On Monday, June 30, 2014 10:24:39 AM UTC-7, William wrote: > > On Mon, Jun 30, 2014 at 9:29 AM, Ondřej Čertík <ondrej...@gmail.com > <javascript:>> wrote: > > Thanks Volker for the tip, that does the job. More comments below: > > Another comment. Evidently Sage uses **Maxima** for > rational_simplify, hence the integer factorization that is required to > do this is not-surprisingly insanely stupidly slow, by modern > standards: >
Another, perhaps simpler, tactic would be to change the Maxima code for integer factorization. This would have the synergistic effect of making any other code that implicitly uses integer factorization in Maxima, faster as well. Also, it could be that the lisp you are using has slow arithmetic compared to some other lisps. Just a guess though. For sqrt(polynomial) it is plausible to do a "square-free" factorization -- not so expensive and reasonable payoff -- for polynomials. > > sage: p = next_prime(10^10); q = next_prime(10^11) > sage: a = sqrt(p^2*q)/p > sage: %time a.rational_simplify() > 1/10000000019*sqrt(10000000038300000037240000001083) > CPU time: 1.12 s, Wall time: 3.52 s > sage: %time factor(p^2*q) > 10000000019^2 * 100000000003 > CPU time: 0.00 s, Wall time: 0.03 s > > Don't even try increasing the size of p, q above... or it'll take forever. > > We should maybe consider sometime make our rational_simplify first > factor or something (I don't know), so it can leverage sage's much > better integer factorization code. > > William > > > > > > On Sun, Jun 29, 2014 at 11:52 PM, John Cremona <john.c...@gmail.com > <javascript:>> wrote: > >> Be careful though: > >> > >> sage: (sqrt(-2)*sqrt(-3)).simplify_radical() > >> -sqrt(3)*sqrt(2) > >> > >> i.e. you cannot use sqrt(a)*sqrt(b)=sqrt(a*b) everywhere without > >> reaching a contradiction. > >> > >> sage: bool( (sqrt(-2)*sqrt(-3)) == sqrt(2)*sqrt(3) ) > >> False > > > > Right, e.g. in sympy: > > > >>>> sqrt(-2)*sqrt(-3) > > -sqrt(6) > > > > > > Coming to the original example, sympy returns > > > >>>> sqrt(3)/sqrt(15) > > sqrt(5)/5 > > > > And Mathematica returns 1/sqrt(5). > > > > > > Here is a more difficult example, in SymPy: > > > >>>> 2**(S(1)/3) * 6**(S(1)/4) > > 2**(7/12)*3**(1/4) > > > > Mathematica returns the same thing. Sage, on the other hand, does not > > simplify this: > > > > sage: 2**(1/3) * 6**(1/4) > > 6^(1/4)*2^(1/3) > > sage: _.rational_simplify() > > 6^(1/4)*2^(1/3) > > > > > > What is interesting though, is that Sage does perform some > > factorization automatically, e.g.: > > > > sage: sqrt(12) > > 2*sqrt(3) > > > > Mathematica and SymPy do the same here. > > > > Here is even better example, in Sage: > > > > sage: sqrt(12)/sqrt(6) > > 1/3*sqrt(6)*sqrt(3) > > sage: _.rational_simplify() > > 1/3*sqrt(6)*sqrt(3) > > > > SymPy or Mathematica: > > > >>>> sqrt(12)/sqrt(6) > > sqrt(2) > > > > So the Sage rule seems to be --- factorize simple sqrt(N) into the > > form a*sqrt(b), do this for all sqrt() in the expression, and then > > just convert 1/sqrt(n) -> sqrt(n)/n, and return the result, after > > canceling common factors. > > > > I think a better rule is to either > > > > a) factor everything, like SymPy or Mathematica > > 2) do not factor anything, and just leave sqrt(12)/sqrt(6) as is. Then > > rational_simplify() should return the result from a). > > > > Ondrej > > > > -- > > You received this message because you are subscribed to the Google > Groups "sage-devel" group. > > To unsubscribe from this group and stop receiving emails from it, send > an email to sage-devel+...@googlegroups.com <javascript:>. > > To post to this group, send email to sage-...@googlegroups.com > <javascript:>. > > Visit this group at http://groups.google.com/group/sage-devel. > > For more options, visit https://groups.google.com/d/optout. > > > > -- > William Stein > Professor of Mathematics > University of Washington > http://wstein.org > -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
