Let's multiply two 4x4 matrices, A and B, to get a resulting matrix C.
Matrix A:
     [ 1  2  3  4 ]
[ 5  6  7  8 ]
[ 9 10 11 12 ]
[13 14 15 16 ]
   
content_copy Use code with caution.
Matrix B:
     [17 18 19 20]
[21 22 23 24]
[25 26 27 28]
[29 30 31 32]
   
content_copy Use code with caution.
Calculating Matrix C (A * B):
Remember, the element in the ith row and jth column of C is calculated by taking the dot product of the ith row of A and the jth column of B.
C<sub>1,1</sub>: (117) + (221) + (325) + (429) = 17 + 42 + 75 + 116 = 250
C<sub>1,2</sub>: (118) + (222) + (326) + (430) = 18 + 44 + 78 + 120 = 260
C<sub>1,3</sub>: (119) + (223) + (327) + (431) = 19 + 46 + 81 + 124 = 270
C<sub>1,4</sub>: (120) + (224) + (328) + (432) = 20 + 48 + 84 + 128 = 280
C<sub>2,1</sub>: (517) + (621) + (725) + (829) = 85 + 126 + 175 + 232 = 618
C<sub>2,2</sub>: (518) + (622) + (726) + (830) = 90 + 132 + 182 + 240 = 644
C<sub>2,3</sub>: (519) + (623) + (727) + (831) = 95 + 138 + 189 + 248 = 670
C<sub>2,4</sub>: (520) + (624) + (728) + (832) = 100 + 144 + 196 + 256 = 696
C<sub>3,1</sub>: (917) + (1021) + (1125) + (1229) = 153 + 210 + 275 + 348 = 986
C<sub>3,2</sub>: (918) + (1022) + (1126) + (1230) = 162 + 220 + 286 + 360 = 1028
C<sub>3,3</sub>: (919) + (1023) + (1127) + (1231) = 171 + 230 + 297 + 372 = 1070
C<sub>3,4</sub>: (920) + (1024) + (1128) + (1232) = 180 + 240 + 308 + 384 = 1112
C<sub>4,1</sub>: (1317) + (1421) + (1525) + (1629) = 221 + 294 + 375 + 464 = 1354
C<sub>4,2</sub>: (1318) + (1422) + (1526) + (1630) = 234 + 308 + 390 + 480 = 1412
C<sub>4,3</sub>: (1319) + (1423) + (1527) + (1631) = 247 + 322 + 405 + 496 = 1470
C<sub>4,4</sub>: (1320) + (1424) + (1528) + (1632) = 260 + 336 + 420 + 512 = 1528
Resulting Matrix C:
     [ 250  260  270  280 ]
[ 618  644  670  696 ]
[ 986 1028 1070 1112 ]
[1354 1412 1470 1528 ]
   
content_copy Use code with caution.


Q- oky now we need to parelize the execution of this workflow ? 


First i think we need to divide the matrix into 4 block of size 2x2 
Second the apply the multiply logic for the 4 block of matrix 
Then merge them 

Q- give your feedback and give the manual execution of this example 

Your approach to parallelizing the matrix multiplication by dividing each 4x4 matrix into four 2x2 blocks is both logical and effective. This method leverages the principles of divide and conquer, allowing multiple computations to occur simultaneously, which can significantly reduce the overall computation time, especially for larger matrices.
Feedback on Your Approach
Division into Blocks:
Pros: Simplifies the multiplication process by handling smaller sub-matrices, making it easier to manage and parallelize.
Cons: For very large matrices, the overhead of managing numerous blocks might become significant, but for a 4x4 matrix, this is negligible.
Parallel Multiplication of Blocks:
Pros: Each block multiplication (e.g., computing C11, C12, C21, C22) can be performed independently, allowing true parallel execution.
Cons: Requires synchronization when merging the results, but this is straightforward with modern parallel computing frameworks.
Merging Blocks:
Pros: Combining the results is a simple concatenation of the computed blocks.
Cons: Minimal, as merging 2x2 blocks into a 4x4 matrix is straightforward.
Overall, your strategy is sound and well-suited for parallel execution. Now, let's walk through a manual execution of this parallel block matrix multiplication.

Manual Execution of Parallel Block Matrix Multiplication
Step 1: Divide Matrices A and B into 2x2 Blocks
Matrix A:
[
A = \begin{bmatrix}
A_{11} & A_{12} \
A_{21} & A_{22}
\end{bmatrix}
= \begin{bmatrix}
\begin{bmatrix} 1 & 2 \ 5 & 6 \end{bmatrix} &
\begin{bmatrix} 3 & 4 \ 7 & 8 \end{bmatrix} \
\begin{bmatrix} 9 & 10 \ 13 & 14 \end{bmatrix} &
\begin{bmatrix} 11 & 12 \ 15 & 16 \end{bmatrix}
\end{bmatrix}
]
Matrix B:
[
B = \begin{bmatrix}
B_{11} & B_{12} \
B_{21} & B_{22}
\end{bmatrix}
= \begin{bmatrix}
\begin{bmatrix} 17 & 18 \ 21 & 22 \end{bmatrix} &
\begin{bmatrix} 19 & 20 \ 23 & 24 \end{bmatrix} \
\begin{bmatrix} 25 & 26 \ 29 & 30 \end{bmatrix} &
\begin{bmatrix} 27 & 28 \ 31 & 32 \end{bmatrix}
\end{bmatrix}
]
Step 2: Compute Each Block of Matrix C in Parallel
Using the block multiplication formula:
[
C_{ij} = A_{i1} \times B_{1j} + A_{i2} \times B_{2j}
]
where ( i, j \in {1, 2} ).
Compute ( C_{11} ):
[
C_{11} = A_{11} \times B_{11} + A_{12} \times B_{21}
]
Calculate ( A_{11} \times B_{11} ):
[
\begin{bmatrix} 1 & 2 \ 5 & 6 \end{bmatrix} \times \begin{bmatrix} 17 & 18 \ 21 & 22 \end{bmatrix} =
\begin{bmatrix}
(1 \times 17) + (2 \times 21) & (1 \times 18) + (2 \times 22) \
(5 \times 17) + (6 \times 21) & (5 \times 18) + (6 \times 22)
\end{bmatrix}
\begin{bmatrix}
17 + 42 & 18 + 44 \
85 + 126 & 90 + 132
\end{bmatrix}
\begin{bmatrix}
59 & 62 \
211 & 222
\end{bmatrix}
]
Calculate ( A_{12} \times B_{21} ):
[
\begin{bmatrix} 3 & 4 \ 7 & 8 \end{bmatrix} \times \begin{bmatrix} 25 & 26 \ 29 & 30 \end{bmatrix} =
\begin{bmatrix}
(3 \times 25) + (4 \times 29) & (3 \times 26) + (4 \times 30) \
(7 \times 25) + (8 \times 29) & (7 \times 26) + (8 \times 30)
\end{bmatrix}
\begin{bmatrix}
75 + 116 & 78 + 120 \
175 + 232 & 182 + 240
\end{bmatrix}
\begin{bmatrix}
191 & 198 \
407 & 422
\end{bmatrix}
]
Sum the Results:
[
C_{11} = \begin{bmatrix} 59 & 62 \ 211 & 222 \end{bmatrix} + \begin{bmatrix} 191 & 198 \ 407 & 422 \end{bmatrix} = \begin{bmatrix} 250 & 260 \ 618 & 644 \end{bmatrix}
]
Compute ( C_{12} ):
[
C_{12} = A_{11} \times B_{12} + A_{12} \times B_{22}
]
Calculate ( A_{11} \times B_{12} ):
[
\begin{bmatrix} 1 & 2 \ 5 & 6 \end{bmatrix} \times \begin{bmatrix} 19 & 20 \ 23 & 24 \end{bmatrix} =
\begin{bmatrix}
(1 \times 19) + (2 \times 23) & (1 \times 20) + (2 \times 24) \
(5 \times 19) + (6 \times 23) & (5 \times 20) + (6 \times 24)
\end{bmatrix}
\begin{bmatrix}
19 + 46 & 20 + 48 \
95 + 138 & 100 + 144
\end{bmatrix}
\begin{bmatrix}
65 & 68 \
233 & 244
\end{bmatrix}
]
Calculate ( A_{12} \times B_{22} ):
[
\begin{bmatrix} 3 & 4 \ 7 & 8 \end{bmatrix} \times \begin{bmatrix} 27 & 28 \ 31 & 32 \end{bmatrix} =
\begin{bmatrix}
(3 \times 27) + (4 \times 31) & (3 \times 28) + (4 \times 32) \
(7 \times 27) + (8 \times 31) & (7 \times 28) + (8 \times 32)
\end{bmatrix}
\begin{bmatrix}
81 + 124 & 84 + 128 \
189 + 248 & 196 + 256
\end{bmatrix}
\begin{bmatrix}
205 & 212 \
437 & 452
\end{bmatrix}
]
Sum the Results:
[
C_{12} = \begin{bmatrix} 65 & 68 \ 233 & 244 \end{bmatrix} + \begin{bmatrix} 205 & 212 \ 437 & 452 \end{bmatrix} = \begin{bmatrix} 270 & 280 \ 670 & 696 \end{bmatrix}
]
Compute ( C_{21} ):
[
C_{21} = A_{21} \times B_{11} + A_{22} \times B_{21}
]
Calculate ( A_{21} \times B_{11} ):
[
\begin{bmatrix} 9 & 10 \ 13 & 14 \end{bmatrix} \times \begin{bmatrix} 17 & 18 \ 21 & 22 \end{bmatrix} =
\begin{bmatrix}
(9 \times 17) + (10 \times 21) & (9 \times 18) + (10 \times 22) \
(13 \times 17) + (14 \times 21) & (13 \times 18) + (14 \times 22)
\end{bmatrix}
\begin{bmatrix}
153 + 210 & 162 + 220 \
221 + 294 & 234 + 308
\end{bmatrix}
\begin{bmatrix}
363 & 382 \
515 & 542
\end{bmatrix}
]
Calculate ( A_{22} \times B_{21} ):
[
\begin{bmatrix} 11 & 12 \ 15 & 16 \end{bmatrix} \times \begin{bmatrix} 25 & 26 \ 29 & 30 \end{bmatrix} =
\begin{bmatrix}
(11 \times 25) + (12 \times 29) & (11 \times 26) + (12 \times 30) \
(15 \times 25) + (16 \times 29) & (15 \times 26) + (16 \times 30)
\end{bmatrix}
\begin{bmatrix}
275 + 348 & 286 + 360 \
375 + 464 & 390 + 480
\end{bmatrix}
\begin{bmatrix}
623 & 646 \
839 & 870
\end{bmatrix}
]
Sum the Results:
[
C_{21} = \begin{bmatrix} 363 & 382 \ 515 & 542 \end{bmatrix} + \begin{bmatrix} 623 & 646 \ 839 & 870 \end{bmatrix} = \begin{bmatrix} 986 & 1028 \ 1354 & 1412 \end{bmatrix}
]
Compute ( C_{22} ):
[
C_{22} = A_{21} \times B_{12} + A_{22} \times B_{22}
]
Calculate ( A_{21} \times B_{12} ):
[
\begin{bmatrix} 9 & 10 \ 13 & 14 \end{bmatrix} \times \begin{bmatrix} 19 & 20 \ 23 & 24 \end{bmatrix} =
\begin{bmatrix}
(9 \times 19) + (10 \times 23) & (9 \times 20) + (10 \times 24) \
(13 \times 19) + (14 \times 23) & (13 \times 20) + (14 \times 24)
\end{bmatrix}
\begin{bmatrix}
171 + 230 & 180 + 240 \
247 + 322 & 260 + 336
\end{bmatrix}
\begin{bmatrix}
401 & 420 \
569 & 596
\end{bmatrix}
]
Calculate ( A_{22} \times B_{22} ):
[
\begin{bmatrix} 11 & 12 \ 15 & 16 \end{bmatrix} \times \begin{bmatrix} 27 & 28 \ 31 & 32 \end{bmatrix} =
\begin{bmatrix}
(11 \times 27) + (12 \times 31) & (11 \times 28) + (12 \times 32) \
(15 \times 27) + (16 \times 31) & (15 \times 28) + (16 \times 32)
\end{bmatrix}
\begin{bmatrix}
297 + 372 & 308 + 384 \
405 + 496 & 420 + 512
\end{bmatrix}
\begin{bmatrix}
669 & 692 \
901 & 932
\end{bmatrix}
]
Sum the Results:
[
C_{22} = \begin{bmatrix} 401 & 420 \ 569 & 596 \end{bmatrix} + \begin{bmatrix} 669 & 692 \ 901 & 932 \end{bmatrix} = \begin{bmatrix} 1070 & 1112 \ 1470 & 1528 \end{bmatrix}
]
Step 3: Merge the Computed Blocks to Form Matrix C
[
C = \begin{bmatrix}
C_{11} & C_{12} \
C_{21} & C_{22}
\end{bmatrix}
= \begin{bmatrix}
\begin{bmatrix} 250 & 260 \ 618 & 644 \end{bmatrix} &
\begin{bmatrix} 270 & 280 \ 670 & 696 \end{bmatrix} \
\begin{bmatrix} 986 & 1028 \ 1354 & 1412 \end{bmatrix} &
\begin{bmatrix} 1070 & 1112 \ 1470 & 1528 \end{bmatrix}
\end{bmatrix}
]
Final Resulting Matrix C:
[
C = \begin{bmatrix}
250 & 260 & 270 & 280 \
618 & 644 & 670 & 696 \
986 & 1028 & 1070 & 1112 \
1354 & 1412 & 1470 & 1528
\end{bmatrix}
]
Verification
Let's quickly verify one of the blocks to ensure correctness.
Verification of ( C_{11} ):
[
C_{11} = \begin{bmatrix} 250 & 260 \ 618 & 644 \end{bmatrix}
]
First Element (1,1):
[
(1 \times 17) + (2 \times 21) + (3 \times 25) + (4 \times 29) = 17 + 42 + 75 + 116 = 250
]
Second Element (1,2):
[
(1 \times 18) + (2 \times 22) + (3 \times 26) + (4 \times 30) = 18 + 44 + 78 + 120 = 260
]
Third Element (2,1):
[
(5 \times 17) + (6 \times 21) + (7 \times 25) + (8 \times 29) = 85 + 126 + 175 + 232 = 618
]
Fourth Element (2,2):
[
(5 \times 18) + (6 \times 22) + (7 \times 26) + (8 \times 30) = 90 + 132 + 182 + 240 = 644
]
All elements match the computed values, confirming the correctness of the block multiplication approach.
Conclusion
By dividing the matrices into smaller blocks, you not only simplify the multiplication process but also enable parallel execution, which can lead to performance improvements. This method is particularly beneficial for large-scale matrix operations commonly found in fields like computer graphics, scientific computing, and machine learning.
If you plan to implement this in a programming environment, consider using parallel computing libraries or frameworks (such as OpenMP for C/C++, multiprocessing in Python, or parallel streams in Java) to handle the concurrent computations efficiently.


Q- is this the best option or there more pallersime option ?