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# These two iterates and the level surfaces e y 1 075 05

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Unformatted text preview: iques for Elliptic Problems 0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Figure 9.3.2: Convergence of the method of steepest descent for the problem of Example 9.3.1. Ellipses have level values of 1, 0.75, 0.5, and 0.25 (outer to inner). The rst two iterations are shown to be converging to the solution at (0 0). 9.3.1 The Conjugate Gradient Method The remedy for the slow convergence encountered with the method of steepest descent is to choose other search directions. With the conjugate gradient method, we choose x( +1) = x( ) + p( ) (9.3.5a) p( ) = r( ) + (x( ) ; x( ;1) ): (9.3.5b) with Thus, the new search direction is a linear combination of the steepest descent search direction and the previous \correction" x( ) ; x( ;1) . The parameters and are to be determined. Using (9.3.5a), we rewrite (9.3.5b) in the form p( ) = r( ) + ;1 p ( ;1) = r( ) + ;1p( ;1) (9.3.5c) 9.3. Conjugate Gradient Methods 35 with the goal of specifying and ;1 so that convergence is as fast as possible. As with steepest descent, we choose to minimize E (x( +1) ). In this case, E (x( +1) ) = E (x( ) + p( ) ) = 1 (x( ) + p( ) )T A(x( ) + p( ) ) ; bT (x( ) + p( ) ): 2 (9.3.6) Di erentiating with respect to E 0 (x( ) + p( ) ) = (x( ) + p( ) )T Ap( ) ; bT p( ) = 0 or, using (9.3.2b) (;r( ) + p( ) A)T p( ) = 0: Thus, ()T () = (r ) ) p ( ) = 0 1 ::: : (9.3.7) (p( )T Ap Let's develop a few properties and relationships that will be interesting in their own right and useful for calculating . First, we'll use (9.3.2b) and (9.3.5a) to write r( +1) = b ; Ax( +1) = b ; A(x( ) + p( ) ) = r( ) ; Ap( ) : (9.3.8) Taking an inner product with p( ) to obtain (r( +1) )T p( ) = (r( ))T p( ) ; (p( ) )T Ap( ) : Using (9.3.7) to eliminate reveals the orthogonality condition (r( +1) )T p( ) = 0: (9.3.9a) Next, take the inner product of (9.3.5c) with r( +1) to obtain (r( +1) )T p( +1) = (r( +1) )T r( +1) + (r( +1) )T p( ) or, using (9.3.9a), (r( +1) )T p( +1) = (r( +1) )T r( +1) : If we select p(0) = r(0)...
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• Spring '14
• JosephE.Flaherty
• Articles with example pseudocode, Gauss–Seidel method, Jacobi method, Iterative method, elliptic problems

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