MCS 471 Runge's Counter Example Against Using High Degree Polynomial Fits

Runge's example is

R(x) = 1/(1+25*x^2), on [-1, +1],

behaves like a polynmial,

R(x) = 1 - 25*x^2 + O(x^4)

near the center x = 0, but behaves like a quadratic hyperbola,

R(x) = 1/(25*x^2) + O(1/(25*x^2)^2)

near the endpoints |x| = 1. Thus, a polynomial fit will have difficulty near the endpoints, where the behavior is certainly not polynomial-like.

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