O Estudo e Implementação do Método das Potências e suas Acelerações de Convergência
Maria de Fátima Oliveira Silva
UFERSA
Ivan Mezzomo
Palavras-chave: Autovalores, Método da Potência, Método dos Mínimos Quadrados.
Resumo
Eigenvalues and eigenvectors are present in different branches of mathematics, including quadratic forms, differential systems, and nonlinear optimization problems, and can be used to solve problems in various fields such as economics, information theory, and structural analysis, among others. To analytically determine the eigenvalues and eigenvectors of a matrix, it is necessary to find the roots of its characteristic polynomial. However, this becomes unfeasible when the polynomial has a very high degree, making it necessary to use iterative numerical methods. One of the most widely used numerical methods for calculating the largest eigenvalue of a matrix is the Power Method (PM). This method, however, is limited to computing the dominant eigenvalue in modulus, provided that it is real. Based on previous studies that identified the second-degree polynomial approximation function of the Least Squares Method (LSM) as the most effective, we conducted an analysis exploring different strategies for accelerating the PM and comparing both methods. The strategies investigated involved using different numbers of initial PM iterations before applying the LSM. The main objective of the acceleration was to reduce the computational cost and the total number of iterations. In this study, the LSM was applied after 6 (LSM 6), 16 (LSM 16), and 26 (LSM 26) iterations of the PM. The matrices used were obtained from the Florida Sparse Matrix Collection repository, being sparse and symmetric. To analyze the effectiveness of the proposed methods, the implementations were carried out in Python version 3.11.5, on a computer with a Ryzen 5 7520u 7th Generation processor, 8 GB of RAM, and Windows 11 64-bit operating system. The stopping criterion was the Relative Error (RE) or 10,000 iterations, with a precision of 10⁻⁵. In all matrices tested, the LSM 6 strategy proved to be more efficient than LSM 16 and LSM 26, considering both the number of iterations and the RE. The results highlight the relevance of the proposed method as an effective strategy to accelerate the convergence of the PM, with LSM 6 being the most efficient in obtaining eigenvalues of different matrices. The objective of this work demonstrates not only computational performance gains but also a solid and easily implementable approach capable of enhancing applications in engineering, scientific simulations, and mathematical modeling. Thus, this study reinforces the importance of integrating classical techniques with modern acceleration strategies, consolidating the LSM as a promising method for future research and practical applications.