<s>
Linear	B-Algorithm
multistep	I-Algorithm
methods	I-Algorithm
are	O
used	O
for	O
the	O
numerical	B-Algorithm
solution	I-Algorithm
of	I-Algorithm
ordinary	I-Algorithm
differential	I-Algorithm
equations	I-Algorithm
.	O
</s>
<s>
Single-step	O
methods	O
(	O
such	O
as	O
Euler	B-Algorithm
's	I-Algorithm
method	I-Algorithm
)	O
refer	O
to	O
only	O
one	O
previous	O
point	O
and	O
its	O
derivative	O
to	O
determine	O
the	O
current	O
value	O
.	O
</s>
<s>
Methods	O
such	O
as	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
take	O
some	O
intermediate	O
steps	O
(	O
for	O
example	O
,	O
a	O
half-step	O
)	O
to	O
obtain	O
a	O
higher	O
order	O
method	O
,	O
but	O
then	O
discard	O
all	O
previous	O
information	O
before	O
taking	O
a	O
second	O
step	O
.	O
</s>
<s>
Multistep	B-Algorithm
methods	I-Algorithm
attempt	O
to	O
gain	O
efficiency	O
by	O
keeping	O
and	O
using	O
the	O
information	O
from	O
previous	O
steps	O
rather	O
than	O
discarding	O
it	O
.	O
</s>
<s>
Consequently	O
,	O
multistep	B-Algorithm
methods	I-Algorithm
refer	O
to	O
several	O
previous	O
points	O
and	O
derivative	O
values	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
linear	B-Algorithm
multistep	I-Algorithm
methods	I-Algorithm
,	O
a	O
linear	O
combination	O
of	O
the	O
previous	O
points	O
and	O
derivative	O
values	O
is	O
used	O
.	O
</s>
<s>
Multistep	B-Algorithm
methods	I-Algorithm
use	O
information	O
from	O
the	O
previous	O
steps	O
to	O
calculate	O
the	O
next	O
value	O
.	O
</s>
<s>
In	O
particular	O
,	O
a	O
linear	B-Algorithm
multistep	I-Algorithm
method	I-Algorithm
uses	O
a	O
linear	O
combination	O
of	O
and	O
to	O
calculate	O
the	O
value	O
of	O
for	O
the	O
desired	O
current	O
step	O
.	O
</s>
<s>
One	O
can	O
distinguish	O
between	O
explicit	B-Algorithm
and	I-Algorithm
implicit	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
Iterative	B-Algorithm
methods	I-Algorithm
such	O
as	O
Newton	O
's	O
method	O
are	O
often	O
used	O
to	O
solve	O
the	O
implicit	O
formula	O
.	O
</s>
<s>
Sometimes	O
an	O
explicit	O
multistep	B-Algorithm
method	I-Algorithm
is	O
used	O
to	O
"	O
predict	O
"	O
the	O
value	O
of	O
.	O
</s>
<s>
The	O
result	O
is	O
a	O
predictor	B-Algorithm
–	I-Algorithm
corrector	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
A	O
simple	O
numerical	O
method	O
is	O
Euler	B-Algorithm
's	I-Algorithm
method	I-Algorithm
:	O
</s>
<s>
Euler	B-Algorithm
's	I-Algorithm
method	I-Algorithm
can	O
be	O
viewed	O
as	O
an	O
explicit	O
multistep	B-Algorithm
method	I-Algorithm
for	O
the	O
degenerate	O
case	O
of	O
one	O
step	O
.	O
</s>
<s>
Euler	B-Algorithm
's	I-Algorithm
method	I-Algorithm
is	O
a	O
one-step	O
method	O
.	O
</s>
<s>
One	O
possibility	O
to	O
resolve	O
this	O
issue	O
is	O
to	O
use	O
the	O
computed	O
by	O
Euler	B-Algorithm
's	I-Algorithm
method	I-Algorithm
as	O
the	O
second	O
value	O
.	O
</s>
<s>
With	O
this	O
choice	O
,	O
the	O
Adams	B-Algorithm
–	I-Algorithm
Bashforth	I-Algorithm
method	I-Algorithm
yields	O
(	O
rounded	O
to	O
four	O
digits	O
)	O
:	O
</s>
<s>
The	O
exact	O
solution	O
at	O
is	O
,	O
so	O
the	O
two-step	O
Adams	B-Algorithm
–	I-Algorithm
Bashforth	I-Algorithm
method	I-Algorithm
is	O
more	O
accurate	O
than	O
Euler	B-Algorithm
's	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Three	O
families	O
of	O
linear	B-Algorithm
multistep	I-Algorithm
methods	I-Algorithm
are	O
commonly	O
used	O
:	O
Adams	B-Algorithm
–	I-Algorithm
Bashforth	I-Algorithm
methods	I-Algorithm
,	O
Adams	O
–	O
Moulton	O
methods	O
,	O
and	O
the	O
backward	B-Algorithm
differentiation	I-Algorithm
formulas	I-Algorithm
(	O
BDFs	O
)	O
.	O
</s>
<s>
The	O
Adams	B-Algorithm
–	I-Algorithm
Bashforth	I-Algorithm
methods	I-Algorithm
are	O
explicit	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
The	O
Adams	B-Algorithm
–	I-Algorithm
Bashforth	I-Algorithm
methods	I-Algorithm
with	O
s	O
=	O
1	O
,	O
2	O
,	O
3	O
,	O
4	O
,	O
5	O
are	O
(	O
;	O
)	O
:	O
</s>
<s>
The	O
Adams	B-Algorithm
–	I-Algorithm
Bashforth	I-Algorithm
method	I-Algorithm
arises	O
when	O
the	O
formula	O
for	O
p	O
is	O
substituted	O
.	O
</s>
<s>
The	O
Adams	B-Algorithm
–	I-Algorithm
Bashforth	I-Algorithm
methods	I-Algorithm
were	O
designed	O
by	O
John	O
Couch	O
Adams	O
to	O
solve	O
a	O
differential	O
equation	O
modelling	O
capillary	O
action	O
due	O
to	O
Francis	O
Bashforth	O
.	O
</s>
<s>
The	O
Adams	O
–	O
Moulton	O
methods	O
are	O
similar	O
to	O
the	O
Adams	B-Algorithm
–	I-Algorithm
Bashforth	I-Algorithm
methods	I-Algorithm
in	O
that	O
they	O
also	O
have	O
and	O
.	O
</s>
<s>
However	O
,	O
the	O
Adams	O
–	O
Moulton	O
methods	O
are	O
implicit	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
By	O
removing	O
the	O
restriction	O
that	O
,	O
an	O
s-step	O
Adams	O
–	O
Moulton	O
method	O
can	O
reach	O
order	O
,	O
while	O
an	O
s-step	O
Adams	B-Algorithm
–	I-Algorithm
Bashforth	I-Algorithm
methods	I-Algorithm
has	O
only	O
order	O
s	O
.	O
</s>
<s>
The	O
Adams	O
–	O
Moulton	O
methods	O
with	O
s	O
=	O
0	O
,	O
1	O
,	O
2	O
,	O
3	O
,	O
4	O
are	O
(	O
;	O
)	O
listed	O
,	O
where	O
the	O
first	O
two	O
methods	O
are	O
the	O
backward	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
and	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
respectively	O
:	O
</s>
<s>
The	O
derivation	O
of	O
the	O
Adams	O
–	O
Moulton	O
methods	O
is	O
similar	O
to	O
that	O
of	O
the	O
Adams	B-Algorithm
–	I-Algorithm
Bashforth	I-Algorithm
method	I-Algorithm
;	O
however	O
,	O
the	O
interpolating	O
polynomial	O
uses	O
not	O
only	O
the	O
points	O
,	O
as	O
above	O
,	O
but	O
also	O
.	O
</s>
<s>
The	O
Adams	O
–	O
Moulton	O
methods	O
are	O
solely	O
due	O
to	O
John	O
Couch	O
Adams	O
,	O
like	O
the	O
Adams	B-Algorithm
–	I-Algorithm
Bashforth	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
The	O
name	O
of	O
Forest	O
Ray	O
Moulton	O
became	O
associated	O
with	O
these	O
methods	O
because	O
he	O
realized	O
that	O
they	O
could	O
be	O
used	O
in	O
tandem	O
with	O
the	O
Adams	B-Algorithm
–	I-Algorithm
Bashforth	I-Algorithm
methods	I-Algorithm
as	O
a	O
predictor-corrector	B-Algorithm
pair	O
;	O
had	O
the	O
same	O
idea	O
.	O
</s>
<s>
The	O
BDF	O
methods	O
are	O
implicit	B-Algorithm
methods	I-Algorithm
with	O
and	O
the	O
other	O
coefficients	O
chosen	O
such	O
that	O
the	O
method	O
attains	O
order	O
s	O
(	O
the	O
maximum	O
possible	O
)	O
.	O
</s>
<s>
These	O
methods	O
are	O
especially	O
used	O
for	O
the	O
solution	O
of	O
stiff	B-Algorithm
differential	I-Algorithm
equations	I-Algorithm
.	O
</s>
<s>
The	O
central	O
concepts	O
in	O
the	O
analysis	O
of	O
linear	B-Algorithm
multistep	I-Algorithm
methods	I-Algorithm
,	O
and	O
indeed	O
any	O
numerical	O
method	O
for	O
differential	O
equations	O
,	O
are	O
convergence	O
,	O
order	O
,	O
and	O
stability	O
.	O
</s>
<s>
More	O
precisely	O
,	O
a	O
multistep	B-Algorithm
method	I-Algorithm
is	O
consistent	O
if	O
the	O
local	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
goes	O
to	O
zero	O
faster	O
than	O
the	O
step	O
size	O
h	O
as	O
h	O
goes	O
to	O
zero	O
,	O
where	O
the	O
local	B-Algorithm
truncation	I-Algorithm
error	I-Algorithm
is	O
defined	O
to	O
be	O
the	O
difference	O
between	O
the	O
result	O
of	O
the	O
method	O
,	O
assuming	O
that	O
all	O
the	O
previous	O
values	O
are	O
exact	O
,	O
and	O
the	O
exact	O
solution	O
of	O
the	O
equation	O
at	O
time	O
.	O
</s>
<s>
A	O
multistep	B-Algorithm
method	I-Algorithm
is	O
said	O
to	O
have	O
order	O
p	O
if	O
the	O
local	O
error	O
is	O
of	O
order	O
as	O
h	O
goes	O
to	O
zero	O
.	O
</s>
<s>
The	O
s-step	O
Adams	B-Algorithm
–	I-Algorithm
Bashforth	I-Algorithm
method	I-Algorithm
has	O
order	O
s	O
,	O
while	O
the	O
s-step	O
Adams	O
–	O
Moulton	O
method	O
has	O
order	O
.	O
</s>
<s>
A	O
linear	B-Algorithm
multistep	I-Algorithm
method	I-Algorithm
is	O
zero-stable	O
for	O
a	O
certain	O
differential	O
equation	O
on	O
a	O
given	O
time	O
interval	O
,	O
if	O
a	O
perturbation	O
in	O
the	O
starting	O
values	O
of	O
size	O
ε	O
causes	O
the	O
numerical	O
solution	O
over	O
that	O
time	O
interval	O
to	O
change	O
by	O
no	O
more	O
than	O
Kε	O
for	O
some	O
value	O
of	O
K	O
which	O
does	O
not	O
depend	O
on	O
the	O
step	O
size	O
h	O
.	O
This	O
is	O
called	O
"	O
zero-stability	O
"	O
because	O
it	O
is	O
enough	O
to	O
check	O
the	O
condition	O
for	O
the	O
differential	O
equation	O
.	O
</s>
<s>
A	O
linear	B-Algorithm
multistep	I-Algorithm
method	I-Algorithm
is	O
zero-stable	O
if	O
and	O
only	O
if	O
the	O
root	O
condition	O
is	O
satisfied	O
.	O
</s>
<s>
Now	O
suppose	O
that	O
a	O
consistent	O
linear	B-Algorithm
multistep	I-Algorithm
method	I-Algorithm
is	O
applied	O
to	O
a	O
sufficiently	O
smooth	O
differential	O
equation	O
and	O
that	O
the	O
starting	O
values	O
all	O
converge	O
to	O
the	O
initial	O
value	O
as	O
.	O
</s>
<s>
This	O
result	O
is	O
known	O
as	O
the	O
Dahlquist	O
equivalence	O
theorem	O
,	O
named	O
after	O
Germund	O
Dahlquist	O
;	O
this	O
theorem	O
is	O
similar	O
in	O
spirit	O
to	O
the	O
Lax	O
equivalence	O
theorem	O
for	O
finite	B-Algorithm
difference	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
Furthermore	O
,	O
if	O
the	O
method	O
has	O
order	O
p	O
,	O
then	O
the	O
global	B-Algorithm
error	I-Algorithm
(	O
the	O
difference	O
between	O
the	O
numerical	O
solution	O
and	O
the	O
exact	O
solution	O
at	O
a	O
fixed	O
time	O
)	O
is	O
.	O
</s>
<s>
To	O
assess	O
the	O
performance	O
of	O
linear	B-Algorithm
multistep	I-Algorithm
methods	I-Algorithm
on	O
stiff	B-Algorithm
equations	I-Algorithm
,	O
consider	O
the	O
linear	O
test	O
equation	O
y'''	O
=	O
λy	O
.	O
</s>
<s>
This	O
polynomial	O
is	O
called	O
the	O
stability	O
polynomial	O
of	O
the	O
multistep	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
If	O
all	O
of	O
its	O
roots	O
have	O
modulus	O
less	O
than	O
one	O
then	O
the	O
numerical	O
solution	O
of	O
the	O
multistep	B-Algorithm
method	I-Algorithm
will	O
converge	O
to	O
zero	O
and	O
the	O
multistep	B-Algorithm
method	I-Algorithm
is	O
said	O
to	O
be	O
absolutely	O
stable	O
for	O
that	O
value	O
of	O
hλ	O
.	O
</s>
<s>
The	O
method	O
is	O
said	O
to	O
be	O
A-stable	B-Algorithm
if	O
it	O
is	O
absolutely	O
stable	O
for	O
all	O
hλ	O
with	O
negative	O
real	O
part	O
.	O
</s>
<s>
The	O
region	O
of	O
absolute	O
stability	O
is	O
the	O
set	O
of	O
all	O
hλ	O
for	O
which	O
the	O
multistep	B-Algorithm
method	I-Algorithm
is	O
absolutely	O
stable	O
.	O
</s>
<s>
For	O
more	O
details	O
,	O
see	O
the	O
section	O
on	O
stiff	B-Algorithm
equations	I-Algorithm
and	O
multistep	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
These	O
two	O
results	O
were	O
proved	O
by	O
Germund	O
Dahlquist	O
and	O
represent	O
an	O
important	O
bound	O
for	O
the	O
order	O
of	O
convergence	O
and	O
for	O
the	O
A-stability	O
of	O
a	O
linear	B-Algorithm
multistep	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
first	O
Dahlquist	O
barrier	O
states	O
that	O
a	O
zero-stable	O
and	O
linear	O
q-step	O
multistep	B-Algorithm
method	I-Algorithm
cannot	O
attain	O
an	O
order	O
of	O
convergence	O
greater	O
than	O
q	O
+	O
1	O
if	O
q	O
is	O
odd	O
and	O
greater	O
than	O
q	O
+	O
2	O
if	O
q	O
is	O
even	O
.	O
</s>
<s>
The	O
second	O
Dahlquist	O
barrier	O
states	O
that	O
no	O
explicit	O
linear	B-Algorithm
multistep	I-Algorithm
methods	I-Algorithm
are	O
A-stable	B-Algorithm
.	O
</s>
<s>
Further	O
,	O
the	O
maximal	O
order	O
of	O
an	O
(	O
implicit	O
)	O
A-stable	B-Algorithm
linear	B-Algorithm
multistep	I-Algorithm
method	I-Algorithm
is	O
2	O
.	O
</s>
<s>
Among	O
the	O
A-stable	B-Algorithm
linear	B-Algorithm
multistep	I-Algorithm
methods	I-Algorithm
of	O
order	O
2	O
,	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
has	O
the	O
smallest	O
error	O
constant	O
.	O
</s>
