On the Laguerre Method for Numerically Inverting Laplace Transforms
On the Laguerre Method for Numerically Inverting Laplace Transforms
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DOI：
10.1287/ijoc.8.4.413
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发表时间：
1996-11
期刊：
影响因子：
--
通讯作者：
J. Abate;G. Choudhury;W. Whitt
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文献类型：
--
作者：
J. Abate;G. Choudhury;W. Whitt
用于数值反转拉普拉斯变换的拉盖尔方法是一种基于 1935 年 Tricomi-Widder 定理的古老方法，该定理表明（在适当的规律性条件下）所需函数可以表示为拉盖尔函数的加权和，其中权重是使用双线性变换从拉普拉斯变换构造的生成函数的系数。我们提出了拉盖尔方法的一个新变体，基于：（i）使用我们之前开发的傅里叶级数方法的变体来计算拉盖尔生成函数的系数，（ii）开发用于缩放的系统方法，以及（iii）当拉盖尔系数不能几何地快速收敛到零时，使用Wynn（epsilon）算法来加速拉盖尔级数的收敛。这些贡献显着扩展了可以通过拉盖尔方法有效反转的变换类别。我们深入了解拉盖尔系数的缓慢收敛并提出补救措施。在加速之前，收敛速度通常可以通过应用达布定理从拉普拉斯变换确定。即使拉盖尔系数以几何速度快速收敛到零，由于舍入误差，也可能很难计算大参数所需的函数。我们通过适当的缩放计算具有较低相对误差的非常小的拉盖尔系数来解决这个问题。我们还针对拉盖尔系数几何快速收敛到零的情况开发了另一种加速技术。我们通过数值例子说明了我们算法的有效性。
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: (i) using our previously developed variant of the Fourier-series method to calculate the coefficients of the Laguerre generating function, (ii) developing systematic methods for scaling, and (iii) using Wynn's (epsilon)-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.