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Derivative Matrices 135
We conclude this section with a few additional remarks. Given the function f ∈
C1 ( Ī), we can evaluate the quantities f(ξ(n) j ), 1 ≤ j ≤ n. After applying the matrix
Dn, we obtain the vector {(Iw,nf) ′ (ξ (n)
i )}1≤i≤n. For n sufficiently large, (Iw,nf) ′ is a
fairly good approximation of f ′ (see (6.6.8)).
Other techniques, such as finite-differences, can be used to compute numerically the
derivative of f. Although they generate derivative matrices with quite good structure
(for instance, they are banded with a narrow bandwidth), the results are in general very
poor when compared with those obtained by the procedure described above. The reason
is that finite-differences are local methods. To compute the derivative at a given point,
these methods use information pertaining to a small neighborhood of the point. In
contrast, spectral methods are global methods. All n nodes in the domain I are applied
in the computation. For example, when f is a polynomial in Pn−1, the values at the
nodes are representative of the whole function in the domain Ī, and the exact derivative
is obtained. The particular distribution of the nodes optimizes the process for a general
f, and the accuracy increases rapidly with the regularity of the function. The reader
may find other persuasive arguments supporting these considerations in boyd(1989),
p.11.
7.3 Boundary conditions in the frequency space
Clearly, the procedure of evaluating the derivative q := p ′ ∈ Pn−1 of a polynomial
p ∈ Pn can be inverted except to within an additive constant. Actually, the derivative
formulas of section 7.1 do not take into consideration the coefficient c0. We need an
extra condition. The most popular approach is to assume that p satisfies p(ξ) = σ ∈ R,
where ξ ∈ Ī. This is a boundary condition when ξ ∈ ∂I, i.e., ξ is an element of the
boundary of Ī. In this case, we can recover the coefficients ci, 1 ≤ i ≤ n, in terms
of the coefficients dj, 0 ≤ j ≤ n − 1, of the expansion q = n−1
j=0 djuj. Finally, c0 is