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One of the attractions of rowprojection methods seen in Chapter 8 is their high degree of
parallelism. In Cimmino™s method, the scalars as well as the new residual vector can © C
be computed in parallel. In the GaussSeidelNE (respectively GaussSeidelNR), it is also
possible to group the unknowns in such a way that any pair of rows (respectively columns)
have disjointed nonzero patterns. Updates of components in the same group can then be
performed in parallel. This approach essentially requires ¬nding a multicolor ordering for
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the matrix (respectively ).
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It is necessary to ¬rst identify a partition of the set into subsets , , d ’F
te i¢it
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hh
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such that the rows (respectively columns) whose indices belong to the same set are
pB CB
structurally orthogonal to each other, i.e., have no nonzero elements in the same column
locations. When implementing a block SOR scheme where the blocking is identical with
that de¬ned by the partition, all of the unknowns belonging to the same set can be )B
updated in parallel. To be more speci¬c, the rows are reordered by scanning those in AB
c
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followed by those in , etc.. Denote by the matrix consisting of the rows belonging to
¨B C
the th block. We assume that all rows of the same set are orthogonal to each other and
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that they have been normalized so that their 2norm is unity. Then a block GaussSeidel
sweep, which generalizes Algorithm 8.1, follows.
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1. Select an initial .
2. For Do:
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4. t pD 
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5. EndDo
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Here, and are subvectors corresponding to the blocking and is a vector of length
C Cz C
the size of the block, which replaces the scalar of Algorithm 8.1. There is parallelism in u©
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each of the steps 3 and 4.
The question that arises is how to ¬nd good partitions . In simple cases, such as CVB
blocktridiagonal matrices, this can easily be done; see Exercise 7. For general sparse ma
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trices, a multicoloring algorithm on the graph of (respectively ) can be em w w
ployed. However, these matrices are never stored explicitly. Their rows can be generated,
used, and then discarded.
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1 Let be a Symmetric Positive De¬nite matrix and consider where is a block
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diagonal of . ¤
Show that is a Symmetric Positive De¬nite matrix. Denote by the associated inner
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