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5.1. Linear Stationary Iterative Methods 209

consequently, the method converges with optimal ω even in cases where
it would not converge for ω = 1. But keep in mind that one needs the
eigenvalues of M to determine ¯ω.
Moreover, we have
¯ ω< 1 ⇔ λ max (M)+ λ min (M) < 0 .
If λ min (M) = −λ max(M), that is, ¯ω = 1, we will achieve an improvement
by relaxation:
(M ¯ω ) < (M) .
Thecaseof ω< 1 is called underrelaxation, whereas in the case of ω> 1
we speak of an overerrelaxation.
In particular, for the Richardson method with the iteration matrix M =
I − A, due to λ min (M)=1 − λ max (A)and λ max (M)=1 − λ min (A), the
optimal ¯ω is given by
2
¯ ω = . (5.32)
λ min (A)+ λ max (A)
Hence
λ max (A) − λ min (A) κ(A) − 1
(M ¯ω )= 1 − ¯ωλ min (A)= = < 1 , (5.33)
λ min (A)+ λ max (A) κ(A)+1
with the spectral condition number of A
λ max (A)
κ(A):=
λ min (A)
(see Appendix A.3).
For large κ(A)wehave
κ(A) − 1 2
(M ¯ω )= ≈ 1 − ,
κ(A)+1 κ(A)
the variable of the proportionality being κ(A). For the example of the
five-point stencil discretization, due to (5.24),

n − 1 π
λ min (A)+ λ max (A)= 4 2 − cos π − cos =8 ,
n n
and thus due to (5.32),
1
¯ ω = .
4
1
Hence the iteration matrix M ¯ω = I− A is identical to the Jacobi iteration:
4
We have rediscovered Jacobi’s method.
By means of (5.33) we can estimate the contraction number, since we
know from (5.24) that
n−1 π
4 1 − cos n π 1+cos n 4n 2
κ(A)= = π ≈ . (5.34)
4 1 − cos π 1 − cos π 2
n n