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150 Polynomial Approximation of Differential Equations
the sequence of vectors such that
(7.6.2) ¯p (k+1)
where I is the identity matrix.
:= (I − R −1 D)¯p (k) + R −1 ¯q, k ∈ N,
Denote by ρ(M) ≥ 0 the spectral radius of the matrix M := I −R −1 D, i.e., the largest
eigenvalue modulus among the, possibly complex, eigenvalues of M. Then we have a
classical convergence result (see for instance isaacson and keller(1966), p.63).
Theorem 7.6.1 - If M admits a diagonal form and ρ(M) < 1, then we have
limk→+∞ ¯p (k) = ¯p, where ¯p is the solution to (7.6.1).
The closer ρ(M) is to zero, the faster the convergence will be, since it depends on the
geometric progression [ρ(M)] k , k ∈ N. Actually, when R = D, we obtain the exact
solution in one step.
Many famous schemes have their root in formula (7.6.2). The simplest is the Richardson
method (see richardson(1910)) obtained by setting R −1 = θI, where θ ∈ R . Let us
indicate by λm ∈ C, 1 ≤ m ≤ n, the eigenvalues of D. If D admits a diagonal form
and the following hypotheses are satisfied:
(7.6.3) Reλm > 0 and 0 < θ <
2 Reλm
, 1 ≤ m ≤ n,
|λm| 2
then ρ(M) < 1, and the Richardson scheme is convergent.
To proceed with our investigation, we need to know more about the eigenvalues of
the matrices introduced in this chapter. This analysis is given in chapter eight.
Despite its simplicity, the Richardson scheme is useful to help understand the basic
principles of iterative methods. Consequently, more elaborate iterative algorithms will
not be discussed.