On the Laguerre Method for Numerically Inverting Laplace Transforms

Abstract

The Laguerre method for numerically inverting Laplace transforms is an old established method based on the 1935 Tricomi-Widder theorem, which shows (under suitable regularity conditions) that the desired function can be represented as a weighted sum of Laguerre functions, where the weights are coefficients of a generating function constructed from the Laplace transform using a bilinear transformation. We present a new variant of the Laguerre method based on: (1) using our previously developed variant of the Fourier-series method to calculate the coefficients of the Laguerre generating function, (2) developing systematic methods for scaling, and (3) using Wynn's ǫ-algorithm to accelerate convergence of the Laguerre series when the Laguerre coefficients do not converge to zero geometrically fast. These contributions significantly expand the class of transforms that can be effectively inverted by the Laguerre method. We provide insight into the slow convergence of the Laguerre coefficients as well as propose a remedy. Before acceleration, the rate of convergence can often be determined from the Laplace transform by applying Darboux's theorem. Even when the Laguerre coefficients converge to zero geometrically fast, it can be difficult to calculate the desired functions for large arguments because of roundoff errors. We solve this problem by calculating very small Laguerre coefficients with low relative error through appropriate scaling. We also develop another acceleration technique for the case in which the Laguerre coefficients converge to zero geometrically fast. We illustrate the effectiveness of our algorithm through numerical examples.

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

“…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.…”

Iterative convergence of passage-time densities in semi-Markov performance models

“…Our tool supports two Laplace transform inversion algorithms, which are briefly outlined below: the Euler technique [12] and the Laguerre method [13] with modifications summarised in [2].…”

“…Typically, we set n = 200. In order to achieve this, however, the scaling method described in [13] must be used to ensure that the Laguerre coefficients have decayed to (near) 0 by n = 200. If this can be accomplished, the inversion of a passage-time density for any number of t-values can be achieved at the fixed cost of calculating 400 truncated summations of the type shown in Eq.…”