[Numpy-discussion] expm

Kevin Jacobs <jacobs@bioinformed.com> bioinformed@gmail....
Fri Jul 20 12:03:09 CDT 2007


On 7/20/07, Anne Archibald <peridot.faceted@gmail.com> wrote:
>
> On 20/07/07, Nils Wagner <nwagner@iam.uni-stuttgart.de> wrote:
> > lorenzo bolla wrote:
> > > hi all.
> > > is there a function in numpy to compute the exp of a matrix, similar
> > > to expm in matlab?
> > > for example:
> > > expm([[0,0],[0,0]]) = eye(2)
> > Numpy doesn't provide expm but scipy does.
> > >>> from scipy.linalg import expm, expm2, expm3
>
> Just as a warning, numpy does provide expm1, but it does something
> different (exp(x)-1, computed directly).
>

On a separate note, I'm working to provide faster and more accurate versions
of sqrtm and expm.  The current versions do not take full advantage of
LAPACK.  Here are some preliminary benchmarks:

Ill-conditioned
----------------
linalg.sqrtm   : error=9.37e-27, 573.38 usec/pass
sqrtm_svd      : error=2.16e-28, 142.38 usec/pass
sqrtm_eig      : error=4.79e-27, 270.38 usec/pass
sqrtm_symmetric: error=1.04e-27, 239.30 usec/pass
sqrtm_symmetric2: error=2.73e-27, 190.03 usec/pass

Well-conditioned
----------------
linalg.sqrtm   : error=1.83e-29, 478.67 usec/pass
sqrtm_svd      : error=8.11e-30, 130.57 usec/pass
sqrtm_eig      : error=4.50e-30, 255.56 usec/pass
sqrtm_symmetric: error=2.78e-30, 237.61 usec/pass
sqrtm_symmetric2: error=3.35e-30, 167.27 usec/pass

Large
----------------
linalg.sqrtm   : error=5.95e-25, 8450081.68 usec/pass
sqrtm_svd      : error=1.64e-24, 151206.61 usec/pass
sqrtm_eig      : error=6.31e-24, 549837.40 usec/pass
sqrtm_symmetric: error=8.55e-25, 177422.29 usec/pass

where:

def sqrtm_svd(x):
  u,s,vt = linalg.svd(x)
  return dot(u,transpose((s**0.5)*transpose(vt)))

def sqrtm_eig(x):
  d,e = linalg.eig(x)
  d = (d**0.5).astype(float)
  return dot(e,transpose(d*e))

def sqrtm_symmetric(x,cond=1e-7):
  d,e = linalg.eigh(x)
  d[d<cond] = 0
  return dot(e,transpose((d**0.5)*e)).astype(float)

def sqrtm_symmetric2(x):
  # Not as robust due to initial Cholesky step
  l=linalg.cholesky(x,lower=1)
  u,s,vt = linalg.svd(l)
  return dot(u,transpose(s*u))

with SciPy linked against ACML.

-Kevin
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