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Derivative Matrices 147
(7.5.3) ˆ d (1)
ij =
⎧
⎪⎨
⎪⎩
n − 1
−
α + 2
α + 1
η (n)
i
i = j = 0,
ˆL (α)
n (η (n)
i ) 1 ≤ i ≤ n − 1, j = 0,
−1
(α + 1) η (n)
j
ˆL (α)
n (η (n)
i )
ˆL (α)
n (η (n)
j )
− α
η (n)
i
2 η (n)
i
ˆL (α)
n (η (n)
j )
η (n)
i
1
− η(n)
j
i = 0, 1 ≤ j ≤ n − 1,
1 ≤ i ≤ n − 1, 1 ≤ j ≤ n − 1, i = j,
1 ≤ i = j ≤ n − 1.
All the coefficients are now computed in terms of the scaled Laguerre functions. Second-
order derivative operators are obtained by squaring ˆ Dn. The corresponding entries are
denoted by ˆ d (2)
ij , 0 ≤ i ≤ n − 1, 0 ≤ j ≤ n − 1.
It is clear that problem (7.4.26) is equivalent to the following one:
(7.5.4)
⎧
n−1
⎪⎨
− ˆ d (2)
ij + 2 ˆ d (1)
ij + (µ − 1)δij ˆp(η (n)
j ) = ˆq(η (n)
i ), 1 ≤ i ≤ n − 1,
⎪⎩
j=0
where ˆq := S (α)
n q.
ˆp(η (n)
0 ) = σ S (α)
n (0),
Hence, we obtain a linear system in the unknowns ˆp(η (n)
j ), 0 ≤ j ≤ n − 1, with the
advantage of having now a matrix which is less affected by rounding errors.
To evaluate the weighted norm of p, it is not necessary to go back to the quantities
p(η (n)
j ), 0 ≤ j ≤ n − 1. We can instead use (3.10.7).
In the Hermite case, the entries in (7.2.5) are modified as follows: