<s>
In	O
the	O
mathematical	O
subfield	O
of	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
numerical	B-Algorithm
stability	I-Algorithm
is	O
a	O
generally	O
desirable	O
property	O
of	O
numerical	B-General_Concept
algorithms	I-General_Concept
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
in	O
numerical	B-General_Concept
algorithms	I-General_Concept
for	O
differential	O
equations	O
the	O
concern	O
is	O
the	O
growth	O
of	O
round-off	B-Algorithm
errors	I-Algorithm
and/or	O
small	O
fluctuations	O
in	O
initial	O
data	O
which	O
might	O
cause	O
a	O
large	O
deviation	O
of	O
final	O
answer	O
from	O
the	O
exact	O
solution	O
.	O
</s>
<s>
Some	O
numerical	B-General_Concept
algorithms	I-General_Concept
may	O
damp	O
out	O
the	O
small	O
fluctuations	O
(	O
errors	O
)	O
in	O
the	O
input	O
data	O
;	O
others	O
might	O
magnify	O
such	O
errors	O
.	O
</s>
<s>
Calculations	O
that	O
can	O
be	O
proven	O
not	O
to	O
magnify	O
approximation	B-Algorithm
errors	I-Algorithm
are	O
called	O
numerically	B-Algorithm
stable	I-Algorithm
.	O
</s>
<s>
One	O
of	O
the	O
common	O
tasks	O
of	O
numerical	B-General_Concept
analysis	I-General_Concept
is	O
to	O
try	O
to	O
select	O
algorithms	O
which	O
are	O
robust	O
–	O
that	O
is	O
to	O
say	O
,	O
do	O
not	O
produce	O
a	O
wildly	O
different	O
result	O
for	O
very	O
small	O
change	O
in	O
the	O
input	O
data	O
.	O
</s>
<s>
Even	O
in	O
this	O
case	O
,	O
there	O
is	O
no	O
guarantee	O
that	O
it	O
would	O
converge	O
to	O
the	O
correct	O
solution	O
,	O
because	O
the	O
floating-point	O
round-off	B-Algorithm
or	O
truncation	B-Algorithm
errors	I-Algorithm
can	O
be	O
magnified	O
,	O
instead	O
of	O
damped	O
,	O
causing	O
the	O
deviation	O
from	O
the	O
exact	O
solution	O
to	O
grow	O
exponentially	O
.	O
</s>
<s>
Consider	O
the	O
problem	O
to	O
be	O
solved	O
by	O
the	O
numerical	B-General_Concept
algorithm	I-General_Concept
as	O
a	O
function	O
mapping	O
the	O
data	O
to	O
the	O
solution	O
.	O
</s>
<s>
The	O
main	O
causes	O
of	O
error	O
are	O
round-off	B-Algorithm
error	I-Algorithm
and	O
truncation	B-Algorithm
error	I-Algorithm
.	O
</s>
<s>
The	O
forward	O
and	O
backward	O
error	O
are	O
related	O
by	O
the	O
condition	B-Algorithm
number	I-Algorithm
:	O
the	O
forward	O
error	O
is	O
at	O
most	O
as	O
big	O
in	O
magnitude	O
as	O
the	O
condition	B-Algorithm
number	I-Algorithm
multiplied	O
by	O
the	O
magnitude	O
of	O
the	O
backward	O
error	O
.	O
</s>
<s>
In	O
many	O
cases	O
,	O
it	O
is	O
more	O
natural	O
to	O
consider	O
the	O
relative	B-Algorithm
error	I-Algorithm
instead	O
of	O
the	O
absolute	B-Algorithm
error	I-Algorithm
.	O
</s>
<s>
Often	O
,	O
we	O
want	O
the	O
error	O
to	O
be	O
of	O
the	O
same	O
order	O
as	O
,	O
or	O
perhaps	O
only	O
a	O
few	O
orders	B-Device
of	I-Device
magnitude	I-Device
bigger	O
than	O
,	O
the	O
unit	B-Algorithm
round-off	I-Algorithm
.	O
</s>
<s>
The	O
usual	O
definition	O
of	O
numerical	B-Algorithm
stability	I-Algorithm
uses	O
a	O
more	O
general	O
concept	O
,	O
called	O
mixed	O
stability	O
,	O
which	O
combines	O
the	O
forward	O
error	O
and	O
the	O
backward	O
error	O
.	O
</s>
<s>
An	O
algorithm	O
is	O
forward	O
stable	O
if	O
its	O
forward	O
error	O
divided	O
by	O
the	O
condition	B-Algorithm
number	I-Algorithm
of	O
the	O
problem	O
is	O
small	O
.	O
</s>
<s>
The	O
above	O
definitions	O
are	O
particularly	O
relevant	O
in	O
situations	O
where	O
truncation	B-Algorithm
errors	I-Algorithm
are	O
not	O
important	O
.	O
</s>
<s>
In	O
other	O
contexts	O
,	O
for	O
instance	O
when	O
solving	O
differential	O
equations	O
,	O
a	O
different	O
definition	O
of	O
numerical	B-Algorithm
stability	I-Algorithm
is	O
used	O
.	O
</s>
<s>
In	O
numerical	B-Algorithm
ordinary	I-Algorithm
differential	I-Algorithm
equations	I-Algorithm
,	O
various	O
concepts	O
of	O
numerical	B-Algorithm
stability	I-Algorithm
exist	O
,	O
for	O
instance	O
A-stability	O
.	O
</s>
<s>
It	O
is	O
important	O
to	O
use	O
a	O
stable	O
method	O
when	O
solving	O
a	O
stiff	B-Algorithm
equation	I-Algorithm
.	O
</s>
<s>
Yet	O
another	O
definition	O
is	O
used	O
in	O
numerical	B-Algorithm
partial	I-Algorithm
differential	I-Algorithm
equations	I-Algorithm
.	O
</s>
<s>
An	O
algorithm	O
for	O
solving	O
a	O
linear	O
evolutionary	O
partial	O
differential	O
equation	O
is	O
stable	O
if	O
the	O
total	O
variation	O
of	O
the	O
numerical	B-General_Concept
solution	I-General_Concept
at	O
a	O
fixed	O
time	O
remains	O
bounded	O
as	O
the	O
step	O
size	O
goes	O
to	O
zero	O
.	O
</s>
<s>
Stability	O
is	O
sometimes	O
achieved	O
by	O
including	O
numerical	B-Algorithm
diffusion	I-Algorithm
.	O
</s>
<s>
Numerical	B-Algorithm
diffusion	I-Algorithm
is	O
a	O
mathematical	O
term	O
which	O
ensures	O
that	O
roundoff	O
and	O
other	O
errors	O
in	O
the	O
calculation	O
get	O
spread	O
out	O
and	O
do	O
not	O
add	O
up	O
to	O
cause	O
the	O
calculation	O
to	O
"	O
blow	O
up	O
"	O
.	O
</s>
<s>
Von	B-Algorithm
Neumann	I-Algorithm
stability	I-Algorithm
analysis	I-Algorithm
is	O
a	O
commonly	O
used	O
procedure	O
for	O
the	O
stability	O
analysis	O
of	O
finite	B-Algorithm
difference	I-Algorithm
schemes	I-Algorithm
as	O
applied	O
to	O
linear	O
partial	O
differential	O
equations	O
.	O
</s>
