References of "Mahony, R"
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See detailOptimization on manifolds : methods and applications
Absil, P.-A.; Mahony, R.; Sepulchre, Rodolphe ULg

in Recent Advances in Optimization and its Applications in Engineering (2010)

Summary. This paper provides an introduction to the topic of optimization on manifolds. The approach taken uses the language of differential geometry, however, we choose to emphasise the intuition of the ... [more ▼]

Summary. This paper provides an introduction to the topic of optimization on manifolds. The approach taken uses the language of differential geometry, however, we choose to emphasise the intuition of the concepts and the structures that are important in generating practical numerical algorithms rather than the technical details of the formulation. There are a number of algorithms that can be applied to solve such problems and we discuss the steepest descent and Newton’s method in some detail as well as referencing the more important of the other approaches. There are a wide range of potential applications that we are aware of, and we briefly discuss these applications, as well as explaining one or two in more detail. [less ▲]

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See detailRiemannian geometry of Grassmann manifolds with a view on algorithmic computation
Absil, P.-A.; Mahony, R.; Sepulchre, Rodolphe ULg

in Acta Applicandae Mathematicae (2004), 80(2), 199-220

We give simple formulas for the canonical metric, gradient, Lie derivative, Riemannian connection, parallel translation, geodesics and distance on the Grassmann manifold of p-planes in R-n. In these ... [more ▼]

We give simple formulas for the canonical metric, gradient, Lie derivative, Riemannian connection, parallel translation, geodesics and distance on the Grassmann manifold of p-planes in R-n. In these formulas, p-planes are represented as the column space of n x p matrices. The Newton method on abstract Riemannian manifolds proposed by Smith is made explicit on the Grassmann manifold. Two applications - computing an invariant subspace of a matrix and the mean of subspaces - are worked out. [less ▲]

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See detailCubically convergent iterations for invariant subspace computation
Absil, P.-A.; Sepulchre, Rodolphe ULg; Van Dooren, P. et al

in SIAM Journal on Matrix Analysis and Applications (2004), 26(1), 70-96

We propose a Newton-like iteration that evolves on the set of fixed dimensional subspaces of R-n and converges locally cubically to the invariant subspaces of a symmetric matrix. This iteration is ... [more ▼]

We propose a Newton-like iteration that evolves on the set of fixed dimensional subspaces of R-n and converges locally cubically to the invariant subspaces of a symmetric matrix. This iteration is compared in terms of numerical cost and global behavior with three other methods that display the same property of cubic convergence. Moreover, we consider heuristics that greatly improve the global behavior of the iterations. [less ▲]

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See detailA Grassmann-Rayleigh quotient iteration for computing invariant subspaces
Absil, P.-A.; Mahony, R.; Sepulchre, Rodolphe ULg et al

in SIAM Review (2002), 44(1), 57-73

The classical Rayleigh quotient iteration (RQI) allows one to compute a one-dimensional invariant subspace of a symmetric matrix A. Here we propose a generalization of the RQl which computes a p ... [more ▼]

The classical Rayleigh quotient iteration (RQI) allows one to compute a one-dimensional invariant subspace of a symmetric matrix A. Here we propose a generalization of the RQl which computes a p-dimensional invariant subspace of A. Cubic convergence is preserved and the cost per iteration is low compared to other methods proposed in the literature. [less ▲]

Detailed reference viewed: 27 (7 ULg)