Fast discrete Laplace transforms

The discrete Laplace transform (DLT) with M inputs and N outputs has a nominal computational cost of O(MN). There are approximate DLT algorithms with O(M+N) cost such that the output errors divided by the sum of the inputs are less than a fixed tolerance η. However, certain important applications of DLTs require a more stringent accuracy criterion, where the output errors divided by the true output values are less than η. We present a fast DLT algorithm combining two strategies. The bottom-up strategy exploits the Taylor expansion of the Laplace transform kernel. The top-down strategy chooses groups of terms in the DLT to include or neglect, based on the whole summand, and not just on the Laplace transform kernel. The overall effort is O(M+N) when the source and target points are very dense or very sparse, and appears to be O((M+N)1.5) in the intermediate regime. Our algorithm achieves the same accuracy as brute-force evaluation, and is typically 10–100 times faster.