END OF AN ERA, FRACTALFORUMS.COM IS CONTINUED ON FRACTALFORUMS.ORG

That would be great!

E is an error symbol. "E = [0,1]*exp(i*]-infinity, +infinity[" is just a way to say "E is the unit disc centered at 0". One doesn't need to worry about it when writing the code. The only formula needed is:

R_n+1 = |B_n|² + 2*(|z_n|*R_n + |A_n|*|C_n|) + 2*|tmax|*(|A_n|*R_n + |B_n|*|C_n|) + |tmax²|*(|C_n|² + 2*|B_n|*R_n) + 2*|tmax³|*|C_n|*R_n + |tmax⁴|*R_n²

R_n+1*|tmax|⁴ is the radius of the disc interval. It is the maximal error that we do by neglecting the higher order terms of the series approximation (truncation error).
R₀ = 0. Simply because there is no truncation error. It remaines equal to zero until the third iteration (or so )


I did some mistakes here. Good news: It should be:
R_n*|tmax|⁴ <= |'|* ||
Because the trucation error is R_n*|tmax|⁴ not R_n.
As it is that formula have to be computed for each pixel because
'=A_n + 2*B_n*t + 3*C_n*t²
is function of t.
But we are interested in its minimal absolute value among all the values of t spanning the zoomed area. A lower bound can be computed using interval arthmetics. I'll come back tomorrow after doing the compuations and checking them.

PS: For deep enought zooms, the bottelneck is mainly in the computation of the references. It is possible to avoid unnecessary computations by reusing the reference until glitches occure. For example, if the reference is centered on the root of a minibrot, one can still use that same reference when zooming near that minibrot. It is also possible to refine a refrence using perturbation (+ Newton).

cool, thanks for your work on this!  i'll [try to] check it out  

for extrapolating your formulas to an arbitrary number of terms, i see the pattern here:

R_n*|tmax|⁴ <= (|A_n| - 2*|B_n|*|tmax| - 3*|C_n|*|tmax|²)* ||

but this one im not sure:

R_n+1 = |B_n|² + 2*(|z_n|*R_n + |A_n|*|C_n|) + 2*|tmax|*(|A_n|*R_n + |B_n|*|C_n|) + |tmax²|*(|C_n|² + 2*|B_n|*R_n) + 2*|tmax³|*|C_n|*R_n + |tmax⁴|*R_n²

maybe i just need to look at it longer

hehe...

    

it works, and it works WELL!  it is taking me farther than the old method of checking when the last term grows larger than the others by a certain amount yada yada, and is even skipping a few more than my experiment with using series acceleration error as the check (which itself was better than the other thing)!  and everything rendering fine as far as i can tell so far.  awesome! 

noting though that my program is currently setup to use a minimum of 8 terms, so hopefully it still bails soon enough when using the same number of terms.

also i wasnt sure what is supposed to be, so i made it the distance between adjacent points.

m = 1
R_n+1 = |A_n|² + 2*|z_n|*R_n + 2*|tmax|*|A_n|*R_n + |tmax²| * R_n²

m = 2
R_n+1 = 2*|z_n|*R_n +2*|A_n|*|B_n| + |tmax|*(2*|A_n|*|R_n|+|B_n|²) + |tmax²|*2*|B_n|*R_n + |tmax³|*R_n²

m = 3 (from above quote)
R_n+1 = |B_n|² + 2*(|z_n|*R_n + |A_n|*|C_n|) + 2*|tmax|*(|A_n|*R_n + |B_n|*|C_n|) + |tmax²|*(|C_n|² + 2*|B_n|*R_n) + 2*|tmax³|*|C_n|*R_n + |tmax⁴|*R_n²

trying the same here  mostly on paper doing boring algebra to get a feel for how it works (results above for m=1,2)

writing with indices helps, addition of letters is awkward - something like
(omitting the n that occurs everywhere seems to make sense too for this part of the algebra scribbling on paper)

cool stuff knighty!


UPDATE
Using the notation from my blog post, https://mathr.co.uk/blog/2016-03-06_simpler_series_approximation.html
Recall the coefficients are


Use "" terms, so the coefficient of the truncated part starting is  (using hat for disc interval), and writing T for tmax, and aliasing and , I think it boils down to this:



note this includes some terms twice multiplied in opposite orders, could be implemented by multiplying by 2 instead of repeated calculations

Thanks.

This looks like it's for just three terms though...

Hm, with R_n depending on |tmax| like that, it makes it harder to cache the series approximation coefficients when zooming around a still-good primary reference...

Running some tests with order m=16 now, seems promising so far it skips at most 1 period of the reference? while other skipping methods allow skipping beyond that.  (I got rid of the off-by-one bugs elsewhere in the code).  Demonstration-quality C++ code attached, note that it detects glitches but doesn't correct them, and location is hard-coded with a zoom sequence.

I plan on updating it with glitch correction and exterior distance estimation once I get the truncation error stuff for the derivative's series approximation sorted.  Equation scribbling ahoy...