<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
the	O
Bulirsch	B-Algorithm
–	I-Algorithm
Stoer	I-Algorithm
algorithm	I-Algorithm
is	O
a	O
method	O
for	O
the	O
numerical	B-Algorithm
solution	I-Algorithm
of	I-Algorithm
ordinary	I-Algorithm
differential	I-Algorithm
equations	I-Algorithm
which	O
combines	O
three	O
powerful	O
ideas	O
:	O
Richardson	B-Algorithm
extrapolation	I-Algorithm
,	O
the	O
use	O
of	O
rational	O
function	O
extrapolation	O
in	O
Richardson-type	O
applications	O
,	O
and	O
the	O
modified	O
midpoint	O
method	O
,	O
to	O
obtain	O
numerical	B-General_Concept
solutions	I-General_Concept
to	O
ordinary	O
differential	O
equations	O
(	O
ODEs	O
)	O
with	O
high	O
accuracy	O
and	O
comparatively	O
little	O
computational	O
effort	O
.	O
</s>
<s>
The	O
idea	O
of	O
Richardson	B-Algorithm
extrapolation	I-Algorithm
is	O
to	O
consider	O
a	O
numerical	B-General_Concept
calculation	I-General_Concept
whose	O
accuracy	O
depends	O
on	O
the	O
used	O
stepsize	O
h	O
as	O
an	O
(	O
unknown	O
)	O
analytic	B-Language
function	I-Language
of	O
the	O
stepsize	O
h	O
,	O
performing	O
the	O
numerical	B-General_Concept
calculation	I-General_Concept
with	O
various	O
values	O
of	O
h	O
,	O
fitting	O
a	O
(	O
chosen	O
)	O
analytic	B-Language
function	I-Language
to	O
the	O
resulting	O
points	O
,	O
and	O
then	O
evaluating	O
the	O
fitting	O
function	O
for	O
h	O
=	O
0	O
,	O
thus	O
trying	O
to	O
approximate	O
the	O
result	O
of	O
the	O
calculation	O
with	O
infinitely	O
fine	O
steps	O
.	O
</s>
<s>
Bulirsch	O
and	O
Stoer	O
recognized	O
that	O
using	O
rational	O
functions	O
as	O
fitting	O
functions	O
for	O
Richardson	B-Algorithm
extrapolation	I-Algorithm
in	O
numerical	O
integration	O
is	O
superior	O
to	O
using	O
polynomial	O
functions	O
because	O
rational	O
functions	O
are	O
able	O
to	O
approximate	O
functions	O
with	O
poles	O
rather	O
well	O
(	O
compared	O
to	O
polynomial	O
functions	O
)	O
,	O
given	O
that	O
there	O
are	O
enough	O
higher-power	O
terms	O
in	O
the	O
denominator	O
to	O
account	O
for	O
nearby	O
poles	O
.	O
</s>
<s>
The	O
modified	O
midpoint	O
method	O
by	O
itself	O
is	O
a	O
second-order	O
method	O
,	O
and	O
therefore	O
generally	O
inferior	O
to	O
fourth-order	O
methods	O
like	O
the	O
fourth-order	B-Algorithm
Runge	I-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
.	O
</s>
