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Eigenvalue Analysis 171
The matrix in (7.4.11) is obtained after eliminating the first and the last unknowns
from system (7.4.10). A (n − 1) × (n − 1) preconditioner for this matrix is obtained
by deleting the first and the last columns and rows from the matrix in (8.4.1). The
preconditioned eigenvalues Λn,m, 1 ≤ m ≤ n−1, coincide with those considered above.
The same preconditioner R in (8.4.1) can be used for the matrix of system (7.4.12).
The resulting preconditioned spectrum does not seem to be much affected by lower-
order terms, provided the coefficients A and B are not too large. As an alternative, one
can use a preconditioner based on a centered finite-difference discretization of the whole
differential operator − d2
dx2 + A d
dx + B (see orszag(1980)).
For other types of boundary conditions we could run into some difficulty. Let
D denote the matrix of system (7.4.13). In this situation, tridiagonal finite-difference
preconditioners seem less effective. All the preconditioned eigenvalues satisfy relation
(8.4.2), except for one of them, which is real, positive, but tends to zero as 1/n 2 . This
badly affects the condition number. A more appropriate (n+1)×(n+1) preconditioner
R := {rij} 0≤i≤n
0≤j≤n
⎪⎨
(8.4.4) rij :=
is obtained as follows:
⎧
−d (1)
ij
−1
h (n)
i ˆ h (n)
i
2
h (n)
i+1h(n) i
−1
h (n)
i+1 ˆ h (n)
i
d (1)
ij
if i = 0, 0 ≤ j ≤ n,
⎪⎩
0 elsewhere.
if 1 ≤ i = j + 1 ≤ n − 1,
+ µ if 1 ≤ i = j ≤ n − 1,
if 1 ≤ i = j − 1 ≤ n − 1,
if i = n, 0 ≤ j ≤ n,
All the resulting preconditioned eigenvalues are now well-behaved. Unfortunately, the
matrix R is sparse but not banded. However, it can be decomposed by a product of