Avoiding dense matrices in pulse propagators

[fidgetspinner]

Dear SpinDynamics forum users,

This question is related to the lecture "Spin Dynamics - Simulation design and coding, part II".

In the lecture (around 45 minutes), it was shown that by taking small time steps the propagator matrix becomes sparse which speeds up the calculations significantly. Out of curiosity, I wanted to check the sparsity of the pulse propagator and to my surprise it was very dense. Therefore, applying the pulse to the initial spin density matrix produces a dense matrix, and also the time evolution becomes slow. Is there something that could be done to avoid this?

Any help is greatly appreciated.

[kuprov]

Alas, only one thing - make the time step shorter. Then the exp(-iHdt) gets closer to the early terms of the Taylor series, which are low powers of a sparse matrix and therefore sparse. Also, check for small elements; those can be dropped.

There are various tricks that avoid matrix exponentiation entirely when only the action on some vector is needed, for example exp(-iHdt)*v, see the discussion in Section 4.9 and Chapter 9 here:

https://link.springer.com/book/10.1007/ ... 31-05607-9

[fidgetspinner]

Thank you Prof. Kuprov for the kind response.