23 red black ordering of the laplacian operator on a

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Unformatted text preview: all real 6= 0. We're now ready to search for the optimal choice of !. Theorem 9.2.5. If a matrix A is consistently ordered then the eigenvalues of M! and of MJ are related by ; (9.2.22) = +1!2 ! 1 : = Proof. From (9.2.21d), we see that the eigenvalues of M! satisfy (D ; !L);1 (1 ; !)D + !U)]q = q where q is the eigenvector of M! corresponding to . Multiplying by (D ; !L), we have (1 ; !)D + !U ; (D ; !L)]q = 0: Multiplying by D;1=! ! D;1(U + L) ; + ! ; 1 I]q = 0: Finally, multiplying by ;1=2 yields D;1 ( 1=2 L + ;1=2 U) ; I]q = 0 9.2. Basic Iterative Solution Methods 21 where satis es (9.2.22). Thus is an eigenvalue of D;1( 1=2 L + ;1=2 U): If A is consistently ordered then the eigenvalues of this matrix are independent of the parameter 1=2 . Thus, we can choose any convenient value of to nd the eigenvalues . In particular, if we choose = 1 then is an eigenvalue of MJ (cf. (9.2.10b). Remark 3. Setting ! = 1 for the Gauss-Seidel method and using (9.2.22), we see that = 1=2 , con rming the relationship between the eigenvalues of MJ and MGS that we found in Example 9.2.5 for the discrete Laplacian. Remark 4. The transformation used in Example 9.2.5 could have also been used to prove this theorem for the discrete Laplacian operator. Let us assume that the eigenvalues of MJ are real. (It su ces to assume that A is symmetric.) Let us also assume that (MJ ) < 1. Then, using Theorem 9.2.4, (MGS ) < 1. Let g( ) = 1=2: (9.2.23) f ( !) = + ! ; 1 ! We sketch f ( !) and g( ) as functions of in Figure 9.2.4. Both halves of g( ) are shown since the eigenvalues of MJ occur in pairs. Thus, if is an eigenvalue of MJ , so is ; . This may be shown for Laplace's equation using the results of Example 9.2.5, but we won't do it here. Let's list several properties of f ( !) and g( ) that can be discerned from (9.2.23) and Figure 9.2.23. 1. f (1 !) = 1, 8 !. 2. The function g( = (MJ ). ) increases linearly with j j. Its largest amplitude occurs when 3. For xed and !, the eigenvalues of M! are gi...
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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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