Graphics Programs Reference
In-Depth Information
f ( x )
h
h
Figure 6.5. Composite Simpson's 1
/
3rule.
x
x 1
x i
x i + 1
x i + 2
x n
a
b
To obtain the composite Simpson's 1
/
3 rule , the integration range( a
,
b ) is divided
into n
1 panels ( n odd)ofwidth h
=
( b
a )
/
( n
1)each, as indicated in Fig. 6.5.
Applying Eq. (a)totwo adjacent panels, wehave
x i + 2
f ( x i + 2 ) ] h
3
f ( x ) dx [ f ( x i )
+
4 f ( x i + 1 )
+
(b)
x i
Substituting Eq. (b) into
b
x n
x i + 2
f ( x ) dx
n
2
f ( x ) dx
=
f ( x ) dx
=
a
x 1
x i
i
=
1
,
3
,...
yields
b
f ( x ) dx I
=
[ f ( x 1 )
+
4 f ( x 2 )
+
2 f ( x 3 )
+
4 f ( x 4 )
+···
(6.10)
a
f ( x n )] h
3
···+
2 f ( x n 2 )
+
4 f ( x n 1 )
+
The composite Simpson's 1
3rule in Eq. (6.10) is perhaps the best-known method of
numerical integration. Its reputationissomewhat undeserved,since the trapezoidal
rule is more robust, and Romberg integrationis moreefficient.
The errorinthe composite Simpson's rule is
/
a ) h 4
180
( b
f (4) (
E
=
ξ
)
(6.11)
fromwhich weconcludethat Eq. (6.10) isexact if f ( x ) is apolynomialofdegree three
or less.
Simpson's 1
3rule requires the number of panels to beeven. If thisconditionis
not satisfied, wecan integrate over the first (or last) three panels with Simpson's 3
/
/
8
rule :
f ( x 4 )] 3 h
8
I
=
[ f ( x 1 )
+
3 f ( x 2 )
+
3 f ( x 3 )
+
(6.12)
and use Simpson's 1
3rule for the remaining panels. The errorinEq. (6.12) is of the
sameorderas in Eq. (6.10).
/
EXAMPLE 6.1
Derive Simpson's 1
/
3rule from Newton-Cotes formulas.
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