Arg, you beat me to it. I have a solution that takes about twice as
long as yours; I didn't you could use _fast_float_ like that:
import math
mypi = float(pi)
def f(x,k):
if x > 0:
tz = math.exp(x*(math.log(x)))
return[x,tz*math.cos(2*mypi*k*x),tz*math.sin(2*mypi*k*x)]
if x < 0:
tz = math.exp(x*(math.log(-x)))
return[x,tz*math.cos(mypi*(2*k+1)*x),tz*math.sin(mypi*(2*k
+1)*x)]
else:
return [0, 1, 0]
dx = float(0.02)
pic = line([[0,1,0],[0,1,0]]);
for k in srange(-3,4,float(1)):
pts = [f(x,k) for x in srange(-4.0,2.0,dx)]
pic = pic + line3d(pts)
-M. Hampton
On May 21, 12:07 pm, "William Stein" <[EMAIL PROTECTED]> wrote:
> On Wed, May 21, 2008 at 9:18 AM, mark mcclure <[EMAIL PROTECTED]> wrote:
>
> > On May 16, 2:44 am, Dan Pillone <[EMAIL PROTECTED]> wrote:
>
> >> Is there a way to plot x^x correctly?
>
> > A lovely article by Mark Meyerson entitled "The x^x Spindle"
> > appeared in Mathematics Magazine back in June of 96. The
> > article shows how to interpret the graph of x^x in 3-space,
> > using the complex values of x^x. This may be plotted in
> > sage as follows:
>
> > def f(x,k):
> > if x != 0:
> > z = exp(x*(maxima.log(x) + 2*I*pi*k))
> > return [x, real(z), imag(z)]
> > else:
> > return [0, 1, 0]
> > dx = 0.02
> > pic = line([[0,1,0],[0,1,0]]);
> > for k in range(-3,4):
> > points = [f(x*dx,k) for x in range(-4/dx,2/dx)]
> > pic = pic + line(points)
> > pic.show(frame_aspect_ratio=[2,1,1], figsize=8)
>
> > The above commands take *way* too long, presumably due to the
> > repeated calls to maxima's complex log function. Sage's log
> > function doesn't work. I was able to perform the loop in
> > maxima, but I was unable to get the result back into sage for
> > plotting.
>
> > The result is quite nice, though. There are countably many
>
> Yep.
>
> Since I know a bit more about how to optimize code in Sage,
> I redid your function but using more tricks. The result draws
> the above in 0.09 seconds (yep!).
>
> See attached or
>
> https://www.sagenb.org/home/pub/1831/
>
> -- William
>
> > different threads, corresponding to the different branches of
> > the complex log; they all spiral about the x-axis. You can
> > see the graph here:
> >https://www.sagenb.org/home/pub/1830/
>
> > The published notebook also shows a plot of x^x for x>0
> > together with points of the form (-p/q)^(-p/q) for odd q.
> > These points all lie where one of the threads pierces the
> > x-re(z) plane.
>
>
>
> Plot_branches_of_x_x_efficiently.sws
> 81KDownload
>
> Picture 3.png
> 41KViewDownload
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