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148 Polynomial Approximation of Differential Equations
(7.5.5)
d ˆ(1) ij := d (1)
ij S(−1/2)
n/2 ([ξ (n)
i ] 2 )
(7.5.6) d ˆ(1) ij := d (1)
ij ξ(n) i S (1/2)
(n−1)/2 ([ξ(n) i ] 2 )
By (1.7.11) and (1.7.12), the above relations imply
(7.5.7)
ˆ d (1)
ij =
⎧
⎪⎨
⎪⎩
ξ (n)
i
ξ (n)
j
ξ (n)
i
ˆH ′ n(ξ (n)
i )
ˆH ′ n(ξ (n)
j )
S (−1/2)
n/2 ([ξ (n)
j ] 2 −1 )
ξ (n)
j S(1/2)
(n−1)/2 ([ξ(n) j ] 2 −1 )
ξ (n)
i
1
− ξ(n)
j
if i = j,
if i = j.
if n is even,
if n is odd,
1 ≤ i ≤ n, 1 ≤ j ≤ n.
Therefore, we transform (7.4.28) to an equivalent problem where the unknown is the
polynomial p multiplied by the scaling function.
7.6 Numerical solution
Various techniques are available for the numerical solution of the systems presented in
the previous sections. Basic algorithms are discussed in all the comprehensive books
on numerical analysis. We mention for instance fox(1964), ralston(1965), isaac-
son and keller(1966), dahlquist and björck(1974), hageman and young(1981),
comincioli(1990).
To take full advantage of the accurate results obtainable with spectral methods,
we always suggest that one operates with programs written in double precision (i.e., 64
bits for a real number storage).
In this book, we are only concerned with the discretization of differential equations
in one variable. For this reason, the dimension n of the matrices is in general small.