Implementations for both parallel block-diagonal-bordered sparse LU
and Choleski factorization have been developed during this research.
The pseudo-code presented in this section focuses on LU factorization,
although Choleski factorization implementations are similar to these
algorithms with modifications to account for the symmetric nature of
the matrices used in Choleski factorization. The
block-diagonal-bordered LU factorization algorithm can be broken into
three component parts as defined in the derivation on available
parallelism in chapter . Pseudo-code representations of
each parallel LU factorization algorithm section are presented
separately in figures
through
. In
particular, each of these figures correspond to the following figure
numbers:
The remaining steps in the parallel algorithm are forward reduction
and backward substitution. The forward reduction algorithm to operate
with the parallel block-diagonal-bordered LU factorization algorithm
can be broken into three component parts, similar to LU factorization.
Pseudo-code representations of each parallel algorithm section are
presented separately in figures through
. In
particular, each of these figures correspond to the following figure
numbers:
The backward substitution algorithm to operate with the parallel
block-diagonal-bordered LU factorization algorithm can be broken into
two component parts, back substitute the last diagonal block then back
substitute the remaining upper triangular matrix. Pseudo-code
representations of each parallel algorithm section are presented
separately in figures and
, respectively for
backward substitution of the last diagonal block and backward
substitution of the diagonal blocks and border.
Figure: Parallel Block-Diagonal-Bordered Sparse LU Factorization Algorithm --- Diagonal Blocks and Border
Figure: Parallel Block-Diagonal-Bordered Sparse LU Factorization Algorithm --- Update the Last Diagonal Block --- Low-Latency Communications Paradigm
Figure: Parallel Block-Diagonal-Bordered Sparse LU Factorization Algorithm --- Update the Last Diagonal Block --- Buffered Communications Paradigm
Figure: Parallel Block-Diagonal-Bordered Sparse LU Factorization Algorithm --- Last Diagonal Block
Figure: Parallel Block-Diagonal-Bordered Sparse Forward Reduction Algorithm --- LU Factorization --- Diagonal Blocks and Border
Figure: Parallel Block-Diagonal-Bordered Sparse Forward Reduction Algorithm --- LU Factorization --- Update the Last Diagonal Block --- Low-Latency Communications Paradigm
Figure: Parallel Block-Diagonal-Bordered Sparse Forward Reduction Algorithm --- LU Factorization --- Update the Last Diagonal Block --- Buffered Communications Paradigm
Figure: Parallel Block-Diagonal-Bordered Sparse Forward Reduction Algorithm --- LU Factorization --- Last Diagonal Block
Figure: Parallel Block-Diagonal-Bordered Sparse Backward Substitution Algorithm --- LU Factorization --- Last Diagonal Block
Figure: Parallel Block-Diagonal-Bordered Sparse Backward Substitution Algorithm --- LU Factorization --- Diagonal Blocks and Border