Re: Why is 2/3 not seen as rational? [was "plotting even function"]

[John B. Thoo]

Henry, I'm glad that I could "contribute" to the list. :-D

Seriously, though, my BIG THANK YOU goes to everyone who participated 
(and to those who may still participate) in this discussion for helping 
me to understand this a little better.  Clearly, numerics has 
subtleties (for me) for which I need to watch.  I have learnt from 
everyone's exchanges.

Cheers.
---John.


On Mar 21, 2005, at 8:17 AM, Henry F. Mollet wrote:

> WOW, I have to go back to school and learn about De Moivre's theorem to
> understand why Octave does what it does. I had blindly believed what
> Octave/Gnuplot was graphing. Had no idea what John Thoo's "simple" 
> question
> would lead to!
> Henry
>
> on 3/21/05 6:51 AM, Mike Miller at address@hidden wrote:
>
> > On Mon, 21 Mar 2005, Paul Kienzle wrote:
> >
> > > The cubed root function is multi-valued and Octave is choosing a
> > > different root than you expect.  Look at -8 for example:
> > >
> > > x^3 + 8 has three roots:
> > >
> > >  octave> roots([1,0,0,8])
> > >  ans =
> > >
> > >  -2.0000 + 0.0000i
> > >   1.0000 + 1.7321i
> > >   1.0000 - 1.7321i
> > >
> > > Octave chooses one of them:
> > >
> > >  octave> (-8).^(1/3)
> > >  ans = 1.0000 + 1.7321i
> >
> >
> > It seems to 'choose' using De Moivre's theorem.
> >
> > For x > 0 and integer n != 0:
> >
> > -x = x * (cos(pi) + sin(pi)*i)
> >
> > (-x)^(1/n) = x^(1/n)*(cos(pi/n)+sin(pi/n)*i)
> >
> > As someone else pointed out, the abs function will force the answer 
> > to be
> > a real integer:
> >
> > - Mapping Function:  abs (Z)
> >     Compute the magnitude of Z, defined as |Z| = `sqrt (x^2 + y^2)'.
> >
> >     For example,
> >
> >          abs (3 + 4i)
> >               => 5
> >
> > So, for x > 0 and integer n != 0,
> >
> > -abs((-x)^(1/n))
> >
> > seems to give the desired answer when n is odd, but that isn't very
> > helpful because -(x)^(1/n) gives the same answer when n is odd.
> >
> > On the other hand, this seems to do what the guy originally wanted 
> > (for
> > scalar integer values of a and b and any real-valued x vector.):
> >
> > abs(rem(a,2))*abs(rem(b,2))*sign(x).*(abs(x).^(a/b)) +
> > (1-abs(rem(a,2)))*abs(x).^(a/b) + 
> > abs(rem(a,2))*(1-abs(rem(b,2)))*x.^(a/b)
> >
> > Mike



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