9.6 Spectral collocation differentiation
The equivalence (9.80) between the discrete truncated expansion and interpolation at Gauss-type points
allows the approximation of the derivative of a function in a very simple way,
Therefore, knowing the values of the function at the collocation points, i.e., the Gauss-type points, we
can construct its interpolant , take an exact derivative thereof, and evaluate the result at the
collocation points to obtain the values of the discrete derivative of at these points. This leads to a
matrix-vector multiplication, where the corresponding matrix elements can be computed once and for
We give the explicit expressions for this differentiation matrix for Chebyshev polynomials both at Gauss
and Gauss–Lobatto points (see, for example, [167, 237]).
with a prime denoting differentiation.
where for and for .