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In-Depth Information
easy to see if we write out
P
iD1
6
f.x
i1
/C
6
f.x
iC
2
/C
6
f.x
iC1
/
as
1
6
f.x
0
/C
4
6
f.x
2
/ C
1
6
f.x
1
/ C
1
6
f.x
1
/C
4
6
f.x
2
2
/C
1
6
f.x
2
/ C
1
6
f.x
2
/C
4
6
f.x
3
2
/C
1
6
f.x
3
/ C
C
1
6
f.x
n1
/C
4
6
f.x
n
2
/C
1
6
f.x
n
/:
This expression can be expressed as two sums plus a contribution from the end
points:
n
X
n
X
1
6
f.x
0
/C
4
f.x
i
2
/ C
2
f.x
i
/C
1
6
f.x
n
/:
6
6
iD1
iD1
A more efficient formula for Simpson's rule is therefore
!
Z
b
X
n1
X
n
f.x/dx
1
6
h
f.a/C f.b/C 2
f.x
i
/ C4
f.x
i
2
/
:
(6.4)
a
iD1
iD1
The corresponding algorithm appears in Algorithm
6.4
. Of course, we could per-
form the hand optimization of incrementing
x
i
instead of using
a C ih
and
a C .i
2
/h
. This saving of one multiplication in each loop can be detected
automatically by a compiler or simply drown in the work required by evaluating
f.x/
.
and
x
i
2
Algorithm 6.4
Optimized Simpson's Rule.
Simpson
(
a
,
b
,
f
,
n
)
h D
ba
n
s
1
D 0
for
i D 1;:::;n1
x D a C ih
s
1
s
1
C f.x/
end for
s
2
D 0
for
i D 1;:::;n
x D a C .i
2
/h
s
2
s
2
C f.x/
end for
s
6
h.f .a/ C f.b/C 2s
1
C 4s
2
/
return
s
