[Haskell-cafe] matrix computations based on the GSL

[Henning Thielemann]

On Thu, 7 Jul 2005, Conal Elliott wrote:

> Maybe we could solve this problem in a simple and general way by working
> with a more abstract notion of linear maps, rather than the matrices
> commonly used to represent linear maps.  Instead of "Matrix n m", where
> n and m are either integers (requiring something like dependent types)
> or type encodings of integers, use "LMap u v", where u and v are vector
> spaces, e.g., R^n and R^m.  This change would eliminate the need for
> dependent types or integer encodings.

I would also dislike integer encodings in the matrix type because it is
very sensible to have other types of indices. I would just do it how it is
solved for Arrays: Two index types for a matrix but the actual matrix size
is not coded in the type. E.g. I would like to multiply a matrix of type
Matrix (Bool, Char) Double with a vector of type Vector Char Double
obtaining a result of type Vector Bool Double.

> Also, it's clear that "LMap R R^n" and "LMap R^n R" are different types
> (for n/=1), so we can distinguish between "row" and "column" vectors
> without creating more types.  Are there other reasons for orienting
> vectors?

My point was that vectors naturally do _not_ represent linear maps at all,
but they are the objects linear maps act on. If I process an audio signal
or an image I can consider it well as vector but why should I consider it
as linear map?

> I can't call these comments a concrete proposal, as it's not clear to me
> how to define LMap and what operations it supports.

The problem will be that there is no canonical representation for linear
maps in general vector spaces. I consider a function of type (Double ->
Double) as an approximation to a real function but we don't have a general
representation for linear operators on real functions.

The remaining problem is that Vector and Linear Map are relative terms.
That is, as you point out, a Linear Map can be a Vector with respect to a
higher order Linear Map. I think in contrast to real mathematics we have
to distinct due to computational restrictions. On the one hand we can work
with a vector space type class which only specifies what basic operations
(namely add and scale) shall be supported by types that want to be called
vectors.  This type class has the full flexibility but no canonical
representation for linear maps. On the other hand we define finite
dimensional vectors (array like) and matrices as representations for
linear maps. For the case of higher order linear maps there should be a
conversion between Matrix and Vector.