### Newton-Hines fractals

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#### superheal

• Fractal Phenom
• Posts: 45

#### Re: Newton-Hines fractals

« Reply #30 on: May 13, 2019, 11:34:42 AM »

• 3f
• Posts: 1928

#### Re: Newton-Hines fractals

« Reply #31 on: May 15, 2019, 08:53:12 AM »
Pure Newton applied to f(c) = "11 iterations of z^2+c". Looks like external rays, wonder why or maybe it's just a coincidence.

• 3f
• Posts: 1344

#### Re: Newton-Hines fractals

« Reply #32 on: May 15, 2019, 09:15:31 AM »
the basins are ray-like (approach infinity in a straight line)
I suspect this is true for any polynomial, M-set-related or not

but I don't think the angles match what you would hope for
If the basins for a root coincided exactly with corresponding external rays, you wouldn't need to trace external rays step by step
you could just use Newton's method starting from a sufficiently large circle at a sufficiently accurate angle

• 3f
• Posts: 1928

#### Re: Newton-Hines fractals

« Reply #33 on: May 16, 2019, 08:16:35 PM »
Halley method update is $$-1/(\frac{f'}{f}- q \frac{f''}{2f'} )$$ with q=1, and q=0 gives you Newton method back.
Image is based on a (4,3) rational function with q=0.9.
Solid red = "escaping" = "diverging". The divergent regions seem to be filled with some Cantor dust of converging points.

• 3f
• Posts: 1928

#### Re: Newton-Hines fractals

« Reply #34 on: May 18, 2019, 07:32:08 AM »
Colorful Halley as above with "wrong sign" q=-1, applied to 11th degree polynomial.

#### marcm200

• Fractal Frankfurter
• Posts: 565

#### Re: Newton-Hines fractals

« Reply #35 on: July 15, 2019, 10:18:26 AM »
I stumbled upon a very interesting paper (I searched the forum for this but couldn't find a reference to it):

Quote
How to find all roots of complex polynomials by Newton’s method
John Hubbard, Dierk Schleicher, Scott Sutherland
Digital Object Identifier
Invent. math. 146, 1–33 (2001)

The result being, that there is a constructible set of finite many complex points so that when starting Newton's method from those points all roots of a polynomial will be found - and that set is only dependent on the degree of the polynomial and not the polynomial itself. So instead of hoping to have found a good enough starting point that will converge to a root (my approach), one could use a precomputed list and scale those accordingly (they work with normalized polynomials, where all the roots lie in the unit disk).

I find those general and constructible results very prize-worthy.

• 3f
• Posts: 1928

#### Re: Newton-Hines fractals

« Reply #36 on: July 15, 2019, 07:52:45 PM »
Thanks, not sure if it's comprehensible but I'll try to read it: https://pi.math.cornell.edu/~hubbard/NewtonInventiones.pdf

• 3f
• Posts: 1928

#### Re: Newton-Hines fractals

« Reply #37 on: July 23, 2019, 04:33:34 AM »
Secant method for rational function with 791 zeros and 791 poles.

• 3f
• Posts: 1928

#### Re: Newton-Hines fractals

« Reply #38 on: October 01, 2019, 08:28:25 PM »
Adding a tiny Hines factor of k=1E-4 to https://fractalforums.org/image-threads/25/gerrit-images/565/msg17007#msg17007 shatters the green non convergent areas into something else. Image is a zoom on biggest green blob from the referenced post.

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