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3.4 Block-Iterative Methods 29
consists of p steps. One block-iteration of the Block-Iteration with the relaxation
parameter λk can be written as:
x k+1 = x k + A T ¯ MB(b − Ax k ), (3.7)
¯MB = ( ¯ D + L) −1
where ¯ D is block-diagonal and L is block-lower triangular and defined as:
⎛
0
⎜
L = ⎜ A2A
⎜
⎝
0
T 1
.
. ..
. .. . ..
⎞
⎟
⎠ ,
⎛
λ
D ¯ ⎜
= ⎝
−1 −1
1 M1 . ..
0
ApA T 1 · · · ApA T p−1 0
The sequence defined by (3.7) converges towards the solution of
A T ¯ MB(b − Ax) = 0.
3.4.2 Symmetric Block-Iteration
0 λ−1 p M −1
p
⎞
⎟
⎠ .(3.8)
In Symmetric Block-Iteration one block-iteration consists of first one blockiteration
of the above Block-Iteration method followed by another block-iteration,
where the blocks appear in reverse order. This gives the algorithm the following
control order t = 1, 2, . . .,p − 1, p, p − 1, . . . 1.
The algorithm for the symmetric block-iteration for x 0 ∈ R n looks as follows:
x k,0 = x k
x k,t = x k,t−1 + λtA T t Mt(b t − Atx k,t−1 ), (3.9)
x k+1 = x k,1 ,
where t = 1, . . .,p − 1, p, p − 1, . . .,1 and x k,1 denotes the last step in (3.9).
One block-iteration of the Symmetric Block-Iteration method can be written in
a general form, where we let
AA T = L + D + L T
be the splitting of AA T into its lower block triangular, block diagonal and upper
block triangular parts. The block-iteration can then be written as:
x k+1 = x k + A T ¯ MSB(b − Ax k ). (3.10)