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

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With respect to void LaEigSolve, DGEEV/DGEEVX seem to return
eigenvalue/eigenvector pairs with the possibility that the eigenvalues might
be complex numbers.  

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

Alternatively, I could have it return a LaGenMatComplex for the eigenvector
component.

Any thoughts on this?

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 Eigenvectors
> 
> 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 always
> > 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 --
> > 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)=inv(A'A)A', or to do the following (sorry for the
> > matlab
> > > style notation) [U,D,V']=svd(A).  pinv(A)=Vpinv(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 equal
> > 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=2e-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 this
> > 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 rank,
> > k, of A can be determined as the number of singular values which exceed
> > 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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