<s>
In	O
mathematics	O
,	O
a	O
linear	B-Algorithm
approximation	I-Algorithm
is	O
an	O
approximation	O
of	O
a	O
general	O
function	O
using	O
a	O
linear	O
function	O
(	O
more	O
precisely	O
,	O
an	O
affine	B-Algorithm
function	I-Algorithm
)	O
.	O
</s>
<s>
They	O
are	O
widely	O
used	O
in	O
the	O
method	O
of	O
finite	B-Algorithm
differences	I-Algorithm
to	O
produce	O
first	O
order	O
methods	O
for	O
solving	O
or	O
approximating	O
solutions	O
to	O
equations	O
.	O
</s>
<s>
The	O
linear	B-Algorithm
approximation	I-Algorithm
is	O
obtained	O
by	O
dropping	O
the	O
remainder	O
:	O
</s>
<s>
For	O
this	O
reason	O
,	O
this	O
process	O
is	O
also	O
called	O
the	O
tangent	B-Algorithm
line	I-Algorithm
approximation	I-Algorithm
.	O
</s>
<s>
Linear	B-Algorithm
approximations	I-Algorithm
for	O
vector	O
functions	O
of	O
a	O
vector	O
variable	O
are	O
obtained	O
in	O
the	O
same	O
way	O
,	O
with	O
the	O
derivative	O
at	O
a	O
point	O
replaced	O
by	O
the	O
Jacobian	O
matrix	O
.	O
</s>
<s>
The	O
period	O
of	O
swing	O
of	O
a	O
simple	O
gravity	O
pendulum	O
depends	O
on	O
its	O
length	O
,	O
the	O
local	O
strength	O
of	O
gravity	O
,	O
and	O
to	O
a	O
small	O
extent	O
on	O
the	O
maximum	O
angle	O
that	O
the	O
pendulum	O
swings	O
away	O
from	O
vertical	O
,	O
,	O
called	O
the	O
amplitude	B-Application
.	O
</s>
<s>
However	O
,	O
if	O
one	O
takes	O
the	O
linear	B-Algorithm
approximation	I-Algorithm
(	O
i.e.	O
</s>
<s>
if	O
the	O
amplitude	B-Application
is	O
limited	O
to	O
small	O
swings	O
,	O
)	O
the	O
period	O
is	O
:	O
</s>
<s>
In	O
the	O
linear	B-Algorithm
approximation	I-Algorithm
,	O
the	O
period	O
of	O
swing	O
is	O
approximately	O
the	O
same	O
for	O
different	O
size	O
swings	O
:	O
that	O
is	O
,	O
the	O
period	O
is	O
independent	O
of	O
amplitude	B-Application
.	O
</s>
<s>
Successive	O
swings	O
of	O
the	O
pendulum	O
,	O
even	O
if	O
changing	O
in	O
amplitude	B-Application
,	O
take	O
the	O
same	O
amount	O
of	O
time	O
.	O
</s>
<s>
If	O
the	O
temperature	O
T	O
does	O
not	O
vary	O
too	O
much	O
,	O
a	O
linear	B-Algorithm
approximation	I-Algorithm
is	O
typically	O
used	O
:	O
</s>
<s>
Because	O
the	O
linear	B-Algorithm
approximation	I-Algorithm
is	O
only	O
an	O
approximation	O
,	O
is	O
different	O
for	O
different	O
reference	O
temperatures	O
.	O
</s>
<s>
When	O
the	O
temperature	O
varies	O
over	O
a	O
large	O
temperature	O
range	O
,	O
the	O
linear	B-Algorithm
approximation	I-Algorithm
is	O
inadequate	O
and	O
a	O
more	O
detailed	O
analysis	O
and	O
understanding	O
should	O
be	O
used	O
.	O
</s>
