Amazoncomexecobidosasin0898714540 for occupying the

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Unformatted text preview: ined are rare, so adaptive computational procedures are generally necessary to approximate a good ω, and the results are often not satisfying. While SOR was a big step forward over the algorithms of the nineteenth century, the second half of the twentieth century saw the development of more robust methods—such as the preconditioned conjugate gradient method (p. 657) and GMRES (p. 655)—that have relegated SOR to a secondary role. D E Example 7.10.7 84 M-matrices are real nonsingular matrices An×n such that aij ≤ 0 for all i = j and A−1 ≥ 0 (each entry of A−1 is nonnegative). They arise naturally in a broad variety of applications ranging from economics (Example 8.3.6, p. 681) to hard-core engineering problems, and, as shown in (7.10.29), they are particularly relevant in formulating and analyzing iterative methods. Some important properties of M-matrices are developed below. T H IG R • A is an M-matrix if and only if there exists a matrix B ≥ 0 and a real number r > ρ(B) such that A = rI − B. (7.10.25) • If A is an M-matrix, then Re (λ) > 0 for all λ ∈ σ (A) . Conversely, all matrices with nonpositive off-diagonal entries whose spectrums are in the right-hand halfplane are M-matrices. (7.10.26) • Principal submatrices of M-matrices are also M-matrices. • If A is an M-matrix, then all principal minors in A are positive. Conversely, all matrices with nonpositive off-diagonal entries whose principal minors are positive are M-matrices. (7.10.28) • If A = M − N is a splitting of an M-matrix for which M−1 ≥ 0, then the linear stationary iteration (7.10.16) is convergent for all initial vectors x(0) and for all right-hand sides b. In particular, Jacobi’s method in Example 7.10.4 (p. 622) converges for all M-matrices. (7.10.29) Y P (7.10.27) O C Proof of (7.10.25). Suppose that A is an M-matrix, and let r = maxi |aii | so that B = rI − A ≥ 0. Since A−1 = (rI − B)−1 ≥ 0, it follows from (7.10.14) in Example 7.10.3 (p. 620) that r > ρ(B). Conversely, if A is any matrix of 84 This terminology was introduced in 1937 by the twentieth-century mathematician Alexand...
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This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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