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Then, the approximate solution of amendment is given by:
f
(
x
)
x
=
x
−
0
(3)
1
0
'
f
(
x
)
0
x
will be got, and it will converge to
If this process is repeated, the subsequence
the real root
x
.
x
x
x
2
0
Fig. 2.
The schematic diagram of extracting a root using Newton iteration
If it is complex plan, the formula of Newton iteration can be given by:
f
'
z
)
z
=
z
−
n
(4)
n
+
1
n
f
(
z
)
n
Most initial points iteration can converge on one root of equation, but few initial
points can not, they are emanative. Therefore, we can set a bigger iterative number
N, when real iterative number exceeds N, the point is not convergent, it escapes.
Because the iterative number is limited, all convergent points are not always
converging on the real roots. So, there is no harm in setting a small value e which is
used to representative the error between convergent point and the real root. If the error
is smaller than e, it is thought convergent.
The steps of Julia set escape time algorithm based on Newton iteration are
described as follows [4]:
(1)
Setting parameters: window size i×j, iterative number N, error e;
(2)
Every point of window do N iterations;
(3)
In iterative process, if the distance between iterative point and root is smaller than
e, the current parameter location is colored. Then, next point goes on.
If the distance is bigger than e, next point goes on still, but, current point is not
colored.
(4)
Until all points of window are done, the process is over.
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