Graphics Programs Reference
In-Depth Information
Weeliminate
f
(
x
) by multiplying the first equationby4 and subtracting itfrom the
second equation. The result is
2
h
2
3
2
hf
(
x
)
f
(
x
)
f
(
x
+
2
h
)
−
4
f
(
x
+
h
)
=−
3
f
(
x
)
−
+
+···
Therefore,
h
2
3
−
f
(
x
+
2
h
)
+
4
f
(
x
+
h
)
−
3
f
(
x
)
f
(
x
)
f
(
x
)
=
+
+···
2
h
or
f
(
x
)
−
+
+
+
−
f
(
x
2
h
)
4
f
(
x
h
)
3
f
(
x
)
(
h
2
)
+
O
(5.8)
2
h
Equation (5.8) iscalled the
second forward finite difference approximation
.
Derivation of finite difference approximationsfor higherderivatives involve
additionalTaylor series. Thus the forward difference approximation for
f
(
x
) utilizes
series for
f
(
x
3
h
); the approximation for
f
(
x
) involves
+
h
),
f
(
x
+
2
h
) and
f
(
x
+
+
+
,
+
+
Taylor expansionsfor
f
(
x
4
h
), etc.As you can see,
the computationsfor high-orderderivatives can become rather tedious. The results
forboth the forward and backward finite differences aresummarizedinTables 5.3a
and 5.3b.
h
),
f
(
x
2
h
)
f
(
x
3
h
) and
f
(
x
f
(
x
)
f
(
x
+
h
)
f
(
x
+
2
h
)
f
(
x
+
3
h
)
f
(
x
+
4
h
)
f
(
x
+
5
h
)
2
hf
(
x
)
−
3
4
−
1
h
2
f
(
x
)
2
−
5
4
−
1
2
h
3
f
(
x
)
−
5
18
−
24
14
−
3
h
4
f
(4)
(
x
)
3
−
14
26
−
24
11
−
2
(
h
2
)
Table 5.3a.
Coefficients of forward finite difference approximationsof
O
f
(
x
−
5
h
)
f
(
x
−
4
h
)
f
(
x
−
3
h
)
f
(
x
−
2
h
)
f
(
x
−
h
)
f
(
x
)
2
hf
(
x
)
1
−
4
3
h
2
f
(
x
)
−
1
4
−
5
2
2
h
3
f
(
x
)
3
−
14
24
−
18
5
h
4
f
(4)
(
x
)
−
−
−
2
11
24
26
14
3
(
h
2
)
Table 5.3b.
Coefficients of backward finite difference approximationsof
O
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