Interpolation  1
17  Interpolation
Motivation
Suppose that you have a polynomial function,
p
(
x
), and you have computed its value at a
set of uniformly spaced points,
x
i
. How would you go about estimating
p
(
a
), where
a
is
between, say,
x
i
and
x
i
+1
?
Having
p(x),
this is, of course, rather elementary.
You would
just evaluate
p(a).
0
5
10
15
20
25
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
Data
x
y
Now suppose that instead of a known function, all you have are a bunch of points (
x
i
,
y
i
)
such as the ones shown above. How do you determine what values should go between the
data points? This is the problem of interpolation, and it arises in virtually every field of
engineering. For example, the data could be from vibration sensors on the Golden Gate
bridge, but you only have enough money for 25 of them. During an earthquake, you
measure the peak vibrations shown above at your sensor locations, but there are
important structural elements between the vibration measurements. You want to estimate
the vibration levels at a number of intermediate points.
Interpolation is one method of solving this problem.
Below we will look at a few
interpolation methods.
Piecewise Linear Interpolation
This is probably the most straightforward technique for interpolation.
Suppose we want to compute the vibration level between sensor numbers 10 and 11.
From the points above we can see that if we draw a straight line between each pair of
points, we will get a value between 0 and 1. The general expression for piecewise linear
interpolation is
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View Full DocumentInterpolation  2
()
j
j
j
j
j
j
y
x
x
x
x
y
y
y
+
−
−
−
=
+
+
int
1
1
int
The point (
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 Spring '08
 Patzek
 Numerical Analysis

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