[R] matrix exponential: M^0

Martin Maechler maechler at stat.math.ethz.ch
Thu Jan 22 10:33:21 CET 2004

>>>>> "PD" == Peter Dalgaard <p.dalgaard at biostat.ku.dk>
>>>>>     on 21 Jan 2004 19:08:38 +0100 writes:

    PD> Martyn Plummer <plummer at iarc.fr> writes:
    >> Calculating the matrix exponential is harder than it
    >> looks (I'm sure Peter knows this). In fact there is a
    >> classic paper by Moler and Van Loan from the 1970s called
    >> "Nineteen dubious ways to calculate the exponential of a
    >> matrix", which they updated last year in SIAM.

    PD> Right (magnificent paper by the way), although I
    PD> actually hadn't heard about the update. As I remember
    PD> it, Octave implements what Moler+v.Loan ends up
    PD> suggesting in the 1978 paper.

The update is actually available online
from http://epubs.siam.org/sam-bin/dbq/article/41801
with the extended title "...., 25 Years Later" .

The extension is 8 pages of text + 1.2 pages of references,
in which (p.42) they say
``The matrix exponential is an important computational tool in
  control theory, so availability of  expm(A)  in early versions
  of Matlab quite possibily contributed to the system's technical
  and commercial success.''

and they also tell how expm() is implemented in Matlab
(scaling, squaring, Padé approximation) which is presumably what
octave does too?


But -- going back to our original subject --

  expm(A) =  ``e ^ A'' := sum_{n=0}^\Inf  A^n / n!

definitely a bit harder than  A^n for (non-negative) integer n ?


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