A FAST WAY TO COMPUTE MATRIX MULTIPLICATION MAE05 Presented by: - PowerPoint PPT Presentation

A FAST WAY TO COMPUTE MATRIX MULTIPLICATION MAE05 Presented by: Forrest Yau Zhen Kit , Jurong Pioneer Junior College 1

HIGHLIGHTS I. Background What led me to research more on this topic? II. Methodology How does my contributions help in this research topic? III. Conclusion What are some future possible directions? 2

I. BACKGROUND 3

NAÏVE MATRIX MULTIPLICATION  It takes 𝑜 3 steps to multiply two 𝑜 × 𝑜 matrices.  Involves a total of 8 multiplication steps for a 2 x 2 matrices.  However, it is not the most optimal method. 4 Can we go faster than 𝒐 𝟒 ?

STRASSEN’S ALGORITHM 5

STRASSEN’S ALGORITHM  𝑁 1 = 𝑏 11 + 𝑏 22 𝑐 11 + 𝑐 22 𝐷 11 = 𝑁 1 + 𝑁 4 − 𝑁 5 + 𝑁 7  𝑁 2 = 𝑏 21 + 𝑏 22 𝑐 11 𝐷 12 = 𝑁 3 + 𝑁 5  𝑁 3 = 𝑏 11 𝑐 12 − 𝑐 22 𝐷 21 = 𝑁 2 + 𝑁 4  𝑁 4 = 𝑏 22 𝑐 21 − 𝑐 11 𝐷 22 = 𝑁 1 − 𝑁 2 + 𝑁 3 + 𝑁 6  𝑁 5 = 𝑏 11 + 𝑏 12 𝑐 22  𝐵 = 𝑏 11 𝑏 12 𝐶 = 𝑐 11 𝑐 12 𝑏 21 𝑏 22 𝑐 21 𝑐 22  𝑁 6 = 𝑏 21 − 𝑏 11 𝑐 11 + 𝑐 12 Where A x B = C= 𝑑 11 𝑑 12  𝑁 7 = 𝑏 12 − 𝑏 22 𝑐 21 + 𝑐 22 6 𝑑 21 𝑑 22

STRASSEN'S ALGORITHM  Involves 7 multiplication steps instead of 8 multiplication steps for a 2 x 2 matrices.  Matrix multiplication algorithm efficiency of O( 𝑜 2.81 ).  Faster compared to the naïve algorithm. 7

CAN WE GO FASTER THAN O( 𝑜 2.81 ) ? 8

II. METHODOLOGY 9

PROPOSED ALGORITHM  Inspiration from Gaussian elimination  Involves back-to-back elimination method to obtain a “zero triangular form”  An attempt to cut down the number of multiplication steps to just 6 10

PERFORMANCE TEST I 11

PERFORMANCE TEST I 12

PERFORMANCE TEST II 13

PERFORMANCE TEST II *Lower the bar, the better 14

III. CONCLUSION 15

THE IMPORTANCE OF FAST MATRIX MULTIPLICATION ALGORITHM  Matrix multiplication has its applications and uses in all areas of study and purposes eg. Big Data, Data Representation.  Cuts down the costs involved. 16

FUTURE DIRECTIONS  Requires further testing of proposed algorithm.  Basis of future research into achieving better efficiency in other 𝑜 × 𝑜 matrices.  Possible use of Master Theorem to determine true complexity. 17

THAT’S THE END OF MY PRESENTATION THANK YOU! 18

A FAST WAY TO COMPUTE MATRIX MULTIPLICATION MAE05 Summary of content covered: • Naïve matrix multiplication & Strassen’s algorithm • Possibility of achieving a more efficient algorithm • Importance of fast matrix multiplication 19

PERFORMANCE TEST II  Benchmarks used to run the algorithm: 1. CPU  Intel(R) Core(TM) i7-7500U CPU @ 2.70GHz 2. GPU  NVIDIA GeForce 940MX 3. RAM 20  8GB