<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
the	O
order	B-Architecture
of	I-Architecture
convergence	I-Architecture
and	O
the	O
rate	B-Architecture
of	I-Architecture
convergence	I-Architecture
of	O
a	O
convergent	B-Algorithm
sequence	I-Algorithm
are	O
quantities	O
that	O
represent	O
how	O
quickly	O
the	O
sequence	O
approaches	O
its	O
limit	O
.	O
</s>
<s>
The	O
rate	B-Architecture
of	I-Architecture
convergence	I-Architecture
is	O
also	O
called	O
the	O
asymptotic	O
error	O
constant	O
.	O
</s>
<s>
In	O
practice	O
,	O
the	O
rate	O
and	O
order	B-Architecture
of	I-Architecture
convergence	I-Architecture
provide	O
useful	O
insights	O
when	O
using	O
iterative	B-Algorithm
methods	I-Algorithm
for	O
calculating	O
numerical	B-General_Concept
approximations	I-General_Concept
.	O
</s>
<s>
If	O
the	O
order	B-Architecture
of	I-Architecture
convergence	I-Architecture
is	O
higher	O
,	O
then	O
typically	O
fewer	O
iterations	O
are	O
necessary	O
to	O
yield	O
a	O
useful	O
approximation	O
.	O
</s>
<s>
Similar	O
concepts	O
are	O
used	O
for	O
discretization	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
The	O
solution	O
of	O
the	O
discretized	B-Algorithm
problem	O
converges	O
to	O
the	O
solution	O
of	O
the	O
continuous	O
problem	O
as	O
the	O
grid	O
size	O
goes	O
to	O
zero	O
,	O
and	O
the	O
speed	B-Architecture
of	I-Architecture
convergence	I-Architecture
is	O
one	O
of	O
the	O
factors	O
of	O
the	O
efficiency	O
of	O
the	O
method	O
.	O
</s>
<s>
However	O
,	O
the	O
terminology	O
,	O
in	O
this	O
case	O
,	O
is	O
different	O
from	O
the	O
terminology	O
for	O
iterative	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
Series	B-Algorithm
acceleration	I-Algorithm
is	O
a	O
collection	O
of	O
techniques	O
for	O
improving	O
the	O
rate	B-Architecture
of	I-Architecture
convergence	I-Architecture
of	O
a	O
series	O
discretization	B-Algorithm
.	O
</s>
<s>
The	O
sequence	O
is	O
said	O
to	O
converge	O
with	O
order	O
to	O
,	O
and	O
with	O
a	O
rate	B-Architecture
of	I-Architecture
convergence	I-Architecture
of	O
,	O
iffor	O
some	O
positive	O
constant	O
if	O
,	O
and	O
if	O
.	O
</s>
<s>
is	O
called	O
linear	B-Architecture
convergence	I-Architecture
if	O
,	O
and	O
the	O
sequence	O
is	O
said	O
to	O
converge	O
Q-linearly	O
to	O
.	O
</s>
<s>
A	O
practical	O
method	O
to	O
calculate	O
the	O
order	B-Architecture
of	I-Architecture
convergence	I-Architecture
for	O
a	O
sequence	O
is	O
to	O
calculate	O
the	O
following	O
sequence	O
,	O
which	O
converges	O
to	O
:	O
</s>
<s>
In	O
addition	O
to	O
the	O
previously	O
defined	O
Q-linear	B-Architecture
convergence	O
,	O
a	O
few	O
other	O
Q-convergence	O
definitions	O
exist	O
.	O
</s>
<s>
Often	O
,	O
however	O
,	O
the	O
"	O
Q-	O
"	O
is	O
dropped	O
and	O
a	O
sequence	O
is	O
simply	O
said	O
to	O
have	O
linear	B-Architecture
convergence	I-Architecture
,	O
quadratic	O
convergence	O
,	O
etc	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
definition	O
of	O
rate	B-Architecture
of	I-Architecture
convergence	I-Architecture
is	O
extended	O
as	O
follows	O
.	O
</s>
<s>
To	O
determine	O
the	O
type	O
of	O
convergence	O
,	O
we	O
plug	O
the	O
sequence	O
into	O
the	O
definition	O
of	O
Q-linear	B-Architecture
convergence	O
,	O
</s>
<s>
Thus	O
,	O
we	O
find	O
that	O
converges	O
Q-linearly	O
and	O
has	O
a	O
convergence	B-Architecture
rate	I-Architecture
of	O
.	O
</s>
<s>
In	O
fact	O
,	O
it	O
is	O
quadratically	B-Architecture
convergent	I-Architecture
.	O
</s>
<s>
A	O
similar	O
situation	O
exists	O
for	O
discretization	B-Algorithm
methods	I-Algorithm
designed	O
to	O
approximate	O
a	O
function	O
,	O
which	O
might	O
be	O
an	O
integral	O
being	O
approximated	O
by	O
numerical	B-Algorithm
quadrature	I-Algorithm
,	O
or	O
the	O
solution	B-Algorithm
of	I-Algorithm
an	I-Algorithm
ordinary	I-Algorithm
differential	I-Algorithm
equation	I-Algorithm
(	O
see	O
example	O
below	O
)	O
.	O
</s>
<s>
The	O
discretization	B-Algorithm
method	I-Algorithm
generates	O
a	O
sequence	O
,	O
where	O
each	O
successive	O
is	O
a	O
function	O
of	O
along	O
with	O
the	O
grid	O
spacing	O
between	O
successive	O
values	O
of	O
the	O
independent	O
variable	O
.	O
</s>
<s>
This	O
is	O
the	O
relevant	O
definition	O
when	O
discussing	O
methods	O
for	O
numerical	B-Algorithm
quadrature	I-Algorithm
or	O
the	O
solution	O
of	O
ordinary	O
differential	O
equations	O
(	O
ODEs	O
)	O
.	O
</s>
<s>
A	O
practical	O
method	O
to	O
estimate	O
the	O
order	B-Architecture
of	I-Architecture
convergence	I-Architecture
for	O
a	O
discretization	B-Algorithm
method	I-Algorithm
is	O
pick	O
step	O
sizes	O
and	O
and	O
calculate	O
the	O
resulting	O
errors	O
and	O
.	O
</s>
<s>
The	O
order	B-Architecture
of	I-Architecture
convergence	I-Architecture
is	O
then	O
approximated	O
by	O
the	O
following	O
formula	O
:	O
</s>
<s>
The	O
error	O
is	O
,	O
more	O
specifically	O
,	O
a	O
global	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
(	O
GTE	O
)	O
,	O
in	O
that	O
it	O
represents	O
a	O
sum	O
of	O
errors	O
accumulated	O
over	O
all	O
iterations	O
,	O
as	O
opposed	O
to	O
a	O
local	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
(	O
LTE	O
)	O
over	O
just	O
one	O
iteration	O
.	O
</s>
<s>
We	O
can	O
solve	O
this	O
equation	O
using	O
the	O
Forward	B-Algorithm
Euler	I-Algorithm
scheme	I-Algorithm
for	I-Algorithm
numerical	I-Algorithm
discretization	I-Algorithm
:	O
</s>
<s>
so	O
converges	O
to	O
with	O
a	O
convergence	B-Architecture
rate	I-Architecture
.	O
</s>
<s>
This	O
sequence	O
converges	O
with	O
order	O
1	O
according	O
to	O
the	O
convention	O
for	O
discretization	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
It	O
is	O
said	O
to	O
converge	O
exponentially	O
using	O
the	O
convention	O
for	O
discretization	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
However	O
,	O
it	O
only	O
converges	O
linearly	O
(	O
that	O
is	O
,	O
with	O
order	O
1	O
)	O
using	O
the	O
convention	O
for	O
iterative	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
Assuming	O
that	O
the	O
relevant	O
derivatives	O
of	O
f	O
are	O
continuous	O
,	O
one	O
can	O
(	O
easily	O
)	O
show	O
that	O
for	O
a	O
fixed	O
point	O
such	O
that	O
,	O
one	O
has	O
at	O
least	O
linear	B-Architecture
convergence	I-Architecture
for	O
any	O
starting	O
value	O
sufficiently	O
close	O
to	O
p	O
.	O
If	O
and	O
,	O
then	O
one	O
has	O
at	O
least	O
quadratic	O
convergence	O
,	O
and	O
so	O
on	O
.	O
</s>
<s>
Many	O
methods	O
exist	O
to	O
increase	O
the	O
rate	B-Architecture
of	I-Architecture
convergence	I-Architecture
of	O
a	O
given	O
sequence	O
,	O
</s>
<s>
Such	O
techniques	O
are	O
in	O
general	O
known	O
as	O
"	O
series	B-Algorithm
acceleration	I-Algorithm
"	O
.	O
</s>
<s>
One	O
example	O
of	O
series	B-Algorithm
acceleration	I-Algorithm
is	O
Aitken	B-Algorithm
's	I-Algorithm
delta-squared	I-Algorithm
process	I-Algorithm
.	O
</s>
<s>
These	O
methods	O
in	O
general	O
(	O
and	O
in	O
particular	O
Aitken	B-Algorithm
's	I-Algorithm
method	I-Algorithm
)	O
do	O
not	O
increase	O
the	O
order	B-Architecture
of	I-Architecture
convergence	I-Architecture
,	O
and	O
are	O
useful	O
only	O
if	O
initially	O
the	O
convergence	O
is	O
not	O
faster	O
than	O
linear	O
:	O
If	O
convergences	O
linearly	O
,	O
one	O
gets	O
a	O
sequence	O
that	O
still	O
converges	O
linearly	O
(	O
except	O
for	O
pathologically	O
designed	O
special	O
cases	O
)	O
,	O
but	O
faster	O
in	O
the	O
sense	O
that	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
if	O
the	O
convergence	O
is	O
already	O
of	O
order	O
2	O
,	O
Aitken	B-Algorithm
's	I-Algorithm
method	I-Algorithm
will	O
bring	O
no	O
improvement	O
.	O
</s>
