<s>
The	O
shifting	B-Algorithm
nth	I-Algorithm
root	I-Algorithm
algorithm	I-Algorithm
is	O
an	O
algorithm	O
for	O
extracting	O
the	O
nth	O
root	O
of	O
a	O
positive	O
real	O
number	O
which	O
proceeds	O
iteratively	O
by	O
shifting	O
in	O
n	O
digits	O
of	O
the	O
radicand	O
,	O
starting	O
with	O
the	O
most	O
significant	O
,	O
and	O
produces	O
one	O
digit	O
of	O
the	O
root	O
on	O
each	O
iteration	O
,	O
in	O
a	O
manner	O
similar	O
to	O
long	B-Algorithm
division	I-Algorithm
.	O
</s>
<s>
At	O
each	O
iteration	O
,	O
the	O
invariant	B-Application
will	O
hold	O
.	O
</s>
<s>
The	O
invariant	B-Application
will	O
hold	O
.	O
</s>
<s>
The	O
first	O
invariant	B-Application
implies	O
that	O
.	O
</s>
<s>
We	O
want	O
to	O
choose	O
so	O
that	O
the	O
invariants	B-Application
described	O
above	O
hold	O
.	O
</s>
<s>
Repeat	O
until	O
desired	O
precision	B-Architecture
is	O
obtained	O
:	O
</s>
<s>
As	O
noted	O
above	O
,	O
this	O
algorithm	O
is	O
similar	O
to	O
long	B-Algorithm
division	I-Algorithm
,	O
and	O
it	O
lends	O
itself	O
to	O
the	O
same	O
notation	O
:	O
</s>
<s>
Note	O
that	O
increasing	O
the	O
base	O
increases	O
the	O
time	O
needed	O
to	O
pick	O
by	O
a	O
factor	O
of	O
,	O
but	O
decreases	O
the	O
number	O
of	O
digits	O
needed	O
to	O
achieve	O
a	O
given	O
precision	B-Architecture
by	O
the	O
same	O
factor	O
,	O
and	O
since	O
the	O
algorithm	O
is	O
cubic	O
time	O
in	O
the	O
number	O
of	O
digits	O
,	O
increasing	O
the	O
base	O
gives	O
an	O
overall	O
speedup	O
of	O
.	O
</s>
