Re: [sage-support] Plot the zeros of a complex function

[William Stein]

On Sun, Jun 9, 2013 at 3:25 PM, <computati...@gmail.com> wrote:
> Suppose I have a complex function f(z) with a continuous family of zeros (e.g., f(z)=z-|z|)
>
> Is there a way to easily plot the set of zeros of f in sage, regardless of how complicated the function f is?
>

You might find complex_plot useful. For example, for

f(z) = z - abs(z)
complex_plot(f, (-3,3), (-3,3))

you'll see a *black line* at the zero set of f(z).

In the notebook you mind find an interact like this useful:


z = var('z')
@interact
def _(f = z-abs(z), B=(2..10)):
show(complex_plot(f, (-B,B), (-B,B)))

Or just click on

http://sagecell.sagemath.org/?q=cdcdd7e5-73b4-4c87-87e2-1be300f86674

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--
William Stein
Professor of Mathematics
University of Washington
http://wstein.org

[computati...@gmail.com]

Thanks, this is exactly what I was looking for!

In fact, I had tried out complex_plot but I must have been using a different color function or something, because the roots were much less apparent to me. Not sure why I couldn't figure this out on my own...

I suppose I have two follow-up questions now:
1. how can I improve the precision of the zero set (it seems to be drawn in low resolution right now)
2. what is the best way to "turn off" the other colors (draw non-zeros as white)

[William Stein]

On Mon, Jun 10, 2013 at 9:09 AM, <computati...@gmail.com> wrote:
> Thanks, this is exactly what I was looking for!
>
> In fact, I had tried out complex_plot but I must have been using a different
> color function or something, because the roots were much less apparent to
> me. Not sure why I couldn't figure this out on my own...
>
> I suppose I have two follow-up questions now:
> 1. how can I improve the precision of the zero set (it seems to be drawn in
> low resolution right now)

Use plot_points:

f(z) = z - abs(z)

complex_plot(f, (-3,3), (-3,3), plot_points=200)

> 2. what is the best way to "turn off" the other colors (draw non-zeros as
> white)

I don't know if this is possible or implemented at present.

[computati...@gmail.com]

Ah, that's unfortunate. Might be a fun summer project to try to implement, if I knew where to start.

On another note: I really like the @interact annotation. I'm messing around with it, because I would like to add another parameter to my plot - an integer representing an index in a discrete family of functions. But I keep getting python errors about my expression not being symbolic when I try to include an integer parameter (like n = var('n'), then passing in n=2). Any tips?

[William Stein]

On Mon, Jun 10, 2013 at 9:24 AM, <computati...@gmail.com> wrote:
> Ah, that's unfortunate. Might be a fun summer project to try to implement,
> if I knew where to start.

1. http://www.sagemath.org/development.html

2. SAGE_ROOT/devel/sage/sage/plot/complex_plot.pyx which I found by
doing search_src('complex_plot')

> On another note: I really like the @interact annotation. I'm messing around
> with it, because I would like to add another parameter to my plot - an
> integer representing an index in a discrete family of functions. But I keep
> getting python errors about my expression not being symbolic when I try to
> include an integer parameter (like n = var('n'), then passing in n=2). Any
> tips?

Just put

n = 2

instead? You have to post code for a more useful answer....

[computati...@gmail.com]

Okay, so the following works:

z = var('z')

n = 2
@interact
def _(f = ((z+1)^n-abs(z^n+1)), B=(2..10)):
    show(complex_plot(f, (-B,B), (-B,B)))

But the following doesn't:

z = var('z')

@interact

def _(f = ((z+1)^n-abs(z^n+1)), n = (2..10), B=(2..10)):
    show(complex_plot(f, (-B,B), (-B,B)))

[William Stein]

Try this:

z,n = var('z,n')

@interact
def _(f = ((z+1)^n-abs(z^n+1)), n = (2..10), B=(2..10)):

f = f.subs(n=n)
show(f)
show(complex_plot(f, (-B,B), (-B,B)))

[computati...@gmail.com]

Looks great, thanks for all the help!
I'll see about messing around with complex_plot a bit to support pure zero plots, if it amounts to something I'll get back in touch...