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Parallel Blocked-Diagonal-Bordered LU Factorization

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 gif. Pseudo-code representations of each parallel LU factorization algorithm section are presented separately in figures gif through gif. In particular, each of these figures correspond to the following figure numbers:

  1. factor the diagonal blocks and border --- figure gif,
  2. update the last diagonal block ---
  3. factor the last diagonal block --- figure gif.

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 gif through gif. In particular, each of these figures correspond to the following figure numbers:

  1. forward reduce the diagonal blocks and border --- figure gif,
  2. update the last diagonal block ---
  3. forward reduce the last diagonal block --- figure gif.

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 gif and gif, 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 



next up previous
Next: Parallel Blocked-Diagonal-Bordered Gauss-Seidel Up: Implementation Pseudo-Code Previous: Implementation Pseudo-Code



David P. Koester
Sun Oct 22 17:27:14 EDT 1995