Series solutions of vector differential equations related to linear systems of algebraic equations

摘要

First order ordinary differential equations are associated to a system of linear algebraic equations with a positive definite matrix. Vector solutions of the differential systems are expressed in the form of infinite series in terms of the system matrix. They represent the parametric equations of the orthogonal trajectories to hypersurfaces defined by a related quadratic functional. The convergence of the series involved is substantially improved for an efficient numerical integration of the system of differential equations. The solution of the corresponding system of algebraic equations is formally expressed as matrix series. Some of the geometric properties of the orthogonal trajectories are shown and a related physical model is also presented.