From: "M. Edward (Ed) Borasky" <znmeb@...>
Date: 2006-07-31T06:49:42+09:00
Subject: Re: Setting precision in Ruby-GSL calculations ?

Nuralanur@aol.com wrote:
> Dear all,
>  
> I'd like to use ruby-gsl for some singular value decompositions, i.e.
> a matrix m should be decomposed into a product of three  matrices,
>  
> m=u*s*v.transposed,
>  
> now,as the documentation says that this algorithm is not implemented
> if the dimensions of m (m1 x m2) are such that m1<m2, I've  implemented
> my own version, re-reading some long-forgotten linear algebra book.
> In the process, I need to find the eigenvectors and the eigenvalues of  
> m*m.transposed,
> but these are printed out and apparently also calculated only to three  
> decimal digits
> by the command
>  
>      eigval, eigvec =  Eigen::symmv(m*m.transposed).
>  
> This causes the matrix u, which is constructed from the values  eigval,
> to have quite strange values for its determinant  (I get values of  1.06, 
> 0.96  for u*u.transpose.det, but u is unitary by definition, i.e.,  
> u*u.transpose.det=1).
>  
> Then, of course, m isn't remotely equal to u*s*v.transposed ....
>  
>  
> Is there a way of setting the precision of the eigenvalue and eigenvector  
> calculations
> any higher ? 
>  
> Thank you very much,
>  
> Best regards,
>  
> Axel    
>   
If m = u*s*v.transposed, what does m.transposed equal? Isn't it 
m.transposed = v*s.transposed*u.transposed?

In other words, can you transpose m ... m.transposed has dimensions m2 x 
m1 ... and take the SVD of m.transposed, and then recover the SVD of m 
from that?

I've forgotten most of my computational linear algebra too. :)

By the way, how large are m1 and m2?