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In numerical methods, like tables, the values of the function are only specified at a discrete number of points! Using interpolation, we can describe or at least approximate the function at every point in space. Interpolation, Linear Interpolation, Error, Interpolating Functions, Exponential Function, Nth Order Polynomial, Data Points, Nodes, Interpolation Points
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CE 341/441 - Lecture 5 -Fall 2004
p. 5.
x^
sin(
x )
CE 341/441 - Lecture 5 -Fall 2004
p. 5.
CE 341/441 - Lecture 5 -Fall 2004
p. 5.
(1)
⇒
(2)
⇒
(3)
⇒
g x
g x
Ax
x^ o
f^
x^ ( o
x^1
f^
x ( 1
g x
o (^
f^
x^ ( o
Ax
o^
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Ax
1
f^
x ( 1
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x^ ( o
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o
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o
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x ( 1
CE 341/441 - Lecture 5 -Fall 2004
p. 5.
f^
x ( 1
f^
x^ ( o
x^1
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) x
1
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CE 341/441 - Lecture 5 -Fall 2004
p. 5.
e x (
f^
x (
g x
e x (
f^
x ( )
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1 (^
g x
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f^
x^ ( o
g x
1 (^
f^
x ( 1
CE 341/441 - Lecture 5 -Fall 2004
p. 5.
(4)
x
f(x
f(x
x^0
x^1
f(x)
g(x)
f(x)
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f^
x (
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(^
df ) ----- dx
x^
x^
x^ o
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d - 2
f d x
x^
x^ o
ξ^
x
x^
CE 341/441 - Lecture 5 -Fall 2004
p. 5.
(8)
⇒
(9)
⇒
(10)
⇒
(11)
f^
x ( )
f^
x (
f^
x^ ( o
f^
x ( 1
f^
x ( )
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x
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1
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x^
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x ( )
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(^
d - 2 f d x
x^
f^
x (
f^
x^ ( o
x 1
x
x^ o
----------------
f^
x (^1
x^
x^ o
x^ o
----------------
x^
x^ o
(^
x^
x 1
(^
d - 2 f d x
2 --------
x^
ξ =
CE 341/441 - Lecture 5 -Fall 2004
p. 5.
⇒ ⇒
where
e x (
f^
x ( )
g x
e x (
f^
x^ ( o
x^1
x
x^ o
f^
x ( 1
x^
x^ o
x^ o
x^
x^ o
(^
x^
x^1
(^
d -
2 f d x
x^
f^
x^ ( o
x^1
x
x^ o
f^
x ( 1
x^
x^ o
x^ o
e x (
x^
x^ o
(^
x^
x 1
(^
d - 2
f d x
2 --------
x^
ξ =
x^ o^
ξ^
x 1
x^ o
x^1 , [^
d 2 f d x
x^
d 2 f d x
x^
d 2
f d x
x^
d 2
f d x
x^
x^ m
x^ o
x^1 +^2
CE 341/441 - Lecture 5 -Fall 2004
p. 5.
max
e x
x^0
x^
x^1
<^
<
h
d 2 f d x
2 --------
x^ m
x^ o^
x^1
h
f^
(^2) ( )
x (
f^
(^2) ( )
x (
CE 341/441 - Lecture 5 -Fall 2004
p. 5.
⇒
g^
e x (
x^
x^ o
(^
x^
x^1
(^
d ) 2 f d x
x^
x^
e^
d ) 2
f d x
x^
=
e^
d 2 f d x
x^
=