Algoritmos para Multiplica\c{c}\~ao Matricial

TL;DR

This paper studies matrix multiplication algorithms, focusing on Strassen and Strassen-Winograd methods, proposing modifications to improve memory usage and execution time, implemented in Julia with benchmarking and visualization tools.

Contribution

The paper introduces modifications to existing Strassen and Strassen-Winograd algorithms to reduce memory and time complexity, with implementation and benchmarking in Julia.

Findings

Modified algorithms reduce memory allocation.

Execution time improved through proposed modifications.

Implementation in Julia demonstrates practical efficiency.

Abstract

The goal of this article is to study algorithms that compute the product between two matrixes, specifically using the ingenuous methods of Strassen and Strassen-Winograd, which will be presented in Section 2. At present, the cited methods are not the most optimal considering the arithmetic complexity of these algorithms (see Table 1). However, changes to the Strassen and Strassen-Winograd methods will be exposed which will result in a reduction in their memory allocation and/or execution time. The algorithms in this study were implemented using the Julia programming language, version 1.9.1, with the aid of the packages Pluto (notebooks), Plots (graphic visualization of the results) and BenchmarkTools (measurement of memory allocation and execution time of the algorithms). -- O objetivo deste artigo \'e estudar algoritmos que computam o produto entre duas matrizes, mais especificamente utilizando os m\'etodos ing\^enuo, de Strassen e de Strassen-Winograd, que ser\~ao apresentados na Se\c{c}\~ao 2. Atualmente, os m\'etodos citados n\~ao s\~ao os mais otimizados considerando a complexidade aritm\'etica de seus algoritmos (vide Tabela 1). No entanto, ser\~ao expostas modifica\c{c}\~oes dos m\'etodos de Strassen e Strassen-Winograd que conseguem reduzir sua aloca\c{c}\~ao de mem\'oria e/ou tempo de execu\c{c}\~ao. Os algoritmos do problema em estudo foram implementados utilizando a linguagem de programa\c{c}\~ao Julia, na vers\~ao 1.9.1, com o aux\'ilio dos pacotes Pluto (notebooks), Plots (visualiza\c{c}\~ao gr\'afica dos resultados) e BenchmarkTools (medi\c{c}\~ao de aloca\c{c}\~ao de mem\'oria e tempo de execu\c{c}\~ao dos algoritmos).