An important class of subproblems in nonlinear equality constrained optimization problems is where the constraint space is a Lie group. Symmetric eigenvalue problems may also be formulated in this manner.
Given a Lie group and a smooth function , the author gives the Newton method on , defined in terms of an appropriate gradient and Hessian of . Quadratic convergence of the method is proved. The method as described depends on a choice of basis vectors for the local tangent space of ; in the next section this method is extended to a coordinate free version.
The coordinate free Newton method is then developed for the symmetric eigenvalue problem. Two numerical examples are presented, and the results of the second compared with those from the shifted QR method.