Matrix multiplication:

Matrix multiplication is a mathematical operation that takes two matrices, say A and B, and produces another matrix C. The element C[i][j] of the resulting matrix is obtained by multiplying the elements of the ith row of matrix A by the corresponding elements of the jth column of matrix B and summing up the results.

Multithreading can be used for the computation of matrix multiplication, where each thread is responsible for computing a subset of the resulting matrix. The strategy involves dividing the workload among threads, each handling a portion of the rows in the resulting matrix.

 

Steps for Multithreaded Matrix Multiplication:

 

Check Compatibility: Ensure that the number of columns in the first matrix is equal to the number of rows in the second matrix.

Initialize Result Matrix: Create a result matrix C with dimensions (number of rows in A) x (number of columns in B).

Determine Thread Workload: Divide the rows of the resulting matrix into equal portions, assigning each portion to a separate thread.

Create Threads: Spawn threads, each responsible for computing the matrix multiplication for its assigned portion of the result matrix.

Thread Execution: Within each thread, perform the matrix multiplication for the assigned rows, calculating the dot product for each element.

Synchronize Threads: Ensure proper synchronization to avoid race conditions, especially when updating shared data (the result matrix).

Wait for Threads to Complete: Wait for all threads to finish their computations before proceeding.

Example:

Matrix A:   | 1  2 |                            Matrix B:    | 5  6 |                 

| 3  4 |                                                | 7  8 |

Step-by-Step Multithreaded Matrix Multiplication:

1. Check Compatibility: Both matrices have 2 columns, so they are compatible for multiplication.

2. Initialize Result Matrix: Create a result matrix C with dimensions 2x2 (matching the rows of A and columns of B).

3. Determine Thread Workload: Divide the rows of C among two threads. Thread 1 will handle the first row of C, and Thread 2 will handle the second row.

4. Create Threads: Spawn two threads: Thread 1 and Thread 2.

5. Thread Execution (Thread 1): Thread 1 calculates the first row of the resulting matrix C.

     - Element C[0][0] = (1*5 + 2*7) = 19

     - Element C[0][1] = (1*6 + 2*8) = 22

6. Thread Execution (Thread 2): Thread 2 calculates the second row of the resulting matrix C.

     - Element C[1][0] = (3*5 + 4*7) = 43

     - Element C[1][1] = (3*6 + 4*8) = 50

 

7. Synchronize Threads: Ensure proper synchronization, especially when updating shared data (result matrix C).

8. Resulting Matrix C:  | 19  22 |

| 43  50 |

 

In this example, we've divided the workload between two threads, each responsible for calculating a portion of the resulting matrix. The final result is the product of matrices A and B, calculated concurrently using multithreading.