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Unformatted text preview: www.amazon.com/exec/obidos/ASIN/0898714540 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 protected] 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 | = max i |aii | j =i |aij | < 1. |aii | Y P 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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