On Fri, 10 Jun 2005, Keean Schupke wrote:
Jan Skibinski's page seems to have disappeared so there is no source I can find for the Orthoganal stuff - besides which I am interested in efficency - and that basically said the algorithms were experimental and use Haskell lists...
I have some private copy: http://www.math.uni-bremen.de/~thielema/Research/LinearAlgebra/
The HaskellDSP defines simple matrix operations, but does not put the operations into the numeric, fractional, and floating classes. This makes the resulting expressions hard to read.
Numeric classes do not exist for readability in the sense of infix operators. If infix operators were the main reason simple operators would suffice. The reason why these operators are methods of classes is that it shall be possible to write mathematical expressions which can be universally used for different types of numbers, such as Float, Double, Rational, Integer and so on. Prelude's class framework is oriented at the scalar types it provides. My concern is: Are matrices a natural extension of these scalar types? What is (fromInteger 1) for the matrix class? The identity matrix or the matrix which consists entirely of ones? Are there non-trivial algorithms written for numbers which can be re-used for matrices without re-writing? If not, why pressing matrices into this framework? You could also define (+), (-), (*) for Strings. But what is the benefit? Are there numerical algorithms which can be re-used for Strings? (Now when I write it, I remember the algorithm for searching a dictionary of strings by the method of false position. But the Prelude numeric class framework wouldn't help here, too, since you need a division of type String -> String -> Double, or so.) If readability in the sense of "writing less" is your main goal then just define infix operators for the functions found in HaskellDSP. (*>) is a good choice for scaling a vector or a matrix, this symbol is already used in numericprelude. You may define (.*) for element-wise multiplication which is already known from MatLab. It is more readable than everywhere (*) because it tells the user something about the types of the operands.
LAPACK does not appear to support higher dimensional matrices... is this right?
Yes, why should it do so? Every finite dimensional operator can be represented by a matrix.
I have found a couple of papers on generalising matrix algebra to higher dimensional matrices... So it seems to be possible...
Certainly, but LAPACK is very low-level. It's your task to break operations on various types of vectors down to basic linear algebra. The point is that every vector from a finite dimensional space can be represented by an array and each linear operator can be represented by a matrix. Everything which is structured differently can be converted to this but it is your task not that of LAPACK or any other linear algebra package.