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146 Polynomial Approximation of Differential Equations
where the ξ (n)
i ’s are the zeroes of Hn. Boundary conditions are replaced by the relation
limx→±∞ P(x) = 0. By the substitution of q(x) := Q(x)ex2, p(x) := P(x)ex2 , x ∈ R,
into (7.4.27) we recover the corresponding problem in Pn−1, i.e.
(7.4.28) −p ′′ (ξ (n)
i ) + 2ξ (n)
i p′ (ξ (n)
i ) + µ p(ξ (n)
i ) = q(ξ (n)
i ), 1 ≤ i ≤ n.
This leads to a non singular n × n linear system in the unknowns p(ξ (n)
j ), 1 ≤ j ≤ n.
7.5 Derivatives of scaled functions
Following the suggestions given in funaro(1990a) and in section 1.6, with the help of
the scaling function (1.6.13), we can in part reduce the problems associated with the
numerical evaluation of high degree Laguerre or Hermite polynomials.
For example, starting from the matrix in (7.2.15), we define a new matrix by setting
(7.5.1)
d ˆ(1) ij := ˜ d (1)
ij S(α) n (η (n)
i )
S (α)
n (η (n)
−1 j ) , 0 ≤ i ≤ n − 1, 0 ≤ j ≤ n − 1.
Let p be a polynomial in Pn−1, n ≥ 1. Then we define ˆp := S (α)
n p. Now, one has
(7.5.2)
n−1
j=0
Thus, the matrix ˆ Dn := { ˆ d (1)
ij } 0≤i≤n−1
0≤j≤n−1
ˆd (1)
ij ˆp(η(n) j ) = S (α)
n (η (n)
i ) p ′ (η (n)
i ), 0 ≤ i ≤ n − 1.
acts on the scaled polynomial ˆp and leads to
its scaled derivative. The values at the nodes of these functions are in general more
appropriate for numerical purposes. Moreover, let us better examine the entries of ˆ Dn.
From (1.6.12), (1.6.13) and (7.5.1), we have