Only a small sample of m matrix theory has been

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Unformatted text preview: write T H D E λx = Bx =⇒ |λ| |x| = |λx| = |Bx| ≤ |B| |x| = B |x| =⇒ (rI − B)|x| ≤ (r − |λ|) |x| =⇒ 0 ≤ |x| ≤ (r − |λ|) (rI − B)−1 |x| =⇒ r − |λ| ≥ 0. (7.10.15) IG R But |λ| = r; otherwise (7.10.15) would imply that |x| (and hence x ) is zero, which is impossible. Thus |λ| < r for all λ ∈ σ (B) , which means ρ (B) < r. Iterative algorithms are often used in lieu of direct methods to solve large sparse systems of linear equations, and some of the traditional iterative schemes fall into the following class of nonhomogeneous linear difference equations. Y P Linear Stationary Iterations O C Let Ax = b be a linear system that is square but otherwise arbitrary. A splitting of A is a factorization A = M − N, where M−1 exists. • Let H = M−1 N (called the iteration matrix), and set d = M−1 b. • For an initial vector x(0)n×1 , a linear stationary iteration is x(k ) = Hx(k − 1) + d, • k = 1, 2, 3, . . . . If ρ(H) < 1, then A is nonsingular and lim x(k ) = x = A−1 b for every initial vector x(0). k→∞ Copyright c 2000 SIAM (7.10.16) (7.10.17) Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.10 Difference Equations, Limits, and Summability http://www.amazon.com/exec/obidos/ASIN/0898714540 621 Proof. To prove (7.10.17), notice that if A = M − N = M(I − H) is a splitting for which ρ(H) < 1, then (7.10.11) guarantees that (I − H)−1 exists, and thus A is nonsingular. Successive substitution applied to (7.10.16) yields x(k ) = Hk x(0) + (I + H + H2 + · · · + Hk−1 )d, so if ρ(H) < 1, then (7.10.9)–(7.10.11) insures that for all x(0), lim x(k ) = (I − H)−1 d = (I − H)−1 M−1 b = A−1 b = x. (7.10.18) It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu k→∞ It’s clear that the convergence rate of (7.10.16) is governed by the size of ρ(H) along with the index of its associated eigenval...
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