The PDSYRK routine performs rank- updates on an symmetric matrix with an column of blocks . After broadcasting rowwise and transposing it, each process updates its local portion of with its own copy of and . The update is complicated by the fact that the globally lower triangular matrix is not necessarily stored in the lower triangular form in the local processes. For details see . The simplest way to do this is to repeatedly update one column of blocks of . However, if the block size is small, this updating process will not be efficient. It is more efficient to update several blocks of columns at a time. The PBLAS routine, PDSYRK efficiently updates by combining several blocks of columns at a time. For details, see .
Figure 10: Performance of the Cholesky factorization on the Intel iPSC/860, Delta, and Paragon.
The effect of the block size on the performance of the Cholesky factorization is shown in Figure 9 on and processors of the Intel Delta. The best performance was obtained at the block size of , but relatively good performance could be expected with the block size of , since the routine updates multiple column panels at a time.
Figure 10 shows the performance of the Cholesky factorization routine. The best performance was attained with the aspect ratio of . The routine ran at 1.8 Gflops for on the iPSC/860; 10.5 Gflops for on the Delta; and 16.9 Gflops for on the Paragon. Since it requires fewer floating point operations () than the other factorizations, it is not surprising that its flop rate is relatively poor.
If is not positive definite, the Cholesky factorization should be terminated in the middle of the computation. As outlined in Section 3.3, a process computes the Cholesky factor from . After computing , process broadcasts a flag to all other processes to stop the computation if is not positive definite. If is guaranteed to be positive definite, the process of broadcasting the flag can be skipped, leading to a corresponding increase in performance.