jerzy.karczmarczuk at info.unicaen.fr jerzy.karczmarczuk at info.unicaen.fr
Sat Sep 30 07:20:37 EDT 2006

```Tamas K Papp writes:

> Henning Thielemann wrote:

>> Actually, laziness allows me to formulate algorithms that look more like
>> the specification of the problem than the solution. E.g., I can formulate
>> the solution of a differential equation in terms of a power series or time
>> series in that way. However I have to put some effort into formulating it
>> in a way that works. E.g. I'm only able to solve such equations if it is
>> possible to express the second derivative in terms of the first and the
>> zeroth one. Computer algebra systems are essentially better here.
>
> In my experience, most people use CAS interactively: they encounter an
> integral or a PDE that's difficult to solve, so they type it into
> Mathematica (which frequently cannot solve it either, then you go
> crazy, numerical, or both ;-).  It is more like a sophisticated
> symbolic calculator with a lot of patterns built in for manipulating
> expressions.

I should have reacted earlier...
*universal* languages. CAS such as Maple, Mupad, and also Mathematica in
a sense are also universal, but they simply have

* enormous libraries permitting to deal with symbolic expressions ;
* Pattern matching/rewriting contraptions useful to manipulate deeply
intricate structures.

All this CAN BE DONE in Haskell as well, but reinventing the wheel is
rarely interesting (sometimes is, though) (*).

Still CAS are no magic, and - getting back to Henning's example: No CAS
will give you the power expansion of a solution of a differential equation
if you can't algorithmize the needed recurrencies. But if you can, then
you can do it in a numerical setting using Haskell or something else.
Haskell allows you to apply laziness in a "perverse" way as well, for
example solving - as a power series - a *singular* equation (Bessel), where
you define the first derivative through the function and its second
derivative, although at first glance this seems mad. But look here,
a quite old stuff... : Theor. Comp. Sci. 187,(1997), pp. 203--219.
on-line :
http://users.info.unicaen.fr/~karczma/arpap/lazysem.pdf
page 4...

Jerzy Karczmarczuk

=======

(*) A reference for your culture:
Jorge Luis Borges, "Pierre Ménard, the Author of Don Quixote". It is a
story about a person who WROTE (not: copied) a book, exactly, letter by
letter, identical to the book of Cervantes. It was identical, but having