Table 9.4: Order 13040 Band Matrix Performance
We consider an order 13040 banded matrix with a bandwidth of 326 under partial row pivoting. For this example, we have compiled timing results for a process grid with random matrices (entries have range 0-10,000), using different values of S on the column distribution (Table 9.4). We indicate timing for A-mode, B-mode, Backsolves and Forward- and Backsolves together (``Solve'' heading). For this example, S=30 saves of the triangular solve cost compared to S=1, or approximately 186 seconds, roughly 6 seconds above the linear optimal. Simultaneously, we incur about 17 seconds additional cost in B-mode, while saving about 93 seconds in the Backsolve. Assuming , in the first above-mentioned objective function, we save about 262 (respectively, 76) seconds. Based on this example, and other experiences, we conclude that this is a successful practical technique for improving overall sparse linear algebra performance. The following example further bolsters this conclusion.