Newton-Type Methods For Simultaneous Matrix Diagonalization

Rima Khouja (1) , Bernard Mourrain (1) , Jean-Claude Yakoubsohn (2)

1 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
2 IMT - Institut de Mathématiques de Toulouse UMR5219

This paper proposes a Newton type method to solve numerically the eigenproblem of several diagonalizable matrices, which pairwise commute. A classical result states that these matrices are simultaneously diagonalizable. From a suitable system of equations associated to this problem, we construct a sequence which converges quadratically towards the solution. This construction is not based on the resolution of linear system as it is the case in the classical Newton method. Moreover, we provide a theoretical analysis of this construction to exhibit a condition to get a quadratic convergence. We also propose numerical experiments, which illustrate the theoretical results. This shows that classical QR method would gain in efficiency incorporating the tests given by the theory.

Mots clés: Simultaneous diagonalization; Newton-type method; eigenproblem; eigenvalues; high precision computation; MSC Classification: 65F15; 65H10; 15A18; 65-04

Domaines: Analyse numérique [math.NA]

Rima Khouja, Bernard Mourrain, Jean-Claude Yakoubsohn. Newton-Type Methods For Simultaneous Matrix Diagonalization. 2021. ⟨hal-03390265v1⟩