Relaxed Newton/Halley

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Relaxed Newton/Halley is a Fractal Formula in lkm3.ufm.

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Hi Gang,

I've added a couple of new formulas to the database:

Super Mandelbrot

and

Relaxed Newton/Halley: This is a calculation formula that extends Newton's method in a few ways. Firstly, it also computes Halley's method (a higher-order version of Newton). You can combine Newton and Halley in a variety of ways and use relaxation on either or both. The formula "solves" the equations z^n = c (typical Newton) and exponential/logarithmic equation that I threw in for fun.

I put two sample parameter sets into lkm3.upr and included one of them below. It looks *much* better rendered, but it goes quickly, even at an anti-aliasing depth of 2.

Kerry Mitchell

Copyright © 2008 by Kerry Mitchell

RelaxedNewton-HalleySample1 { 
fractal: 
  title="Relaxed Newton-Halley sample 1" width=800 height=600 layers=1
  credits="Kerry Mitchell;12/22/2008"
layer: 
  caption="Layer 1" opacity=100 method=multipass
  mapping: center=12.77869337/6.5465480455 magn=0.36921562 angle=9.8477
formula: 
  maxiter=27 percheck=off filename="lkm3.ufm" 
  entry="relaxed-newton-halley"
  p_bailin=1E-12 p_pixeltype="initial z" p_initialz=3/0
  p_root=1/0 p_ztype=change p_eqntype="roots of c" p_power=3/0
  p_solntype=Newton p_weightn=1/0 p_weighth=0/1
  p_relaxcoord=rectangular
  p_relaxrect=2.333398283040441305782961/1.615583436541993183120363
  p_relaxcenter=1/0 p_relaxradius=1 p_relaxdegree=60
  p_relaxradian=1 p_relaxm=1 p_relaxn=3 p_pifac=0
  p_relaxrect2=1/0 f_relaxfn1=ident f_relaxfn2=ident
  p_printrelax=no 
inside: 
  transfer=linear repeat=no
  solid=4278190335 filename="lkm.ucl" entry="statistics"
  p_vartype=imag/real p_stattype=mean 
outside: 
  transfer=none solid=4278190335 filename="lkm3.ucl" entry="basic-3"
  p_vartype=z p_const=0/0 p_colorby=iterations f_fofvar=sqrt
gradient: 
  smooth=yes index=0 color=0 index=100 color=13610927
  index=300 color=16678656 index=399 color=16777215 
opacity:
 smooth=no index=0 opacity=255
}