altering Newton’s method

The usual scheme for Newton’s method is z_{n+1} = z_{n} - \frac{f(z_{n})}{f'(z_{n})}. But what if we just choose some random g(z) for \frac{f(z_{n})}{f'(z_{n})}?

Set g(z) = \frac{\cos(e^{z})}{e^{\sin(z)}}
Now, as far as differential equations go, solving
\frac{f(z)}{f'(z)} = \frac{\cos(e^{z})}{e^{\sin(z)}}
is pretty tricky.

But I can still generate a fractal from it:

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y=xc^x+x+1

I saw solving y=xc^{x}+x+1 on m.se today, and decided that it was ripe for a Newton’s method fractal, so, here you go:

It uses the following python code:


#!/usr/bin/python

from mpmath import *
import pylab

def g(z,x):
return x*(z**x) + x + 1

def newt(z):
x = z
for i in range(1,30):
x = x - g(z,x)/((z**x)*(1.0+x*fp.log(z))+1.0)
return x

fp.cplot(lambda z: newt(z), [-5.0,5.0], [-5.0,5.0], verbose=True, points=800000)