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{\large \bf Activities Report for June --- September 1993} \\
{\large David Koester}
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My activities for the reporting period June --- September 1993 has focused
on three areas:
\begin{enumerate}
\item documenting work in progress,
\item examining ordering techniques for block-diagonal-bordered form sparse matrices,
\item integrating a parallel dense matrix solver into my block-diagonal-bordered sparse matrix solver to factor the nearly dense last block.
\end{enumerate}
All efforts in these three areas support both my Ph.D. research and
the NPAC --- Niagara Mohawk contract.

In order to document my work in progress, two NPAC technical reports
are in the process of being written:
\begin{enumerate}
\item {\em Parallel LU Factorization of Block-Diagonal Bordered Sparse Matrices}, 
\item {\em Ordering Irregular Matrices for Parallel Block-Diagonal-Bordered LU Factorization}.  
\end{enumerate} 
The first report is awaiting data to be collected in the latest
implementation of my parallel block-bordered diagonal sparse matrix
solver, while additional editing is required for the second paper.
Significant portions of these papers will be used in my thesis.

Direct block-diagonal-bordered sparse matrix solvers exhibit distinct
advantages when compared to current parallel direct sparse matrix
solvers.  Task assignments for numerical factorization on
distributed-memory multi-processors depend only on the assignment of
independent blocks to processors and the processor assignments of data
in the last block.  My research into block-diagonal-bordered form
sparse matrix solvers has developed a new three-step preprocessing
phase that requires special ordering techniques to determine the
block-diagonal-bordered form and to perform load-balancing to ensure
efficient parallel operations.  The applicability of two prospective
ordering techniques to order power systems sparse matrices into
block-diagonal-bordered form with uniform distribution of work for all
processors has been examined using power systems data from the
Boeing-Harwell sparse matrix series.  These two ordering techniques
are based on recursive spectral bisection and node-tearing nodal
analysis --- a form of diakoptics.  Node-tearing nodal analysis has
distinct advantages over the technique based on recursive spectral
bisection.  In order to determine the extent of the applicability of
the node-tearing nodal analysis technique to general irregular sparse
matrices, node-tearing nodal analysis has been extensively exercised
on various sparse matrices in the Boeing-Harwell series, in addition
to actual electrical power system network data obtained from Niagara
Mohawk.

A poster session covering my matrix ordering research as been accepted
at the Scalable Parallel Libraries Conference 6-8 October 1993 at
Mississippi State University.  The poster and accompanying paper will
describe the problems with limited scalability in sparse matrix
solvers due to the limited parallelization available in real-life
sparse matrices.  This phenomenon is clearly visible in animations
developed from series of ordered sparse matrix representations.

My last, and most recent task, has been to examine efficient parallel
dense LU factorization techniques to be used to factor the last block
in a block-diagonal-bordered sparse matrix.  Ordering research has
shown that the last block will be nearly dense after considering
fillin due to factorization, so a dense matrix solver will be the most
efficient technique available to factor this portion of the matrix.
Previous NPAC research into FORTRAN benchmarks has been utilized to
develop an efficient parallel dense matrix solver for the last-block
in the sparse matrix.  My software implementation is currently under
development, although development effort will be significantly
decreased because large portions of the code will directly utilize
existing SAXPY routines from the LAPACK software library.

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