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17 Interpolation

# 17 Interpolation - 17 Interpolation Motivation Suppose that...

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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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Interpolation - 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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17 Interpolation - 17 Interpolation Motivation Suppose that...

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