This is a Julia translation of the MATLAB implementation of Al-Mohy and Higham's
function for computing expm(A)*v when A is sparse, without explicitly computing expm(A).
The original code can be found at Matlabcentral File Exchange, and the theory is explained in the following article:
Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators, Awad H. Al-Mohy and Nicholas J. Higham, SIAM Journal on Scientific Computing 2011 33:2, 488-511. (preprint)
expmv(t,A,v)Although both the ExpmV.jl and Expokit.jl implementations are in the early stages of development (ExpmV.jl is a direct translation of MATLAB code, and Expokit.jl is not fully optimized), here are some crude benchmarks (using Benchmark.jl) that indicate large gains over the dense expm. Source can be found in test\benchmark.jl.
Benchmark 1: Complex matrix with density of 0.0191, dimension 100, 100 trials
| Row | Function | Average | Relative | Replications |
|---|---|---|---|---|
| 1 | Expmv.jl |
0.0213879 | 5.24362 | 100 |
| 2 | Expokit.jl |
0.00407885 | 1.0 | 100 |
| 3 | Julia's dense expm |
0.0686303 | 16.8259 | 100 |
Benchmark 2: Complex matrix with density of 0.1913, dimension 100, 100 trials
| Row | Function | Average | Relative | Replications |
|---|---|---|---|---|
| 1 | Expmv.jl |
0.00602035 | 1.74028 | 100 |
| 2 | Expokit.jl |
0.00345941 | 1.0 | 100 |
| 3 | Julia's dense expm |
0.0275519 | 7.96434 | 100 |
Benchmark 3: Complex matrix with density of 0.3651, dimension 100, 100 trials
| Row | Function | Average | Relative | Replications |
|---|---|---|---|---|
| 1 | Expmv.jl |
0.0105414 | 1.97621 | 100 |
| 2 | Expokit.jl |
0.00533413 | 1.0 | 100 |
| 3 | Julia's dense expm |
0.0354484 | 6.64558 | 100 |
Clearly the current ExpmV.jl implementation needs to be looked at more carefully (See Issue #1 in particular, which appears to cause a bottle neck due to my incomplete implementation)
Released under the BSD 2-clause license used in Al-Mohy and Higham's original code.