Algorithms for Parameter Selection in the Weeks Method for Inverting the Laplace Transform

The Weeks method is one of the most efficient numerical techniques for inverting the Laplace transform, provided two free parameters in the Laguerre expansion on which the method is based are chosen judiciously. However, there appear to be no theoretical estimates or numerical algorithms for computing these parameters. The two algorithms presented in this paper fill that gap. Both algorithms aim to minimize a theoretical error estimate, and both are implemented with the FFT. The first algorithm is restricted to a certain class of transforms and the user is expected to provide information on the singularities of the transform. The second algorithm, though more expensive, is completely general---no singularity information is required. In challenging numerical tests, both algorithms successfully predicted values of the parameters that are near optimal.