<s>
The	O
Bogacki	B-Algorithm
–	I-Algorithm
Shampine	I-Algorithm
method	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
,	O
that	O
was	O
proposed	O
by	O
Przemysław	B-Algorithm
Bogacki	I-Algorithm
and	O
Lawrence	O
F	O
.	O
Shampine	O
in	O
1989	O
.	O
</s>
<s>
The	O
Bogacki	B-Algorithm
–	I-Algorithm
Shampine	I-Algorithm
method	I-Algorithm
is	O
a	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
of	O
order	O
three	O
with	O
four	O
stages	O
with	O
the	O
First	O
Same	O
As	O
Last	O
(	O
FSAL	O
)	O
property	O
,	O
so	O
that	O
it	O
uses	O
approximately	O
three	O
function	O
evaluations	O
per	O
step	O
.	O
</s>
<s>
It	O
has	O
an	O
embedded	O
second-order	O
method	O
which	O
can	O
be	O
used	O
to	O
implement	O
adaptive	B-Algorithm
step	I-Algorithm
size	I-Algorithm
.	O
</s>
<s>
The	O
Bogacki	B-Algorithm
–	I-Algorithm
Shampine	I-Algorithm
method	I-Algorithm
is	O
implemented	O
in	O
the	O
ode3	O
for	O
fixed	O
step	O
solver	O
and	O
ode23	O
for	O
a	O
variable	O
step	O
solver	O
function	O
in	O
MATLAB	B-Language
.	O
</s>
<s>
Low-order	O
methods	O
are	O
more	O
suitable	O
than	O
higher-order	O
methods	O
like	O
the	O
Dormand	B-Algorithm
–	I-Algorithm
Prince	I-Algorithm
method	I-Algorithm
of	O
order	O
five	O
,	O
if	O
only	O
a	O
crude	O
approximation	O
to	O
the	O
solution	O
is	O
required	O
.	O
</s>
<s>
The	O
Butcher	O
tableau	O
for	O
the	O
Bogacki	B-Algorithm
–	I-Algorithm
Shampine	I-Algorithm
method	I-Algorithm
is	O
:	O
</s>
<s>
Then	O
,	O
one	O
step	O
of	O
the	O
Bogacki	B-Algorithm
–	I-Algorithm
Shampine	I-Algorithm
method	I-Algorithm
is	O
given	O
by	O
:	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
is	O
a	O
third-order	O
approximation	O
,	O
so	O
the	O
difference	O
between	O
and	O
can	O
be	O
used	O
to	O
adapt	B-Algorithm
the	I-Algorithm
step	I-Algorithm
size	I-Algorithm
.	O
</s>
