Spectral differentiation

From chapter 9, instead of using local node set, use all of the nodes for global interpolation. The convergence is spectral if $f$ has infinitely many derivatives on the interval (carries through to the derivatives).

Chebyshev differentiation matrix is a dense matrix with entries:

$$D_{00} = \frac{2n^2 +1}6, \quad D_{nn} = -\frac{2n^2 +1}6\\ D_{ij} = \begin{cases} -\frac{x_i}{2(1-x^2_i)}, & i - j,\\ \frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, & i \neq j, \end{cases}$$