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From: tet@uvaee.UUCP (Thomas E. Tkacik)
Newsgroups: net.math
Subject: Re: Is the Mandelbrot set a fiction??
Message-ID: <467@uvaee.UUCP>
Date: Wed, 2-Oct-85 10:15:51 EDT
Article-I.D.: uvaee.467
Posted: Wed Oct  2 10:15:51 1985
Date-Received: Fri, 4-Oct-85 03:25:25 EDT
References: <418@aero.ARPA> <646@petsd.UUCP> <221@epicen.UUCP> <480@aero.ARPA>
Reply-To: tet@uvaee.UUCP (Thomas E. Tkacik)
Organization: EE Dept., U of Virginia, Charlottesville
Lines: 36

>I'm the one that started this Mandelbrot discussion. The reason I wondered
>about the reality of it is for certain numbers near the boundary, it is
>near impossible on a finite machine to carry thru the iterations without
>the error propagation overwhelming the result. Of course, there are isolated
>cases where you can prove a number is within the set, and it is easy to
>show that certain numbers are outside the set if the number of iterations is
>small enough. The really tough ones are those that require more than 20 or so
>iterations, becuase by then the error has complely swamped the result.
>
I have written a program to generate Mandelbrot's set, and have used it
magnify some areas by a linear factor of about 100,000.  The results
have been spectacular.  It has been obvious by just looking at the resulting
pictures that roundoff errors have not completely swamped the results.
To be sure, there are probably a few mistaken points in the pictures,
but nothing extreme.
To generate these pictures, I had to increase the iterations in some of the
more extreme cases to 2048.  I did expect lots of roundoff errors, so I used
double precision (on a VAX that's 64 bits). 
>
>I did work out a way where you can generate a large number of numbers in the
>set when given just a few--but that doesn't help establish whether a number
>picked at random is in or out of the set. The way it works is: Suppose a number
>c1 is in the set. Then the number c2=c1**2+c1 is also in the set, and likewise
>
According to what you said above, this should also only work for the first
20 or so iterations before roundoff errors swamp the result.  This would not
generate many points, and would give points outside any region of intrest,
so I think we are stuck doing this one pixel at a time.
I agree that roundoff error will set a limit to what can be seen in the set,
but I think that is a limit that has not yet been reached,
(at least by me :-) .)

Has anyone else generated pictures with very high magnifications, and/or
many iterations.  I would like to hear of your results.
                                Tom Tkacik
      ...!decvax!mcnc!ncsu!uvacs!uvaee!tet