Perturbation equations

The perturbation calculation algorithm in Ultra Fractal can dramatically speed up deep-zooming, allowing an image to complete in minutes rather than hours or even days. You can add perturbation support to your own formulas by providing the perturbation equations. Reworking the fractal equation to perturbation equations does require some (high school level) mathematics, and unfortunately not all fractal types are suitable for perturbation calculations.

Let's take the Mandelbrot formula as an example, written in pseudo-formula language with z' for the next value of z:

z' = z^2 + c

This is the reference orbit. To calculate another pixel on the screen, we use this orbit and deviate from it slightly. Let's call this orbit z2 and its location c2:

z2' = z2^2 + c2

Rewrite this using dz for the difference between z2 and z, and dc for the difference between c2 and c:

z2 = z + dz
c2 = c + dc
z' + dz' = (z + dz)^2 + c + dc

Now rewrite this to get a formula for the next value of dz, based on the current values of dz and z:

z^2 + c + dz' = z^2 + 2*z*dz + dz^2 + c + dc
dz' = 2*z*dz + dz^2 + dc

Note that the z^2 and c terms have cancelled out on both sides, leaving us with an equation where all terms are multiplied by dz. Because dz starts out small (first 0, then dc which is small because we're zoomed in far), all terms are small as well, so they can be calculated using regular double precision. For this to work, it's essential that there is no "large" z term left. For example, you might be tempted to write the equation like this:

dz' = (z + dz)^2 - z^2 + dc  ; Does not work

That does not work because the magnitude of z is typically around 1, and the magnitude of dz is around 1/magnification which is small. With deep zooming, regular floating-point precision does not provide enough significant decimals to add the two without losing the value of dz.

In short, if your formula can be reduced so all terms are small (in the order of dz or dc), it will work with perturbation calculations.

Sometimes, only certain parameter combinations for the formula will produce working perturbation equations. For example, with the standard Mandelbrot formula, only positive integer powers like 2, 3 and 4 can be reduced, but a power like 2.5 or a complex power does not reduce. In this case, use the perturb setting to specify when the perturbation equations are valid.

To express the perturbation equations in the formula language, add perturbinit and perturbloop sections, and use the #dz and #dpixel predefined symbols:

PerturbMandelbrot {
init:
  z = 0
loop:
  z = sqr(z) + #pixel
bailout:
  |z| < 4
perturbinit:
  #dz = 0
perturbloop:
  #dz = 2 * #z * #dz + sqr(#dz) + #dpixel
}

Notes

The bailout section from the regular formula is also used for perturbation calculations. The #z value is calculated by adding the reference z value and the current #dz value.
The #dz predefined symbol is equal to the dz value used in the equations above, and #dpixel is equal to dc.
To keep your formula compatible with earlier versions of Ultra Fractal (for example if you publish it in the formula database), use compiler directives and test for the VER60 symbol. See the Mandelbrot formula in Standard.ufm for an example.
See the SuperFractalThing web site for the original whitepaper by K.I. Martin that introduced this algorithm.

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