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Eigenvalue Analysis 169
Next, following orszag(1980), we consider the matrix R := {rij} 0≤i≤n , with entries
0≤j≤n
⎪⎨
(8.4.1) rij :=
⎧
1 if i = j = 0 or i = j = n,
−1
h (n)
i ˆ h (n)
i
2
h (n)
i+1h(n) i
−1
h (n)
i+1 ˆ h (n)
i
⎪⎩
0 elsewhere.
if 1 ≤ i = j + 1 ≤ n − 1,
if 1 ≤ i = j ≤ n − 1,
if 1 ≤ i = j − 1 ≤ n − 1,
The matrix R represents the second-order centered finite-differences operator defined
on the grid points η (n)
j , 0 ≤ j ≤ n. If f is a given regular function, the matrix R
applied to the vector {f(η (n)
j )}0≤j≤n furnishes an approximation of −f ′′ (η (n)
i ),
1 ≤ i ≤ n − 1. We remark that this kind of discretization is of local type (see section
7.2), hence it does not possess the good convergence behavior of spectral methods, even
if the nodes of a Gauss formula are involved. However, R is tridiagonal, thus, the
computation of R −1 can be performed in a fast way. Therefore, R satisfies the first
requirement to be a good preconditioning matrix. Now, we must study the spectrum of
R −1 D. We denote by Λn,m, 0 ≤ m ≤ n, the eigenvalues obtained after preconditioning.
Two of them, say Λn,0 and Λn,n, are equal to 1. Numerical experiments indicate the
existence of a constant c > 1 (depending on α and β) such that, for any n ≥ 2, one
has
(8.4.2) 1 ≤ |Λn,m| < c, 1 ≤ m ≤ n − 1.
Comparing this with (8.3.6), the improvement is impressive. Furthermore, the Λn,m’s
are real and positive and c is in general less than 2.5 . We can be more precise in the
Chebyshev case. In fact, in haldenwang, labrosse, abboudi and deville(1984),