Cálculo numérico de autovalores, decomposição em valores singulares e aplicações

Abstract

This work presents a study of numerical methods used for computing eigenvalues of real matrices, with a focus on the combination of the reduction to Hessenberg form and the QR method with Wilkinson’s shift. Theoretical foundations are first reviewed, including similarity transformations, diagonalization, the Schur factorization, and spectral properties. Next, Householder reflections and their role in the partial triangularization of matrices are described, a fundamental step for accelerating iterative eigenvalue algorithms. In the second part, classical iterative methods are analyzed — the power method, inverse power method, inverse iteration with shift, and the Rayleigh quotient iteration — culminating in the formulation of the QR method with shift, together with the deflation mechanism. Finally, the two phases are implemented and applied to a numerical example constructed with predefined eigenvalues and small perturbations. The complete iterative process is documented, highlighting the rapid convergence of the method, the efficiency of Wilkinson’s shift, and the stability of deflation.