<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
given	O
a	O
square	O
grid	O
in	O
one	O
or	O
two	O
dimensions	O
,	O
the	O
five-point	B-Algorithm
stencil	I-Algorithm
of	O
a	O
point	O
in	O
the	O
grid	O
is	O
a	O
stencil	B-Algorithm
made	O
up	O
of	O
the	O
point	O
itself	O
together	O
with	O
its	O
four	O
"	O
neighbors	O
"	O
.	O
</s>
<s>
It	O
is	O
used	O
to	O
write	O
finite	B-Algorithm
difference	I-Algorithm
approximations	O
to	O
derivatives	B-Algorithm
at	O
grid	O
points	O
.	O
</s>
<s>
It	O
is	O
an	O
example	O
for	O
numerical	B-Algorithm
differentiation	I-Algorithm
.	O
</s>
<s>
The	O
first	B-Algorithm
derivative	I-Algorithm
of	O
a	O
function	O
f	O
of	O
a	O
real	O
variable	O
at	O
a	O
point	O
x	O
can	O
be	O
approximated	O
using	O
a	O
five-point	B-Algorithm
stencil	I-Algorithm
as	O
:	O
</s>
<s>
Alternatively	O
,	O
apply	O
Richardson	B-Algorithm
extrapolation	I-Algorithm
to	O
the	O
central	B-Algorithm
difference	I-Algorithm
approximation	I-Algorithm
to	O
on	O
grids	O
with	O
spacing	O
2h	O
and	O
h	O
.	O
</s>
<s>
Evaluating	O
the	O
derivatives	B-Algorithm
of	O
the	O
five	O
Lagrange	O
polynomials	O
at	O
gives	O
the	O
same	O
weights	O
as	O
above	O
.	O
</s>
<s>
This	O
stencil	B-Algorithm
is	O
often	O
used	O
to	O
approximate	O
the	O
Laplacian	O
of	O
a	O
function	O
of	O
two	O
variables	O
:	O
</s>
<s>
From	O
the	O
3	O
point	O
stencils	B-Algorithm
for	O
the	O
second	O
derivative	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
with	O
respect	O
to	O
x	O
and	O
y	O
:	O
</s>
