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Eigenvalue Analysis 173
Now, we have a bidiagonal preconditioner. Though the spread of eigenvalues of the
preconditioned matrix is reduced, we are still far from the results of the previous section.
The eigenvalues of R −1 D are complex with positive real part, but they approach the
imaginary axis with increasing n.
A more effective preconditioner was proposed in funaro(1987). First introduce the
linear operator T : R n → R n . For any polynomial p ∈ Pn−1, the operator T maps
the vector {p(ξ (n)
j )}1≤j≤n into the vector {p(η (n)
i )}1≤i≤n. The way to perform this
transformation is described by formula (4.4.1). The coefficients l (n)
j (η (n)
i ), 1 ≤ i ≤ n,
1 ≤ j ≤ n, are the entries of the n × n matrix relative to the mapping T. Then, we
define Z := {zij} 0≤i≤n , to be the (n + 1) × (n + 1) matrix
0≤j≤n
⎧
1
⎪⎨
if i = j = 0,
(8.5.2) zij :=
⎪⎩
0 elsewhere.
l (n)
j (η (n)
i ) if 1 ≤ i ≤ n, 1 ≤ j ≤ n,
We claim that ZR is a good preconditioning matrix for D. We remark that Z is a
full matrix. On the other hand, we can find explicitly the entries of Z −1 . Indeed, one
easy checks the relation
(8.5.3) p(ξ (n)
i ) =
n
j=1
which is true for any p ∈ Pn−1.
p(η (n)
j )
1 + η(n)
j
1 + ξ (n)
i
˜ l (n)
j (ξ (n)
i ), 1 ≤ i ≤ n,
Clearly, (8.5.3) is the inverse of the transformation T. Noting that (ZR) −1 = R −1 Z −1 ,
we can evaluate ¯p → (ZR) −1 ¯p with a cost proportional to n 2 . The computing time
is further reduced, in the Chebyshev case, by using the FFT (see section 4.4). Let us
examine the eigenvalues Λn,m, 0 ≤ m ≤ n, of (ZR) −1 D. We have a precise answer in
the Chebyshev case. Actually, one has
(8.5.4) Λn,0 = 1, Λn,m = m
Therefore, we get 1 ≤ Λn,m < π
2 , 0 ≤ m ≤ n.
π sin 2n
sin mπ , 1 ≤ m ≤ n.
2n