**Unformatted text preview: **this ordering feature can aﬀect the performance of the algorithm, it was the
object of much study at one time. Furthermore, when core memory is a concern,
Gauss–Seidel enjoys an advantage because as soon as a new component xi (k ) is
computed, it can immediately replace the old value xi (k − 1), whereas Jacobi
requires all old values in x(k − 1) to be retained until all new values in x(k )
have been determined. Something that both algorithms have in common is that
diagonal dominance in A guarantees global convergence of each method. D
E T
H IG
R Problem: Explain why diagonal dominance in A is suﬃcient to guarantee
convergence of the Gauss–Seidel method for all initial vectors x(0) and for all
right-hand sides b .
Solution: Show ρ (H) < 1. Let (λ, z) be any eigenpair for H, and suppose
that the component of maximal magnitude in z occurs in position m. Write
(D − L)−1 Uz = λz as λ(D − L)z = Uz, and write the mth row of this latter
equation as λ(d − l) = u, where Y
P d = amm zm , O
C l=− Diagonal dominance |amm | >
|u| + |l| = j <m amj zj , and u=− j <m
j =m |amj | and |zj | ≤ |zm | for all j yields amj zj ≤ |zm | amj zj + amj zj .
j >m j >m |amj | +
j <m |amj |
j >m < |zm ||amm | = |d| =⇒ |u| < |d| − |l|. This together with λ(d − l) = u and the backward triangle inequality (Example
5.1.1, p. 273) produces the conclusion that
|λ| = |u|
|u|
≤
< 1,
|d − l|
|d| − |l| and thus ρ(H) < 1. Note: Diagonal dominance in A guarantees convergence for both Jacobi and
Gauss–Seidel, but diagonal dominance is a rather severe condition that is often Copyright c 2000 SIAM Buy online from SIAM
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624
Chapter 7
Eigenvalues and Eigenvectors
http://www.amazon.com/exec/obidos/ASIN/0898714540 It is illegal to print, duplicate, or distribute this material
Please report violations to [email protected] not present in applications. For example the linear system in Example 7.6.2
(p. 563) that results from discretizing Laplace...

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