A new method of hierarchically partitioning the numerical calculations associated with the finite-element analysis of rotating electrical machinery is set forth. In this method, the device is partitioned geometrically and hierarchically into its constituent subsystems. Each Newton-Raphson iteration in a magnetically nonlinear device involves the hierarchical solution of relatively compact linear algebraic equations. This is contrasted with conventional approaches involving the formation or updating and solution of a single but large-dimensional sparse equation. In addition to an improvement in the computational efficiency, this approach facilitates efficient management of rotation. The proposed approach is demonstrated to require less than one-fourth the computation time, while producing identical results to those of a conventional formulation when solving for cogging torque and back electromotive force (back-EMF) of a permanent-magnet synchronous machine. The numerical results are also shown to be in agreement with the measured data.