117 to write ak2 ak1 ak1 k1 k1 therefore k1 ak2 2 c1

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Prove that if there exists a matrix norm such that limk→∞ Ak = 0. A &lt; 1, then 7.10.7. By examining the iteration matrix, compare the convergence of Jacobi’s method and the Gauss–Seidel method for each of the following coeﬃcient matrices with an arbitrary right-hand side. Explain why this shows that neither method can be universally favored over the other. ⎛ ⎞ ⎛ ⎞ 1 2 −2 2 −1 1 A1 = ⎝ 1 1 1⎠. A2 = ⎝ 2 2 2⎠. 22 1 −1 −1 2 Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.10 Diﬀerence Equations, Limits, and Summability http://www.amazon.com/exec/obidos/ASIN/0898714540 It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu 7.10.8. Let A = 2 −1 0 −1 2 −1 0 −1 2 639 (the ﬁnite-diﬀerence Example 1.4.1, p. 19). (a) Verify that A satisﬁes the special case conditions given in Example 7.10.6 that guarantee the validity of (7.10.24). (b) Determine the optimum SOR relaxation parameter. (c) Find the asymptotic rates of convergence for Jacobi, Gauss– Seidel, and optimum SOR. (d) Use x(0) = (1, 1, 1)T and b = (2, 4, 6)T to run through several steps of Jacobi, Gauss–Seidel, and optimum SOR to solve Ax = b until you can see a convergence pattern. D E 7.10.9. Prove that if ρ (Hω ) &lt; 1, where Hω is the iteration matrix for the SOR method, then 0 &lt; ω &lt; 2. Hint: Use det (Hω ) to show |λk | ≥ |1 − ω | for some λk ∈ σ (Hω ) . T H 7.10.10. Show that the spectral radius of the Jacobi iteration matrix for the discrete Laplacian Ln2 ×n2 described in Example 7.6.2 (p. 563) is ρ (HJ ) = cos π/(n + 1). IG R 7.10.11. Consider a scalar sequence {α1 , α2 , α3 , . . .} and the associated Ces`ro a sequence of averages {µ1 , µ2 , µ3 , . . .}, where µn = (α1 +α2 +· · ·+αn )/n. Prove that if {αn } converges to α, then {µn } also converges to α. Y P Note: Like scalars, a vector sequence {vn } in a ﬁnite-dimensional space converges to v if and only if...
View Full Document

This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

Ask a homework question - tutors are online