Jaime Villate <villate at fe.up.pt> writes:
> On 11/01/2012 09:40 AM, Rupert Swarbrick wrote:
>> What I*don't* know is whether it's possible to do this automatically. I
>> can see that one could split things up for rational functions by
>> computing the denominator and using allroots(), but I suppose that isn't
>> really generic enough for someone to have been interested to write it.
> Hi,
> in the case of plot2d, I think that a better approach would be to stop
> sending point coordinates to Gnuplot when those points fall outside
> the domain (defined by [x,x1,x2] and [y,y1,y2]) and start a new branch
> when the points fall again within the domain. I have already done
> something similar for plotdf and I will attempt the same for a future
> plot2d, unless somebody has a better idea?
Ah, that does make sense. I suppose you still get the problem that a
narrow pole could "evade detection". A bit of thought gives the
following slightly contrived example:
f(x) = signum(x) * abs(x)^(-1/n)
The larger n is, the "thinner" the pole will be. Assuming that 0 isn't a
grid point, suppose that the grid points x-, x+ either side of zero both
have absolute value greater than 1/k for some k > 0. Then f(x-) >
-k^(1/n) and f(x+) < k^(1/n).
Assuming that y1 < 0 < y2, in order that neither falls out of your y
range, I just need n such that log(k)/n < log(min(-y1, y2)), which is
doable as long as both are bigger than 1.
So, yeah, ensuring that my example breaks the scheme you suggest did
take a little bit of calculation. Also, it's manifestly not a rational
function, so it's not like my suggestion of "pole-hunting" would solve
things any better.
Does anyone know whether there's been serious attempts to solve this
problem more generally? I suppose there's no hope of a good solution for
all smooth functions, but maybe something based on Laurent series could
deal sensibly with functions that have (say) a positive convergence
radius at all points. (We'd be plotting over a compact interval, of
course, so we only need finitely many such expansions) I haven't thought
about this very seriously though, and I'm a topologist rather than an
analyst by training, so maybe I've missed something obvious?
Rupert
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