-Verlag, New York (1974)
• Peter Valko's Web Page, Numerical Inversion of the Laplace Transform (2007)
• Peter Valko, Vladimir Vojta, The List (more than 1500 publications) (2007)

Prof. Dr. Helmut Weber, May 2012
Hochschule RheinMain University of Applied Sciences
Department Design/Computer Science/Media