A researcher tabulates atmospheric density at equally spaced altitudes and fits a…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Approximating a function: interpolation and differentiation

Problem

A researcher tabulates atmospheric density at

21

equally spaced altitudes and fits a degree-

20

polynomial through them, then argues: "Interpolation is a linear map that reproduces the data exactly at the nodes, and every extra node is another exact constraint. So the error between nodes can only get smaller as nodes are added. Twenty-one equally spaced nodes must therefore interpolate Runge's function

1/(1+25x^2)

better than eleven such nodes." Find the error in this reasoning. Support the correction using the Lebesgue constants

\Lambda_{10} = 29.9

and

\Lambda_{20} = 1.1\times10^4

for equally spaced nodes, and the measured maximum errors on Runge's function of

1.92

n=10

and

59.8

n=20

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