Summary
Given a set of 
data points, interpolation by a polynomial of degree 
can be bad when 
is large.
Illustration of the Runge Phenomenon
Summary
Given a set of data points, interpolation by a polynomial of degree can be bad when is large.
Define a function
As an example, consider the smooth function 
.
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(1) |
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Interpolate at 5 equally-spaced points
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(2) |
| > | ![`:=`(xpts, [seq(`+`(`-`(1), `/`(`*`(`*`(2, `+`(i, `-`(1)))), `*`(`+`(npts, `-`(1))))), i = 1 .. npts)])](images/Runge_13.gif){kind=small} |
![[-1, -`/`(1, 2), 0, `/`(1, 2), 1]](images/Runge_14.gif){kind=small} |
(3) |
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![[-1., -.5000000000, 0., .5000000000, 1.]](images/Runge_16.gif){kind=small} |
(4) |
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![[0.7692307692e-1, .2500000000, 1., .2500000000, 0.7692307692e-1]](images/Runge_18.gif){kind=small} |
(5) |
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| > | ![`:=`(DataPoints, [seq([xpts[i], ypts[i]], i = 1 .. npts)]); -1](images/Runge_19.gif){kind=small} |
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(6) |
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A cubic spline fit does much better
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Interpolate at a larger number of equally-spaced points
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(7) |
| > | ![`:=`(xpts, [seq(`+`(`-`(1), `/`(`*`(`*`(2, `+`(i, `-`(1)))), `*`(`+`(npts, `-`(1))))), i = 1 .. npts)]); -1](images/Runge_34.gif){kind=small} |
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| > | ![`:=`(DataPoints, [seq([xpts[i], ypts[i]], i = 1 .. npts)]); -1](images/Runge_37.gif){kind=small} |
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A cubic spline fit does much better
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