Simplifications of 3*sqrt(2)/sqrt(3)/sqrt(6)

Am Freitag, den 03.07.2009, 14:55 -0400 schrieb Stavros Macrakis:
> On Fri, Jul 3, 2009 at 1:48 PM, Robert Dodier
> <robert.dodier at gmail.com> wrote:
>         > And I think always calling factor for these cases may not be
>         such a good
>         > idea.  What if the number is the product of two fairly large
>         primes or
>         > even a prime?  Maxima will spend a lot of time trying to
>         find the factors.
>         
>         
>         I don't have a very strong opinion about it, but I am leaning
>         towards
>         omitting the call to factor from simplification, and moving it
>         into
>         radcan or something like that, on the general theory that we
>         should
>         try to avoid potentially expensive operations in
>         simplification
> 
> I agree that general simplification limit the amount of time it takes
> on integer factorization, and that an explicit command (e.g. radcan)
> should be called for larger cases.
> 
> Here's how I'd limit it...  suppose we think of a 'largish Maxima
> calculation' as involving 1000 factorizations (that's a lot), and we
> don't want more than say 10 seconds of CPU to be spent on
> factorization for this simplification. Then we have a budget of about
> 10mS per factorization, which on my machine would allow integers up to
> about 10^13.  This is a worst-case analysis, but 10^13 should in any
> case be adequate for most work.

The factorization of integers which are a base of an exponentiation is
already done in the Maxima routine simpnrt. The algorithm in simpnrt
does the factorization only on a base of the list of small prime numbers
up to 10000.

Therefore, I have proposed the same scheme to factorize integers in TMS.
We only loose computation time, because some work is doubled. The reason
is, that simpnrt has already all factors by hand, but does not use these
factors completely. This could be improved, if we agree about the
usefulness of simplification like the expression
3*sqrt(2)/sqrt(3)/sqrt(6) --> 1.

The function $factor will not work in general for big numbers. Try e.g.
factor(prev_prime(100!)*prev_prime(200!)). After a long time I have got
a Lisp Error.

Dieter Kaiser