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168 Polynomial Approximation of Differential Equations
(8.3.7) (R −1 D)¯p = ¯q ∗ , where ¯q ∗ := R −1 ¯q.
Of course (8.3.7) is equivalent to (7.6.1). The basic difference is that, by an appropriate
choice of the matrix R, we can modify κ(R −1 D) to get a well-behaved system. Thus,
we first compute the vector ¯q ∗ , then we solve (8.3.7) with an iterative approach. In
this context, R is named preconditioning matrix (or preconditioner). Such a matrix
must fulfill two requirements. First, R must be easily invertible, using an inexpensive
algorithm. Second, the condition number of R −1 D has to be as close as possible to 1.
The choice R = I (which leads to the Richardson method) satisfies the first restriction
but not the latter. The choice R = D gives κ(R −1 D) = 1, but the computation of
R −1 brings us back to the initial problem. The desired R is the right balance between
these two extreme cases.
The next several sections describe how to construct preconditioners for the matrices
previously analyzed.
8.4 Preconditioners for second-order operators
We begin by studying problem (7.4.9). The case of first-order derivatives is more delicate
and is considered later.
Let D denote the (n + 1) × (n + 1) matrix corresponding to the system (7.4.9)
(in (7.4.10) we have an example for n = 3). We recall that D coincides with the
second derivative matrix − ˜ D 2 n (see section 7.2), except for two rows, which have been
suitably modified to take care of the boundary conditions.
As remarked in section 8.3, the condition number of D grows at least as n 4 . We
are concerned with finding an appropriate preconditioner R for D. To this end, for
any n ≥ 2, we define h (n)
j := η (n)
j − η(n)
j−1 , 1 ≤ j ≤ n, and ˆ h (n)
j := 1
2 (η(n)
j+1 − η(n)
j−1 ),
1 ≤ j ≤ n − 1.