h p p22 1 1 mj mj 2 mj 22 1

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Unformatted text preview: tral radius (M! ) as a function of ! for SOR iteration. Proof. With the various symmetries (cf. Strikwerda 5], Section 13.3), it su ces to consider positive values of . With this choice and our interest in calculating the spectral radius of M! , we clearly want the positive sign in (9.2.24a). Thus, in summary, we have 1. = 1 when ! = 0, h i2 p 2 + 1 ; ! , 0 < ! ! , and 2. = ! =2 + (! =2) ~ 3. j j = ! ; 1, ! < !. ~ The values of increase with thus, the spectral radius (M! ) occurs when = (MJ ), as given by (9.2.25b). The minimum value of (M! ) occurs at ! corresponding to ~ = (MJ ), as given by (9.2.25a). The spectral radius (M! ) is displayed as a function of ! in Figure 9.2.5. Several corollaries follow from Theorem 9.2.6 and we'll list one. Corollary 9.2.1. Under the conditions of Theorem 9.2.6, the SOR method converges for ! 2 (0 2) when (MJ ) < 1. 9.2. Basic Iterative Solution Methods 25 Proof. From (9.2.25) (M!OP T ) = !OPT ; 1 = p22 ; 1: 1 + 1 ; (MJ ) Thus, 0 < (M!OP T ) < 1 when 0 < (MJ ) < 1. Additional inspection of (9.2.25b) reveals that d (M! )=d! is non-positive when ! < !OPT and unity when ! > !OPT (Figure 9.2.5). Since (M! ) = 1 at ! = 0 2, we conclude that (M! ) < 1 for 0 < ! < 2. An examination of (9.2.25) and Figure 9.2.5 reveals that !OPT 2 (1 2). Increasing (MJ ) increases !OPT and, in particular, !OPT ! 2 as (MJ ) ! 1. From Figure 9.2.5, we also see that overestimating !OPT by a given amount increases (M! ) less than underestimating it by the same amount. Example 9.2.8. Let us solve the problem of Example 9.2.4 using SOR iteration. With J = K = 3 and x = y = 1=4, we obtain the Gauss-Seidel method (9.2.16) as ) ^( Ujk +1) = 1 (Uj(+1 k + Uj(;+1) + Uj( k)+1 + Uj( k+1) ): 1k ;1 4 With relaxation, we compute ( ( ^( Ujk +1) = !Ujk +1) + (1 ; !)Ujk ) : We'll explicitly eliminate the intermediate variable to obtain the SOR method as ( ) ( Ujk +1) = ! (Uj(+1 k + Uj(;+1) + Uj( k)+1 + Uj( k+1) ) + (1 ; !)Ujk ) : 1k ;1 4 Using (9.2...
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.

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