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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
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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 [email protected] 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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