RTR Eigensolvers Package
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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 package.


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:

Eigenvalue-specific applications of these methods can be found at:


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.


Version Date .zip .tgz
0.3.0 19 Mar 2008 rtresgev-0.3.zip rtresgev-0.3.tgz
Major: Added support for complex Hermitian problems.
  • Improved error messages for incorrect calling syntax.
  • RTRESGEV: asymptotic computation of rho corrected for case where tCG max dimension is exceeded.
  • Minor user-transparent changes improving code readability.
  • Cleaned up some commenting in auxiliary routines.
0.2.0 7 Mar 2008 rtresgev-0.2.zip rtresgev-0.2.tgz
Major: Optional subspace acceleration.
minor: Cleaned up formatting, consistent output/verbosity between RTRESGEV and IRTRESGEV.
0.1.0 16 Feb 2008 rtresgev-0.1.zip rtresgev-0.1.tgz
Initial public release. Provides RTR and IRTR solvers for the symmetric generalized eigenvalue problem.


The authors of the codes are:


Funding for this work came in part from:

  • National Science Foundation Award 032944: "Collaborative Research: Model Reduction of Dynamical Systems for Real Time Control"
  • National Science Foundation Award 9912415: "Efficient Algorithms for Large Scale Dynamical Systems"

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