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More on the Weeks method for the numerical inversion of the Laplace transform. (English) Zbl 0659.65138

The Weeks-Tricomi approach to the numerical inversion of the Laplace transform F(s) involves the numerical evaluation of the series \(f(t)=\sum_{s}C_ s(\sigma,b,f)L_ s(bt)\) where L is a Laguerre polynomial and b and \(\sigma\) are incidental parameters satisfying \(b>0\) and \(\sigma >\sigma_ 0\), the Laplace convergence index. The authors present an elegant proof of the following result, which characterizes the optimum value of b to use in terms of \(\sigma\) and the singularities of F(s). This is: “Of the set of circles which enclose all singularities of F(s) but exclude \(\sigma\), let the one which subtends the smallest angle at \(\sigma\) have tangent segment of length \(\ell\). Then \(2\ell\) is the value of b for which the series converges most rapidly.” Unfortunately, this is helpful only in the comparatively straightfoward situations in which the locations of the singularities of F(s) is available and all these happen to lie within a finite region of the complex plane.
Reviewer: J.N.Lyness

MSC:

65R10 Numerical methods for integral transforms
45L05 Theoretical approximation of solutions to integral equations
44A10 Laplace transform

Software:

Algorithm 662
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References:

[1] Garbow, B.S., Giunta, G., Lyness, J.N., Murli, A.: Algorithm 662: A Fortran software package for the numerical inversion of the Laplace transform based on Weeks’ method. ACM Trans. Math. Soft.14, 171-176 (1988) · Zbl 0709.65505 · doi:10.1145/45054.214375
[2] Giunta, G., Laccetti, G., Rizzardi, M.: On computing the abscissa of convergence of a Laplace transform function. Publ. Dip. di Matematica e Applicazioni, Univ. di Napoli, n. 41 (1986) · Zbl 0727.65117
[3] Giunta, G., Lyness, J.N., Murli, A.: An implementation of Weeks’ method for the inverse Laplace transform problem. Argonne National Laboratory, ANL/MCS TM-39, October 1984 · Zbl 0642.65086
[4] Lyness, J.N., Giunta, G.: A modification of the Weeks method for numerical inversion of the Laplace transform. Math. of Comput.47, 313-322 (1986) · Zbl 0611.65088 · doi:10.1090/S0025-5718-1986-0842138-1
[5] Tricomi, F.: Transformazione di Laplace e polinomi di Laguerre. Rend. Acc. Naz. Lincei 2, 232-239 (1935) · Zbl 0011.20403
[6] Weeks, T.: Numerical inversion of Laplace transforms using Laguerre functions. J. ACM13, 419-429 (1986) · Zbl 0141.33401 · doi:10.1145/321341.321351
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