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. From another perspective, the integers are encodings of the vector space types, but a very restricted subset of vector space types. If we take the encoding out of the interface, we might get a much mor general library, allowing for linear maps between *any* vector spaces, not just ones of the form R^n. For instance, linear maps are vector spaces also, so we can handle "LMap u (LMap u v)" (the range type of the second derivative of a u->v function). 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? 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. It may be worth exploring, though. Cheers, - Conal -----Original Message----- From: Lennart Augustsson Sent: Thursday, July 07, 2005 5:44 AM To: David Roundy Cc: haskell-cafe@haskell.org Subject: Re: [Haskell-cafe] matrix computations based on the GSL David Roundy wrote:
The issue is that Haskell (as far as I understand, and noone has suggested anything to the contrary) doesn't have a sufficiently powerful type system to represent matrices or vectors in a statically typed way. It would be wonderful if we could represent matrix multiplication as
matrix_mul :: Matrix n m -> Matrix m o -> Matrix n o
Actually, Haskell does allow you to do that. But the syntax of the types gets pretty horrendous. I'm sure Oleg will show you how any moment now. :) -- Lennart