Re: complexity of numerical software

Jaap Spies wrote: 
Jean-Claude Arbaut wrote:

Jaap Spies wrote:

Jean-Claude Arbaut wrote:



Yes. We have the same question!



Okay. I thought we were talking about code like SLATEC, NSWC, CMLIB, etc. They have (almost) nothing to do with algorithms to find subsets, which I would classify with discrete or combinatorics algorithms.


We were discussing the cost of a writing a system (a new CAS), not
writing new "standard" libraries. In the "new" system there will be
numerical programs using the well tested basic libraries. You have to
see my remarks in this light!

When I write sqrt(2) in a CAS, or writing code that uses sqrt() I do not
want to be bothered by the gory details of the underlying "low-level"
libraries.

This explains my statement 

I you write a new CAS, it would be silly to use only those "standard"
libraries: they compute with IEEE754, hence the precision is very
limited for a CAS. You'd probably prefer a multiprecision library,
but I don't know of a standard one... GMP and others only compute
with integers, but you need approximate values of many transcendental
functions.

You'll also need to write special algorithms for that, since the problem
of computing with high precision numbers or with rationnals have its own difficulties: among others, how do you reduce the size of numbers (to
increase speed) appearing in computations, like the solution of a linear system of equations. These are also numerical computations, though not
the classical "number crunching" BLAS.

Oh, and just an idea: what is *really* a CAS ? I have already used Maple, Mathematica and MuPAD, and then I discovered GAP: no more
"formal" computations, but "true" mathematical objects :-) I mean,
it looks like in Maple, the elementary object is an "expression",
which can be almost anything. In GAP, there are no expressions, only
values: polynomials, finite groups, matrices, are _values_. That makes
a great difference I think. Maybe I'm wrong...
.

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