37 to eliminate reveals the orthogonality condition r

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Unformatted text preview: , then we may write the above expression as (r( ) )T p( ) = (r( ) )T r( ) = 0 1 ::: : (9.3.9b) 36 Solution Techniques for Elliptic Problems Let us next expand (9.3.6) while using (9.3.1) and (9.3.2b) to show that E (x( +1) ) = E (x( ) ) ; (r( ) )T p( ) + (1=2) 2 (p( ) )T Ap( ) : Using (9.3.7) and (9.3.9b) () T ()2 1 () T ()2 E (x( +1) ) = E (x( ) ) ; 2 (r( ) )T p (] ) = E (x( ) ) ; 1 (r( ) )T r (] ) : (9.3.10) 2 (p ) Ap (p ) Ap Having (9.3.10), we see that the error E (x( +1) ) is decreased most rapidly when (p( ) )T Ap( ) is minimal. This will be our criterion for determining . Using (9.3.5c), we have (p( ) )T Ap( ) = (r( ) + ;1 p( ;1) )T A(r( ) + ;1p( ;1) ): Minimizing with respect to ;1 gives (r( ))T Ap( ;1) ;1 = ; (p( ;1) )T Ap( ;1) or reindexing r( +1) T Ap( ) = ; ((p( ) )) Ap( ) T = 0 1 ::: : (9.3.11) Using (9.3.5c) (p( +1) )T Ap( ) = (r( +1) )T Ap( ) + (p( ) )T Ap( ) which, upon use of (9.3.11), reveals (p( +1) )T Ap( ) = 0 = 0 1 ::: : (9.3.12a) Thus, the search directions are orthogonal with respect to a strain energy inner product. We usually call this a conjugacy condition and say that the search directions are conjugate. Using (9.3.12a) with (9.3.5c), we have (p( ) )T Ap( ) = (r( ) )T Ap( ) + ;1 (p( ;1) )T Ap( ) = (r( ) )T Ap( ) : Combining this result with (9.3.8) yields (r( +1) )T r( ) = (r( ) )T r( ) ; (p( ) )T Ar( ) = (r( ) )T r( ) ; (p( ) )T Ap( ) : 9.3. Conjugate Gradient Methods 37 Using (9.3.7) and (9.3.9b), we nd the orthogonality relation (r( +1) )T r( ) = 0: (9.3.12b) Equation (9.3.11) can be put in a slightly simpler form by using (9.3.8) and (9.3.12b) to obtain (r( +1) )T r( +1) = (r( +1) )T r( ) ; (r( +1) )T Ap( ) = ; (r( +1) )T Ap( ) : Using this with (9.3.11) and (9.3.7) ( +1) )T ( +1) ( +1) T ( +1) = 1 (r ( ) T r ( ) = (r ( )) T r ( ) : (p ) r (p ) Ap Finally, using (9.3.9b) ( +1) )T r( +1) = (r (r( ) )T r( ) : We summarize our ndings as a theorem. (9.3.12c) Theorem 9.3.1. The residuals and search directions of the conjugate gradient method satisfy (r( ) )T...
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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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