APPLICATION OF WEEKS METHOD FOR THE NUMERICAL INVERSION OF THE LAPLACE TRANSFORM TO THE MATRIX EXPONENTIAL

Communications in Mathematical Sciences

Volumn 3, Issue 3, 2005, Pages 335-372

  Kano, Patrick O ^a   Brio, Moysey ^a   Moloney, Jerome V ^a  

^a UNIVERSITY OF ARIZONA   (United States)

Author keywords

Matrix exponential; Nonparaxial beam propagation equation; Weeks method

Indexed keywords

EID: 41249092819     PISSN: 15396746     EISSN: 19450796     Source Type: Journal    
DOI: 10.4310/CMS.2005.v3.n3.a4     Document Type: Article

References (43)

1. [1] J. Abate, G. Choudhury and W. Whitt, On the Laguerre method for numerically inverting Laplace transforms, INFORMS Journal on Computing, 8, 413-427, 1996.
2. [2] J. Abate and P. P. Valkó, Multi-precision Laplace transform inversion, Int. J. Numer. Meth.Engng., 60, 5, 979-993, 2004.
3. [3] G. Arfken and H. Weber, Mathematical Methods for Physicists, Harcourt Academic Press, San Diego, 5th edition, 2001.
4. [4] David H. Bailey, Y. Hida, X. S. Li and B. Thompson, ARPREC: An Arbitrary Precision Computing Package, http://crd.lbl/gov/~dhbailey/mpdist/
5. [5] Richard Bellman, Numerical Inversion of the Laplace Transform: Applications to Biology,Economics, Engineering, and Physics, American Elsevier Pub. Co., New York, 1966.
6. [6] A. J. Davies and D. Crann, Parallel Laplace transform methods for boundary element solutions of diffusion-type problems, Electronic Journal of Boundary Elements, 2, 231-238, 2002.
7. [7] B. Davies, Integral Transforms and Their Applications, Springer-Verlag, New York, 3rd edition,2002.
8. [8] P. Davies and N. Higham, A Schur-Parlett algorithm for computing matrix functions, SIAMJ. Matrix Anal. Appl., 25, 2, 464-485, 2003.
9. [9] F. R. De Hoog, J. H. Knight and A. N. Stokes, An improved method for numerical inversionof Laplace transforms, SIAM J. Sci. Stat. Comput., 3, 357-366, 1982.
10. [10] Gustav Doetsch, Guide to the Applications of the Laplace and Z-Transforms, 2nd Englishedition, Van Nostrand Reinhold Company, London, 1971.
11. [11] D. G. Duffy, On the numerical inversion of Laplace transforms: Comparison of three new methods on characteristic problems from applications, ACM Trans. Math. Software, 19, 333-359, 1993.
12. [12] G. Giunta, G. Lacetti and M. R. Rizzadri, More on the Weeks method for the numerical inversion of the Laplace transform, Numer. Math., 54, 193-200, 1988.
13. [13] G. Giunta, A. Murli and G. Schmid, An analysis of bilinear transform polynomial methods of inversion of Laplace transforms, Numer. Math., 69, 269-282, 1995.
14. [14] G. Goloub and Charles Van Loan, Matrix Computations, John Hopkins University Press, Bal-timore, 1983.
15. [15] Urs. Graf, Applied Laplace Transforms and Z-Transforms for Scientists and Engineers: A Computational Approach Using a Mathematica Package, Birkhauser Verlag, Basel, 2004.
16. [16] Stephen Greedy, Advances in the spectral index method for the analysis of photonic integrated circuits, Doctoral Dissertation, Univ. of Nottingham, 2002.
17. [17] W. Gropp, E. Lusk and A. Skjellum, Using MPI: Portable Parallel Programming With the Message-Passing Interface, The MIT Press, London, 1995.
18. [18] J. R. Higgins, Completeness and Basis Functions of Sets of Special Functions, CambridgeUniv. Press, Cambridge, 1977.
19. [19] Nicholas Higham, Stable iterations for the matrix square root, Numerical Algorithms, 15, 227-242, 1997.
20. [20] G. Lifante, Integrated Photonics: Fundamentals, John Wiley and Sons, San Francisco, 2003.
21. [21] S. H. Lui, Computation of pseudospectra by continuation, SIAM J. Sci. Comp., 18, 567-573,1997.
22. [22] MATLAB http://www.mathworks.com
23. [23] C. Moler and C. Van Loan, Nineteen dubious ways to compute the exponential of a matrix,twenty-five years later, SIAM Review, 45, 1, 3-49, 2003.
24. [24] A. Murli and M. Rizzardi, Algorithm 682. Talbot's method for the Laplace inversion problem,ACM Trans. Math. Soft., 16, 158-168, 1990.
25. [25] C. O'Cinneide, Euler summation for Fourier series and Laplace transform inversion, StochasticModels, 13, 315-337, 1997.
26. [26] B. N. Parlett, A recurrence among the elements of functions of triangular matrices, LinearAlgebra Appl., 14, 117-121, 1976.
27. [27] E. A. Patent, J. J. G. M van der Tol, J. J. M Binsma and M. K. Smit, Conditions for optimum performance of unbalanced MMI couplers, Optics Express, 9, 7, 2001.
28. [28] M. Popescu and W. Shyy, Assessment of dispersion-relation-preserving and space-time CE/SE schemes for wave equations, Numerical Heat Transfer, 42, 93-118, 2002.
29. [29] W. H. Press, S. A. Teukolsky, W. Vetterling and B. Flannery, Numerical Recipes In C: The Art of Scientific Computing, Cambridge University Press, 1992.
30. [30] E. A. Sudicky, The Laplace transform Galerkin technique for efficient time-continuous solution of solute transport in double-porosity media, Geoderma, 46, 209-232, 1990.
31. [31] A. Talbot, The accurate numerical inversion of Laplace transforms, J. Inst. Maths. Applics.,23, 97-120, 1979.
32. [32] Hillel. Tal-Ezer, Spectral methods in time for hyperbolic equations, SIAM J. Numer. Anal., 23,1, 11-26, 1986.
33. [33] L. Trefethen and D. Bau III, Numerical Linear Algebra, SIAM, Philadelphia, 1997.
34. [34] L. Trefethen, Computation of pseudospectra, Acta Numerica, 8, 247-295, 1999.
35. [35] L. Trefetehn, Spectral Methods in MATLAB, SIAM, Philadelphia, 2000.
36. [36] W. T. Weeks, Numerical inversion of the Laplace transform using Laguerre functions, J. Assoc.Comput. Mach., 13, 419-429, 1966.
37. [37] J. A. C. Weideman, Algorithms for parameter selection in the Weeks method for inverting the Laplace transform, SIAM J. Sci. Comput., 21, 1, 111-128, 1999.
38. [38] J. Wimp, Sequence Transformations and Their Applications, Academic Press, New York, 1981.
39. [39] T. G. Wright and L. N. Trefethen, Large-scale computation of pseudospectra using ARPACK and Eigs, SIAM J. Sci. Comp., 23, 591-605, 2001.
40. [40] P. P. Valko and B. L. Vojta, The list, http://pumpjack.tamu.edu/~valko, 2001.
41. [41] P. P. Valkó and J. Abate, Comparison of sequence accelerators for the Gaver method of nu-merical Laplace transform inversion, Comput. Math. Appl., 48, 629-636, 2004.
42. [42] Junji. Yamauchi, Propagating Beam Analysis of Optical Waveguides, Research Studies Press, Philadelphia, 2003.
43. [43] David. Yevick, The application of complex Pade approximants to vector field propagation, IEEE Photonics Technology Letters, 12, 12, 1041-1135, 2000.