Demonstrate (by use of examples) that approximations by high order interpolating polynomials are unstable.
Interpolating polynomials of higher degree tend to be very oscillatory and peaked, especially near the endpoints of the interval.
which can be proved by Runge's phenomenon,
For
the case of the Runge function, interpolated at equidistant points,
each of the two multipliers in the upper bound for the
approximation error grows to infinity with n. Although often used
to explain the Runge phenomenon, the fact that the upper bound of
the error goes to infinity does not necessarily imply, of course,
that the error itself also diverges with n.
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