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Example 7.10.4 81 Jacobi’s method is produced by splitting A = D − N, where D is the
diagonal part of A (we assume each aii = 0 ), and −N is the matrix containing
the oﬀ-diagonal entries of A. Clearly, both H = D−1 N and d = D−1 b can be
formed with little eﬀort. Notice that the ith component in the Jacobi iteration
x(k ) = D−1 Nx(k − 1) + D−1 b is given by It is illegal to print, duplicate, or distribute this material
Please report violations to email@example.com xi (k ) = bi − j =i aij xj (k − 1) /aii . (7.10.19) This shows that the order in which the equations are considered is irrelevant
and that the algorithm can process equations independently (or in parallel).
For this reason, Jacobi’s method was referred to in the 1940s as the method of
simultaneous displacements. D
E Problem: Explain why Jacobi’s method is guaranteed to converge for all initial
vectors x(0) and for all right-hand sides b when A is diagonally dominant as
deﬁned and discussed in Examples 4.3.3 (p. 184) and 7.1.6 (p. 499). T
H Solution: According to (7.10.17), it suﬃces to show that ρ(H) < 1. This follows
by combining |aii | > j =i |aij | for each i with the fact that ρ(H) ≤ H ∞
(Example 7.1.4, p. 497) to write
ρ(H) ≤ H Example 7.10.5 ∞ IG
R = max
i j |aij |
|aii | j =i |aij |
|aii | Y
82 The Gauss–Seidel method is the result of splitting A = (D − L) − U, where
D is the diagonal part of A ( aii = 0 is assumed) and where −L and −U
contain the entries occurring below and above the diagonal of A, respectively.
The iteration matrix is H = (D − L)−1 U, and d = (D − L)−1 b. The ith entry
in the Gauss–Seidel iteration x(k ) = (D − L)−1 Ux(k − 1) + (D − L)−1 b is O
C xi (k ) = bi − j <i aij xj (k ) − j >i aij xj (k − 1) /aii . (7.10.20) This shows that Gauss–Seidel determines xi (k ) by using the newest possible
information—namely, x1 (k ), x2 (k ), . . . , xi−1 (k ) in the curr...
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