<s>
In	O
mathematics	O
and	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
an	O
adaptive	B-Algorithm
step	I-Algorithm
size	I-Algorithm
is	O
used	O
in	O
some	O
methods	O
for	O
the	O
numerical	B-Algorithm
solution	I-Algorithm
of	I-Algorithm
ordinary	I-Algorithm
differential	I-Algorithm
equations	I-Algorithm
(	O
including	O
the	O
special	O
case	O
of	O
numerical	B-Algorithm
integration	I-Algorithm
)	O
in	O
order	O
to	O
control	O
the	O
errors	O
of	O
the	O
method	O
and	O
to	O
ensure	O
stability	B-Algorithm
properties	I-Algorithm
such	O
as	O
A-stability	O
.	O
</s>
<s>
Using	O
an	O
adaptive	B-Algorithm
stepsize	I-Algorithm
is	O
of	O
particular	O
importance	O
when	O
there	O
is	O
a	O
large	O
variation	O
in	O
the	O
size	O
of	O
the	O
derivative	O
.	O
</s>
<s>
For	O
example	O
,	O
when	O
modeling	O
the	O
motion	O
of	O
a	O
satellite	O
about	O
the	O
earth	O
as	O
a	O
standard	O
Kepler	O
orbit	O
,	O
a	O
fixed	O
time-stepping	O
method	O
such	O
as	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
may	O
be	O
sufficient	O
.	O
</s>
<s>
Romberg	B-Algorithm
's	I-Algorithm
method	I-Algorithm
and	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
–	I-Algorithm
Fehlberg	I-Algorithm
are	O
examples	O
of	O
a	O
numerical	B-Algorithm
integration	I-Algorithm
methods	O
which	O
use	O
an	O
adaptive	B-Algorithm
stepsize	I-Algorithm
.	O
</s>
<s>
For	O
simplicity	O
,	O
the	O
following	O
example	O
uses	O
the	O
simplest	O
integration	O
method	O
,	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
;	O
in	O
practice	O
,	O
higher-order	O
methods	O
such	O
as	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
are	O
preferred	O
due	O
to	O
their	O
superior	O
convergence	O
and	O
stability	B-Algorithm
properties	I-Algorithm
.	O
</s>
<s>
We	O
write	O
,	O
where	O
is	O
the	O
error	O
in	O
the	O
numerical	B-General_Concept
solution	I-General_Concept
.	O
</s>
<s>
Let	O
us	O
now	O
apply	O
Euler	B-Algorithm
's	I-Algorithm
method	I-Algorithm
again	O
with	O
a	O
different	O
step	O
size	O
to	O
generate	O
a	O
second	O
approximation	O
to	O
y( tn+1	O
)	O
.	O
</s>
<s>
Take	O
the	O
new	O
step	O
size	O
to	O
be	O
one	O
half	O
of	O
the	O
original	O
step	O
size	O
,	O
and	O
apply	O
two	O
steps	O
of	O
Euler	B-Algorithm
's	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Since	O
we	O
have	O
to	O
apply	O
Euler	B-Algorithm
's	I-Algorithm
method	I-Algorithm
twice	O
,	O
the	O
local	O
error	O
is	O
(	O
in	O
the	O
worst	O
case	O
)	O
twice	O
the	O
original	O
error	O
.	O
</s>
<s>
This	O
technique	O
is	O
called	O
Richardson	B-Algorithm
extrapolation	I-Algorithm
.	O
</s>
<s>
A	O
drawback	O
is	O
that	O
the	O
step	O
size	O
may	O
become	O
prohibitively	O
small	O
,	O
especially	O
when	O
using	O
the	O
low-order	O
Euler	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Similar	O
methods	O
can	O
be	O
developed	O
for	O
higher	O
order	O
methods	O
,	O
such	O
as	O
the	O
4th-order	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Adaptive	B-Algorithm
stepsize	I-Algorithm
methods	O
that	O
use	O
a	O
so-called	O
'	O
embedded	O
 '	O
error	O
estimate	O
include	O
the	O
Bogacki	B-Algorithm
–	I-Algorithm
Shampine	I-Algorithm
,	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
–	I-Algorithm
Fehlberg	I-Algorithm
,	O
Cash	B-Algorithm
–	I-Algorithm
Karp	I-Algorithm
and	O
Dormand	B-Algorithm
–	I-Algorithm
Prince	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
The	O
ODE	O
solver	O
in	O
many	O
programming	O
languages	O
uses	O
this	O
procedure	O
as	O
the	O
default	O
strategy	O
for	O
adaptive	B-Algorithm
stepsize	I-Algorithm
control	O
,	O
which	O
adds	O
other	O
engineering	O
parameters	O
to	O
make	O
the	O
system	O
more	O
stable	O
.	O
</s>
