Numerical Inversion of Laplace Transforms Using Laguerre Functions

Abstract

A method is described for the numerical inversion of Laplace transforms, in which the inverse is obtained as an expansion in terms of orthonormal Laguerre functions. In order for this to be accomplished, the given Laplace transform is expanded in terms of the Laplace transforms of the orthonormal Laguerre functions. The latter expansion is then reduced to a cosine series whose approximate expansion coefficients are obtained by means of trigonometric interpolation. The computational steps have been arranged to facilitate automatic digital computation, and numerical illustrations have been given.

Gram matrix of a Laguerre model: application to model reduction of irrational transfer function

“…Compute the Laguerre spectra of successive integrals and derivatives or their approximations using (12) or (13) and (8) or (10). 2. Compute the elements of the main diagonal of the Gram matrix using (5) or in practice the following truncated sums…”

“…Remark: It will be noted that the described procedures (8) and (12) or (10) and (13) give the Laguerre spectra of repeated strict derivatives and integrals. Therefore, in model reduction application, the reduced model of F (s) is provably stable by a Lyapunov method if the Gram matrix is computed on Ω or on Ω= f 1 , f 2 ,..., f q ,..., f r+1 and if the model reduction procedure is based on the approximation of either first or last function of the set by the others [16].…”

“…From G (s), the first N = 100 coefficients of a Laguerre model with α =2 .42 is computed using a well-known technique [9][10][11][12] of experimentation, however, we have coded our own version of the algorithm using a discrete cosine transform. The (not crucial) choice α =2.42 for the computation of the Laguerre spectrum of G (s) is obtained using the optimization method described in [19].…”

“…Here we consider systems described by Laguerre models. Indeed the Laguerre functions have shown their large potential in numerous applications as in signal analysis and parameter identification [4], system identification [5,6], approximation of finite or infinite-dimensional system [7,8], numerical inversion of the Laplace transform [9][10][11][12], industrial control [13]... This is more particularly in the context of the numerical inversion of infinite-dimensional Laplace transfer functions that we have based our works.…”

“…In practice the Laguerre series (3) will be truncated and the Laguerre coefficients c n will be evaluated by a well-known technique based on discrete Fourrier transform computation [9][10][11][12]. Now, let F i (s) denote the Laplace transform of the function f i (t)a n d{c i,n } n≥0 its Laguerre spectrum.…”