In 2008 Absil et al. published a book with optimization methods in Riemannian manifolds. The authors developed steepest descent, Newton, trust-region and conjugate gradients methods using an approximation of the geodesic called retraction. In this paper we present implementations of the of steepest descent method of Absil et al. using Matlab software. We show the implementation and numerical results to minimize the Rayleigh quotient on the unit sphere and the Brockett function on the Stiefel manifold that are strongly related to eigenvalue problems in computational Linear Algebra. Moreover, we introduce nonconvex functions studied in the paper of Papa Quiroz et al.(2008), verifying that for these functions the method also has good performance.
Report 1, FCNM-UNAC, 2015