Re: [Lapackpp-devel] Pseudoinverse, Eigenvalues and Eigenvectors

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Am Mittwoch, 30. M=E4rz 2005 22:40 schrieb Jacob \(Jack\) Gryn:
> With respect to void LaEigSolve, DGEEV/DGEEVX seem to return
> eigenvalue/eigenvector pairs with the possibility that the eigenvalues
> might be complex numbers.

Right, that's what happens with general (non-symmetric) matrices.

> Maybe it would be better to implement it using dgesdd?  This may need a
> transpose of the V matrix though, which is probably bad.

No, SVD is something different.

> Alternatively, I could have it return a LaGenMatComplex for the eigenvect=
or
> component.
>
> Any thoughts on this?

Yes, returning a LaGenMatComplex is probably better. However, the conversio=
n=20
from two real-valued vectors to one complex-valued will have to be done in=
=20
Lapackpp.

Christian

>
> Jack
>
> > -----Original Message-----
> > From: Jacob (Jack) Gryn [mailto:jg...@cs...]
> > Sent: March 30, 2005 12:37 PM
> > To: 'Christian Stimming'
> > Cc: 'lap...@li...'
> > Subject: RE: [Lapackpp-devel] Pseudoinverse, Eigenvalues and Eigenvecto=
rs
> >
> > I'll work on LaLUInverseIP(LaGenMatDouble &A, LaVectorLongInt &PIV) and
> > LaLUInverseIP(LaGenMatDouble &A, LaVectorLongInt &PIV, LaVectorDouble
> > &work)
> >
> > How about adding
> > LaInverse(const LaGenMatDouble &A, LaGenMatDouble &invA)
> > LaInverseIP(LaGenMatDouble &A)
> >
> > For those who want to do it all in one shot?
> >
> > I'll also work on
> >
> > void LaEigSolve(const LaGenMatDouble &A, LaVectorDouble &eigvals,
> > LaGenMatDouble &eigvec)
> >
> > Jack
> >
> > > -----Original Message-----
> > > From: lap...@li...
> > > [mailto:lapackpp-devel- ad...@li...] On Behalf Of
> > > Christian Stimming
> > > Sent: March 30, 2005 3:55 AM
> > > To: Jacob (Jack) Gryn
> > > Cc: lap...@li...
> > > Subject: Re: [Lapackpp-devel] Pseudoinverse, Eigenvalues and
> >
> > Eigenvectors
> >
> > > Hi,
> > >
> > > Jacob (Jack) Gryn schrieb:
> > > > Apparently Lapack has a function named DGETRI for obtaining the
> >
> > inverse
> >
> > > of a
> > >
> > > > function.  Would it be reasonable to create a function called inv()
> > > > in LaGenMatDouble that makes use of DGETRI?  (If not somewhere
> > > > else?).
> > >
> > > Ok, I didn't know of DGETRI. Yes, that's reasonable to be implemented.
> > > However, the input to DGETRI is not any general matrix but the input =
is
> > > the result of the DGETRF operation, which in lapack++ is available e.=
g.
> > > in LUFactorizeIP in linslv.cc and laslv.h (huh, obviously I forgot to
> > > add the leading "La" to the name; sorry).
> > >
> > > Therefore I would suggest to add a function like
> > > LaLUInverseIP(LaGenMatDouble &A, LaVectorLongInt &PIV) to
> > > laslv.h/linslv.cc, where the documentation clearly says that the input
> > > variables *must* come from LUFactorizeIP. I think that's better than
> > > adding an inv() method to LaGenMatDouble, because that one would alwa=
ys
> > > have to compute both dgetrf and dgetri but the fact that these are two
> > > separate steps should be visible to lapackpp's users, IMHO.
> > >
> > > You can decide for yourself what to do with the WORK array of DGETRI =
=2D-
> > > if this function is called several times, then you can gain a
> > > significant speedup if the application re-uses the same WORK array
> > > instead of allocating a new one each time (really!). Probably you can
> > > add two functions for the Inverse, one with the WORK array as argument
> > > and the other one without, where the latter internally allocates a new
> > > WORK array and then calls the former function.
> > >
> > > > In addition, I would like to build a pseudoinverse function as well
> >
> > (not
> >
> > > > implemented in lapack).  The two options would be to do it according
> >
> > to
> >
> > > the
> > >
> > > > definition pinv(A)=3Dinv(A'A)A', or to do the following (sorry for =
the
> > >
> > > matlab
> > >
> > > > style notation) [U,D,V']=3Dsvd(A).  pinv(A)=3DVpinv(D)U', where to =
get
> > >
> > > pinv(D),
> > >
> > > > just find the inverse of each non-zero value in the diagonal matrix.
> > > >
> > > > The second way apparently is more computationally efficient, but a
> >
> > loop
> >
> > > with
> > >
> > > > if statements to get the pseudoinverse of D may take longer than
> > > > using lapack to calculate the pseudoinverse by definition.
> > >
> > > I'm still quite hesistant to add functions concerning "the inverse" to
> > > a numerical library (the exception being cases like above, when there
> > > already exists a LAPACK function for this). Really, all the numerics
> > > guy's faces turn red and they start sweating once you tell them you
> > > want to calculate the inverse :-))
> > > http://www.netlib.org/lapack/lug/node26.html#1232
> > >
> > > But seriously, I found for myself that the definition of "the useful
> > > pseodo-inverse" depends much more on your actual application than you
> > > might think, which means the calculation of the pseudoinverse should =
be
> > > done in application code instead of lapackpp code.
> > >
> > > The SVD way of calculating it is a good example, because when you want
> > > to calculate pinv(D) by inverting "each non-zero value", you have to
> > > think twice what "non-zero" actually means. Does it mean "exactly equ=
al
> > > to the floating point number 0.0"? Probably not, but rather "smaller =
in
> > > absolute value than some value epsilon". If epsilon is the machine's
> > > precision, then it's approx. eps=3D2e-16 for doubles. But for your
> > > application it might turn out that a much better value is something
> > > smaller err larger, e.g. 1e-8 or 1e-4. But the whole point is that th=
is
> > > really really really depends on your application. Also on
> > > http://www.netlib.org/lapack/lug/node53.html it says that there is no
> > > explicit LAPACK function for the pseudoinverse but "The effective ran=
k,
> > > k, of A can be determined as the number of singular values which exce=
ed
> > > a suitable threshold." So I would ask you not to implement such a
> > > function in lapackpp.
> > >
> > > > Secondly, if I want to obtain the eigenvalues and eigenvectors of a
> > >
> > > general
> > >
> > > > matrix, is DGEEV the correct function to implement?  Should I create
> > > > a function to add to EigSlv.cc ?
> > >
> > > Yes, and yes. (Or maybe dgeevx?) For dgeev/dgeevx (whichever seems
> > > easier for you) you can add a function right away. Just go ahead.
> > >
> > > Thanks,
> > >
> > > Christian
> > >
> > >
> > >
> > >
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