On 10/14/06, A. M. Archibald <[EMAIL PROTECTED]> wrote:
<snip>
Well, that would work. On the other hand, it seems overly pessimistic from my experiments here. What seems to be a better guide is rank reduction. For instance, I can do a perfectly decent fit to Greg's data with single precision, degree 10, and ~1000 data points with rcond = ~5e-7 (effectively single precision precision). Degree 11 blows up entirely even though it loses rank. It is also true that rcond=1e-3 fails for degree 11 even though rank is strongly reduced. What looks to be taking place is roundoff error in evaluating the polynomial in single precision in the presence of higher order terms that still belong to the reduced basis functions and rank reduction is a good indicator of this. Bear in mind that I am now normalizing x by dividing it by its largest element, which futzes with the condition number. The condition number of the unscaled fit doesn't bear thinking about.
Hmmm, I wonder if we have a dictionary of precisions indexed by dtype somewhere?
I see some folks bitchin' at it when I google. There are remarkably few hits, though. For the moment I am going with the smaller rcond numbers and raising an error on rank reduction. I suspect something similar should be done for pinv.
Chuck
On 14/10/06, Charles R Harris <[EMAIL PROTECTED]> wrote:
>
>
> On 10/13/06, A. M. Archibald <[EMAIL PROTECTED] > wrote:
> > On 13/10/06, Tim Hochberg <[EMAIL PROTECTED]> wrote:
> > > Charles R Harris wrote:
>
<snip>
Numerical Recipes (http://www.nrbook.com/a/bookcpdf/c15-4.pdf )
recommend setting rcond to the number of data points times machine
epsilon (which of course is different for single/double). We should
definitely warn the user if any singular value is below s[0]*rcond (as
that means that there is effectively a useless basis function, up to
roundoff).
Well, that would work. On the other hand, it seems overly pessimistic from my experiments here. What seems to be a better guide is rank reduction. For instance, I can do a perfectly decent fit to Greg's data with single precision, degree 10, and ~1000 data points with rcond = ~5e-7 (effectively single precision precision). Degree 11 blows up entirely even though it loses rank. It is also true that rcond=1e-3 fails for degree 11 even though rank is strongly reduced. What looks to be taking place is roundoff error in evaluating the polynomial in single precision in the presence of higher order terms that still belong to the reduced basis functions and rank reduction is a good indicator of this. Bear in mind that I am now normalizing x by dividing it by its largest element, which futzes with the condition number. The condition number of the unscaled fit doesn't bear thinking about.
Hmmm, I wonder if we have a dictionary of precisions indexed by dtype somewhere?
I don't get the impression that the warnings module is much tested; I
had similar headaches.
I see some folks bitchin' at it when I google. There are remarkably few hits, though. For the moment I am going with the smaller rcond numbers and raising an error on rank reduction. I suspect something similar should be done for pinv.
Chuck
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