Geodesic
On the convergence of a new Rayleigh quotient method with applications to large eigenproblems
Electronic transactions on numerical analysis, Tome 7 (1998), pp. 182-189
In this paper we propose a variant of the Rayleigh quotient method to compute an eigenvalue and corresponding eigenvectors of a matrix. It is based on the observation that eigenvectors of a matrix with eigenvalue zero are also singular vectors corresponding to zero singular values. Instead of computing eigenvector approximations by the inverse power method, we take them to be the singular vectors corresponding to the smallest singular value of the shifted matrix. If these singular vectors are computed exactly the method is quadratically convergent. However, exact singular vectors are not required for convergence, and the resulting method combined with Golub-Kahan-Krylov bidiagonalization looks promising for enhancement/refinement methods for large eigenvalue problems.
Classification : 15A42, 65F15, 65H15
Keywords: Rayleigh quotient method, singular value decomposition, large eigenproblem 
@article{ETNA_1998__7__a2,
     author = {O'Leary,  D.P. and Stewart,  G.W.},
     title = {On the convergence of a new {Rayleigh} quotient method with applications to large eigenproblems},
     journal = {Electronic transactions on numerical analysis},
     pages = {182--189},
     year = {1998},
     volume = {7},
     zbl = {0912.65028},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_1998__7__a2/}
}
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O'Leary,  D.P.; Stewart,  G.W. On the convergence of a new Rayleigh quotient method with applications to large eigenproblems. Electronic transactions on numerical analysis, Tome 7 (1998), pp. 182-189. http://geodesic.mathdoc.fr/item/ETNA_1998__7__a2/