Performance of Xxxxxxxx’s Algorithm Sample Clauses

Performance of Xxxxxxxx’s Algorithm. At the “bottom level” of the recursive algorithm we used the point LU fac- torization algorithm from LAPACK [1]. The matrix multiplications were carried out using an implementation of Strassen’s matrix-matrix multiplica- tion routine, using a CMSSL routine (Version 3.0) for matrix-matrix multi- plication at the bottom level; see [7] for details. No stabilization was added in these timing-tests. Both A and b consist of elements from a rectangular distribution in the interval [−2, 2]. Computation of the approximate inverse is carried out in single precision. On the CM–200, this is advantageous because single precision computations are twice as fast as double precision computations. On the CM–5, they are performed with the same speed, so on this machine there is no reason to switch to single precision. The remaining tasks are performed in double pre- cision: x˜ = A˜−1 b is computed, and then x˜ is refined using standard iterative refinement. No extended precision is used in computing the residuals. Test were carried out with various levels of recursion, and the results are shown as the crosses in Figure 5.1. The number p at each cross is the recursion level, meaning that the smallest block size was n/2p × n/2p. The execution times are measured in seconds, and they include the time for a fixed number of 5 steps of iterative improvement. We also show the performance of the CMSSL LU factorization routine, including forward and back substitutions. For comparison, Figure 5.1 also shows the execution time of the point algorithm from LAPACK (that was used at the bottom level of Strassen’s algorithm) when applied to the full matrix. We make the following remarks about the results: • Xxxxxxxx’s method is able to speed up significantly the performance of the underlying point LU factorization algorithm. • The performance of Xxxxxxxx’s method depends greatly on the number of recursion levels. • The optimal number of recursions depends on the matrix order n. The dependence of the optimal recursion level on the matrix size is not surprising, as the overhead increases for each recursion level. The optimal number of recursion levels is very difficult to predict a priori, since per- formance of the algorithm depends on several factors, such as the number of matrix-matrix multiplications and additions, and the nearest-neighbor communication. Regarding the first issue, we note that Xxxxxxxx’s matrix-matrix multi- plication algorithm seems to perform better on the MasPar MP–1, w...
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