[comp.specification] Re^2: Linear Algebra and Specifications

steve@hubcap.clemson.edu ("Steve" Stevenson) (07/04/90)

jwb@cepmax.ncsu.EDU (John W. Baugh Jr.) writes:

>Of course, I know that numerical analysis and linear algebra are
>"intimately intertwined."  If you re-read my post you'll see that
>I'm talking about _specifying_matrix_operators_.   My question
>regards the level at which such operators are specified.  For
>example, if I write a program that happens to make use of matrices,
>I could take advantage of the "nice" algebraic properties of
>real numbers, and hence the matrices defined over them, to reason
>about my program.

Linear operators (and hence matrix operators) don't just come about, since
they're related to the problem at hand. Are you asking about how the
modeling is done? Remember that matrices are but one way of specifying an
operator. For example, for years, Schrodinger's equation was thought
to be independent of another quantum system (the name escapes me) which
was done in matrices. Eventually they were shown to be equivalent; 
Schrodinger's formulation won out on other grounds. Another example is
the relationship between differential and integral equations. The
algebraic properties are there, see Wilkerson and a myriad of others who
have worked on these problems.

>  Of course, matrices of finite-precision numbers
>don't share these "nice" properties, which might encourage one to
>specify even more details, namely a floating point representation
>for matrix elements.

They are there, but you have to reason on norms and bounds. Don't see
why you are even interested in floating point operations if you're in
the spec. business.

> The trouble is that pretty soon I'm at an
>"implementation" level, and I might just as well be writing
>Fortran code (since I'm specifying all these details anyway).
>The dramatic difference between these two levels of specification
>is the "chasm" I'm referring to.  So, my question is: has any work
>been done in the area of specifying matrix operators?

And so I ask you, what problem and what matrix operators? The model to
be worked IS the specification. Whether you use matrices or not is
dependent on a lot of things. Give me an example.
-- 
===============================================================================
Steve (really "D. E.") Stevenson           steve@hubcap.clemson.edu
Department of Computer Science,            (803)656-5880.mabell
Clemson University, Clemson, SC 29634-1906