Ultra-Fast Matrix Multiplication: An Empirical Analysis of Highly Optimized Vector Algorithms

33 Matrices play an important role in mathematics and computer science, but more importantly, they are ubiquitous in our daily lives as they are instrumental in the efficient manipulation and storage of digital data. Matrix multiplication is essential not only in graph theory but also in applied fields, such as computer graphics and digital signal processing (DSP). DSP chips are found in all cell phones and digital cameras, as matrix operations are the processes by which DSP chips are able to digitize sounds or images so that they can be stored or transmitted electronically. Fast matrix multiplication is still an open problem, but implementation of existing algorithms [5] is a more common area of development than the design of new algorithms [6]. Strassen’s algorithm is an improvement over the naive algorithm in the case of multiplying two 2×2 matrices, because it uses only seven scalar multiplications as opposed to the usual eight. Even though it has been shown that Strassen’s algorithm is optimal for two-by-two matrices [6], there have been asymptotic improvements to the algorithm for very large matrices. Thus, the search for improvements over Strassen’s algorithm for smaller matrices is still being conducted. Even Strassen’s algorithm is not considered an efficient reduction as it requires the size of the multiplicand matrices to be large powers of two. The following two sections present the naive, Winograd’s, and Strassen’s algorithms along with discussions of the theoretical bounds for each algorithm. We then present a confirmation of the theoretical study on the running times of the algorithms, followed by the results of an empirical study. In the final two sections, we present an improved Hybrid algorithm, which incorporates Strassen’s asymptotical advantage with Winograd’s practical performance, and discuss the stunning performance of the AltiVecoptimized Strassen’s algorithm. Naive Algorithm