### Fixed Point Iteration

```Experiments with MATLAB
Mandelbrot Set
Roger Jang (張智星)
CSIE Dept, National Taiwan University
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http://mirlab.org/jang
Mandelbrot Set
• The Fractal Geometry of Nature
– By Benoit Mandelbrot, 1982
• Mandelbrot Set
– Appeared on the cover of Scientific American in 1985,
which is about the time that computer graphical
displays were first becoming widely available
– Has stimulated deep research topics in mathematics
and has also been the basis for an uncountable
number of graphics projects, hardware demos, and
Web pages
2
Convergence for Iterations of
Complex Numbers
• Iteration:
zk 1  zk2 , k  0,1,...
• Consider the region in
the complex plane
consisting of the initial
values for which the
iteration remain
bounded as k
approaches infinity.
• Alternatively:
z | lim z
0
n 
n

is bounded with zk 1  zk2 , k  0,1,...
3
Definition of the Mandelbrot set
•
z | lim z
0
n 
n

is bounded with zk 1  zk2  z0 , k  0,1,...
– Red: Mandelbrot Set
– Black: Region of rich
structure
4
Examples
• Point inside the set
– z0 = .25-.54i generates a
cycle of length four
– Verification
z0 = .25-.54i; z = 0;
z=z^2+z0
• Point outside the set
– z0 = .22-.54i generates
an unbounded trajectory
– Manual test
z0 = .22-.54i; z = 0;
z=z^2+z0
5
Details of the Fringe
• Criterion for divergence
– As soon as z satisfies
abs(z)>2, subsequent
iterations essentially
square the value of abs(z)
and diverge.
• Display
– The number of iterations
required for z to escape the
the basis for showing the
detail in the fringe.
– We can then use different
colors to represent the
above iteration count.
6
Implementation Minutes
• Code snippet
function k = M(z0,
maxCount)
z = 0;
k = 0;
while abs(z)<2 && k<
maxCount
z = z^2+z0;
k = k + 1;
end
• Observations
– The value returned by this
function is between 1 and
maxCount.
– If the value is maxCount,
then z0 is in the set.
– We can use the value as an
index into a color map of
size maxCount-by-3.
7
Demos
•
mandelbrot
– thumbnail icons of the twelve regions
•
mandelbrot(r)
– r=1~12 starts with r-th interesting
regions outside the Mandelbrot set.
– Titles of the plots
•
•
•
•
•
•
•
•
•
•
•
•
•
r=1full
r=2mini mandelbrot
r=3plaza
r=4seahorses
r=5west wing
r=6dueling dragons
r=7buzzsaw
r=8nebula
r=9vortex1
r=10vortex2
r=11vortex3
r=12geode (deep detail)
mandelbrot(center,width,grid,depth,c
mapindx)
8
“The Valley of the Seahorses”
• Commands
– mandelbrot(4)
– mandelbrot(-.7700.1300i, 0.1, 1024, 512)
• Fun thing to try
– Try “spinmap(5, 1)”!
-0.08
-0.09
-0.1
-0.11
-0.12
-0.13
-0.14
-0.15
-0.16
-0.17
-0.18
-0.82
-0.81
-0.8
-0.79
-0.78
-0.77
-0.76
-0.75
-0.74
9
-0.73
-0.72
“Buzzsaw”
• Commands
– mandelbrot(7)
– mandelbrot(0.00164372
1971153+0.8224676332
98876i, 4.0e-11, 1024,
2048, 2)
• Observations
– It’s as small as a the
cross-section of a hair
– Try “spinmap(5, 1)”!
– Self-similarity!
-11
2
0.001643721971153 + 0.822467633298876i
x 10
1
0
-1
-2
-2
-1
0
1
10
2
-11
x 10
“Vortex”
• Commands
-12
– mandelbrot(9)
– mandelbrot(1.749759145130366460.00000000368513796i,
6.0e-12, 1024, 2048, 2)
• Observations
– It’s as small as a the
cross-section of a hair
– Try “spinmap(5, 1)”!
3
-1.7497591451303665 - 0.0000000036851380i
x 10
2
1
0
-1
-2
-3
-3
-2
-1
0
1
2
11
3
-12
x 10
“Geode”
• Commands
– mandelbrot(12)
– mandelbrot(0.28692299
709-0.01218247138i,
6.0e-10, 2048, 4096, 1)
-10
3
0.28692299709000 - 0.01218247138000i
x 10
2
1
• Observations
– Try “spinmap(5, 1)”!
– Self-similarity!
0
-1
-2
-3
-3
-2
-1
0
1
2
12
3
-10
x 10
```