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I have a high-school student doing a modeling project using interpolation. (Hopefully someone understands what I mean without having to write out explicit examples, as I only have screenshots.)

When fitting polynomials of order (n-1) to n points, I understand that it will always pass through the points but sometimes the behaviour of the polynomial between the points is rather chaotic. I found that it's often easier to just split the curve into several polynomials and fit these individual.

What aspect of the distribution of the points causes these deep troughs and peaks in comparison to the smooth curve between the points?

Cliff
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    I don't have time to write a full discussion of this topic right now. Suffice it to say that a full answer to this question requires graduate level numerical analysis, and is still rather unsatisfactory because it is rather vague about precise details. In the meantime, I will refer you to the term "Runge phenomenon" to give you some things to look up. – Ian Dec 12 '14 at 01:54
  • Thanks for the quick response, I'll take at look at the "Runge phenomenon" – Cliff Dec 12 '14 at 02:08
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    To be short : a polynomial of degree $n-1$ may have $(n-2)$ minima and maxima. – Claude Leibovici Dec 13 '14 at 13:30

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