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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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