[Haskell] small boys performance

[Andrzej Jaworski]

I am glad you are interested Dan.

I would give Haskell a clean start, particularly in the light of what Johan
has just said.  Calling external CAS engine (Matlab style) makes little
sense since tensor work is done in external packages and Haskell has its own
interesting libraries that could be turned into engine, e. g. Algebraic
Domain Constructor (DoCon) based on functors, which guarantee higher level
of needed abstraction than any other CAS but it unfortunately does not
handle non-commutative algebra. If somebody extends this I will eat my
previous mail:-)

Calculating the components of a tensor is a straightforward process (this
can be done with CAS) but manipulation of indicial tensor formulas is a
different animal. Using symbolic indices on tensors simplifies tensor
expressions, so for example A_{ij}S^{ij}-0, if (A) S are symmetric or
anti-symmetric allowing for more complicated properties like linear and
non-linear identities. Then preserving the indices in tensor symbolic form
gives you - to use Haskell jargon - mach more expressive power. Such as
typing an action functional to find the corresponding equations of motion.
With indexed objects as symbolic indices you can generate component-wise
calculations from symbolic input. Numerical indices will not allow for this.

I do not intend to bore anybody with differential geometry but as I was
pushed that far let me add that if Haskell was made to handle Riemannian
geometry it could be useful in next generation machine learning research
where logic, probability and geometry meet.

Regards,
-Andrzej