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(
h
2
)errorin
Note that the truncation erroris now
O
(
h
), which is not as goodas the
O
central difference approximations.
Wecan derive the approximationsfor higherderivatives in the same manner. For
example, Eqs. (a) and (c) yield
f
(
x
+
2
h
)
−
2
f
(
x
+
h
)
+
f
(
x
)
f
(
x
)
=
+
O
(
h
)
(5.7)
h
2
The third and fourth derivatives can be derivedinasimilar fashion. The results are
shown in Tables 5.2a and 5.2b.
f
(
x
)
f
(
x
+
h
)
f
(
x
+
2
h
)
f
(
x
+
3
h
)
f
(
x
+
4
h
)
hf
(
x
)
−
1
1
h
2
f
(
x
)
1
−
2
1
h
3
f
(
x
)
−
−
1
3
3
1
h
4
f
(4)
(
x
)
1
−
4
6
−
4
1
Table 5.2a.
Coefficients of forward finite difference approximations
of
O
(
h
)
f
(
x
−
4
h
)
f
(
x
−
3
h
)
f
(
x
−
2
h
)
f
(
x
−
h
)
f
(
x
)
hf
(
x
)
−
1
1
h
2
f
(
x
)
1
−
2
1
h
3
f
(
x
)
−
1
3
−
3
1
h
4
f
(4)
(
x
)
1
−
4
6
−
4
1
Table 5.2b.
Coefficients of backward finite difference approximations
of
O
(
h
)
Second Noncentral Finite Difference Approximations
Finite difference approximationsof
(
h
) are not popular duetoreasonsthat will be
explained shortly. The commonpractice istouse expressionsof
O
(
h
2
). To obtain
noncentral difference formulasofthisorder, wehavetoretain moreterms in the
Taylor series.As an illustration, we will derive the expression for
f
(
x
).Westart with
Eqs. (a) and (c), which are
O
h
2
2
h
3
6
h
4
24
hf
(
x
)
f
(
x
)
f
(
x
)
f
(4)
(
x
)
f
(
x
+
h
)
=
f
(
x
)
+
+
+
+
+···
4
h
3
3
2
h
4
3
2
hf
(
x
)
2
h
2
f
(
x
)
f
(
x
)
f
(4)
(
x
)
f
(
x
+
2
h
)
=
f
(
x
)
+
+
+
+
+···
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