Re: matrix library ?

From: lums_at_[hidden]
Date: 2001-03-13 08:14:22

It is important here to make the distinction between a linear algebra 
library and an array library.  Blitz++ is an array library.  MTL 
(e.g.) is a linear algebra library.  One would expect (as is the 
case), that linear algebra operations are not as easy in an array 
library as they would be in a linear algebra library.

Array libraries and linear algebra libraries are easily confused 
since linear operators (matrices) are commonly represented (at least 
in the dense case) as two-dimensional arrays.

This distinction becomes quite obvious when developing layers of 
abstraction and concepts for a linear algebra library.  Formally (and 
this comes from the precise definition that mathematicians have 
developed over the years), a linear algebra consists of a linear 
vector space (which is also formally defined) plus a set of linear 
transformations that can be applied to that space.  I think the 
relationships between these entities has been well specified (both by 
mathematicians and more relevant to a C++ library, by the generic 
programming group at RPI).  Arrays do not appear anywhere at this 
layer.  In fact, classic two dimensional arrays are only a small part 
of what is required in the representation layer of a linear algebra 
library.

I have spent a good part of my professional career thinking about 
these issues and they are quite non-trivial when you really start 
digging into them.  It does not seem difficult at first -- and 
indeed, the world is full of not-very-good "matrix libraries" because 
they are so easy to construct.  Most such libraries I would not call 
linear algebra libraries because most such libraries do not have any 
kind of conceptual analysis of the linear algebra problem domain (so 
I use the name "matrix library").

At the risk of tacky self-promotion, I would recommend Jeremy Siek's 
MS thesis (about MTL) as an example of the kinds of analysis (and 
implementation) that are required for a real linear algebra library:
http://www.lsc.nd.edu/downloads/research/mtl/papers/thesis.pdf

--- In boost_at_y..., Toon Knapen <toon_at_s...> wrote:
> Dave Steffen wrote:
> >
> >  Not at all - this is more or less how Blitz++ does it, although 
the
> >  syntax isn't quite as clean. It looks something like
> > 
> >  A = sum(B(i,k) * C(k,j), k); // A_ij = B_ik * C_jk
> >
> This syntax makes it also difficult to write y = Ax +b 
> (A matrix ; y,x,b vectors)
> 
> Blitz++ is indeed a superb library but this notations makes
> it difficult to use if for linear algebra.
> 
> t