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216 5. Iterative Methods for Systems of Linear Equations

where M is the iteration matrix of the basic iteration. This matrix should
be normal, for example, such that
p k (M) 2 = (p k (M))

holds. Then we have the obvious estimation

e
e

(0)

(k)
≤ p k (M)e (0)
≤ p k (M)
(0)
≤ (p k (M)) e . (5.42)
2 2 2 2 2
If the method is to be defined in such a way that

(p k (M)) = max |p k (λ)| λ ∈ σ(M)

is minimized by choosing p k , then the knowledge of the spectrum σ(M)is
necessary. Generally, instead of this, we assume that suitable supersets are
known: If σ(M)is real and
a ≤ λ ≤ b for all λ ∈ σ(M) ,
then, due to

e

(k)
≤ max p k (λ) e ,

(0)
2 2
λ∈[a,b]
it makes sense to determine the polynomials p k as a solution of the
minimization problem on [a, b],
max |p k (λ)|→ min for all p ∈P k with p(1) = 1 . (5.43)
λ∈[a,b]
In the following sections we will introduce methods with an analogous
convergence behaviour, without control parameters necessary for their
definition.
For further information on semi-iterative methods see, for example, [16,
Chapter 7].

Exercises

5.1 Investigate Jacobi’s method and the Gauss–Seidel method for solving
the linear system of equations Ax = b with respect to their convergence if
we have the following system matrices:
   
1 2 −2 2 −11
1
(a) A =  1 1 1  , (b) A =  2 2 2  .
2
2 2 1 −1 −12
5.2 Prove the consistency of the SOR method.

5.3 Prove Theorem 5.6, (1).