Restarted Arnoldi method for Laplace transforms and complete Bernstein functions

This repository contains an implementation of the restarted Arnoldi method for Laplace transforms and complete Bernstein functions based on [1], [2]. In particular, the function laplace_restarting computes an approximation to F(A)*b, where F is a Laplace transform or a complete Bernstein function.

Usage

In addition to A, b and a target accuracy, the user needs to provide the function handle f which is

• the inverse Laplace transform of F (if F is a Laplace transform) or
• the density (wrt Lebesgue measure) of the Lévy measure in the Lévy–Khintchine representation of F (if F is a Bernstein function).

For detailed information on how to use laplace_restarting, see its function header and the examples below. Note that MATLAB R2019b or later is required.

Examples

The folder experiments contains several exemplary scripts:

• The scripts in experiments/comparisons contains the examples presented in [1],[2]. For Hermitian A, they also compare the performance of laplace_restarting to the two-pass Lanczos method.
• The script in experiments/error_bounds demonstrates the computation of a posteriori error bounds as in [2], [3].

The examples need access to the files in solutions, which need to be downloaded using git annex get or manually via https://uni-wuppertal.sciebo.de/s/3lltZM3WdNrYCLT .

How to cite

Please use the original paper [1] to cite this package.

References

[1] A. Frommer, K. Kahl, M. Schweitzer, and M. Tsolakis: Krylov subspace restarting for matrix Laplace transforms, SIAM J. Matrix Anal. Appl., 44 (2023), pp. 693–717, doi: 10.1137/22M1499674

[2] M. Tsolakis: Efficient Computation of the Action of Matrix Rational Functions and Laplace transforms (Doctoral Thesis), Bergische Universität Wuppertal, Germany, 2023, doi: 10.25926/BUW/0-106

[3] A. Frommer, and M. Schweitzer: Error bounds and estimates for Krylov subspace approximations of Stieltjes matrix functions, BIT, 56 (2016), pp. 865–892, doi: 10.1007/s10543-015-0596-3