Course Project On :
Strassen's Matrix Multiplication
Under The Guidance Of:
Prof.
Subodh Kumar
Presented By:
Gaurav Jain
Lalchand

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 )

Strassens’s Matrix Multiplication

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

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

Strassens’s Matrix Multiplication
Ref : Accelerating High Performance Applications with CUDA and MPI

Why MPI + CUDA ?..
➢
Equations naturally suitable for CUDA environment
➢
Incapability of CUDA : No inter GPU communication.
➢
MPI : Data distributing mechanism
➢
CUDA : Main Execution Engine

MPI + CUDA

Steps Performed
➢
➢
➢
Divide the input matrix into four equal parts
Send the appropiate part to the corresponding process
Each process compute the corresponding equation


Node Contains GPU
Use kernels on their own GPU to compute result

Steps Performed
➢
➢
➢
➢
Divide the input matrix into four equal parts
Send the appropiate part to the corresponding process
Each process compute the corresponding equation
➢
Process will send their result to the head process of equation
➢
➢
➢
All Heads collect data
Head will compute C's equation
All head send their partial result to master node
Master will combine & display the result

Detailed Description – Step 1
P
1
P
5
= (A
11
+ A
22
)(B
11
+B
22
)
= (A
11
+ A
12
) * B
22
P
6
P
2
= (A
21
= (A
21
- A
+ A
11
22
) * B
) * (B
11
11
+ B
12
)
P
7
P
3
= A
11
= (A
12
- A
* (B
22
12
- B
) * (B
21
22
)
+ B
22
)
P
4
= A
22
* (B
21
- B
11
)

Detailed Description – Step 2
P
1
, P
5
P
2
, P
6
P
3
, P
7
P
4

Detailed Description – Step 3
P
1
, P
5
P
2
, P
6
P3 , P7
P
4
Declare
Result

Experimental Result - 1

Experimental Result - 2

Experimental Result - 3

Accelerating High Performance Applications with CUDA and MPI :
N. P . Karunadasa & D. N. Ranasinghe
Strassen’s Matrix Multiplication on GPUs : Junjie Li , Sanjay Ranka

Thanks