Lecture 4 - Computer Science Department - University of Kentucky

On the other hand, define
We expand F(a+2h) as
Basic Simpson’s Rule
F (
x
)
= x f
(
t
)
dt
a
∫
5
2 4
3 2
4
(4) 2
5
(5)
F ( a
+ 2 h
) =
F ( a
)
+
2
hF '( a
)
+
2
h F
"( a
)
+
h F
"'( a
)
+
h
f
( a
)
+
h F
( a
)
+L
3 3 5!
Note that F’=f, F(a)=0, F”=f’, F”’=f”, we have
∫ + 5
a 2h
2 4 3 2 4 2 5 (4)
f(
x)
dx=
2hf(
a)
+ 2h
f'(
a)
+ h f"(
a)
+ h f"'(
a)
+ h f ( a)
+L
a
3 3 5⋅4!
From the previous page, we have
h
[ f ( a)
+ 4 f ( a + h)
+ f ( a + 2h)]
=
3
2 4 3
2hf
( a)
+ 2h
f '(
a)
+ h f "( a)
3
+ 2 4 20 5 (4
h f "'(
a)
+ h f
) ( a)
+L
3 3⋅
4!
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