numpy (matrix solver) - python vs. matlab

[Russ P.]

On May 3, 10:30 am, someone <newsbo... at gmail.com> wrote:
> On 05/02/2012 11:45 PM, Russ P. wrote:
>
>
>
> > On May 2, 1:29 pm, someone<newsbo... at gmail.com>  wrote:
>
> >>> If your data starts off with only 1 or 2 digits of accuracy, as in your
> >>> example, then the result is meaningless -- the accuracy will be 2-2
> >>> digits, or 0 -- *no* digits in the answer can be trusted to be accurate.
>
> >> I just solved a FEM eigenvalue problem where the condition number of the
> >> mass and stiffness matrices was something like 1e6... Result looked good
> >> to me... So I don't understand what you're saying about 10 = 1 or 2
> >> digits. I think my problem was accurate enough, though I don't know what
> >> error with 1e6 in condition number, I should expect. How did you arrive
> >> at 1 or 2 digits for cond(A)=10, if I may ask ?
>
> > As Steven pointed out earlier, it all depends on the precision you are
> > dealing with. If you are just doing pure mathematical or numerical
> > work with no real-world measurement error, then a condition number of
> > 1e6 may be fine. But you had better be using "double precision" (64-
> > bit) floating point numbers (which are the default in Python, of
> > course). Those have approximately 12 digits of precision, so you are
> > in good shape. Single-precision floats only have 6 or 7 digits of
> > precision, so you'd be in trouble there.
>
> > For any practical engineering or scientific work, I'd say that a
> > condition number of 1e6 is very likely to be completely unacceptable.
>
> So how do you explain that the natural frequencies from FEM (with
> condition number ~1e6) generally correlates really good with real
> measurements (within approx. 5%), at least for the first 3-4 natural
> frequencies?
>
> I would say that the problem lies with the highest natural frequencies,
> they for sure cannot be verified - there's too little energy in them.
> But the lowest frequencies (the most important ones) are good, I think -
> even for high cond number.

Did you mention earlier what "FEM" stands for? If so, I missed it. Is
it finite-element modeling? Whatever the case, note that I said, "If
you are just doing pure mathematical or numerical work with no real-
world measurement error, then a condition number of
1e6 may be fine." I forgot much more than I know about finite-element
modeling, but isn't it a purely numerical method of analysis? If that
is the case, then my comment above is relevant.

By the way, I didn't mean to patronize you with my earlier explanation
of orthogonal transformations. They are fundamental to understanding
the SVD, and I thought it might be interesting to anyone who is not
familiar with the concept.