Re: evaluation-- one or many levels, your thoughts?

• To: mathgroup at smc.vnet.net
• Subject: [mg117031] Re: evaluation-- one or many levels, your thoughts?
• From: David Bailey <dave at removedbailey.co.uk>
• Date: Tue, 8 Mar 2011 05:34:57 -0500 (EST)
• References: <ikq8gs\$7vl\$1@smc.vnet.net> <ikt5i5\$5\$1@smc.vnet.net> <il2d46\$61n\$1@smc.vnet.net>

```On 07/03/2011 10:47, Richard Fateman wrote:
> On 3/5/2011 3:07 AM, David Bailey wrote:
> ..
>
>

> e.g.
> x1=x2
> x2= Sum[a[i]*v^i,{i,1,1000}]
>
> x3=x1  is, or should be, fast  whether it really assigns literally x2
> or the Sum.
It is too fast to measure - I have just tested it! Mathematica seems to
be VERY good at tracking which parts of expressions need evaluating. I

>
> 2. You are mimicking the execution of a program in a language like
> FORTRAN, but are doing everything symbolically so as to find bugs,
> and relate the computed values to expected mathematical formulas.
> The fact that x=x+1   produces, for x, a value of "x+1" means that
> x was not properly initialized and you have found a bug in the FORTRAN
> program.
>
If I wanted to do this, I'd probably replace X by a sequence of
variables representing the successive values, so we would end up with
something like
X3=X2+1

I'd certainly not represent the operation x=x+1 by that Mathematica
expression!

Before you object that this would generate an impossible number of
variables, it is worth pointing out that such a simulation would
probably also generate some horrendously large expressions, if performed
for many cycles.

It is interesting that you need to resort to such an obscure example!

I think complete evaluation is desirable because there is no clear
distinction between programming variables and algebraic variables.
If you write

x=f[i,j,l,m] (where f stands for a complicated function)

you clearly want f to evaluate if i is a loop variable, but not if none
of the variables have changed since the expression has been created - in
other words, Mathematica would need to track variable changes, whether
it performed exhaustive evaluation or not!

More generally, I think any CAS system has to contain compromises,
particularly when it is also meant to a wider role as a numerical
calculator. When I first encountered Mathematica, the exhaustive
evaluation paradigm surprised me, but it is clearly implemented in an
efficient way, so I am happy with it. Furthermore, there is absolutely
no way in which you could change this now, so any discussion seems a
little pointless.

David Bailey
http://www.dbaileyconsultancy.co.uk

```

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