Re: Generalized Eigenvector and Singular Value Decompositions

• To: mathgroup at christensen.cybernetics.net
• Subject: [mg306] Re: [mg304] Generalized Eigenvector and Singular Value Decompositions
• From: rickli at aut.ee.ethz.ch (Martin Rickli)
• Date: Fri, 9 Dec 1994 11:41:52 +0200

```[...]
>
>Being a good consumer, I want instant gratification and, therefore, was
>hoping that some kind soul out there had run into similar needs and had
>developed the requisite Mathematica code. Otherwise, I fear that I shall have
>to dig into the numerical aspects myself. Since the approaches which jump to
>mind feature SVDs it appears that I would have to start with generating a
>more efficient SVD. :(
>
>Mayhaps, if any of y'all are concerned about the numerical matrix algebra
>aspects of Mma, you could drop a note to WRI and explain the urgency of their
>working on my problems. ;)
>
>Any help would be much appreciated,
>
>Mark.
>---
>Mark Kotanchek
>Signal Processing Dept - 363 ASB
>Applied Research Lab/Penn State
>P.O. Box 30
>State College, PA 16804
>
>e-mail: kotanchek at psu.edu (NeXTmail)
>TEL:    (814)863-0682
>FAX:    (814)863-0753

Yes, we are very concerned about numerical matrix algebra aspects of Mma.
Whenever possible we use Matlab which can ONLY do numerical matrix algebra.
The built-in algorithms can be expected to be state of the art and its fast
thanks to numerically optimized techniques.

These days I tripped over a bug in SingularValues which seams to appear
only in the Mac version of Mma (v2.2).
Try the following:

mat = {{1, 1}, {-Sqrt[2], Sqrt[2]}};

SingularValues[ SetPrecision[mat,19] ] (* Mac MachinePrecision *)
SingularValues[ SetPrecision[mat,20] ]

This matrix shouldn't pose numerical difficulties but SingularValues fails
to converge if calculations are done with machine precision numbers. With
higher precision numbers the result is correct.
As I said above, this seams to appear only in the Mac version of Mma.

Another problem related to matrix algebra is with Dot.

Dot[{{0}}, {{1.2}}] results in {{0.}} instaed of {{0}}

This might lead to unexpected results/problems.
This happens both in Mac and Unix-Sparc versions of Mma.

May be we should promote the existing link between Mma and Matlab (as well
as Matlab -> Mma). If you're content with machine precision results that's
the way to go IMHO (besides easier matrix notation).

I'm not affiliated with Mathworks (Matlab), but a frequent user of the program.

Your opinions are welcome.
Sincerely

______________________________________________________________________
Martin Rickli                           E-mail: Rickli at aut.ee.ethz.ch
Automatic Control Laboratory
ETH Zurich                              Physikstr. 3, ETL K12
CH-8092 Zurich,  Switzerland

```

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