<s>
The	O
explicit	B-Algorithm
methods	I-Algorithm
are	O
those	O
where	O
the	O
matrix	O
is	O
lower	B-Algorithm
triangular	I-Algorithm
.	O
</s>
<s>
The	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
is	O
first	O
order	O
.	O
</s>
<s>
The	O
(	O
explicit	B-Algorithm
)	O
midpoint	B-Algorithm
method	I-Algorithm
is	O
a	O
second-order	O
method	O
with	O
two	O
stages	O
(	O
see	O
also	O
the	O
implicit	O
midpoint	B-Algorithm
method	I-Algorithm
below	O
)	O
:	O
</s>
<s>
Heun	B-Algorithm
's	I-Algorithm
method	I-Algorithm
is	O
a	O
second-order	O
method	O
with	O
two	O
stages	O
.	O
</s>
<s>
It	O
is	O
also	O
known	O
as	O
the	O
explicit	B-Algorithm
trapezoid	O
rule	O
,	O
improved	B-Algorithm
Euler	I-Algorithm
's	I-Algorithm
method	I-Algorithm
,	O
or	O
modified	B-Algorithm
Euler	I-Algorithm
's	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Ralston	O
's	O
third-order	O
method	O
is	O
used	O
in	O
the	O
embedded	O
Bogacki	B-Algorithm
–	I-Algorithm
Shampine	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
"	O
original	O
"	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
embedded	O
methods	O
are	O
designed	O
to	O
produce	O
an	O
estimate	O
of	O
the	O
local	O
truncation	O
error	O
of	O
a	O
single	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
step	O
,	O
and	O
as	O
result	O
,	O
allow	O
to	O
control	O
the	O
error	O
with	O
adaptive	B-Algorithm
stepsize	I-Algorithm
.	O
</s>
<s>
The	O
simplest	O
adaptive	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
involves	O
combining	O
Heun	B-Algorithm
's	I-Algorithm
method	I-Algorithm
,	O
which	O
is	O
order	O
2	O
,	O
with	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
,	O
which	O
is	O
order	O
1	O
.	O
</s>
<s>
The	O
Fehlberg	B-Algorithm
method	I-Algorithm
has	O
two	O
methods	O
of	O
orders	O
1	O
and	O
2	O
.	O
</s>
<s>
The	O
Bogacki	B-Algorithm
–	I-Algorithm
Shampine	I-Algorithm
method	I-Algorithm
has	O
two	O
methods	O
of	O
orders	O
2	O
and	O
3	O
.	O
</s>
<s>
The	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
–	I-Algorithm
Fehlberg	I-Algorithm
method	I-Algorithm
has	O
two	O
methods	O
of	O
orders	O
5	O
and	O
4	O
;	O
it	O
is	O
sometimes	O
dubbed	O
RKF45	B-Algorithm
.	O
</s>
<s>
The	O
coefficients	O
here	O
allow	O
for	O
an	O
adaptive	B-Algorithm
stepsize	I-Algorithm
to	O
be	O
determined	O
automatically	O
.	O
</s>
<s>
Cash	O
and	O
Karp	O
have	O
modified	O
Fehlberg	B-Algorithm
's	O
original	O
idea	O
.	O
</s>
<s>
The	O
backward	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
is	O
first	O
order	O
.	O
</s>
<s>
The	O
implicit	O
midpoint	B-Algorithm
method	I-Algorithm
is	O
of	O
second	O
order	O
.	O
</s>
<s>
It	O
is	O
the	O
simplest	O
method	O
in	O
the	O
class	O
of	O
collocation	B-Algorithm
methods	I-Algorithm
known	O
as	O
the	O
Gauss-Legendre	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
It	O
is	O
a	O
symplectic	B-Algorithm
integrator	I-Algorithm
.	O
</s>
<s>
The	O
Crank	B-Algorithm
–	I-Algorithm
Nicolson	I-Algorithm
method	I-Algorithm
corresponds	O
to	O
the	O
implicit	O
trapezoidal	B-Algorithm
rule	I-Algorithm
and	O
is	O
a	O
second-order	O
accurate	O
and	O
A-stable	O
method	O
.	O
</s>
<s>
These	O
methods	O
are	O
based	O
on	O
the	O
points	O
of	O
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
quadrature	I-Algorithm
.	O
</s>
<s>
The	O
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
method	I-Algorithm
of	O
order	O
four	O
has	O
Butcher	O
tableau	O
:	O
</s>
<s>
The	O
Gauss	B-Algorithm
–	I-Algorithm
Legendre	I-Algorithm
method	I-Algorithm
of	O
order	O
six	O
has	O
Butcher	O
tableau	O
:	O
</s>
<s>
Diagonally	O
Implicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
(	O
DIRK	O
)	O
formulae	O
have	O
been	O
widely	O
used	O
for	O
the	O
numerical	O
solution	O
of	O
stiff	O
initial	O
value	O
problems	O
;	O
</s>
<s>
The	O
simplest	O
method	O
from	O
this	O
class	O
is	O
the	O
order	O
2	O
implicit	O
midpoint	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Kraaijevanger	O
and	O
Spijker	O
's	O
two-stage	O
Diagonally	O
Implicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
:	O
</s>
<s>
Qin	O
and	O
Zhang	O
's	O
two-stage	O
,	O
2nd	O
order	O
,	O
symplectic	O
Diagonally	O
Implicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
:	O
</s>
<s>
Pareschi	O
and	O
Russo	O
's	O
two-stage	O
2nd	O
order	O
Diagonally	O
Implicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
:	O
</s>
<s>
This	O
Diagonally	O
Implicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
is	O
A-stable	O
if	O
and	O
only	O
if	O
.	O
</s>
<s>
Qin	O
and	O
Zhang	O
's	O
Diagonally	O
Implicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
corresponds	O
to	O
Pareschi	O
and	O
Russo	O
's	O
Diagonally	O
Implicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
with	O
.	O
</s>
<s>
Two-stage	O
2nd	O
order	O
Diagonally	O
Implicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
:	O
</s>
<s>
Again	O
,	O
this	O
Diagonally	O
Implicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
is	O
A-stable	O
if	O
and	O
only	O
if	O
.	O
</s>
<s>
Crouzeix	O
's	O
two-stage	O
,	O
3rd	O
order	O
Diagonally	O
Implicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
:	O
</s>
<s>
Crouzeix	O
's	O
three-stage	O
,	O
4th	O
order	O
Diagonally	O
Implicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
:	O
</s>
<s>
Three-stage	O
,	O
3rd	O
order	O
,	O
L-stable	O
Diagonally	O
Implicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
:	O
</s>
<s>
Nørsett	O
's	O
three-stage	O
,	O
4th	O
order	O
Diagonally	O
Implicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
has	O
the	O
following	O
Butcher	O
tableau	O
:	O
</s>
<s>
All	O
are	O
implicit	B-Algorithm
methods	I-Algorithm
,	O
have	O
order	O
2s2	O
and	O
they	O
all	O
have	O
c1	O
=	O
0	O
and	O
cs	O
=	O
1	O
.	O
</s>
<s>
Unlike	O
any	O
explicit	B-Algorithm
method	I-Algorithm
,	O
it	O
's	O
possible	O
for	O
these	O
methods	O
to	O
have	O
the	O
order	O
greater	O
than	O
the	O
number	O
of	O
stages	O
.	O
</s>
<s>
The	O
Lobatto	O
IIIA	O
methods	O
are	O
collocation	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
The	O
second-order	O
method	O
is	O
known	O
as	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
:	O
</s>
<s>
The	O
Lobatto	O
IIIB	O
methods	O
are	O
not	O
collocation	B-Algorithm
methods	I-Algorithm
,	O
but	O
they	O
can	O
be	O
viewed	O
as	O
discontinuous	O
collocation	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
The	O
Lobatto	O
IIIC	O
methods	O
also	O
are	O
discontinuous	O
collocation	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
The	O
Lobatto	O
IIIC*	O
method	O
for	O
is	O
sometimes	O
called	O
the	O
explicit	B-Algorithm
trapezoidal	I-Algorithm
rule	I-Algorithm
.	O
</s>
<s>
Radau	O
methods	O
are	O
fully	O
implicit	B-Algorithm
methods	I-Algorithm
(	O
matrix	O
A	O
of	O
such	O
methods	O
can	O
have	O
any	O
structure	O
)	O
.	O
</s>
<s>
The	O
first	O
order	O
Radau	O
method	O
is	O
similar	O
to	O
backward	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
.	O
</s>
