Update README.md

Author: yangyang14641

README.md

@@ -127,8 +127,8 @@ for (i = 0; i < n; i++)
    * | Stage | Algorithm Operation |Computing and  Communication Analysis|
      |-------|---------------------|---------------------|
      |   0   | 1. Process <img src="https://tex.s2cms.ru/svg/P_%7Bij%7D" alt="P_{ij}">  has <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BA_%7Bij%7D%7D" alt="\mathbf{A_{ij}}">, <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BB_%7Bij%7D%7D" alt="\mathbf{B_{ij}}">  but needs <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BA_%7Bii%7D%7D" alt="\mathbf{A_{ii}}"> (for each index <img src="https://tex.s2cms.ru/svg/i" alt="i">) <br/> 2. Process <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BP_%7Bii%7D%7D" alt=\mathbf{P_{ii}}> broadcast <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BA_%7Bii%7D%7D" alt="\mathbf{A_{ii}}"> across process mesh row <img src="https://tex.s2cms.ru/svg/i" alt="i"> <br/> 3. Process <img src="https://tex.s2cms.ru/svg/P_%7Bij%7D" alt="P_{ij}"> computes <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BC_%7Bij%7D%3DA_%7Bii%7DB_%7Bij%7D%7D" alt="\mathbf{C_{ij}=A_{ii}B_{ij}}"> | Communication in Broadcast Operation: <br/> <img src="https://tex.s2cms.ru/svg/%5Cleft(%20q-1%20%5Cright)%20%20%5Ctimes%20q%20%5Ctimes%20%5Cfrac%7BM%7D%7Bq%7D%20%5Ctimes%20%5Cfrac%7BK%7D%7Bq%7D" alt="\left( q-1 \right)  \times q \times \frac{M}{q} \times \frac{K}{q}"> <br/> Computing in each process: <br/> <img src="https://tex.s2cms.ru/svg/%5Cfrac%7BM%7D%7Bq%7D%20%5Ctimes%20%5Cfrac%7BK%7D%7Bq%7D%20%5Ctimes%20%5Cfrac%7BN%7D%7Bq%7D" alt="\frac{M}{q} \times \frac{K}{q} \times \frac{N}{q}"> <br/> Computing in total: <br/> <img src="https://tex.s2cms.ru/svg/%5Cleft(%20%5Cfrac%7BM%7D%7Bq%7D%20%5Ctimes%20%5Cfrac%7BK%7D%7Bq%7D%20%5Ctimes%20%5Cfrac%7BN%7D%7Bq%7D%20%5Cright)%20q%20%5Ctimes%20q" alt="\left( \frac{M}{q} \times \frac{K}{q} \times \frac{N}{q} \right) q \times q"> |
-     |   1   | 1. <img src="https://tex.s2cms.ru/svg/P_%7Bij%7D" alt="P_{ij}"> has <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BA_%7Bij%7D%7D" alt="\mathbf{A_{ij}}"> and <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BB_%7Bij%7D%7D" alt="\mathbf{B_{ij}}"> but needs <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BA_%7Bii%2B1%7D%7D" alt="\mathbf{A_{ii+1}}"> and <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BB_%7Bi%2B1j%7D%7D" alt="\mathbf{B_{i+1j}}"> <br/> 1.1 Shift the <img src="https://tex.s2cms.ru/svg/j-th" alt="j-th"> block column of <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BB_%7Bij%7D%7D" alt="\mathbf{B_{ij}}"> by one block up (block <img src="https://tex.s2cms.ru/svg/0" alt="0"> goes to block <img src="https://tex.s2cms.ru/svg/q-1" alt="q-1">) (period) <br/> 1.2 <img src="https://tex.s2cms.ru/svg/P_%7Bii%2B1%7D" alt="P_{ii+1}"> broadcast <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BA_%7Bii%2B1%7D%7D" alt="\mathbf{A_{ii+1}}">  across process mesh row <img src="https://tex.s2cms.ru/svg/i" alt="i"> <br/> 2. Process <img src="https://tex.s2cms.ru/svg/P_%7Bij%7D" alt="P_{ij}"> Compute <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BC_%7Bij%7D%3DC_%7Bij%7D%2BA_%7Bii%2B1%7DB_%7Bi%2B1j%7D%7D" alt="\mathbf{C_{ij}=C_{ij}+A_{ii+1}B_{i+1j}}"> | Communication in shift operation: <br/> <img src="https://tex.s2cms.ru/svg/%5Cleft(%20q%20%5Ctimes%20q%20%5Cright)%20%5Ctimes%20%5Cleft(%20%5Cfrac%7BK%7D%7Bq%7D%20%5Ctimes%20%5Cfrac%7BN%7D%7Bq%7D%20%5Cright)" alt="\left( q \times q \right) \times \left( \frac{K}{q} \times \frac{N}{q} \right)"> <br/> Communication in broadcast operation: <br/> <img src="https://tex.s2cms.ru/svg/%5Cleft%5B%5Cleft(%20q-1%20%5Cright)%20%5Ctimes%20q%5Cright%5D%20%5Ctimes%20%5Cleft(%20%5Cfrac%7BM%7D%7Bq%7D%20%5Ctimes%20%5Cfrac%7BK%7D%7Bq%7D%5Cright)" alt="\left[\left( q-1 \right) \times q\right] \times \left( \frac{M}{q} \times \frac{K}{q}\right)"> <br/> Communication in total: <br/> <img src="https://tex.s2cms.ru/svg/%5Cleft(%20q%20%5Ctimes%20q%20%5Cright)%20%5Ctimes%20%5Cleft(%20%5Cfrac%7BK%7D%7Bq%7D%20%5Ctimes%20%5Cfrac%7BN%7D%7Bq%7D%5Cright)%20%2B%20%5Cleft%5B%20%5Cleft(%20q-1%20%5Cright)%5Ctimes%20q%20%5Cright%5D%20%5Ctimes%20%5Cright(%20%5Cfrac%7BM%7D%7Bq%7D%20%5Ctimes%20%5Cfrac%7BK%7D%7Bq%7D%5Cleft)" alt="\left( q \times q \right) \times \left( \frac{K}{q} \times \frac{N}{q}\right) + \left[ \left( q-1 \right)\times q \right] \times \right( \frac{M}{q} \times \frac{K}{q}\left)"> <br/> Computing in each process: <br/> <img src="https://tex.s2cms.ru/svg/%5Cfrac%7BM%7D%7Bq%7D%20%5Ctimes%20%5Cfrac%7BK%7D%7Bq%7D%20%5Ctimes%20%5Cfrac%7BN%7D%7Bq%7D" alt="\frac{M}{q} \times \frac{K}{q} \times \frac{N}{q}"> <br/> Computing in total: <br/> <img src="https://tex.s2cms.ru/svg/%5Cleft(%20%5Cfrac%7BM%7D%7Bq%7D%20%5Ctimes%20%5Cfrac%7BK%7D%7Bq%7D%20%5Ctimes%20%5Cfrac%7BN%7D%7Bq%7D%20%5Cright)%20%5Ctimes%20q%20%5Ctimes%20q" alt="\left( \frac{M}{q} \times \frac{K}{q} \times \frac{N}{q} \right) \times q \times q"> |
-     |   2   | 1. <img src="https://tex.s2cms.ru/svg/P_%7Bij%7D" alt="P_{ij}"> has <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BA_%7Bij%7D%7D" alt="\mathbf{A_{ij}}">  and <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BB_%7Bij%7D%7D" alt="\mathbf{B_{ij}}">  but needs <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BA_%7Bii%2B2%7D%7D" alt="\mathbf{A_{ii+2}}">  and <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BB_%7Bi%2B2j%7D%7D" alt="\mathbf{B_{i+2j}}"> <br/> 1.1 Shift the <img src="https://tex.s2cms.ru/svg/j-th" alt="j-th"> block column of <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BB_%7Bij%7D%7D" alt="\mathbf{B_{ij}}"> by one block up (block <img src="https://tex.s2cms.ru/svg/0" alt="0"> goes to block <img src="https://tex.s2cms.ru/svg/q-1" alt="q−1">) (period) <br/> 1.2 <img src="https://tex.s2cms.ru/svg/P_%7Bii%2B2%7D" alt="P_{ii+2}"> broadcast <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BA_%7Bii%2B2%7D%7D" alt="\mathbf{A_{ii+2}}">  across process mesh row <img src="https://tex.s2cms.ru/svg/i" alt="i"> <br/> 2. Process <img src="https://tex.s2cms.ru/svg/P_%7Bij%7D" alt="P_{ij}">  Compute <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BC_%7Bij%7D%3DC_%7Bij%7D%2BA_%7Bii%2B2%7DB_%7Bi%2B2j%7D%7D" alt="\mathbf{C_{ij}=C_{ij}+A_{ii+2}B_{i+2j}}"> | Communication in shift operation: <br/> <img src="https:">(q × q)×(K/q×N/q) <br/> Communication in broadcast operation: <br/> <img src="https:">[(q−1)× q]×(M/q×K/q) <br/> Communication in total: <br/> <img src="https:">[(q−1)× q]×(M/q×K/q) <br/> Computing in each process: <br/> <img src="https:">M/q×K/q×N/q <br/> Computing in total:<br/><img src="https:">(M/q×K/q×N/q)× q × q |
+     |   1   | 1. <img src="https://tex.s2cms.ru/svg/P_%7Bij%7D" alt="P_{ij}"> has <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BA_%7Bij%7D%7D" alt="\mathbf{A_{ij}}"> and <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BB_%7Bij%7D%7D" alt="\mathbf{B_{ij}}"> but needs <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BA_%7Bii%2B1%7D%7D" alt="\mathbf{A_{ii+1}}"> and <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BB_%7Bi%2B1j%7D%7D" alt="\mathbf{B_{i+1j}}"> <br/> 1.1 Shift the <img src="https://tex.s2cms.ru/svg/j-th" alt="j-th"> block column of <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BB_%7Bij%7D%7D" alt="\mathbf{B_{ij}}"> by one block up (block <img src="https://tex.s2cms.ru/svg/0" alt="0"> goes to block <img src="https://tex.s2cms.ru/svg/q-1" alt="q-1">) (period) <br/> 1.2 <img src="https://tex.s2cms.ru/svg/P_%7Bii%2B1%7D" alt="P_{ii+1}"> broadcast <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BA_%7Bii%2B1%7D%7D" alt="\mathbf{A_{ii+1}}">  across process mesh row <img src="https://tex.s2cms.ru/svg/i" alt="i"> <br/> 2. Process <img src="https://tex.s2cms.ru/svg/P_%7Bij%7D" alt="P_{ij}"> Compute <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BC_%7Bij%7D%3DC_%7Bij%7D%2BA_%7Bii%2B1%7DB_%7Bi%2B1j%7D%7D" alt="\mathbf{C_{ij}=C_{ij}+A_{ii+1}B_{i+1j}}"> | Communication in shift operation: <br/> <img src="https://tex.s2cms.ru/svg/%5Cleft(%20q%20%5Ctimes%20q%20%5Cright)%20%5Ctimes%20%5Cleft(%20%5Cfrac%7BK%7D%7Bq%7D%20%5Ctimes%20%5Cfrac%7BN%7D%7Bq%7D%20%5Cright)" alt="\left( q \times q \right) \times \left( \frac{K}{q} \times \frac{N}{q} \right)"> <br/> Communication in broadcast operation: <br/> <img src="https://tex.s2cms.ru/svg/%5Cleft%5B%5Cleft(%20q-1%20%5Cright)%20%5Ctimes%20q%5Cright%5D%20%5Ctimes%20%5Cleft(%20%5Cfrac%7BM%7D%7Bq%7D%20%5Ctimes%20%5Cfrac%7BK%7D%7Bq%7D%5Cright)" alt="\left[\left( q-1 \right) \times q\right] \times \left( \frac{M}{q} \times \frac{K}{q}\right)"> <br/> Communication in total: <br/> <img src="https://tex.s2cms.ru/svg/%5Cleft(%20q%20%5Ctimes%20q%20%5Cright)%20%5Ctimes%20%5Cleft(%20%5Cfrac%7BK%7D%7Bq%7D%20%5Ctimes%20%5Cfrac%7BN%7D%7Bq%7D%5Cright)%20%2B%20%5Cleft%5B%20%5Cleft(%20q-1%20%5Cright)%5Ctimes%20q%20%5Cright%5D%20%5Ctimes%20%5Cleft(%20%5Cfrac%7BM%7D%7Bq%7D%20%5Ctimes%20%5Cfrac%7BK%7D%7Bq%7D%5Cright)" alt="\left( q \times q \right) \times \left( \frac{K}{q} \times \frac{N}{q}\right) + \left[ \left( q-1 \right)\times q \right] \times \left( \frac{M}{q} \times \frac{K}{q}\right)"> <br/> Computing in each process: <br/> <img src="https://tex.s2cms.ru/svg/%5Cfrac%7BM%7D%7Bq%7D%20%5Ctimes%20%5Cfrac%7BK%7D%7Bq%7D%20%5Ctimes%20%5Cfrac%7BN%7D%7Bq%7D" alt="\frac{M}{q} \times \frac{K}{q} \times \frac{N}{q}"> <br/> Computing in total: <br/> <img src="https://tex.s2cms.ru/svg/%5Cleft(%20%5Cfrac%7BM%7D%7Bq%7D%20%5Ctimes%20%5Cfrac%7BK%7D%7Bq%7D%20%5Ctimes%20%5Cfrac%7BN%7D%7Bq%7D%20%5Cright)%20%5Ctimes%20q%20%5Ctimes%20q" alt="\left( \frac{M}{q} \times \frac{K}{q} \times \frac{N}{q} \right) \times q \times q"> |
+     |   2   | 1. <img src="https://tex.s2cms.ru/svg/P_%7Bij%7D" alt="P_{ij}"> has <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BA_%7Bij%7D%7D" alt="\mathbf{A_{ij}}">  and <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BB_%7Bij%7D%7D" alt="\mathbf{B_{ij}}">  but needs <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BA_%7Bii%2B2%7D%7D" alt="\mathbf{A_{ii+2}}">  and <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BB_%7Bi%2B2j%7D%7D" alt="\mathbf{B_{i+2j}}"> <br/> 1.1 Shift the <img src="https://tex.s2cms.ru/svg/j-th" alt="j-th"> block column of <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BB_%7Bij%7D%7D" alt="\mathbf{B_{ij}}"> by one block up (block <img src="https://tex.s2cms.ru/svg/0" alt="0"> goes to block <img src="https://tex.s2cms.ru/svg/q-1" alt="q−1">) (period) <br/> 1.2 <img src="https://tex.s2cms.ru/svg/P_%7Bii%2B2%7D" alt="P_{ii+2}"> broadcast <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BA_%7Bii%2B2%7D%7D" alt="\mathbf{A_{ii+2}}">  across process mesh row <img src="https://tex.s2cms.ru/svg/i" alt="i"> <br/> 2. Process <img src="https://tex.s2cms.ru/svg/P_%7Bij%7D" alt="P_{ij}">  Compute <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BC_%7Bij%7D%3DC_%7Bij%7D%2BA_%7Bii%2B2%7DB_%7Bi%2B2j%7D%7D" alt="\mathbf{C_{ij}=C_{ij}+A_{ii+2}B_{i+2j}}"> | Communication in shift operation: <br/> <img src="https:" alt="\left( q \times q \right) \times \left( \frac{K}{q} \times \frac{N}{q} \right)"> <br/> Communication in broadcast operation: <br/> <img src="https://tex.s2cms.ru/svg/%5Cleft%5B%5Cleft(%20q-1%20%5Cright)%20%5Ctimes%20q%20%5Cright%5D%20%5Ctimes%20%5Cleft(%20%5Cfrac%7BM%7D%7Bq%7D%20%5Ctimes%20%5Cfrac%7BK%7D%7Bq%7D%20%5Cright)" alt="\left[\left( q-1 \right) \times q \right] \times \left( \frac{M}{q} \times \frac{K}{q} \right)"> <br/> Communication in total: <br/> <img src="https://tex.s2cms.ru/svg/%5Cleft%5B%5Cleft(%20q-1%20%5Cright)%20%5Ctimes%20q%20%5Cright%5D%20%5Ctimes%20%5Cleft(%20%5Cfrac%7BM%7D%7Bq%7D%20%5Ctimes%20%5Cfrac%7BK%7D%7Bq%7D%5Cright)" alt="\left[\left( q-1 \right) \times q \right] \times \left( \frac{M}{q} \times \frac{K}{q}\right)"> <br/> Computing in each process: <br/> <img src="https://tex.s2cms.ru/svg/%5Cfrac%7BM%7D%7Bq%7D%20%5Ctimes%20%5Cfrac%7BK%7D%7Bq%7D%20%5Ctimes%20%5Cfrac%7BN%7D%7Bq%7D" alt="\frac{M}{q} \times \frac{K}{q} \times \frac{N}{q}"> <br/> Computing in total:<br/><img src="https://tex.s2cms.ru/svg/%5Cleft(%5Cfrac%7BM%7D%7Bq%7D%20%5Ctimes%20%5Cfrac%7BK%7D%7Bq%7D%20%5Ctimes%20%5Cfrac%7BN%7D%7Bq%7D%5Cright)%20%5Ctimes%20q%20%5Ctimes%20q" alt="\left(\frac{M}{q} \times \frac{K}{q} \times \frac{N}{q}\right) \times q \times q"> |
      |  ...  |         ...         |  |
      | q-2-i | <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BC_%7Bij%7D%3DC_%7Bij%7D%2BA_%7Biq-2%7DB_%7Bq-2j%7D%7D" alt="\mathbf{C_{ij}=C_{ij}+A_{iq-2}B_{q-2j}}"> |  |
      | q-1-i | <img src="https://tex.s2cms.ru/svg/%5Cmathbf%7BC_%7Bij%7D%3DC_%7Bij%7D%2BA_%7Biq-1%7DB_%7Bq-1j%7D%7D" alt="\mathbf{C_{ij}=C_{ij}+A_{iq-1}B_{q-1j}}"> |  |