numerical noise with exponential matrix (expm())

Some one told me that that numerical noise normally exists with the expm() function. The noise value is normally with a magnitude of 10^-19. Is that the truth? If it is, how can I eliminate the noise, as in my calculation the result need to be calculated with some numbers with similar magnitude.

[John D'Errico]

Sorry, but it is time to take a basic numerical analysis class. Or read Cleve's 19 dubious ways .

You CANNOT eliminate that noise working in floating point arithmetic. Trying to do the impossible is something only asked of a grad student, but rarely achieved.