|Re: [eigen] Matrix exponential|
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On Wed, 6 May 2009, Benoit Jacob wrote:
The complexity of the present approach is said to be roughly 20n^3
which is huge, though, and in practice, the most common case is
exponentiating a skew-adjoint matrix (think exp(itH) where H is a
hamiltonian, think Stone's theorem...), whose diagonalization boils
down to diagonalization of a selfadjoint matrix. So it'd be very
important to make sure the selfadjointeigensolver is robust enough
that it can always safely be used without convergence issues!
In the symmetric case, the cost of the scaling-and-squaring algorithm
drops to half because the cost of matrix multiplications and solves drops
to half (at least theoretically). So then it's comparable to the symmetric
QR algorithm. However, my guess is that it's better (stability-wise) to
use the spectral decomposition. Some time I want to experiment with this -
I don't know of any previous work.
I don't know whether complex numbers make a difference here.