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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 http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 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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