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40 Polynomial Approximation of Differential Equations
In the numerical computations, the following algorithm is generally adopted. For
each zero, an initial guess is obtained as described above (for the Jacobi case the average
between the upper and lower bound in the estimates (3.1.1) or (3.1.2)-(3.1.3) can be
taken into account). A few iterations of a Newton method is then sufficient to determine
accurately the zero. The point values of the polynomial and its derivative can be
computed from (1.1.2) and (1.1.3).
Concerning the case of Laguerre (or Hermite), when n grows, rounding errors occur
in the procedure, due to the sharp oscillations exhibited by the polynomials. Consider-
able improvements are obtained operating with scaled Laguerre (or Hermite) functions.
Of course, the values of the zeroes are the same, but the derivatives in the neighborhood
of the zeroes are now moderate.
Similar arguments hold for the determination of the zeroes of the derivatives of
orthogonal polynomials. An initial approximation can obtained by considering that
each zero η (n)
k , 1 ≤ k ≤ n − 1, of u′ n lies between two consecutive zeroes of un. In
alternative, one can use relations (1.3.6), (1.6.6), (1.7.8).
As it will be clear in the following sections, for n ≥ 1, in the Jacobi case it is
convenient to set η (n)
0 := −1 and η (n)
n := 1 (though these are not in general zeroes).
Particularly interesting are the situations where an explicit expression is known, i.e.
(3.1.11) η (n)
k
(3.1.12) η (n)
k
(3.1.13) η (n)
k
(3.1.14) η (n)
k
= −cos kπ
n
0 ≤ k ≤ n, if α = β = − 1
2 ,
2k + 1
1
= −cos π 1 ≤ k ≤ n − 1, if α = β =
2n + 2 2 ,
2k + 1
1
= −cos π 1 ≤ k ≤ n, if α = , β = −1
2n + 1 2 2 ,
= −cos
2k
1
π 0 ≤ k ≤ n − 1, if α = −1 , β =
2n + 1 2 2 .