<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
the	O
condition	B-Algorithm
number	I-Algorithm
of	O
a	O
function	O
measures	O
how	O
much	O
the	O
output	O
value	O
of	O
the	O
function	O
can	O
change	O
for	O
a	O
small	O
change	O
in	O
the	O
input	O
argument	O
.	O
</s>
<s>
Very	O
frequently	O
,	O
one	O
is	O
solving	O
the	O
inverse	O
problem	O
:	O
given	O
one	O
is	O
solving	O
for	O
x	O
,	O
and	O
thus	O
the	O
condition	B-Algorithm
number	I-Algorithm
of	O
the	O
(	O
local	O
)	O
inverse	O
must	O
be	O
used	O
.	O
</s>
<s>
In	O
linear	B-General_Concept
regression	I-General_Concept
the	O
condition	B-Algorithm
number	I-Algorithm
of	O
the	O
moment	B-Algorithm
matrix	I-Algorithm
can	O
be	O
used	O
as	O
a	O
diagnostic	O
for	O
multicollinearity	O
.	O
</s>
<s>
The	O
condition	B-Algorithm
number	I-Algorithm
is	O
an	O
application	O
of	O
the	O
derivative	B-Algorithm
,	O
and	O
is	O
formally	O
defined	O
as	O
the	O
value	O
of	O
the	O
asymptotic	O
worst-case	O
relative	O
change	O
in	O
output	O
for	O
a	O
relative	O
change	O
in	O
input	O
.	O
</s>
<s>
The	O
condition	B-Algorithm
number	I-Algorithm
is	O
frequently	O
applied	O
to	O
questions	O
in	O
linear	B-Language
algebra	I-Language
,	O
in	O
which	O
case	O
the	O
derivative	B-Algorithm
is	O
straightforward	O
but	O
the	O
error	O
could	O
be	O
in	O
many	O
different	O
directions	O
,	O
and	O
is	O
thus	O
computed	O
from	O
the	O
geometry	O
of	O
the	O
matrix	B-Architecture
.	O
</s>
<s>
More	O
generally	O
,	O
condition	B-Algorithm
numbers	I-Algorithm
can	O
be	O
defined	O
for	O
non-linear	O
functions	O
in	O
several	O
variables	O
.	O
</s>
<s>
A	O
problem	O
with	O
a	O
low	O
condition	B-Algorithm
number	I-Algorithm
is	O
said	O
to	O
be	O
well-conditioned	B-Algorithm
,	O
while	O
a	O
problem	O
with	O
a	O
high	O
condition	B-Algorithm
number	I-Algorithm
is	O
said	O
to	O
be	O
ill-conditioned	B-Algorithm
.	O
</s>
<s>
In	O
non-mathematical	O
terms	O
,	O
an	O
ill-conditioned	B-Algorithm
problem	O
is	O
one	O
where	O
,	O
for	O
a	O
small	O
change	O
in	O
the	O
inputs	O
(	O
the	O
independent	O
variables	O
)	O
there	O
is	O
a	O
large	O
change	O
in	O
the	O
answer	O
or	O
dependent	O
variable	O
.	O
</s>
<s>
The	O
condition	B-Algorithm
number	I-Algorithm
is	O
a	O
property	O
of	O
the	O
problem	O
.	O
</s>
<s>
Some	O
algorithms	O
have	O
a	O
property	O
called	O
backward	B-Algorithm
stability	I-Algorithm
;	O
in	O
general	O
,	O
a	O
backward	O
stable	O
algorithm	O
can	O
be	O
expected	O
to	O
accurately	O
solve	O
well-conditioned	B-Algorithm
problems	I-Algorithm
.	O
</s>
<s>
Numerical	B-General_Concept
analysis	I-General_Concept
textbooks	O
give	O
formulas	O
for	O
the	O
condition	B-Algorithm
numbers	I-Algorithm
of	O
problems	O
and	O
identify	O
known	O
backward	O
stable	O
algorithms	O
.	O
</s>
<s>
As	O
a	O
rule	O
of	O
thumb	O
,	O
if	O
the	O
condition	B-Algorithm
number	I-Algorithm
,	O
then	O
you	O
may	O
lose	O
up	O
to	O
digits	O
of	O
accuracy	O
on	O
top	O
of	O
what	O
would	O
be	O
lost	O
to	O
the	O
numerical	O
method	O
due	O
to	O
loss	O
of	O
precision	B-Architecture
from	O
arithmetic	O
methods	O
.	O
</s>
<s>
However	O
,	O
the	O
condition	B-Algorithm
number	I-Algorithm
does	O
not	O
give	O
the	O
exact	O
value	O
of	O
the	O
maximum	O
inaccuracy	O
that	O
may	O
occur	O
in	O
the	O
algorithm	O
.	O
</s>
<s>
Note	O
that	O
this	O
is	O
before	O
the	O
effects	O
of	O
round-off	B-Algorithm
error	I-Algorithm
are	O
taken	O
into	O
account	O
;	O
conditioning	O
is	O
a	O
property	O
of	O
the	O
matrix	B-Architecture
,	O
not	O
the	O
algorithm	O
or	O
floating-point	B-Algorithm
accuracy	O
of	O
the	O
computer	O
used	O
to	O
solve	O
the	O
corresponding	O
system	O
.	O
</s>
<s>
In	O
particular	O
,	O
one	O
should	O
think	O
of	O
the	O
condition	B-Algorithm
number	I-Algorithm
as	O
being	O
(	O
very	O
roughly	O
)	O
the	O
rate	O
at	O
which	O
the	O
solution	O
x	O
will	O
change	O
with	O
respect	O
to	O
a	O
change	O
in	O
b	O
.	O
</s>
<s>
Thus	O
,	O
if	O
the	O
condition	B-Algorithm
number	I-Algorithm
is	O
large	O
,	O
even	O
a	O
small	O
error	O
in	O
b	O
may	O
cause	O
a	O
large	O
error	O
in	O
x	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
if	O
the	O
condition	B-Algorithm
number	I-Algorithm
is	O
small	O
,	O
then	O
the	O
error	O
in	O
x	O
will	O
not	O
be	O
much	O
bigger	O
than	O
the	O
error	O
in	O
b	O
.	O
</s>
<s>
The	O
condition	B-Algorithm
number	I-Algorithm
is	O
defined	O
more	O
precisely	O
to	O
be	O
the	O
maximum	O
ratio	O
of	O
the	O
relative	B-Algorithm
error	I-Algorithm
in	O
x	O
to	O
the	O
relative	B-Algorithm
error	I-Algorithm
in	O
b	O
.	O
</s>
<s>
Assuming	O
that	O
A	O
is	O
a	O
nonsingular	O
matrix	B-Architecture
,	O
the	O
error	O
in	O
the	O
solution	O
A−1b	O
is	O
A−1e	O
.	O
</s>
<s>
When	O
the	O
condition	B-Algorithm
number	I-Algorithm
is	O
exactly	O
one	O
(	O
which	O
can	O
only	O
happen	O
if	O
A	O
is	O
a	O
scalar	O
multiple	O
of	O
a	O
linear	O
isometry	O
)	O
,	O
then	O
a	O
solution	O
algorithm	O
can	O
find	O
(	O
in	O
principle	O
,	O
meaning	O
if	O
the	O
algorithm	O
introduces	O
no	O
errors	O
of	O
its	O
own	O
)	O
an	O
approximation	O
of	O
the	O
solution	O
whose	O
precision	B-Architecture
is	O
no	O
worse	O
than	O
that	O
of	O
the	O
data	O
.	O
</s>
<s>
However	O
,	O
it	O
does	O
not	O
mean	O
that	O
the	O
algorithm	O
will	O
converge	B-Algorithm
rapidly	O
to	O
this	O
solution	O
,	O
just	O
that	O
it	O
will	O
not	O
diverge	O
arbitrarily	O
because	O
of	O
inaccuracy	O
on	O
the	O
source	O
data	O
(	O
backward	O
error	O
)	O
,	O
provided	O
that	O
the	O
forward	O
error	O
introduced	O
by	O
the	O
algorithm	O
does	O
not	O
diverge	O
as	O
well	O
because	O
of	O
accumulating	O
intermediate	O
rounding	B-Algorithm
errors	I-Algorithm
.	O
</s>
<s>
The	O
condition	B-Algorithm
number	I-Algorithm
may	O
also	O
be	O
infinite	O
,	O
but	O
this	O
implies	O
that	O
the	O
problem	O
is	O
ill-posed	B-Algorithm
(	O
does	O
not	O
possess	O
a	O
unique	O
,	O
well-defined	O
solution	O
for	O
each	O
choice	O
of	O
data	O
;	O
that	O
is	O
,	O
the	O
matrix	B-Architecture
is	O
not	O
invertible	O
)	O
,	O
and	O
no	O
algorithm	O
can	O
be	O
expected	O
to	O
reliably	O
find	O
a	O
solution	O
.	O
</s>
<s>
The	O
definition	O
of	O
the	O
condition	B-Algorithm
number	I-Algorithm
depends	O
on	O
the	O
choice	O
of	O
norm	O
,	O
as	O
can	O
be	O
illustrated	O
by	O
two	O
examples	O
.	O
</s>
<s>
If	O
is	O
normal	B-Algorithm
,	O
then	O
where	O
and	O
are	O
maximal	O
and	O
minimal	O
(	O
by	O
moduli	O
)	O
eigenvalues	O
of	O
respectively	O
.	O
</s>
<s>
The	O
condition	B-Algorithm
number	I-Algorithm
with	O
respect	O
to	O
L2	O
arises	O
so	O
often	O
in	O
numerical	O
linear	B-Language
algebra	I-Language
that	O
it	O
is	O
given	O
a	O
name	O
,	O
the	O
condition	B-Algorithm
number	I-Algorithm
of	O
a	O
matrix	B-Architecture
.	O
</s>
<s>
If	O
is	O
the	O
matrix	B-Architecture
norm	O
induced	O
by	O
the	O
(	O
vector	O
)	O
norm	O
and	O
is	O
lower	B-Algorithm
triangular	I-Algorithm
non-singular	O
(	O
i.e.	O
</s>
<s>
recalling	O
that	O
the	O
eigenvalues	O
of	O
any	O
triangular	B-Algorithm
matrix	I-Algorithm
are	O
simply	O
the	O
diagonal	O
entries	O
.	O
</s>
<s>
The	O
condition	B-Algorithm
number	I-Algorithm
computed	O
with	O
this	O
norm	O
is	O
generally	O
larger	O
than	O
the	O
condition	B-Algorithm
number	I-Algorithm
computed	O
relative	O
to	O
the	O
Euclidean	O
norm	O
,	O
but	O
it	O
can	O
be	O
evaluated	O
more	O
easily	O
(	O
and	O
this	O
is	O
often	O
the	O
only	O
practicably	O
computable	O
condition	B-Algorithm
number	I-Algorithm
,	O
when	O
the	O
problem	O
to	O
solve	O
involves	O
a	O
non-linear	O
algebra	O
,	O
for	O
example	O
when	O
approximating	O
irrational	O
and	O
transcendental	O
functions	O
or	O
numbers	O
with	O
numerical	B-General_Concept
methods	I-General_Concept
)	O
.	O
</s>
<s>
If	O
the	O
condition	B-Algorithm
number	I-Algorithm
is	O
not	O
significantly	O
larger	O
than	O
one	O
,	O
the	O
matrix	B-Architecture
is	O
well-conditioned	B-Algorithm
,	O
which	O
means	O
that	O
its	O
inverse	O
can	O
be	O
computed	O
with	O
good	O
accuracy	O
.	O
</s>
<s>
If	O
the	O
condition	B-Algorithm
number	I-Algorithm
is	O
very	O
large	O
,	O
then	O
the	O
matrix	B-Architecture
is	O
said	O
to	O
be	O
ill-conditioned	B-Algorithm
.	O
</s>
<s>
Practically	O
,	O
such	O
a	O
matrix	B-Architecture
is	O
almost	O
singular	O
,	O
and	O
the	O
computation	O
of	O
its	O
inverse	O
,	O
or	O
solution	O
of	O
a	O
linear	O
system	O
of	O
equations	O
is	O
prone	O
to	O
large	O
numerical	O
errors	O
.	O
</s>
<s>
A	O
matrix	B-Architecture
that	O
is	O
not	O
invertible	O
is	O
often	O
said	O
to	O
have	O
a	O
condition	B-Algorithm
number	I-Algorithm
equal	O
to	O
infinity	O
.	O
</s>
<s>
Alternatively	O
,	O
it	O
can	O
be	O
defined	O
as	O
,	O
where	O
is	O
the	O
Moore-Penrose	O
pseudoinverse	B-Algorithm
.	O
</s>
<s>
For	O
square	O
matrices	O
,	O
this	O
unfortunately	O
makes	O
the	O
condition	B-Algorithm
number	I-Algorithm
discontinuous	O
,	O
but	O
it	O
is	O
a	O
useful	O
definition	O
for	O
rectangular	O
matrices	O
,	O
which	O
are	O
never	O
invertible	O
but	O
are	O
still	O
used	O
to	O
define	O
systems	O
of	O
equations	O
.	O
</s>
<s>
Condition	B-Algorithm
numbers	I-Algorithm
can	O
also	O
be	O
defined	O
for	O
nonlinear	O
functions	O
,	O
and	O
can	O
be	O
computed	O
using	O
calculus	O
.	O
</s>
<s>
The	O
condition	B-Algorithm
number	I-Algorithm
varies	O
with	O
the	O
point	O
;	O
in	O
some	O
cases	O
one	O
can	O
use	O
the	O
maximum	O
(	O
or	O
supremum	O
)	O
condition	B-Algorithm
number	I-Algorithm
over	O
the	O
domain	B-Algorithm
of	O
the	O
function	O
or	O
domain	B-Algorithm
of	O
the	O
question	O
as	O
an	O
overall	O
condition	B-Algorithm
number	I-Algorithm
,	O
while	O
in	O
other	O
cases	O
the	O
condition	B-Algorithm
number	I-Algorithm
at	O
a	O
particular	O
point	O
is	O
of	O
more	O
interest	O
.	O
</s>
<s>
The	O
condition	B-Algorithm
number	I-Algorithm
of	O
a	O
differentiable	O
function	O
in	O
one	O
variable	O
as	O
a	O
function	O
is	O
.	O
</s>
<s>
Most	O
elegantly	O
,	O
this	O
can	O
be	O
understood	O
as	O
(	O
the	O
absolute	O
value	O
of	O
)	O
the	O
ratio	O
of	O
the	O
logarithmic	O
derivative	B-Algorithm
of	O
,	O
which	O
is	O
,	O
and	O
the	O
logarithmic	O
derivative	B-Algorithm
of	O
,	O
which	O
is	O
,	O
yielding	O
a	O
ratio	O
of	O
.	O
</s>
<s>
This	O
is	O
because	O
the	O
logarithmic	O
derivative	B-Algorithm
is	O
the	O
infinitesimal	O
rate	O
of	O
relative	O
change	O
in	O
a	O
function	O
:	O
it	O
is	O
the	O
derivative	B-Algorithm
scaled	O
by	O
the	O
value	O
of	O
.	O
</s>
<s>
Note	O
that	O
if	O
a	O
function	O
has	O
a	O
zero	O
at	O
a	O
point	O
,	O
its	O
condition	B-Algorithm
number	I-Algorithm
at	O
the	O
point	O
is	O
infinite	O
,	O
as	O
infinitesimal	O
changes	O
in	O
the	O
input	O
can	O
change	O
the	O
output	O
from	O
zero	O
to	O
positive	O
or	O
negative	O
,	O
yielding	O
a	O
ratio	O
with	O
zero	O
in	O
the	O
denominator	O
,	O
hence	O
infinite	O
relative	O
change	O
.	O
</s>
<s>
The	O
last	O
term	O
is	O
the	O
difference	B-Algorithm
quotient	I-Algorithm
(	O
the	O
slope	O
of	O
the	O
secant	O
line	O
)	O
,	O
and	O
taking	O
the	O
limit	B-Algorithm
yields	O
the	O
derivative	B-Algorithm
.	O
</s>
<s>
Condition	B-Algorithm
numbers	I-Algorithm
of	O
common	O
elementary	O
functions	O
are	O
particularly	O
important	O
in	O
computing	O
significant	B-Architecture
figures	I-Architecture
and	O
can	O
be	O
computed	O
immediately	O
from	O
the	O
derivative	B-Algorithm
;	O
see	O
significance	O
arithmetic	O
of	O
transcendental	O
functions	O
.	O
</s>
<s>
Condition	B-Algorithm
numbers	I-Algorithm
can	O
be	O
defined	O
for	O
any	O
function	O
mapping	O
its	O
data	O
from	O
some	O
domain	B-Algorithm
(	O
e.g.	O
</s>
<s>
an	O
-tuple	O
of	O
real	O
numbers	O
)	O
into	O
some	O
codomain	B-Algorithm
(	O
e.g.	O
</s>
<s>
an	O
-tuple	O
of	O
real	O
numbers	O
)	O
,	O
where	O
both	O
the	O
domain	B-Algorithm
and	O
codomain	B-Algorithm
are	O
Banach	O
spaces	O
.	O
</s>
<s>
The	O
condition	B-Algorithm
number	I-Algorithm
of	O
at	O
a	O
point	O
(	O
specifically	O
,	O
its	O
relative	O
condition	B-Algorithm
number	I-Algorithm
)	O
is	O
then	O
defined	O
to	O
be	O
the	O
maximum	O
ratio	O
of	O
the	O
fractional	O
change	O
in	O
to	O
any	O
fractional	O
change	O
in	O
,	O
in	O
the	O
limit	B-Algorithm
where	O
the	O
change	O
in	O
becomes	O
infinitesimally	O
small	O
:	O
</s>
<s>
where	O
is	O
a	O
norm	O
on	O
the	O
domain/codomain	O
of	O
.	O
</s>
<s>
where	O
denotes	O
the	O
Jacobian	O
matrix	B-Architecture
of	O
partial	O
derivatives	B-Algorithm
of	O
at	O
,	O
and	O
is	O
the	O
induced	O
norm	O
on	O
the	O
matrix	B-Architecture
.	O
</s>
