<s>
In	O
mathematics	O
,	O
a	O
stiff	B-Algorithm
equation	I-Algorithm
is	O
a	O
differential	O
equation	O
for	O
which	O
certain	O
numerical	B-General_Concept
methods	I-General_Concept
for	O
solving	O
the	O
equation	O
are	O
numerically	B-Algorithm
unstable	I-Algorithm
,	O
unless	O
the	O
step	O
size	O
is	O
taken	O
to	O
be	O
extremely	O
small	O
.	O
</s>
<s>
It	O
has	O
proven	O
difficult	O
to	O
formulate	O
a	O
precise	O
definition	O
of	O
stiffness	B-Algorithm
,	O
but	O
the	O
main	O
idea	O
is	O
that	O
the	O
equation	O
includes	O
some	O
terms	O
that	O
can	O
lead	O
to	O
rapid	O
variation	O
in	O
the	O
solution	O
.	O
</s>
<s>
When	O
integrating	O
a	O
differential	O
equation	O
numerically	B-General_Concept
,	O
one	O
would	O
expect	O
the	O
requisite	O
step	O
size	O
to	O
be	O
relatively	O
small	O
in	O
a	O
region	O
where	O
the	O
solution	O
curve	O
displays	O
much	O
variation	O
and	O
to	O
be	O
relatively	O
large	O
where	O
the	O
solution	O
curve	O
straightens	O
out	O
to	O
approach	O
a	O
line	O
with	O
slope	O
nearly	O
zero	O
.	O
</s>
<s>
In	O
order	O
for	O
a	O
numerical	B-General_Concept
method	I-General_Concept
to	O
give	O
a	O
reliable	O
solution	O
to	O
the	O
differential	O
system	O
sometimes	O
the	O
step	O
size	O
is	O
required	O
to	O
be	O
at	O
an	O
unacceptably	O
small	O
level	O
in	O
a	O
region	O
where	O
the	O
solution	O
curve	O
is	O
very	O
smooth	O
.	O
</s>
<s>
The	O
phenomenon	O
is	O
known	O
as	O
stiffness	B-Algorithm
.	O
</s>
<s>
Such	O
systems	O
are	O
thus	O
known	O
as	O
stiff	B-Algorithm
systems	I-Algorithm
.	O
</s>
<s>
We	O
seek	O
a	O
numerical	B-General_Concept
solution	I-General_Concept
that	O
exhibits	O
the	O
same	O
behavior	O
.	O
</s>
<s>
One	O
of	O
the	O
most	O
prominent	O
examples	O
of	O
the	O
stiff	B-Algorithm
ordinary	I-Algorithm
differential	I-Algorithm
equations	I-Algorithm
(	O
ODEs	O
)	O
is	O
a	O
system	O
that	O
describes	O
the	O
chemical	O
reaction	O
of	O
Robertson	O
:	O
</s>
<s>
Here	O
,	O
the	O
stiffness	B-Algorithm
arises	O
from	O
the	O
coexistence	O
of	O
very	O
slow	O
and	O
very	O
fast	O
reactions	O
.	O
</s>
<s>
To	O
solve	O
them	O
,	O
the	O
software	O
packages	O
KPP	O
and	O
Autochem	B-Application
can	O
be	O
used	O
.	O
</s>
<s>
In	O
this	O
section	O
we	O
consider	O
various	O
aspects	O
of	O
the	O
phenomenon	O
of	O
stiffness	B-Algorithm
.	O
</s>
<s>
"	O
Phenomenon	O
"	O
is	O
probably	O
a	O
more	O
appropriate	O
word	O
than	O
"	O
property	O
"	O
,	O
since	O
the	O
latter	O
rather	O
implies	O
that	O
stiffness	B-Algorithm
can	O
be	O
defined	O
in	O
precise	O
mathematical	O
terms	O
;	O
it	O
turns	O
out	O
not	O
to	O
be	O
possible	O
to	O
do	O
this	O
in	O
a	O
satisfactory	O
manner	O
,	O
even	O
for	O
the	O
restricted	O
class	O
of	O
linear	O
constant	O
coefficient	O
systems	O
.	O
</s>
<s>
We	O
shall	O
also	O
see	O
several	O
qualitative	O
statements	O
that	O
can	O
be	O
(	O
and	O
mostly	O
have	O
been	O
)	O
made	O
in	O
an	O
attempt	O
to	O
encapsulate	O
the	O
notion	O
of	O
stiffness	B-Algorithm
,	O
and	O
state	O
what	O
is	O
probably	O
the	O
most	O
satisfactory	O
of	O
these	O
as	O
a	O
"	O
definition	O
"	O
of	O
stiffness	B-Algorithm
.	O
</s>
<s>
J	O
.	O
D	O
.	O
Lambert	O
defines	O
stiffness	B-Algorithm
as	O
follows	O
:	O
</s>
<s>
If	O
a	O
numerical	B-General_Concept
method	I-General_Concept
with	O
a	O
finite	O
region	O
of	O
absolute	O
stability	B-Algorithm
,	O
applied	O
to	O
a	O
system	O
with	O
any	O
initial	O
conditions	O
,	O
is	O
forced	O
to	O
use	O
in	O
a	O
certain	O
interval	O
of	O
integration	O
a	O
step	O
length	O
which	O
is	O
excessively	O
small	O
in	O
relation	O
to	O
the	O
smoothness	O
of	O
the	O
exact	O
solution	O
in	O
that	O
interval	O
,	O
then	O
the	O
system	O
is	O
said	O
to	O
be	O
stiff	O
in	O
that	O
interval	O
.	O
</s>
<s>
There	O
are	O
other	O
characteristics	O
which	O
are	O
exhibited	O
by	O
many	O
examples	O
of	O
stiff	B-Algorithm
problems	I-Algorithm
,	O
but	O
for	O
each	O
there	O
are	O
counterexamples	O
,	O
so	O
these	O
characteristics	O
do	O
not	O
make	O
good	O
definitions	O
of	O
stiffness	B-Algorithm
.	O
</s>
<s>
Nonetheless	O
,	O
definitions	O
based	O
upon	O
these	O
characteristics	O
are	O
in	O
common	O
use	O
by	O
some	O
authors	O
and	O
are	O
good	O
clues	O
as	O
to	O
the	O
presence	O
of	O
stiffness	B-Algorithm
.	O
</s>
<s>
A	O
linear	O
constant	O
coefficient	O
system	O
is	O
stiff	O
if	O
all	O
of	O
its	O
eigenvalues	O
have	O
negative	O
real	O
part	O
and	O
the	O
stiffness	B-Algorithm
ratio	O
is	O
large	O
.	O
</s>
<s>
Stiffness	B-Algorithm
occurs	O
when	O
stability	B-Algorithm
requirements	O
,	O
rather	O
than	O
those	O
of	O
accuracy	O
,	O
constrain	O
the	O
step	O
length	O
.	O
</s>
<s>
Stiffness	B-Algorithm
occurs	O
when	O
some	O
components	O
of	O
the	O
solution	O
decay	O
much	O
more	O
rapidly	O
than	O
others	O
.	O
</s>
<s>
The	O
origin	O
of	O
the	O
term	O
"	O
stiffness	B-Algorithm
"	O
has	O
not	O
been	O
clearly	O
established	O
.	O
</s>
<s>
Significant	O
difficulties	O
can	O
occur	O
when	O
standard	O
numerical	B-General_Concept
techniques	I-General_Concept
are	O
applied	O
to	O
approximate	O
the	O
solution	O
of	O
a	O
differential	O
equation	O
when	O
the	O
exact	O
solution	O
contains	O
terms	O
of	O
the	O
form	O
,	O
where	O
is	O
a	O
complex	O
number	O
with	O
negative	O
real	O
part	O
.	O
</s>
<s>
These	O
are	O
all	O
examples	O
of	O
a	O
class	O
of	O
problems	O
called	O
stiff	O
(	O
mathematical	O
stiffness	B-Algorithm
)	O
systems	O
of	O
differential	O
equations	O
,	O
due	O
to	O
their	O
application	O
in	O
analyzing	O
the	O
motion	O
of	O
spring	O
and	O
mass	O
systems	O
having	O
large	O
spring	O
constants	O
(	O
physical	O
stiffness	B-Algorithm
)	O
.	O
</s>
<s>
(	O
while	O
"	O
large	O
"	O
is	O
not	O
a	O
clearly-defined	O
term	O
,	O
but	O
the	O
larger	O
the	O
above	O
quantities	O
are	O
,	O
the	O
more	O
pronounced	O
will	O
be	O
the	O
effect	O
of	O
stiffness	B-Algorithm
.	O
)	O
</s>
<s>
Equation	O
behaves	O
quite	O
similarly	O
to	O
a	O
simple	O
exponential	O
,	O
but	O
the	O
presence	O
of	O
the	O
term	O
,	O
even	O
with	O
a	O
small	O
coefficient	O
,	O
is	O
enough	O
to	O
make	O
the	O
numerical	B-General_Concept
computation	I-General_Concept
very	O
sensitive	O
to	O
step	O
size	O
.	O
</s>
<s>
The	O
behaviour	O
of	O
numerical	B-General_Concept
methods	I-General_Concept
on	O
stiff	B-Algorithm
problems	I-Algorithm
can	O
be	O
analyzed	O
by	O
applying	O
these	O
methods	O
to	O
the	O
test	O
equation	O
subject	O
to	O
the	O
initial	O
condition	O
with	O
.	O
</s>
<s>
This	O
solution	O
approaches	O
zero	O
as	O
when	O
If	O
the	O
numerical	B-General_Concept
method	I-General_Concept
also	O
exhibits	O
this	O
behaviour	O
(	O
for	O
a	O
fixed	O
step	O
size	O
)	O
,	O
then	O
the	O
method	O
is	O
said	O
to	O
be	O
A-stable	B-Algorithm
.	O
</s>
<s>
A	O
numerical	B-General_Concept
method	I-General_Concept
that	O
is	O
L-stable	O
(	O
see	O
below	O
)	O
has	O
the	O
stronger	O
property	O
that	O
the	O
solution	O
approaches	O
zero	O
in	O
a	O
single	O
step	O
as	O
the	O
step	O
size	O
goes	O
to	O
infinity	O
.	O
</s>
<s>
A-stable	B-Algorithm
methods	O
do	O
not	O
exhibit	O
the	O
instability	O
problems	O
as	O
described	O
in	O
the	O
motivating	O
example	O
.	O
</s>
<s>
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
applied	O
to	O
the	O
test	O
equation	O
take	O
the	O
form	O
,	O
and	O
,	O
by	O
induction	O
,	O
.	O
</s>
<s>
The	O
function	O
is	O
called	O
the	O
stability	B-Algorithm
function	O
.	O
</s>
<s>
This	O
motivates	O
the	O
definition	O
of	O
the	O
region	O
of	O
absolute	O
stability	B-Algorithm
(	O
sometimes	O
referred	O
to	O
simply	O
as	O
stability	B-Algorithm
region	O
)	O
,	O
which	O
is	O
the	O
set	O
.	O
</s>
<s>
The	O
method	O
is	O
A-stable	B-Algorithm
if	O
the	O
region	O
of	O
absolute	O
stability	B-Algorithm
contains	O
the	O
set	O
,	O
that	O
is	O
,	O
the	O
left	O
half	O
plane	O
.	O
</s>
<s>
Consider	O
the	B-Algorithm
Euler	I-Algorithm
methods	I-Algorithm
above	O
.	O
</s>
<s>
The	O
region	O
of	O
absolute	O
stability	B-Algorithm
for	O
this	O
method	O
is	O
thus	O
which	O
is	O
the	O
disk	O
depicted	O
on	O
the	O
right	O
.	O
</s>
<s>
The	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
is	O
not	O
A-stable	B-Algorithm
.	O
</s>
<s>
The	O
value	O
of	O
z	O
when	O
taking	O
step	O
size	O
is	O
,	O
which	O
is	O
outside	O
the	O
stability	B-Algorithm
region	O
.	O
</s>
<s>
However	O
,	O
with	O
step	O
size	O
,	O
we	O
have	O
which	O
is	O
just	O
inside	O
the	O
stability	B-Algorithm
region	O
and	O
the	O
numerical	O
results	O
converge	O
to	O
zero	O
,	O
albeit	O
rather	O
slowly	O
.	O
</s>
<s>
This	O
region	O
contains	O
the	O
left	O
half-plane	O
,	O
so	O
the	O
trapezoidal	B-Algorithm
method	I-Algorithm
is	O
A-stable	B-Algorithm
.	O
</s>
<s>
In	O
fact	O
,	O
the	O
stability	B-Algorithm
region	O
is	O
identical	O
to	O
the	O
left	O
half-plane	O
,	O
and	O
thus	O
the	O
numerical	B-General_Concept
solution	I-General_Concept
of	O
converges	O
to	O
zero	O
if	O
and	O
only	O
if	O
the	O
exact	O
solution	O
does	O
.	O
</s>
<s>
Nevertheless	O
,	O
the	O
trapezoidal	B-Algorithm
method	I-Algorithm
does	O
not	O
have	O
perfect	O
behavior	O
:	O
it	O
does	O
damp	O
all	O
decaying	O
components	O
,	O
but	O
rapidly	O
decaying	O
components	O
are	O
damped	O
only	O
very	O
mildly	O
,	O
because	O
as	O
.	O
</s>
<s>
This	O
led	O
to	O
the	O
concept	O
of	O
L-stability	B-Algorithm
:	O
a	O
method	O
is	O
L-stable	O
if	O
it	O
is	O
A-stable	B-Algorithm
and	O
as	O
.	O
</s>
<s>
The	O
trapezoidal	B-Algorithm
method	I-Algorithm
is	O
A-stable	B-Algorithm
but	O
not	O
L-stable	O
.	O
</s>
<s>
The	O
implicit	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
is	O
an	O
example	O
of	O
an	O
L-stable	O
method	O
.	O
</s>
<s>
Explicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
have	O
a	O
strictly	B-Algorithm
lower	I-Algorithm
triangular	I-Algorithm
coefficient	O
matrix	O
and	O
thus	O
,	O
their	O
stability	B-Algorithm
function	O
is	O
a	O
polynomial	O
.	O
</s>
<s>
It	O
follows	O
that	O
explicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
cannot	O
be	O
A-stable	B-Algorithm
.	O
</s>
<s>
The	O
stability	B-Algorithm
function	O
of	O
implicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
is	O
often	O
analyzed	O
using	O
order	O
stars	O
.	O
</s>
<s>
The	O
order	O
star	O
for	O
a	O
method	O
with	O
stability	B-Algorithm
function	O
is	O
defined	O
to	O
be	O
the	O
set	O
.	O
</s>
<s>
A	O
method	O
is	O
A-stable	B-Algorithm
if	O
and	O
only	O
if	O
its	O
stability	B-Algorithm
function	O
has	O
no	O
poles	O
in	O
the	O
left-hand	O
plane	O
and	O
its	O
order	O
star	O
contains	O
no	O
purely	O
imaginary	O
numbers	O
.	O
</s>
<s>
The	O
method	O
is	O
A-stable	B-Algorithm
if	O
all	O
solutions	O
of	O
the	O
recurrence	O
relation	O
converge	O
to	O
zero	O
when	O
.	O
</s>
<s>
The	O
region	O
of	O
absolute	O
stability	B-Algorithm
for	O
a	O
multistep	B-Algorithm
method	I-Algorithm
of	O
the	O
above	O
form	O
is	O
then	O
the	O
set	O
of	O
all	O
for	O
which	O
all	O
such	O
that	O
satisfy	O
.	O
</s>
<s>
Again	O
,	O
if	O
this	O
set	O
contains	O
the	O
left	O
half-plane	O
,	O
the	O
multi-step	O
method	O
is	O
said	O
to	O
be	O
A-stable	B-Algorithm
.	O
</s>
<s>
It	O
does	O
not	O
include	O
all	O
the	O
left	O
half-plane	O
(	O
in	O
fact	O
it	O
only	O
includes	O
the	O
real	O
axis	O
between	O
)	O
so	O
the	O
Adams	B-Algorithm
–	I-Algorithm
Bashforth	I-Algorithm
method	I-Algorithm
is	O
not	O
A-stable	B-Algorithm
.	O
</s>
<s>
Explicit	O
multistep	B-Algorithm
methods	I-Algorithm
can	O
never	O
be	O
A-stable	B-Algorithm
,	O
just	O
like	O
explicit	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
Implicit	O
multistep	B-Algorithm
methods	I-Algorithm
can	O
only	O
be	O
A-stable	B-Algorithm
if	O
their	O
order	O
is	O
at	O
most	O
2	O
.	O
</s>
<s>
The	O
latter	O
result	O
is	O
known	O
as	O
the	O
second	O
Dahlquist	O
barrier	O
;	O
it	O
restricts	O
the	O
usefulness	O
of	O
linear	B-Algorithm
multistep	I-Algorithm
methods	I-Algorithm
for	O
stiff	B-Algorithm
equations	I-Algorithm
.	O
</s>
<s>
An	O
example	O
of	O
a	O
second-order	O
A-stable	B-Algorithm
method	O
is	O
the	O
trapezoidal	B-Algorithm
rule	I-Algorithm
mentioned	O
above	O
,	O
which	O
can	O
also	O
be	O
considered	O
as	O
a	O
linear	B-Algorithm
multistep	I-Algorithm
method	I-Algorithm
.	O
</s>
