Nonlinear Eigenvalue Problems

Alan Edelman

Department of Mathematics / Laboratory for Computer Science Massachusetts Institute of Technology


Abstract

We may increase the scope of the algebraic eigenvalue problem by focusing on the geometrical properties of the eigenspaces. In our work,we have created template software for variational problems defined on sets of subspaces. Areas where such problems arise include electronic structures computation, eigenvalue regularization, control theory, signal processing and graphics camera calibration. The geometry also provides the numerical analyst theoretical insight into the common mathematical structure underlying eigenvalue algorithms. Suppose that one has any real valued function F(Y) that one wishes to optimize over Y such that Y'Y=I, where Y' is either transpose or Hermitian transpose as appropriate. We describe a template designed to optimize such an F taking advantage of any ``Grassmann'' symmetries such an F may have. Joint work with Ross Lippert.