![]() ![]() The most common result of an overflow is that the least significant representable digits of the result are stored the result is said to wrap around the maximum (i.e. In computer programming, an integer overflow occurs when an arithmetic operation attempts to create a numeric value that is outside of the range that can be represented with a given number of digits – either higher than the maximum or lower than the minimum representable value. ![]() This is wrapping in contrast to saturating. All digits are set to the maximum 9 and the next increment of the white digit causes a cascade of carry-over additions setting all digits to 0, but there is no higher digit (1,000,000s digit) to change to a 1, so the counter resets to zero. a single algorithm for computing all quadratic Julia sets does not exist.Integer overflow can be demonstrated through an odometer overflowing, a mechanical version of the phenomenon. there are many algorithms ( each has it's own numerical problems).many types of Julia sets ( like parabolic = hard to compute or Cremer = impossible to compute ( up to now) ).One can also use interval arithmethic, it will not change the numerical errors caused by computations but the images will be more accurate because of errors caused by scanning the plane Distance between dwell bands is decreasing when getting toward Julia set so it can be used as a measure of desired precision dynamically adjusting precision ( and other parameters) when going near Julia set ( zoom in or point close to Julia set ).choosing bigger resolution ( subpixel accuracy ).Rounding (and other) errors will be always in numerical computations, but they can be smaller by : Using inverse iteration makes "easy" to find them. If you want to make numerical aproximation of Julia set ( not filled-in Julia set) then then points of Julia set are repellers of forward iteration ( = hard to find). Here are a few examples of numerical problems : It will give you a clear visual perception of the gap between mathematics as it is and what really happens with your computer calculations. Now plot the points for everything I've mentioned above - not the iterations, but the points themselves. This approach is nice because it shows that points on the unit circle converge piecewise to the unit circle. You can switch to polar coordinates when the starting point c=(0,0). To convince yourself, open a spreadsheet, and set up a couple of columns for unit circle points, and do the tedious iterative calculations:Ĭomputationally, nothing behaves well on the unit circle. The problem is exactly with round-off error. They either run off to infinity (with the postman) or converge to (0,0). However, when performing computer calculations, most the points that should remain on the unit circle do not behave well due to round-off errors. Points inside the Julia set converge to (0,0) points outside diverge to infinity points on the Julia set stay at the unit circle. It is a nice starting place and the most elementary means of generating a Julia set. ![]() Falconer demonstrates z= z*z+c where c=(0,0) as a nice starting place. The boundary is referred to as the Julia set by Kenneth Falconer in Fractals: A Short Introduction (NY: Oxford University Press, 2013). ![]() (They may or may not manifest themselves elsewhere, but my efforts were principally at the boundary.) When calculating Julia Sets and performing iterative operations with real numbers, round-off errors manifest themselves at the boundary of the Julia set. And, of course, the Julia set of a rational function may very well be the whole Riemann sphere. The book by Braverman and Yampolsky cited in Alan's answer contains a theorem showing that certain quadratic functions of the form $e^z+z^2$ have uncomputable Julia sets. Note that the claims above assume hyperbolic behavior and, in fact, there are definitely Julia sets that cannot be computed. I don't have any specific references but a google search for Julia set shadowing turns up a number of scholarly hits that might be relevant. Second, even near the Julia set, the Shadowing lemma guarantees that our computer generates actual orbits, though perhaps not exactly the orbit we intended. As the Julia set is the complement of the Fatou set, it works well there as well. Thus, the escape time algorithm is essentially a test for membership of the Fatou set, which works well. The generic point lies in the Fatou set which is intrinsically stable. That is, the Julia set (where the chaos happens) is a compact, nowhere dense subset of the Riemann sphere. I believe there are two main forces at work that generally mitigate the problem of round off error in this context.įirst, the predominate behavior under the iteration of a polynomial (or even generally a rational function) is stability. Computer generated images of Julia sets tend to agree strongly with theory so we expect there must be some reason. ![]()
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