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Derivative Matrices 129
Similarly, for the Chebyshev and Laguerre cases, we respectively get
(7.1.16) c (1)
i
(7.1.17) c (1)
i
⎧
0 if i = n,
=
⎪⎨ 2ncn
⎪⎩
c (1)
i+2 + 2(i + 1)ci+1 if
if i = n − 1,
1 ≤ i ≤ n − 2,
1
2 c(1)
2 + c1 if i = 0,
=
⎧
⎨ 0 if i = n,
⎩
0 ≤ i ≤ n − 1.
c (1)
i+1 − ci+1 if
The Hermite case has already a simplified form given by (7.1.13).
7.2 Derivative matrices in the physical space
Here we consider Pn−1, n ≥ 2, to be generated by the basis of Lagrange polynomials
{l (n)
j }1≤j≤n introduced in section 3.2. In view of (3.2.2), the derivative of a polynomial
p ∈ Pn is obtained by evaluating
(7.2.1)
In particular, we have
d
p =
dx
(7.2.2) p ′ (ξ (n)
i ) =
n
j=1
n
j=1
p(ξ (n)
j ) d
dx l(n)
j .
d (1)
ij p(ξ(n)
j ), 1 ≤ i ≤ n,