[Haskell-cafe] matrix computations based on the GSL

[Henning Thielemann]

On Wed, 29 Jun 2005, Henning Thielemann wrote:

> On Wed, 29 Jun 2005, Jacques Carette wrote:
>
> > This is called operator overloading.  It is completely harmless because
> > you can tell the two * apart from their type signatures.  It is a
> > complete and total waste of time to use two different symbols to mean
> > the same thing.
>
> But the multiplication in A*x already needs multi-parameter type classes.
> :-)

I'm uncertain about how who want to put the different kinds of
multiplication into one method, even with multi-parameter type classes.
You need instances

 (*) :: Matrix -> Matrix -> Matrix
 (*) :: RowVector -> Matrix -> RowVector
 (*) :: Matrix -> ColumnVector -> ColumnVector
 (*) :: RowVector -> ColumnVector -> Scalar
 (*) :: ColumnVector -> RowVector -> Matrix
 (*) :: Scalar -> RowVector -> RowVector
 (*) :: RowVector -> Scalar -> RowVector
 (*) :: Scalar -> ColumnVector -> ColumnVector
 (*) :: ColumnVector -> Scalar -> ColumnVector

but you have to make sure that it is not possible to write an expression
which needs
 (*) :: Matrix -> RowVector -> RowVector

Further you need
 transpose :: RowVector -> ColumnVector
 transpose :: ColumnVector -> RowVector
 transpose :: Matrix -> Matrix
and you must forbid, say
 transpose :: RowVector -> RowVector

Not to mention many special (linear) operators which need to be prepared
for both column and row vectors, like Fourier transform, component
permutation, convolution, which essentially do the same on both types.


I assume that you mean with "common linear algebra idioms" transforms like
the following one. (x and y are column vectors)
 (x * y^T)^2
   = x * y^T * x * y^T
   = x * (y^T * x) * y^T    | because y^T * x is "scalar"
   = (y^T * x) * (x * y^T)
 (Analogously this can be done with bra-ket notation, i.e. <y| and |x>,
then transposition must be adjoint)
 Now your concerns are that this becomes too complicated if the
distinction between row and column vectors is dropped and instead
different operators are used?


> > I should know better than to get into a discussion like this with
> > someone who believes they singlehandedly know better than tens of
> > thousands of mathematicians...

Is the design discussion related to linear algebra for Maple documented
somewhere? For Modula-3 such a discussion was recorded on video tape (then
got lost :-( ) and was later edited in "Systems programming with Modula-3"
by Greg Nelson. It's fun to read and very informative and I haven't seen
such a discussion for other languages.