Roots of unity
Hans Peter Würmli
wurmli at freesurf.ch
Sun Jan 20 17:12:43 CET 2002
On Sun, 20 Jan 2002 15:38:14 +0100 (CET)
"Richard B. Kreckel" <kreckel at thep.physik.uni-mainz.de> wrote:
To: ginac-list at thep.physik.uni-mainz.de
Subject: Re: Roots of unity
> > > which is correct, if one simplifies 1/2+(-27/2+3/2*sqrt(93))^(-1)-1/18*sqrt(93) to 1
> > This is already way cool, isn't it? But, to "simplify" is not
> > well-defined. Such computations should not be done automatically at the
> > level of the anonymous avaluator (i.e. classwhatever::eval()).
> Oops, I just realized that this particular case above is even more trivial
> and what I wrote doesn't apply to it yet. Simply calling normal() will
> get the job done, since there is only one sqrt() in there.
I had tried with normal(), but now I realise that it makes a difference whether I use sqrt(2) or pow(2, numeric(1,2)).
If I take the same expression, but the numeric symbol replaced by, say, x, the same simplification does not take place.
I of course fully agree with you that "simplification" is not well defined terminology. On the other hand, you could still try to represent, say,
1/x^(1/2) as x^(1/2)/x
i.e. having all radicals of symbols in the numerator. Provided a didn't make a mistake, the above "simplification" can be accomplished generally. Take N(x^(1/n)) and D(x^(1/n)) two polynomials, regarding x^(1/n) as a symbol (where n is the lcm of the denominators of all exponents). If I can accomplish a full partial fraction representation of N/D, then I can use
(x^(1/n) - a) * (x^((n-1)/n)/a^(n-1) + x^((n-2)/n)/a^(n-2) + ... + 1) = x/a^(n-1) -a
If I look at the algorithms for symbolic integration, I have at least the hope that one could do a lot without having to find all roots of the denominator.
By the way: thank you for the references. If I have time (which is doubtful) I will consult them. I tried the little program for the roots of a third order polynomial rather to get used to GiNaC and to learn how one would, if one wanted, extend GiNaC to have algebraic structures and operations on them.
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