\quick" Implementation of Block Lu Algorithms on the Cm-200

The CMSSL library only includes a limited amount of mathematical algorithms. Hence, when writing code for the Connection Machine one often has to implement LAPACK-style routines already developed for other architec-tures, in the hope that acceptable performance can thus be obtained relatively quickly. Due to the massively parallel structure of the CM, algorithms with a serial or global structure|such as LU factorization|tend to yield poor performance (global here meaning that elements do not only interact with elements in their neighborhood). The purpose of this note is partly to show what performance one can typically obtain when implementing a global algorithm on the CM-200 in a relatively limited period of time, and partly to examine the pros and contras for using a block algorithm. The testing has been performed on the LU factorization in normal as well as in a blocked version. The implementation has been performed by the use of BLAS level 3 equivalent routines and the results have been compared to the LU factorization present in the CMSSL library. The implementation and optimization of both the normal and blocked version have altogether been carried out within 14 days. The obtained performance is very disappointing: only 4% of the CMSSL routine for large matrices. The implementation of the LU factorization is done by walking down the diagonal and subtracting outer products along the way. The outer products are calculated by means of spreads in the unoptimized version and by the use of a mask and scans in the optimized version. The bottleneck of the operation is undoubtedly the spread/scan operations since these demand a tremendous amount of communication and are thus very time consuming even though the communication is along a single axis. Pivoting and equilibration are not implemented. Since the functions need global communication the best results have been obtained by choosing a :NEWS layout. The solver is implemented in a way similar to the factorization. The backward as well as the forward substitution is performed be means of spreads (unoptimized) and scans (optimized). In the optimized version care is taken not to create unnecessary temporaries. Random test systems lead to the performance shown in Fig. 1. The timings shown are elapsed times computed on a 8K CM-200 using double precision. The timings show a complexity|for large N |of N 2:6. For the factorization the optimized version is a little faster than the unoptimized (typically a factor of two)