On 10/2/2013 11:52 AM, Robert Dodier wrote:
> On 2013-10-02, Richard Fateman <fateman at gmail.com> wrote:
>
>> On 10/2/2013 12:06 AM, Robert Dodier wrote:
>>> If we can't even solve a simple equation unambiguously,
>> You are missing the point. We have solved the equation and offered all
>> the roots.
> I dunno. There are at most three distinct solutions to solve(z^3 = x, z).
> There's no reason for solve to return something that could be
> interpreted as representing more than three values.
The reason for this is that it is clumsy to express the roots this way:
Let q be any of the three specific cube roots of x.
The three solutions are then q, q*exp(2*%pi*%i/3), q*exp(-2*%pi*%i/3).
I am pretty sure that solve computes them that way, and then turns the
answers
over to some simplification program. For contrast, see solve(z^11=x,z);
So solve computes the right number of solutions.
>
> (-1)^(1/6) and expressions like it are unambiguous only if it is agreed
> which root is represented by that -- e.g. always the principal root,
It seems to me that it is not possible to identify the principal root,
in general, so we can't
do much with this. To illustrate:
let x = r*exp(%i*theta) where for exposition here we agree that r is
known to be positive.
Of course we don't know this at all and it affect the result, but I'm in
a generous mood.
Then the principal kth root of x would be r^(1/k) *
exp(%i*theta/k), with the condition
that -%pi< theta <= %pi. Which we also don't know, and it affects
the solution too.
We could be told, after the fact and after choosing a principal root,
that theta=45.
> never any other, and never the whole set of roots.
So for the solution of a quadratic equation you would just return a
principal root?
I don't understand this.
> For better or worse,
> Maxima is, to the best of my knowledge, not consistent about that, so
> until Maxima is consistent, a user can't assume an unambiguous result.
domain:real, (-8)^(1/3) returns -2
domain:complex returns 2*(-1)^(1/3). Is this helpful? Consistent?
I think that a thorough reading of material on branch cuts and complex
mappings would
clear some of the air; seeing how it does or does not apply to the more
general
situation where we have too many variables is daunting.
We would like subst(a,b,simplify(f(b))) to be equal to
simplify(f(a)) but we won't get this
to happen generally.
RJF
>
> best
>
> Robert Dodier
>
> _______________________________________________
> Maxima mailing list
> Maxima at math.utexas.edu
> http://www.math.utexas.edu/mailman/listinfo/maxima