RTRESGEV contains implementations of Riemannian Trust-Region
methods for the computation of Extreme Generalized Symmetric Eigenvalues.
This software provides two separate eigenvalue solvers (or, xolvers, as some prefer): one based on the Riemannian
trust-region (RTR) method and the other based on the implicit Riemannian trust-region (IRTR) method. Both of these are
block, preconditioned eigensolvers with provably robust global convergence properties and superlinear local convergence.
RTRESGEV currently provides implementations in MATLAB, with implementations of
the IRTR solvers in Trilinos/Anasazi in the Trilinos 9 release (Oct 2008).
These are specially tailored implementations of the RTR methods for this
problem; for more general implementations applicable to a wider variety of
Riemannian optimization problems, see the GenRTR
The solvers implement the RTR and IRTR Riemannian optimization solvers applied to the problem of minimizing the
generalized Rayleigh quotient over the Grassmann manifold. Given a matrix pencil (A,B), where A is
symmetric/Hermitian and B is symmetric/Hermitian positive definite, the solvers compute the leftmost (i.e., smallest algebraic)
eigenvalues of (A,B). To compute the rightmost eigenvalues, just submit the pencil (-A,B).
The RTR and IRTR solvers are described for general Riemannian optimization in the following papers:
The software is intended to be as flexible as possible with respect to the storage/application of the operators
describing the eigenvalue problem and the preconditioner. This is accomplished through the (optional) use of
MATLAB function handles.
Users also have the option of passing the eigenvalue operators via dense or sparse MATLAB matrices. For specific information
on usage, please consider downloading the software or visiting the documentation.