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38 Polynomial Approximation of Differential Equations
Figure 3.1.1 - Legendre zeroes for n = 7 and n = 10.
Zeroes in the Chebyshev case - Due to (1.5.1) the zeroes are those of ultraspherical
polynomials with α = β = −1 2 . Hence, they are exactly given by (3.1.4). On the
other hand, thanks to equality (1.5.6), they can be also obtained from the roots of the
equation: cos nθ = 0, n ≥ 1, θ ∈]0,π[. In figure 3.1.2 we give the distribution of the
Chebyshev zeroes for n = 7 and n = 10.
Figure 3.1.2 - Chebyshev zeroes for n = 7 and n = 10.
Zeroes in the Laguerre case - According to tricomi(1954), p.153, and gatteschi(1964),
a good approximation of the zeroes of Laguerre polynomials is obtained by the following
procedure. For any n ≥ 1, we first find the roots of the equation
(3.1.8) y (n)
k
− sin y(n)
k
3 n − k + 4
= 2π , 1 ≤ k ≤ n.
2n + α + 1
Then we set ˆy (n)
k := cos 1
2y(n) 2 k , and define for 1 ≤ k ≤ n,