Research on fast matrix multiplication algorithm

Abstract

This research mainly focuses on fast matrix multiplication algorithms. Fast matrix multiplication is one of the most fundamental problems in computer science. The fast matrix multiplication algorithm differs from conventional matrix multiplication in that it offers a faster computational approach that can perform the operation in less than O(n3) time complexity. This algorithm provides a more efficient method for multiplying matrices, significantly reducing the computational requirements. The Laser method, developed by Coppersmith and Winograd, is an algorithm for matrix multiplication that does not involve direct computation. It establishes a relationship between matrix multiplication and tensors and simplifies the operation by finding an intermediate tensor that is computationally manageable. This method applies a series of simplification operations to determine an upper bound on the computational complexity of matrix multiplication. However, as matrices become larger, the computational and memory requirements increase, posing challenges for practical implementation. This research will present the main ideas and performance of the Laser method and discuss the improvements made to the Laser method, including refined analysis and asymmetric hashing techniques. Additionally, it highlights the need for further exploration, such as parallel computing and optimization strategies, to enhance the efficiency of matrix multiplication algorithms. Furthermore, this research will also provide a prospectus for the future of matrix multiplication algorithms, such as the practical implementation of the Laser method.