Graphics Programs Reference
In-Depth Information
h
2
2!
h
3
3!
h
4
4!
hf
(
x
)
f
(
x
)
f
(
x
)
f
(4)
(
x
)
f
(
x
−
h
)
=
f
(
x
)
−
+
−
+
−···
(b)
(2
h
)
2
2!
(2
h
)
3
3!
(2
h
)
4
4!
2
hf
(
x
)
f
(
x
)
f
(
x
)
f
(4)
(
x
)
f
(
x
+
2
h
)
=
f
(
x
)
+
+
+
+
+···
(c)
(2
h
)
2
2!
(2
h
)
3
3!
(2
h
)
4
4!
2
hf
(
x
)
f
(
x
)
f
(
x
)
f
(4)
(
x
)
f
(
x
−
2
h
)
=
f
(
x
)
−
+
−
+
−···
(d)
We also record the sums and differences of the series:
h
4
12
f
(4)
(
x
)
h
2
f
(
x
)
f
(
x
+
h
)
+
f
(
x
−
h
)
=
2
f
(
x
)
+
+
+···
(e)
h
3
3
2
hf
(
x
)
f
(
x
)
f
(
x
+
h
)
−
f
(
x
−
h
)
=
+
+···
(f )
4
h
4
3
4
h
2
f
(
x
)
f
(4)
(
x
)
+
+
−
=
+
f
(
x
2
h
)
f
(
x
2
h
)
2
f
(
x
)
+
+···
(g)
8
h
3
3
4
hf
(
x
)
f
(
x
)
f
(
x
+
2
h
)
−
f
(
x
−
2
h
)
=
+
+···
(h)
Note that the sumscontain only evenderivatives, while the differences retain just the
odd derivatives.Equations(a)-(h) can be viewedassimultaneousequationsthatcan
be solved for various derivatives of
f
(
x
). The number of equations involved and the
number of termskept in each equationdepend on the order of the derivative and the
desireddegree of accuracy.
First Central Difference Approximations
The solution of Eq. (f ) for
f
(
x
) is
h
2
6
f
(
x
+
h
)
−
f
(
x
−
h
)
f
(
x
)
f
(
x
)
=
−
−···
2
h
Keeping only the first term on the right-hand side, wehave
+
−
−
f
(
x
h
)
f
(
x
h
)
f
(
x
)
(
h
2
)
=
+
O
(5.1)
2
h
which iscalled the
first central difference approximation
for
f
(
x
). The term
(
h
2
)
O
reminds usthat the truncation errorbehaves as
h
2
.
FromEq. (e) weobtain
h
2
12
f
(4)
(
x
)
f
(
x
+
h
)
−
2
f
(
x
)
+
f
(
x
−
h
)
f
(
x
)
=
+
+···
h
2
or
f
(
x
+
h
)
−
2
f
(
x
)
+
f
(
x
−
h
)
f
(
x
)
(
h
2
)
=
+
O
(5.2)
h
2
Search WWH ::
Custom Search