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Unformatted text preview: ’s equation on a square is not
diagonally dominant (e.g., look at the ﬁfth row in the 9 × 9 system on p. 564).
But such systems are always positive deﬁnite (Example 7.6.2), and there is a
classical theorem stating that if A is positive deﬁnite, then the Gauss–Seidel
iteration converges to the solution of Ax = b for every initial vector x(0). The
same cannot be said for Jacobi’s method, but there are matrices (the M-matrices
of Example 7.10.7, p. 626) having properties resembling positive deﬁniteness for
which Jacobi’s method is guaranteed to converge—see (7.10.29). D
E Example 7.10.6 The successive overrelaxation (SOR) method improves on Gauss–Seidel
by introducing a real number ω = 0, called a relaxation parameter, to form
the splitting A = M − N, where M = ω −1 D − L and N = (ω −1 − 1)D + U.
As before, D is the diagonal part of A ( aii = 0 is assumed) and −L and −U
contain the entries occurring below and above the diagonal of A, respectively.
Since M−1 = ω (D − ω L)−1 = ω (I − ω D−1 L)−1 , the SOR iteration matrix is
−1 Hω = M −1 N = (D − ω L) T
H −1 (1 − ω )D + ω U = (I − ω D IG
R −1 L) (1 − ω )I + ω D−1 U , and the k th SOR iterate emanating from (7.10.16) is x(k ) = Hω x(k − 1) + ω (I − ω D−1 L)−1 D−1 b. (7.10.21) This is the Gauss–Seidel iteration when ω = 1. Using ω > 1 is called overrelaxation, while taking ω < 1 is referred to as underrelaxation. Writing (7.10.21) in
the form (I − ω D−1 L)x(k ) = (1 − ω )I + ω D−1 U x(k − 1) + ω D−1 b and considering the ith component on both sides of this equality produces Y
C xi (k ) = (1 − ω )xi (k − 1) + ω
aij xj (k ) −
aij xj (k − 1) . (7.10.22)
j >i The matrix splitting approach is elegant and unifying, but it obscures the simple
idea behind SOR. To understand the original motivation, write the Gauss–Seidel
iterate in (7.10.20) as xi (k ) = xi (k − 1) + ck , where ck is the “correction term”
ck = 1
aii n aij xj (k ) −
j <i aij xj...
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