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
all:
with
We give the explicit expressions for this differentiation matrix for Chebyshev polynomials both at Gauss
and Gauss–Lobatto points (see, for example, [167, 237
]).
Chebyshev–Gauss.
with a prime denoting differentiation.
Chebyshev–Gauss–Lobatto.
where
for
and
for
.