## 5 Results and Analysis
### WPK1 and WPK2
#### 5.3.2 Aggregate Throughput and Speedup  
Table 4 provides for all eight schemes the statistical information such as _mean_, standard deviation (_sd_), variance (_var_), maximum (_max_), and minimum (_min_) of the GFLOP/s metric of the SpMV kernel for the 26 benchmark matrices. The values are listed in descending order of the _mean_ value. We observe that SURAA provides the highest _mean_ GFLOP/s of 15.75. The next best scheme in terms of the _mean_ throughput is CSR5, it has the second highest _mean_ GFLOP/s, 14.28, followed by 13.65 for WPK2. In terms of the _max_ throughout, SURAA again leads the table with the highest value of 36.7077 GFLOP/s, followed by SELL-P with the second best value of 26.8 GFLOP/s, followed by CSR5. Based on the _mean_ values, WPK2 comes third after SURAA and CSR5. As noted earlier, WPK1 and WPK2 were unable to execute some of the matrices in the benchmark dataset and their results have not been included in computing _mean_ and other qualities in the table.  
In order to calculate the aggregate performance of SURAA, we calculate aggregate speedup as in Equation (5).  
\[aggregate\_speedup=\frac{\sum_{i=1}^{26}\text{SURAA}_{i}}{\sum_{i=1}^{26}reference \_scheme_{i}}. \tag{5}\]  
Figure 8 indicates the aggregate GFLOP/s achieved for SURAA and other tools. These GFLOP/s are aggregated over all the matrices for each scheme and do not imply to be the GFLOP/s for a single GPU at one time. The intention of providing the aggregate throughput is to highlight the aggregate performance gain or loss that could be achieved by the use of a single SpMV computation scheme  
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Scheme** & **Mean** & **sd** & **var** & **max** & **min** \\ \hline SURAA & 15.757504 & 6.345914889 & 40.27063578 & 36.70778163 & 5.650293069 \\ CSR5 & 14.28352308 & 4.213901396 & 17.75696497 & 23.0126 & 7.17599 \\ WPK2 & 13.65709 & 4.444209 & 19.75099 & 22.78246 & 5.016615 \\ HYB & 10.51479136 & 2.494732444 & 6.223689965 & 17.53628655 & 7.179585746 \\ CSR & 9.82140064 & 5.373886198 & 28.9341673 & 22.99131028 & 0.083364399 \\ WPK1 & 8.717564 & 4.462194 & 19.02874 & 17.58582 & 0.02361821 \\ SELL-P & 7.867543077 & 6.535661011 & 42.71486485 & 26.8 & 0.0223 \\ BCSR & 2.721219 & 3.399639 & 11.55754 & 12.95043 & 0.313091 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Statistical information on the throughput in Giga Floating Point Operations per second (GFLOP/s) metric of the SpMV kernel for the 26 benchmark matrices.  
for matrices from various domains. SURAA has the highest throughput compared to all the other seven techniques, i.e., CSR, HYB, BSR, CSR5, WPK1, WPK2, and SELL-P. For our benchmark suite of 26 matrices, we achieve an aggregate throughput of 409.7 GFLOP/s compared to the next best CSR5 (371.37 GFLOP/s), HYB (273.38 GFLOP/s), CSR (255.42 GFLOP/s), WPK2 (218.51 GFLOP/s), SELL-P (204.56 GFLOP/s), WPK1 (165.163 GFLOP/s), and BSR (70.751 GFLOP/s). The reason for low aggregate throughput for WPK1 anf WPK2 is that, as noted earlier, they were unable to execute some of the matrices in the benchmark dataset. It means that SURAA has an aggregate speedup of 1.1\(\times\) compared to CSR5, 1.49\(\times\) compared to HYB, 1.60\(\times\) compared to CSR, 1.87\(\times\) compared with WPK2, 2.0\(\times\) compared to SELL-P, 2.47\(\times\) compared to WPK1, and 5.79\(\times\) compared to BSR for the given 26 matrices.