More matrix exponential methods

[jarlebring]

Wish-list / todo-list for more matrix exponential methods

• An implementation of A. H. Al-Mohy and N. J. Higham, A new scaling and squaring algorithm
  for the matrix exponential, SIAM J. Matrix Anal. Appl., 31(3), (2009) This is basically the same as Higham 2005, but with a more accurate backward error estimate which in practice leads to less multiplications. The method behind expm in recent versions of matlab.
• expmpoly: http://personales.upv.es/~jorsasma/software/expmpol.m described in Boosting the computation of the matrix exponential, Appl. Math. Comput., 2018 Requires only multiplications - no inverse. Main computation code can be generated with graph_exp_sid in GraphMatFun.jl.
• ExpMethodArbitraryPrecision. Arbitrary precision scaling-and-squaring: https://epubs.siam.org/doi/10.1137/18M1228876?mobileUi=0

[ChrisRackauckas]

Nice! Yeah, an arbitrary precision one would be very nice for the ecosystem. We've very much been missing it

[danielwe]

Adding this method to the list, which appears to be the expv and phiv version of the 2009 reference above. Currently used in Matlab for their expmv. Abstract claim:

Experimental comparisons with MATLAB codes based on Krylov subspace, Chebyshev polynomial, and Laguerre polynomial methods show the new algorithm to be sometimes much superior in terms of computational cost and accuracy."

• Al-Mohy, Awad H., and Nicholas J. Higham. “Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators.” SIAM Journal on Scientific Computing 33, no. 2 (January 2011): 488–511. https://doi.org/10.1137/100788860.

[danielwe]

Existing but old implementation of the above here: https://github.com/marcusps/ExpmV.jl. New StaticArrays-specific implementation here: https://github.com/jecs/StaticExpmv.jl.

[jarlebring]

The methods in this setting are to compute the martix exp(A). The algorithm expmv is for exp(matrix)v where v is a vector (okay it can also be a tall skinny matrix). expmv belongs to a different method class, where you cannot use the squaring tricks we use in scaling-and-squaring. An implementation of expmv in SciML would probably be appreciated, but probably fit better in a setting independent of the above methods.

[danielwe]

ExponentialUtilities.jl already contains expv and phiv methods, no? See https://docs.sciml.ai/ExponentialUtilities/stable/#expv-and-phiv, as well as the final line in the readme.