Runges Phenomenon

Runge's Phenomenon ? ¶

$http://math.boisestate.edu/~wright/courses/m297/RungePhenom.gif?width=500$

In the mathematical field of numerical analysis, Runge's phenomenon is a problem that occurs when using polynomial interpolation with polynomials of high degree.

http://upload.wikimedia.org/wikipedia/commons/f/f9/Rungesphenomenon.png?width=400

[PNG external image]

The red curve is the Runge function.
The blue curve is a 5th-order interpolating polynomial (using six equally-spaced interpolating points).
The green curve is a 9th-order interpolating polynomial (using ten equally-spaced interpolating points).
At the interpolating points, the error between the function and the interpolating polynomial is (by definition) zero. Between the interpolating points (especially in the region close to the endpoints 1 and −1), the error between the function and the interpolating polynomial gets worse for higher-order polynomials.

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