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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 http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 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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