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Consider the seven-point data set

          t                  y
  -0.97492791218182   0.09519417591355
  -0.78183148246803   0.14059547049600
  -0.43388373911756   0.34691493861656
  -0.00000000000000   1.00000000000000
   0.43388373911756   0.34691493861656
   0.78183148246803   0.14059547049600
   0.97492791218182   0.09519417591355

are the seven Chebyshev points

and the corresponding value of the Runge function

Fitting a degree-6 polynomial through the seven point data set, using piecewise linear and spline interpolants are shown in Figure [*].

Figure: Different interpolants for the data set. Polynomial interpolant is order 6 (upper plot) and oscillates. Piecewise linear interplation (middle) is non-smooth, while spline interpolation (lower plot) is smooth and approximates the function very well.

Adrian Sandu 2001-08-26