Matrix Transpose and Block Case

Tuning Strategy

1. Transpose matrix B to ensure cache line aligned addresses, and iterate blocks in small areas to reduce cache misses.
2. Determine the block size based on the actual cache size and environment configuration.

Figure 1 Matrix transpose

Figure 2 Matrix transpose and block code

Procedure

1. Run the block_transpose_B_matmult case whose matrix determinant is 8192.

   1	./matmul 8192 4

   Command output:

   1 2 3

   Size is 8192, Matrix multiplication method is: 4, Check correctness is: 0 Initialization time = 2.787273s Matrix multiplication time = 3.711554s

   When the matrix determinant is 8192, the parallel computing takes approximately 3.7s.

2. Create a roofline task for the block_transpose_B_matmult case whose matrix determinant is 8192.

   1	devkit tuner roofline -o block_transpose_B_matmult_8192 -m region ./matmul 8192 4

   Command output:

   1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

   Note: 1. Roofline task is currently only supported on the 920 platform. 2. The application must be a binary file in ELF format, and read permissions are required to detect the format of the application. 3. Roofline task collection needs to ensure the application has finished running. 4. The estimated time of roofline collection is about 3 * application estimated time. 5. Roofline analysis is available only on physical machines. 6. You can learn about the roofline profiling method by looking at document /usr/local/devkit/tuner/docs/ROOFLINE_KNOW_HOW.MD RFCOLLECT: Start collection for ./matmul RFCOLLECT: Launch application to collect performance metrics of ./matmul Size is 8192, Matrix multiplication method is: 4, Check correctness is: 0 Initialization time = 2.794598s ROOFLINE_EVENTS are initialized. Matrix multiplication time = 3.743286s RFCOLLECT: Launch application to do binary instrumentation of ./matmul Size is 8192, Matrix multiplication method is: 4, Check correctness is: 0 Initialization time = 8.353251s Matrix multiplication time = 3.849523s RFCOLLECT: Launch benchmarks for measuring roofs RFCOLLECT: Processing all collected data RFCOLLECT: Result is captured at /matrix_multiplication/rfcollect-20240506-195201.json RFCOLLECT: Run "rfreport /matrix_multiplication/rfcollect-20240506-195201.json" to get report. Get roofline report ... The roofline json report: /matrix_multiplication/block_transpose_B_matmult_8192.json The roofline html report: /matrix_multiplication/block_transpose_B_matmult_8192.html

3. View the block_transpose_B_matmult_8192.html report.
   Figure 3 block_transpose_B_matmult_8192.html
   

   In this case, Parallel Threads of roofs is 128, Elapsed Time is 3.646 seconds, GFLOP Count is 1168.231, and Performance is 320.399 GFLOPS.

Tuning Result

The internal block loop has extra computation amount of 6.25% and the GFLOP count increases from 1099.512 to 1168.231. However, the overall performance increases by 191.9% and the end-to-end performance is greatly improved. See the following table.

1. The L1, L2, L3, and DDR points are located in the Compute and Memory Bound area in Figure 3.
2. The memory bottlenecks lie in L1 and L2, mainly in L2.
3. The cache line utilization is improved over Matrix Transpose Case. The computation density (FLOP/BYTE) sequence is: L1 ≈ L2 < L3 < DDR.