GenRTR Riemannian Optimization Package

Generic RTR

GenRTR is the Generic Riemannian Trust-Region package. GenRTR is a MATLAB package for the optimization of functions defined on Riemannian manifolds. GenRTR currently provides two Riemannian trust-region methods:

  • the Riemannian Trust-Region (RTR) method (more info)
  • the Implicit Riemannian Trust-Region (IRTR) method (more info)

While the current emphasis concerns trust-region methods, the framework is suitable for the implementation of any retraction-based optimization method. Future plans involve the expansion of the framework to other optimization methods.


The optimization methods utilize MATLAB function handles to access user-provided routines for the objective function, gradient, Hessian, retraction, etc. This enables the optimization solvers to be coded in a generic manner and applied to minimization of arbitrary functions on arbitrary Riemannian manifolds. It also allows the encapsulation of a problem into a single driver (e.g., rtreig), as opposed to a collection of files (e.g., f.m, grad.m).

GenRTR is released under a modified BSD open-source license, available here.


The package distribution currently contains drivers for the following applications:

  • Computation of the full symmetric eigenvalue decomposition over the orthogonal group (rtreig, rtreig2)
  • Computation of the dominant singular value decomposition over the Stiefel manifold (rtrdsvd, irtrdsvd)
  • Computation of extreme eigenspaces of a symmetric definite matrix pencil over the Grassmann manifold (rtresgev, irtresgev, tmesgev)


The authors of the code 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"

Related software

  • The RTRESGEV package provides specialized versions of the RTR-based eigensolvers.