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166 Polynomial Approximation of Differential Equations
from which one can easily deduce κ(D) ≥ 1.
This quantity gives an idea of the distribution of the eigenvalues of D in the complex
plane. When κ(D) is large, we generally expect scattered eigenvalues with considerable
variations in their magnitude. On the other hand, when κ(D) is close to 1, the moduli
of the eigenvalues are gathered together in a small interval. We can be more precise for
symmetric matrices. In this situation, we have the relation
(8.3.4) κ(D) = π(D) := max1≤m≤n |λm|
min1≤m≤n |λm| ,
where the λm’s are the eigenvalues of D.
When D is not symmetric we have the inequality: 1 ≤ π(D) ≤ κ(D).
Nearly all the matrices analyzed in the sequel are not symmetric. Nevertheless, for
these matrices, the quantities π(D) and κ(D) display more or less the same behavior.
So that, allowing ourselves an abuse of terminology, the two concepts will often be
interchanged.
To better understand the utility of the condition number, we return to the iterative
method described in section 7.6. The speed of convergence of the scheme (7.6.2) depends
on the spectral radius ρ(M), where M := I − R −1 D. In the Richardson method we
have
(8.3.5) ρ(M) = max |1 − θλm|,
1≤m≤n
where the λm’s are the eigenvalues of D, satisfying the conditions in (7.6.3).
If the eigenvalues are clustered around a particular real value, then we can find θ such
that ρ(M) is near zero. This will result in a fast convergence. On the contrary, if
the eigenvalues are scattered, the parameter θ cannot control all of them, and ρ(M)
will be dangerously close to 1. In short, we have to beware of matrices with a large
condition number. These are called ill-conditioned matrices.
Unfortunately, all the matrices considered in section 7.4 are ill-conditioned. As an
example, we examine the eigenvalue problem (8.1.1). Since the λn,m’s are distinct, the
corresponding matrix admits a diagonal form. From figures 8.1.1 to 8.1.4, it is clear that