10 times faster on a simple plain matrix multiplication. Amazing. With optimizing only memory access. I did not realise, before this series of articles, that memory can take a such place in optimisation. Thanks a lot, it's so interesting and well written.
Memory part 5: What programmers can do
Posted Oct 27, 2007 14:15 UTC (Sat) by bartoldeman (guest, #4205) [Link]
It is even 20 times faster if you use ATLAS 3.8: http://math-atlas.sourceforge.net/ Its DGEMM routine does the job in around 708,000,000 cycles, another factor of 2 faster (my other numbers, also on a Core 2 (a Duo, but single threaded), were very similar to Ulrich's so I can state this with some confidence). Of course there's been a lot of research and tweaking to obtain this score. ATLAS' SUMMARY.LOG reports for DGEMM: Performance: 4846.05MFLOPS (302.88 percent of of detected clock rate) and this is on a 1.6GHz Core 2 Duo, not 2.66GHz! GOTOBLAS may also be worth looking at, for comparison. It looks more into TLB misses than ATLAS does.
matrix multiply optimization
Posted Oct 27, 2007 21:44 UTC (Sat) by giraffedata (subscriber, #1954) [Link]
Only about a factor of 5 is due to memory optimization. The vector instruction optimization is simply applying more CPU horsepower.What I can't figure out is how the transposition speeds anything up. The article points out that it removes 1000 non-sequential accesses per column from the multiplication loop, but I see that same 1000 non-sequential accesses per column added to the transposition loop.
matrix multiply optimization
Posted Oct 27, 2007 21:55 UTC (Sat) by bartoldeman (guest, #4205) [Link]
Because transposition uses O(N^2) accesses and multiplication O(N^3). The accesses in the transposition are more expensive but there are N times fewer than in the multiplication...
matrix multiply optimization
Posted Oct 27, 2007 22:41 UTC (Sat) by giraffedata (subscriber, #1954) [Link]
Aha. Perfectly clear now. The article neglects to explain this; I'd probably say, "the original traverses mul2 in this expensive nonsequential way 1000 times, whereas the improved version does it only once."
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