High-Precision Deep Zoom



The Bat Fractal resulted from a bug while I was trying to get Newton's method for the third roots of unity working. I've since developed it a little more and made some really nice images from it. These are the original ones from when it first popped up.

Karatsuba Test Error

This was the result of one of my initial tests of the Karatsuba multiplication method. Still had some more work to do....

Loss of Precision 1
Loss of Precision 2

These aren't really software bugs, but just examples of how things can go wrong. They show loss of numerical precision at only a modestly deep zoom into a secant method fractal (10-9). The top one shows a hint of the problem starting to develop, and the bottom one, at slightly higher magnification, is totally messed up.

Complex Exponential Bug

This is the result of my first attempt at writing the code for raising a complex number to a complex power. The test is using Newton's method for the cubic roots of 1. It turned out there was a typo in the code for the exp function.

The way to this, by the way, if you're curious, is to use

za = ea log z

First convert z from its cartesian form z=x+iy to its polar form z=reiu

r = (x2+y2)1/2

u = arctan (x/y)

The use the definitions of exp and log:

log z = log r + iu

Multiply that by a and call the product w=a log z. Put w in polar form w=Weis and stick that into the definition of exp:

exp w = eW (cos s + i sin s)