Lecture 4 - Computer Science Department - University of Kentucky

Let h be the interval length
Error Analysis (5)
x i h
∫
+ f
(x)dx =
[f (x
)
+
f (x
x i 2
1
)] −
h
12
=
i i+1
f"(
ξ
)
1 3
Sum over all subintervals to get the composite trapezoid rule
b
−1
x
−1
−1
i
∫ = ∑ ∫
+ 1
f ( x)
dx f ( x)
x = ∑ [ f ( xi
) + f ( xi
+ 1)]
− ∑
a
n
i=
0
x
i
h
2
n
i=
0
3
h
12
n
i=
0
f "( ξ )
i
Note that h=(b-a)/n, we use Intermediate-Value Theorem of Continuous
Functions,
3 1
1
h n −
n−
b − a 2 ⎡1
− ∑ f "( ξi
) = − h
12 i
12
⎢ ∑
= 0 ⎣n
i
=
0
⎤
f "( ξi
) ⎥ ⎦
i i ⎦
= −
b − a
12
h
2
f
"( ζ )
18