<s>
The	O
Newton	B-Algorithm
fractal	I-Algorithm
is	O
a	O
boundary	O
set	O
in	O
the	O
complex	O
plane	O
which	O
is	O
characterized	O
by	O
Newton	O
's	O
method	O
applied	O
to	O
a	O
fixed	O
polynomial	O
or	O
transcendental	O
function	O
.	O
</s>
<s>
It	O
is	O
the	O
Julia	B-Language
set	I-Language
of	O
the	O
meromorphic	O
function	O
which	O
is	O
given	O
by	O
Newton	O
's	O
method	O
.	O
</s>
<s>
In	O
this	O
way	O
the	O
Newton	B-Algorithm
fractal	I-Algorithm
is	O
similar	O
to	O
the	B-Algorithm
Mandelbrot	I-Algorithm
set	I-Algorithm
,	O
and	O
like	O
other	O
fractals	O
it	O
exhibits	O
an	O
intricate	O
appearance	O
arising	O
from	O
a	O
simple	O
description	O
.	O
</s>
<s>
It	O
is	O
relevant	O
to	O
numerical	B-General_Concept
analysis	I-General_Concept
because	O
it	O
shows	O
that	O
(	O
outside	O
the	O
region	O
of	O
quadratic	O
convergence	O
)	O
the	O
Newton	O
method	O
can	O
be	O
very	O
sensitive	O
to	O
its	O
choice	O
of	O
start	O
point	O
.	O
</s>
<s>
An	O
open	O
set	O
for	O
which	O
the	O
iterations	O
converge	O
towards	O
a	O
given	O
root	O
or	O
cycle	O
(	O
that	O
is	O
not	O
a	O
fixed	O
point	O
)	O
,	O
is	O
a	O
Fatou	B-Language
set	I-Language
for	O
the	O
iteration	O
.	O
</s>
<s>
The	O
complementary	O
set	O
to	O
the	O
union	O
of	O
all	O
these	O
,	O
is	O
the	O
Julia	B-Language
set	I-Language
.	O
</s>
<s>
The	O
Fatou	B-Language
sets	I-Language
have	O
common	O
boundary	O
,	O
namely	O
the	O
Julia	B-Language
set	I-Language
.	O
</s>
<s>
Therefore	O
,	O
each	O
point	O
of	O
the	O
Julia	B-Language
set	I-Language
is	O
a	O
point	O
of	O
accumulation	O
for	O
each	O
of	O
the	O
Fatou	B-Language
sets	I-Language
.	O
</s>
<s>
It	O
is	O
this	O
property	O
that	O
causes	O
the	O
fractal	O
structure	O
of	O
the	O
Julia	B-Language
set	I-Language
(	O
when	O
the	O
degree	O
of	O
the	O
polynomial	O
is	O
larger	O
than	O
2	O
)	O
.	O
</s>
<s>
The	O
special	O
choice	O
corresponds	O
to	O
the	O
Newton	B-Algorithm
fractal	I-Algorithm
.	O
</s>
<s>
When	O
is	O
outside	O
this	O
disk	O
,	O
the	O
fixed	O
points	O
are	O
locally	O
unstable	O
,	O
however	O
the	O
map	O
still	O
exhibits	O
a	O
fractal	O
structure	O
in	O
the	O
sense	O
of	O
Julia	B-Language
set	I-Language
.	O
</s>
<s>
If	O
is	O
a	O
polynomial	O
of	O
degree	O
,	O
then	O
the	O
sequence	O
is	O
bounded	B-Algorithm
provided	O
that	O
is	O
inside	O
a	O
disk	O
of	O
radius	O
centered	O
at	O
.	O
</s>
<s>
More	O
generally	O
,	O
Newton	O
's	O
fractal	O
is	O
a	O
special	O
case	O
of	O
a	O
Julia	B-Language
set	I-Language
.	O
</s>
<s>
The	O
Nova	O
fractal	O
invented	O
in	O
the	O
mid	O
1990s	O
by	O
Paul	O
Derbyshire	O
,	O
is	O
a	O
generalization	O
of	O
the	O
Newton	B-Algorithm
fractal	I-Algorithm
with	O
the	O
addition	O
of	O
a	O
value	O
at	O
each	O
step	O
:	O
</s>
<s>
In	O
order	O
to	O
implement	O
the	O
Newton	B-Algorithm
fractal	I-Algorithm
,	O
it	O
is	O
necessary	O
to	O
have	O
a	O
starting	O
function	O
as	O
well	O
as	O
its	O
derivative	O
function	O
:	O
</s>
<s>
The	O
above-defined	O
functions	O
can	O
be	O
translated	O
in	O
pseudocode	B-Language
as	O
follows	O
:	O
</s>
