Xref: utzoo sci.math.num-analysis:586 sci.math:10143 comp.graphics:10208 comp.theory:412
Path: utzoo!utgpu!jarvis.csri.toronto.edu!cs.utexas.edu!usc!apple!motcsd!xdos!doug
From: doug@xdos.UUCP (Doug Merritt)
Newsgroups: sci.math.num-analysis,sci.math,comp.graphics,comp.terminals.tty5620,comp.theory
Subject: Re: How much precision does Mandelbrot need
Message-ID: <674@xdos.UUCP>
Date: 3 Mar 90 18:17:27 GMT
References: <8602@cbnewsh.ATT.COM>
Reply-To: doug@xdos.UUCP (Doug Merritt)
Followup-To: sci.math.num-analysis
Organization: Hunter Systems, Mountain View CA (Silicon Valley)
Lines: 31

In article <8602@cbnewsh.ATT.COM> wcs@cbnewsh.ATT.COM (Bill Stewart 201-949-0705 erebus.att.com!wcs) writes:
>How much precision does it take to calculate Mandelbrot sets correctly?
> [...] Since Mandelbrot sets depend very
>strongly on initial conditions, it would seem that loss of precision
>would quasi-randomly cause different results near any boundaries.

That's precisely the case. The net effect is that the deeper you zoom
into the set, the more precision you need. If your zoom factor is only
1x, then you need very little precision. Using fixed point works fine
up to a certain point. I haven't figured it out precisely, but I guess
a rough rule of thumb might be that you need

	# bits of precision = log_base_2(zoom * screen_res * iterations)

This guess gives a maximum zoom factor of about 4K for a 1K (per edge) pixel
screen at 1K iterations using 32 bit scaled integers, which seems like
it's in the right general ballpark.

Note that the number of iterations required for good resolution also
increases as you zoom in further. 256 iterations is ok for a zoom of 2x,
but 1K is needed for deeper zooms, and even more for really deep zooms.
An extremely crude rule of thumb for this might be something like

	iterations = log_base_2(zoom) * 128 + 128

This is all just to give you a general feel for it; there are probably
far better estimates available.
	Doug
-- 
Doug Merritt		{pyramid,apple}!xdos!doug
Member, Crusaders for a Better Tomorrow		Professional Wildeyed Visionary