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174 Polynomial Approximation of Differential Equations
The improvement is explained by noting that the finite-differences operator R is a
good approximation of the operator Z−1D, which maps the values p(η (n)
j ), 0 ≤ j ≤ n,
into the values p ′ (ξ (n)
i ) ≈ (p(η (n)
i ) − p(η (n)
i−1 ))/h(n) i , 1 ≤ i ≤ n. Thus, R−1 (Z−1D) is
very close to the identity operator. For different values of α and β, the results are
similar.
With minor modifications, the same preconditioner can be used to deal with the
matrix of system (7.4.4). For more details, we refer the reader to funaro and got-
tlieb(1988).
Though the results obtained with the preconditioning matrix ZR are impressive,
for more general problems, such as (7.4.7), the use of a full preconditioner is not rec-
ommended. For example, consider A ≡ 1 in (7.4.7). The matrix ZR + I turns
out to be an appropriate preconditioner for D + I, i.e. κ((ZR + I) −1 (D + I)) is
close to 1. Nevertheless, we cannot cheaply invert ZR + I. An alternative is to ap-
proximate the operator Z by a banded matrix ˆ Z. In this way ˆ ZR + I will be
also banded. This idea has been developed in funaro and rothman(1989). In that
paper, the operator T is substituted by a mapping ˆ T, which extrapolates the point
values p(η (n)
i ), 1 ≤ i ≤ n, with the help of second-degree polynomials. The resulting
preconditioner is four-diagonal and the eigenvalues of the matrix ( ˆ ZR + I) −1 (D + I)
are complex, but clustered in a neighborhood of the real unity. The preconditioned
condition number seems to be suitable for numerical applications.
8.6 Convergence of the eigenvalues
We are now ready to consider the first application of spectral methods to solve several
problems in physics. For example, let us analyze the following eigenvalue problem:
(8.6.1)
⎧
⎨ −φ ′′ = λ φ in I =] − 1,1[,
⎩
φ(−1) = φ(1) = 0.