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p
2
.a/
D
f.a/;
p
2
.c/
D
f.c/;
(1.49)
p
2
.b/
D
f.b/:
In order to determine A, B and C such that these conditions hold, it is common
to introduce so-called “divided differences”. By using these, A, B and C can
be expressed uniquely in terms of f.a/, f.c/ and f.b/. That is, however, a bit
technical, and we refer the interested reader to, e.g., Conte and de Boor [10].
When A, B and C are determined, p
2
.x/ can be integrated and it turns out that
we get
Z
b
b
a
6
p
2
.x/ dx
D
Œf .a/
C
4f .c /
C
f.b/
a
and hence Simpson's rule is
Z
b
b
a
6
f.x/dx
Œf .a/
C
4f .c /
C
f.b/:
(1.50)
a
Redo (b) above using (
1.50
). Discuss the relative error for the midpoint rule, the
trapezoidal rule, and Simpson's rule for this problem.
(g) Let
b
a
n
h
D
for an integer n>0and define
x
i
D
a
C
ih
for i
D
0;1;2;:::;n.
Use (
1.50
) to approximate
Z
x
i
f.x/dx and use this result to derive the
x
i 1
composite Simpson's rule
Z
b
n
X
f.x
i 1
/
C
4f .x
i 1=2
/
C
f.x
i
/
;
h
6
f.x/dx
a
i D1
where
1
2
.x
i 1
C
x
i
/:
x
i 1=2
D
(h) Redo (d) for the composite Simpson's rule.
