its deals with how parallel programming can be achieved. its just an example of parallel programming.

3. General introduction Basic Matrix Multiplication Suppose we want to multiply two matrices of size N x N : for example A x B = C . C 11 = a 11 b 11 + a 12 b 21 C 12 = a 11 b 12 + a 12 b 22 C 21 = a 21 b 11 + a 22 b 21 C 22 = a 21 b 12 + a 22 b 22 2x2 matrix multiplication can be accomplished in 8 multiplication. (2 log 2 8 =2 3 )

4. Time analysis Strassens’s Matrix Multiplication P 1 = (A 11 + A 22 )(B 11 +B 22 ) P 2 = (A 21 + A 22 ) * B 11 P 3 = A 11 * (B 12 - B 22 ) P 4 = A 22 * (B 21 - B 11 ) P 5 = (A 11 + A 12 ) * B 22 P 6 = (A 21 - A 11 ) * (B 11 + B 12 ) P 7 = (A 12 - A 22 ) * (B 21 + B 22 )

5. C 11 = P 1 + P 4 - P 5 + P 7 C 12 = P 3 + P 5 C 21 = P 2 + P 4 C 22 = P 1 + P 3 - P 2 + P 6 Time analysis

8. * Consider Multiplication of 4x4 matrix: A11 A12 A13 A14 A21 A22 A23 A24 A31 A32 A33 A34 A41 A42 A43 A44 B11 B12 B13 B14 B21 B22 B23 B24 B31 B32 B33 B34 B41 B42 B43 B44 C11 C12 C13 C14 C21 C22 C23 C24 C31 C32 C33 C34 C41 C42 C43 C44

9.  = The sub matrix of size 2x2 can be computed as follows: Now c11 ,c12, c21,c22 can be computed as follows: C 11 = P 1 + P 4 - P 5 + P 7 C 12 = P 3 + P 5 C 21 = P 2 + P 4 C 22 = P 1 + P 3 - P 2 + P 6 P 1 = (A 11 + A 22 )(B 11 +B 22 ) P 2 = (A 21 + A 22 ) * B 11 P 3 = A 11 * (B 12 - B 22 ) P 4 = A 22 * (B 21 - B 11 ) P 5 = (A 11 + A 12 ) * B 22 P 6 = (A 21 - A 11 ) * (B 11 + B 12 ) P 7 = (A 12 - A 22 ) * (B 21 + B 22 ) A11 A12 A21 A22 B11 B12 B21 B22 C11 C12 C21 C22

10. In similar way we can calculate all the multiplications. And same steps we can perform on any dimension of matrix. Results : Input Size(n) No. of multiplication In sequential algo. No. of multiplication In the implemented algo. No. of multiplication In Strassen’s 2 8 7 7 4 64 56 49 8 512 448 343 16 4096 3584 2401 . . . . . . . . p p 3 (7/8)*p 3 n 2.807

11. Applying above algorithm when size is not in the power of 2 Steps required is as follows Step 1) Make the size to nearest power of 2 Step 2) Make the value of all extra rows to be zero Step 3) Make the value of all extra columns to be zero Step 4) Then apply above algorithm Suppose the dimension of given matrix is 3 then Nearest of 3 is 4 which is power of 2 Hence matrix becomes A11 A12 A13 0 A21 A22 A23 0 A31 A32 A33 0 0 0 0 0

12. Efficiency calculation: Time analysis : If the value of dimension is not in the power of 2 then first make it to the nearest power of 2 then start proceeding…… T(2)=7 ( from base condition of strassen’s multiplication) T(N)= (7/8)*N 3 Hence total no. of multiplication required is = (7 * n 3 )/8 Which is less then (n 3 ).

13. Proposed Approach 2: Matrix multiplication by creating process for each computation is as follows: A matrix can be divided in to number of parts and each part can be multiply by creating a separate process. A new process can be created by using fork () system call. Different steps involved in this approach are Step 1 : Partitioning a) Divide matrices into rows b) Each primitive task has corresponding rows of three matrices Step 2 : Communication a) Each task must eventually see every row of B b) Organize tasks into a ring Step 3: Agglomeration and mapping a) Fixed number of tasks, each requiring same amount of computation b) Regular communication among tasks c) Strategy: Assign each process a contiguous group of rows

14. Examples: Let in the first case total no. of process to be created is equal to total no. of rows in the matrix i.e. total no. of process = n where n is the dimension of matrix. Task1 Task2 Task3 Task4 A11 A12 A13 A14 A21 A22 A23 A24 A31 A32 A33 A34 A41 A42 A43 A44

15. Let in the second case total no. of process to be created is n/2. Task1 Task2 Task3 Task4

17. Parent process Child process 1 Child process 2 Child process 3 Child process 4 Child process p Fork Fork Fork Fork Fork

19. Prototype for process creation: Prototype for process control: #include <sys/types.h> #include <unistd.h> pid_t fork(void); Returns: 0 in child, process ID of child in parent, -1 on error. #include <sys/wait.h> pid_t wait(int * status_p ); pid_t waitpid(pid_t child _ pid , int * status _ p , int options ); Returns: process ID if OK, 0 (for some variations of wait), -1 on error .

20. Efficiency: In the first case since we are creating a new process to multiply a row of 1 st matrix to all columns of 2 nd matrix. And same operation is performing for all row of 1 st matrix. Hence we are achieving concurrent programming. Hence efficiency increases much more. In the second case we are creating a new process to multiply two rows of 1 st matrix to all columns of 2 nd matrix one by one. And same operation is performing for all sets of 2 rows. Hence here also we are achieving concurrent programming. Hence efficiency increases rapidly.

24. BIBLIOGRAPHY 1)Unix system programming using c++ - Terrence Chan 2) Introduction to the design & analysis of algorithms – Anany Levitin 3) The C Programming Language - Brian W.Kernighan && Dennnis M.Ritchie, 2 nd edition

25. THANK YOU Presented By –PRAMIT KUMAR