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130 Polynomial Approximation of Differential Equations
where
(7.2.3) d (1)
ij :=
d
dx l(n)
j (ξ (n)
i ), 1 ≤ i ≤ n, 1 ≤ j ≤ n.
Since the polynomials p and p ′ are uniquely determined by their values at the nodes
ξ (n)
i , 1 ≤ i ≤ n, the linear transformation (7.2.2) is equivalent to performing an exact
derivative in the space Pn−1. The associated n × n matrix Dn := {d (1)
ij
} 1≤i≤n , allows
1≤j≤n
us to pass from the the vector {p(ξ (n)
j )}1≤j≤n to the vector {p ′ (ξ (n)
i )}1≤i≤n, with a
computational cost proportional to n 2 .
The next step is to give an explicit expression to the entries d (1)
ij , 1 ≤ i ≤ n,
1 ≤ j ≤ n. Computing the derivatives of the Lagrange polynomials in (3.2.4) yields
(7.2.4)
d
dx l(n)
j
(x) = u′ n(x)(x − ξ (n)
j ) − un(x)
u ′ n(ξ (n)
j ) (x − ξ (n)
j ) 2
After evaluation of (7.2.4) at the nodes, one obtains
(7.2.5) d (1)
ij =
⎧
⎪⎨
⎪⎩
u ′ n(ξ (n)
i )
u ′ n(ξ (n)
j )
u ′′ n(ξ (n)
i )
2 u ′ n(ξ (n)
i )
ξ (n)
i
1
− ξ(n)
j
, x ∈ I, x = ξ (n)
j , 1 ≤ j ≤ n.
if i = j,
if i = j.
To get (7.2.5) one recalls that un(ξ (n)
i ) = 0, 1 ≤ i ≤ n. The diagonal entries (i.e.,
i = j) are obtained by noting that
(7.2.6) lim
x→ξ (n)
i
d
dx l(n)
i (x) = lim
x→ξ (n)
i
u ′′ n(x)
2 u ′ n(ξ (n)
i ) = u′′ n(ξ (n)
i )
2 u ′ n(ξ (n)
, 1 ≤ i ≤ n.
i )
Recalling that un is solution of a Sturm-Liouville problem, we can express the values
u ′′ n(ξ (n)
i ), 1 ≤ i ≤ n, in terms of u ′ (ξ (n)
i ), 1 ≤ i ≤ n. This gives