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Each formula and error is for one sub-interval with N 1 data/node/integration points. We can extend the interval to [a,b] by summing integrals and errors (error increases by one order). Numerical Integration, Simpson’s 1/3 Rule, Error, Extended Simpson’s 1/3 Rule, Newton Cotes Closed Formulae, Newton Cotes Open Formulae, Lagrange Quadratic Interpolation
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CE 341/441 - Lecture 18 - Fall 2004
p. 18.
o
2
79
f
x (
h
x
2
x
o
f
0
f
1
x
0
x
1 f
(x)
g(x)
f
2
x
2
h
h
x
0
x
1
=h
x
2
=2h
coordinate shift
CE 341/441 - Lecture 18 - Fall 2004
p. 18.
⇒ ⇒ ⇒
g x
f
o
o
x (
f
^1
^1
x (
f
^2
^2
x (
o
x (
x
x
1
x
x
2
x
o
x
1
x
o
x
2
o
x (
x
2
hx
h
2
h
2
^1
x (
x
x
o
x
x
2
x
1
x
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x
1
x
2
^1
x (
hx
x
2
h
2
^2
x (
x
x
o
x
x
1
x
2
x
o
x
2
x
1
^2
x (
x
2
hx
h
2
CE 341/441 - Lecture 18 - Fall 2004
p. 18.
⇒
h --- 3
f
o
f
^1
f
^2
f
x (
x
h --- 3
f
o
f
^1
f
^2
d
0 2
h
f
x (
f
o
f
^1
f
^2
x
1
h
f
x (
f
^1
x
h
f
1 (
)
1
x
h
2
f
2 (
)
1
x
h
3
f
3 (
)
1
x
h
4
f
4 (
)
1
O x
h
5
f
o
f
^1
h f
1 (
)
1
h
2
f
^1
2 (
)
h
3 6
f
3 (
)
1
h
4
f
4 (
)
1
O h
5
f
^1
f
^1
f
^2
f
^1
h f
1 (
)
1
h
2
f
2 (
)
1
h
3 ----^6
f
3 (
)
1
h
4
f
4 (
)
1
O h
5
CE 341/441 - Lecture 18 - Fall 2004
p. 18.
⇒ ⇒
f
^1
x
h
f
1 (
)
1
x
h
2
f
2 (
)
1
x
h
3
f
3 (
)
1
x
h
4
f
4 (
)
1
O x
h
5
x d
0 2
h
h --- 3
f
^1
h f
1 (
)
1
h
2
f
2 (
)
1
h
3 6
f
3 (
)
1
h
4
f
^1
4 (
)
O h
5
f
^1
f
^1
h f
^1
1 (
)
h
2
f
^1
2 (
)
h
3 6
f
3 (
)
1
h
4
f
4 (
)
1
O h
5
h
f
^1
h
2
h
2
f
1 (
)
1
h
3
h
3
f
2 (
)
1
h
4
h
4
f
3 (
)
1
h
5
h
5
f
4 (
)
1
O h
6
h --- 6 3
f
^1
h
2
f
2 (
)
1
h
4
f
4 (
)
1
O h
5
CE 341/441 - Lecture 18 - Fall 2004
p. 18.
80
f a
0
f
1
f
2
f
N b
x
f
(x)
f
3
f
4
f
(x)
sub-int. 1
h
h
sub-int. 2
h
h
b
a
CE 341/441 - Lecture 18 - Fall 2004
p. 18.
⇒
⇒
f
x (
x d
h --- 3
f
o
f
^1
f
^2
f
^2
f
^3
f
^4
f
^4
f
^5
f
^6
f
N
4
f
N
3
f
N
2
f
N
2
f
N
1
f
N
a b
,
[
]
h --- 3
f
o
f
^1
f
^2
f
^3
f
^4
f
^5
f
N
2
f
N
1
f
N
a b
, [
]
h --^3
f a
f b
f a
ih
+
f a
ih
+
i^
2 2
,
N =
2
i^
1 2
,
N =
1
a b
,
[
]
CE 341/441 - Lecture 18 - Fall 2004
p. 18.
⇒
⇒
h
a b
h
b
a
b
a
a b
,
[
]
h
5 ----- 90
b
a
f
4 (
)
a b
,
[
]
h
4
-------- 180
b
a
f
4 (
)
CE 341/441 - Lecture 18 - Fall 2004
p. 18.
f
0
f
1
f
2
f
N
h
= x
0
x
s
x
1
x
2
= x
N
x
E
g(x)
f
x (
x d
x
S x
α
h w
o
f
o
w
1
f
^1
w
2
f
^2
w
N
f
N
f
i
f
x
i
(
x
i
x
S
ih
h
x
N
x
o
CE 341/441 - Lecture 18 - Fall 2004
p. 18.
a b
g x
Exact for
f
2 (
)
f
4 (
)
f
4 (
)
f
6 (
)
CE 341/441 - Lecture 18 - Fall 2004
p. 18.
82
N f
x (
g x
x
o
x
1
,
[
]
x
1
x
2
,
[
]
f
0
f
1
f
2
x
0
x
1
x
g(x)^2
f
(x)
E[x
0
,x
1
E[x
1
,x
2
x
1
CE 341/441 - Lecture 18 - Fall 2004
p. 18.
Error compared to closed
interval
Interval size compared to
closed interval
f
x (
x d
x
S x
α
h w
0
f
^0
w
1
f
^1
w
N
f
N
α
w
i
i
h
3
f
2 (
)
h
5
f
4 (
)
h
5
f
4 (
)
CE 341/441 - Lecture 18 - Fall 2004
p. 18.
a b
