[boost] Re: Boost Mathematicians

From: David B. Held (dheld_at_[hidden])
Date: 2004-07-06 10:24:49

"David Abrahams" <dave_at_[hidden]> wrote in message
news:uoemxdu50.fsf_at_boost-consulting.com...
> Walter Landry <wlandry_at_[hidden]> writes:
>
> > David Abrahams <dave_at_[hidden]> wrote:
> >> * prod should be named operator*; this is a linear algebra library
> >>   after all.
> >
> > Actually, operator* can be thought of as either an inner product or
> > an outer product.  I'm not familiar with the innards of uBLAS, but
> > that is a problem common to most linear algebra libraries.
>
> IIUC in the domain of linear algebra when mathemeticians  write
>
>    AB
>
> for 2D matrices A and B, they mean ordinary matrix multiplication (I
> think inner product) yielding another 2D matrix, and that's what
> operator* should mean in a linear algebra library.  If you want an
> outer product yielding a 4D matrix I think you're basically outside
> the domain of linear algebra.

Actually, taking a look at mathworld.wolfram.com and a few other
sites, I have reached the tentative conclusion that outer products
can only be formed over vectors, not matrices.  Matrix multiplication
is well-defined and unambiguous, whereas vector multiplication
comes in at least four varieties: dot product, cross product (also
called the "outer product"), perpendicular dot product, and vector
direct product.

Hence, it would make sense that operator*(matrix, matrix) would
mean "matrix multiplication".  However, operator*(vector, vector)
is the problem child.  Using the most literal symbolism, it seems
that it should mean "dot product".  It is unfortunate that there is no
operatorX in C++, and that such user-defined operators cannot
be created.  In this case, I don't see a problem with Mr. Virgilio's
suggestion to use operator^ for cross product, except possibly
precedence rules and/or the fact that there is such a thing as a
"wedge product" which uses roughly the same symbol.  However,
it is not specific to linear algebra, so it shouldn't be a significant
problem.

Dave

---
Outgoing mail is certified Virus Free.
Checked by AVG anti-virus system (http://www.grisoft.com).
Version: 6.0.710 / Virus Database: 466 - Release Date: 6/23/2004