Untitled - Cdm.unimo.it

134 Polynomial Approximation of Differential Equations
The matrices (7.2.13) and (7.2.14) have been presented for instance in gottlieb,
hussaini and orszag(1984), together with the entries of the matrices corresponding
to the second derivatives.
{ ˜ d (1)
ij
For the Laguerre Gauss-Radau case, the entries of the n × n matrix ˜ Dn =
} 0≤i≤n−1 , are obtained by (3.2.11). These are
0≤j≤n−1
(7.2.15) ˜ d (1)
ij =
⎧
⎪⎨
⎪⎩
n − 1
−
α + 2
n! Γ(α + 2) L (α)
n (η (n)
i )
Γ(n + α + 1) η (n)
i
−Γ(n + α + 1)
n! Γ(α + 2) η (n)
j L(α) n (η (n)
j )
L (α)
n (η (n)
i )
L (α)
n (η (n)
j )
η (n)
i
2 η (n)
i
− α
η (n)
i
1
− η(n)
j
For the computation one has to recall (3.1.19) and (1.6.1).
i = j = 0,
1 ≤ i ≤ n − 1, j = 0,
i = 0, 1 ≤ j ≤ n − 1,
i = j 1 ≤ i ≤ n − 1, 1 ≤ j ≤ n − 1,
1 ≤ i = j ≤ n − 1.
In all the examples discussed, the algorithm to perform the derivative in the physical
space corresponds to a matrix-vector multiplication. Therefore, its cost is proportional
to n 2 . The matrices Dn and ˜ Dn, in the various cases, are full and do not display
any particular property. This means that in general we cannot improve the procedure
of evaluating a derivative, as suggested in the previous section. Nevertheless, for the
Chebyshev case, a faster algorithm exists. In view of the results in section 4.3, when n is
a power of 2, we can go from the physical space to the frequency space (and conversely)
by applying the FFT with a cost proportional to nlog 2n. Therefore, we can use (7.1.8),
the cost of which is only proportional to n. This observation makes the Chebyshev
Lagrange basis preferable, when large values of n are taken into account.