BLOCK M-MATRICES AND COMPUTATION OF INVARIANT TORI

In this work a generalization of nonsingular M-matrices to block matrices is proposed, where positivity of numbers is replaced by positive definiteness of blocks. In addition, the outer diagonal blocks are multiples of the identity. This generalization is arrived at by studying the matrices that arise from discretizing linear first-order systems of partial differential equations (PDEs) where each equation has the same principal part. These PDEs occur in the study of invariant tori of dynamical systems. In this paper, a first-order discretization of these PDEs is investigated and block M-matrix properties are used to establish stability and error estimates. To obtain higher-order convergence, an error expansion is proved, which legitimates Richardson's extrapolation. Some of the numerical and algorithmic aspects of the proposed discretization are discussed and briefly contrasted to others. Some numerical examples to illustrate the theory and to highlight the interplay between attractivity and smoothness of the tori versus accuracy of the approximation are also presented.