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Eigenvalue Analysis 167
the magnitude of the eigenvalues grows with n. It is especially for the eigenvalues with
the largest modulus that the ratio between the real part and the imaginary part is small.
According to (7.6.3), this yields a very restrictive condition on θ. For instance, in the
Chebyshev case, the parameter θ is required to be proportional to 1/n 2 to achieve
convergence when applying the Richardson method. This makes the spectral radius in
(8.3.5) very close to 1. Consequently, the Richardson scheme is very ineffective. The
situation is worse for the Legendre case.
Concerning second-order or fourth-order problems, no improvements are observed.
Although the eigenvalues are in general real and positive, they are quite scattered. For
instance, referring to problem (8.2.1), it is not difficult to obtain the estimate
(8.3.6) c1 ≤ |λn,m| ≤ c2 n 4 , 1 ≤ m ≤ n − 1,
where c1 and c2 are positive constants.
Moreover, assuming that the λn,m’s are real and in increasing order, we find that λn,1
converges to a real number for n → +∞, while λn,n−1 grows like n 4 . Sharper
estimates are given in weideman and trefethen(1988) in the Chebyshev case, and
in vandeven (1990) in the Legendre case. For the proof of (8.3.6), when −1 < α < 1,
−1 < β < 1, we refer the reader to section 8.6.
Thus, the condition number of the matrix corresponding to the system (7.4.9) is ex-
pected to be very large. The minimal spectral radius obtained from (8.3.5) is ρ =
(λn,n−1 − λn,1)/(λn,n−1 + λn,1), relative to the choice θ := 2/(λn,n−1 + λn,1) (see
isaacson and keller(1966), p.84). Therefore, ρ is very close to 1, and the con-
vergence of the scheme is unbelievably slow. The situation becomes extremely bad
for fourth-order problems. In heinrichs(1989), the author suggests a way to replace
problem (7.4.9) with an equivalent one, whose matrix has a reduced condition number.
Such an improvement is still insufficient when applied within the framework of iterative
techniques.
It is time now to introduce the idea of preconditioning. We note that the scheme
(7.6.2) can be interpreted as a byproduct of the Richardson method with θ = 1, applied
to the system