Fast discrete Laplace transforms

The discrete Laplace transform (DLT) with $M$ inputs and $N$ outputs has a nominal computational cost of $O\left(MN\right)$. 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 $\eta$. 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 $\eta$. 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 $O\left({(M+N)}^{1.5}\right)$ in the intermediate regime. Our algorithm achieves the same accuracy as brute-force evaluation, and is typically 10–100 times faster.