05.07.1
Chapter 05.07
Higher Order Interpolation is a Bad Idea
After reading this chapter, you should be able to:
1. Understand why higher order interpolation is a bad idea
Example
Peter: “Dr. Kaw, what is this you were talking about in class that higher order interpolation
is a bad idea? More points, more accuracy; isn’t that the way it works?”
Kaw: “Come on in. In 1901, Runge wanted to show that higher order interpolation is a bad
idea. He took this function, )251/(1)( 2
xxf in the domain [-1,1].”
Table 1. Six equidistant points of )251/(1)( 2
xxf
x
y
-1 0.038462
-0.6 0.1
-0.2 0.5
0.2 0.5
0.6 0.1
1 0.038462
Let us choose 6 points equidistantly between –1 and 1 as given in Table 1. You can
interpolate these 6 data points by a 5th order polynomial. In Figure 1, I am then plotting the
fifth order polynomial and the original function. See the oscillations in the interpolating
polynomial. The polynomial does go through the six points, but at many other points it does
not even come close to the original function. Just look at 85.0
x, the value of the function
is 0.052459, but the fifth order polynomial gives you a value of –0.055762. That is a
whopping 206.30 % relative error and also note the opposite sign.
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