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MathGroup Archive 2002

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RE: RE: RE: Generating Two Unit Orthogonal Vectors

  • To: mathgroup at smc.vnet.net
  • Subject: [mg36498] RE: [mg36476] RE: [mg36448] RE: Generating Two Unit Orthogonal Vectors
  • From: "DrBob" <drbob at bigfoot.com>
  • Date: Mon, 9 Sep 2002 00:29:45 -0400 (EDT)
  • Reply-to: <drbob at bigfoot.com>
  • Sender: owner-wri-mathgroup at wolfram.com

Daniel,

I don't mean to be overly critical of WRI's documentation -- it's very
good, as such things go.  Nor do I like to overlook chances to make it
better.  Do you?

Your own comments below point out that we should expect vectors
resulting from NullSpace to be "orthonormal by the usual
conjugate-symmetric inner product on C".  But that's not spelled out
(documented) for dummies like myself who don't know that's a natural
result of singular values decomposition.

The "mission" of NullSpace (or any function) is to adhere to
documentation, so reasonable persons may differ on whether orthogonality
is a feature we should depend on.

The mission of documentation is to tell us what to expect.  When it
doesn't, the result is that we spend all this time discussing issues
online, trying to figure things out.  A simple "don't depend on
orthogonal results" would be nice, if that's the intent.

In any case, I just spent ten minutes LOOKING for implementation notes
for NullSpace, and have not found any.  Searching for "implementation
notes" doesn't help and there's no link from NullSpace.  What use is
documentation I can't find?

In general, I don't like Mathematica's quirky Help Browser, in which I
cannot search for anything that's not indexed.  Every other help engine
on my computer (and there are hundreds) allows me to search for words,
and that's exactly what I need in order to find all mentions of
NullSpace.

Bobby Treat

-----Original Message-----
From: danl at wolfram.com [mailto:danl at wolfram.com] 
To: mathgroup at smc.vnet.net
Subject: [mg36498] Re: [mg36476] RE: [mg36448] RE: Generating Two Unit Orthogonal
Vectors


DrBob wrote:
> 
> Wouldn't it be nice if NullSpace's behavior were DOCUMENTED?
Otherwise,
> it's futile to give it approximate numbers expecting any particular
> behavior.  Even if it always works, it may not work in the next
version
> of Mathematica.
> 
> Bobby

The expected, and documented, behavior is that the output should be a
basis for the null space, that is, solutions of the homogeneous matrix
equation A.x==0. If this were to stop working then that would be a
serious bug. Is this the behavior you mean?

The implementation notes of the manual mention that approximate
NullSpace is based on a singular values decomposition. This in fact
gives resulting vectors that are orthonormal by the usual
conjugate-symmetric inner product on C (though these are now not
"normal" to the original vector in this same inner product, unless they
are real-valued). But this basis-orthogonality is not part of the
mission of NullSpace and moreover should not become part of it. Hence
that particular (and implementation dependent) aspect of NullSpace
should not become documented.

Daniel Lichtblau
Wolfram Research




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