2.42 Halley
(type=halley)
The Halley map is an algorithm used to find the roots of polynomial
equations by successive "guesses" that converge on the correct value as
you feed the results of each approximation back into the formula. It
works very well -- unless you are unlucky enough to pick a value that is
on a line BETWEEN two actual roots. In that case, the sequence explodes
into chaos, with results that diverge more and more wildly as you
continue the iteration.
This fractal type shows the results for the polynomial Z(Z^a - 1), which
has a+1 roots in the complex plane. Use the <T>ype command and enter
"halley" in response to the prompt. You will be asked for a parameter,
the "order" of the equation (an integer from 2 through 10 -- 2 for Z(Z^2
- 1), 7 for Z(Z^7 - 1), etc.). A second parameter is the relaxation
coefficient, and is used to control the convergence stability. A number
greater than one increases the chaotic behavior and a number less than
one decreases the chaotic behavior. The third parameter is the value
used to determine when the formula has converged. The test for
convergence is ||Z(n+1)|^2 - |Z(n)|^2| < epsilon. This convergence test
produces the whisker-like projections which generally point to a root.