Comparative numerical laboratory
Interpolation: Equispaced vs Chebyshev Nodes — Chapter 3
Interpolate the Runge-type function and compare node choices for the same degree.
\[
f(x)=\frac{1}{1+25x^2}, p_N(x_i)=f(x_i).
\]
Methods compared
| Method | Convenience | Inconvenience / warning |
|---|---|---|
| Equispaced interpolation | Simple nodes; easy to understand. | Can produce endpoint oscillations for high degree. |
| Chebyshev interpolation | Controls endpoint growth and reduces Runge oscillations. | Nodes are nonuniform; implementation is less elementary. |
| Piecewise thinking | Splines avoid one huge polynomial. | Need smoothness and boundary choices. |
Illustrative plot / animation
Numerical diagnostics
What the student must compare
- maximum error
- endpoint behavior
- node distribution
- visual oscillation