**Unformatted text preview: **), where HGS
is the Gauss–Seidel iteration matrix. For example, the discrete Laplacian Ln2 ×n2
in Example 7.6.2 (p. 563) satisﬁes the special case conditions, and the spectral
radii of the iteration matrices associated with L are Y
P Jacobi: ρ (HJ ) = cos πh
≈ 1 − (π 2 h2 /2)
2
Gauss–Seidel: ρ (HGS ) = cos πh
≈ 1 − π 2 h2 ,
1 − sin πh
SOR: ρ Hωopt =
≈ 1 − 2πh,
1 + sin πh (see Exercise 7.10.10), O
C where we have set h = 1/(n + 1). Examining asymptotic rates of convergence
reveals that Gauss–Seidel is twice as fast as Jacobi on the discrete Laplacian
because RGS = − log10 cos2 πh = −2 log10 cos πh = 2RJ . However, optimal
SOR is much better because 1 − 2πh is signiﬁcantly smaller than 1 − π 2 h2 for
even moderately small h. The point is driven home by looking at the asymptotic
rates of convergence for h = .02 ( n = 49) as shown below:
Jacobi: RJ ≈ .000858,
Gauss–Seidel: RGS = 2RJ ≈ .001716,
SOR: Ropt ≈ .054611 ≈ 32RGS = 64RJ .
83 This special case was developed by the contemporary numerical analyst David M. Young, Jr.,
who produced much of the SOR theory in his 1950 Ph.D. dissertation that was directed by
Garrett Birkhoﬀ at Harvard University. The development of SOR is considered to be one of the
major computational achievements of the ﬁrst half of the twentieth century, and it motivated
at least two decades of intense eﬀort in matrix computations. Copyright c 2000 SIAM Buy online from SIAM
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626
Chapter 7
Eigenvalues and Eigenvectors
http://www.amazon.com/exec/obidos/ASIN/0898714540
In other words, after things settle down, a single SOR step on L (for h = .02)
is equivalent to about 32 Gauss–Seidel steps and 64 Jacobi steps! It is illegal to print, duplicate, or distribute this material
Please report violations to [email protected] Note: In spite of the preceding remarks, SOR has limitations. Special cases
for which the optimum ω can be explicitly determ...

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