Re: matrix library ?

From: lums_at_[hidden]
Date: 2001-03-14 09:31:44

--- In boost_at_y..., "David Abrahams" <abrahams_at_m...> wrote:

> So, "Matrix Template Library" is a misnomer?

Touche'!  Yes, the name was picked when I was still giddy after 
learning about the STL and slowly learning about generic programming 
general.  It was also named before our concept hierarchy was very 
well defined.

> Okay, but what use is a matrix library without easy-to-use/build 
linear
> algebra operations? That is, why would anyone want to make an "array
> library" as you define the term?

Lots of uses I would imagine.  C++ has built-in arrays (and 
valarray), after all, that people use for lots of things.  Some 
scientific things would include representing regularly discretized 
domains.  What I was getting at here is that the empasis (in my mind) 
of an array library is storage and data manipulation.  The emphasis 
in a linear algebra library is in the (abstract) linear algebra 
operations.  One would certainly build a linear algebra library on 
top of an array 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").
> 
> Can you try to make this a bit more concrete? Examples
> illustrating/contrasting the strengths of each type of library 
would help a
> lot.

An array library is primarily concerned with storage, data layout, 
data manipulation.  It's operations would primarily be geared towards 
data access, providing views of the data, re-organizing the data.

A linear algebra library would be concerned with mathemetically based 
concepts and linear algebra operations using those concepts.  For 
instance you would have an operation in a linear algebra library that 
looks like:

template <class LinearOperator, class VectorSpaceX, class 
VectorSpaceY>
apply(LinearOperator A, VectorSpace x, VectorSpace y)

where this realizes y <- A*x

This provides linear algebra functionality -- arrays are nowhere to 
be found.  In fact, if A is sparse, it will be a matrix, but it won't 
be a rectangular 2D array.  A might not be an explicitly represented 
matrix of any kind -- it could simply be a function that returns the 
result A*x without using a stored matrix (e.g., a matrix-free 
operator).

Is this answering your question?

These definitions are just one attempt and there are certain to be 
other interpretations.