Unformatted text preview: +1) uses the same manipulations.
4. Storage is needed for A, M (possibly in factored form), x( ) , p( ) , r( ), z( ) , and
Ap( ) . Relative to the conjugate gradient procedure, additional storage is needed
for M and z( ) . Storage costs are still modest relative to those of direct methods.
5. In addition to the matrix multiplication (Ap( ) ) and the two inner products per step
required by the conjugate gradient method, a linear equations solution (Mz( +1) =
r( +1) ) is required 44 Solution Techniques for Elliptic Problems
6. Using (9.3.13) for the conjugate gradient method
(p( ) )T Ap( ) = 0 (~( ) )T ~( ) = 0
rr 6= : Using (9.3.18a, 9.3.18b), we obtain the \orthogonality conditions"
(r( ) )T M;1 r( ) = 0 (p( ) )T Ap( ) = 0 6= (9.3.20) 7. When M is positive de nite
(r( ) )T z( ) = (r( ))T M;1 r( ) > 0:
Thus, values of can always be obtained and the procedure does not fail. Let us select some preconditionings, beginning with some choices based on iterative
strategies. It will be convenient to write the basic xed-point strategy (9.2.2) in the form
Mx( +1) = Nx( ) + b (9.3.21a) ^^
A=M;N (9.3.21b) where Comparing (9.3.21) with (9.2.10b), (9.2.15b), and (9.2.21d), we have
Jacobi iteration: ^
MJ = D ^
NJ = L + U (9.3.22) Gauss-Seidel iteration: ^
MGS = D ; L ^
NGS = U (9.3.23) SOR Iteration:
M! = ! (D ; !L) ^
N! = 1 ; ! D + U:
! (9.3.24) 9.3. Conjugate Gradient Methods 45 Recall that D is the diagonal part, L is the negative of the lower traingular part, and
U is the negative of the upper triangular part of A (cf. (9.2.8)). Let us also include
symmetric successive over relaxation (SSOR) in our study. As discussed in Section 9.2,
SSOR takes two SOR sweeps with the unknowns placed in reverse order on the second
sweep. Using (9.3.24) and (9.3.21), the rst step of the SOR procedure is
(D ; !L)x( +1=2) = (1 ; !)D + !U]x( ) + !b: (9.3.25a) Reversing the sweep direction on the second step yields
(D ; !U)x( +1) = (1 ; !)D + !L]x( +1=2) + !b: (9.3.25b) The intermediate solution x( +1=2) can be eliminated to obtain a scheme...
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.
- Spring '14