56 research outputs found
Two-sided Grassmann-Rayleigh quotient iteration
The two-sided Rayleigh quotient iteration proposed by Ostrowski computes a
pair of corresponding left-right eigenvectors of a matrix . We propose a
Grassmannian version of this iteration, i.e., its iterates are pairs of
-dimensional subspaces instead of one-dimensional subspaces in the classical
case. The new iteration generically converges locally cubically to the pairs of
left-right -dimensional invariant subspaces of . Moreover, Grassmannian
versions of the Rayleigh quotient iteration are given for the generalized
Hermitian eigenproblem, the Hamiltonian eigenproblem and the skew-Hamiltonian
eigenproblem.Comment: The text is identical to a manuscript that was submitted for
publication on 19 April 200
A geometric Newton method for Oja's vector field
Newton's method for solving the matrix equation runs
up against the fact that its zeros are not isolated. This is due to a symmetry
of by the action of the orthogonal group. We show how
differential-geometric techniques can be exploited to remove this symmetry and
obtain a ``geometric'' Newton algorithm that finds the zeros of . The
geometric Newton method does not suffer from the degeneracy issue that stands
in the way of the original Newton method
Lagrange Multipliers and Rayleigh Quotient Iteration in Constrained Type Equations
We generalize the Rayleigh quotient iteration to a class of functions called
vector Lagrangians. The convergence theorem we obtained generalizes classical
and nonlinear Rayleigh quotient iterations, as well as iterations for tensor
eigenpairs and constrained optimization. In the latter case, our generalized
Rayleigh quotient is an estimate of the Lagrange multiplier. We discuss two
methods of solving the updating equation associated with the iteration. One
method leads to a generalization of Riemannian Newton method for embedded
manifolds in a Euclidean space while the other leads to a generalization of the
classical Rayleigh quotient formula. Applying to tensor eigenpairs, we obtain
both an improvements over the state-of-the-art algorithm, and a new
quadratically convergent algorithm to compute all complex eigenpairs of sizes
typical in applications. We also obtain a Rayleigh-Chebyshev iteration with
cubic convergence rate, and give a clear criterion for RQI to have cubic
convergence rate, giving a common framework for existing algorithms
A new approach to numerical algorithms
In this paper we developed a new Lanczos
algorithm on the Grassmann manifold.
This work comes in the wake of the article by
A. Edelman, T. A. Arias and S. T. Smith,
“The geometry of algorithms with
orthogonality constraints
An algorithm to compute the polar decomposition of a 3 × 3 matrix
We propose an algorithm for computing the polar decomposition of a 3 × 3 real matrix that is based on the connection between orthogonal matrices and quaternions. An important application is to 3D transformations in the level 3 Cascading Style Sheets specification used in web browsers. Our algorithm is numerically reliable and requires fewer arithmetic operations than the alternative of computing the polar decomposition via the singular value decomposition
Experimental String Field Theory
We develop efficient algorithms for level-truncation computations in open
bosonic string field theory. We determine the classical action in the universal
subspace to level (18,54) and apply this knowledge to numerical evaluations of
the tachyon condensate string field. We obtain two main sets of results. First,
we directly compute the solutions up to level L=18 by extremizing the
level-truncated action. Second, we obtain predictions for the solutions for L >
18 from an extrapolation to higher levels of the functional form of the tachyon
effective action. We find that the energy of the stable vacuum overshoots -1
(in units of the brane tension) at L=14, reaches a minimum E_min = -1.00063 at
L ~ 28 and approaches with spectacular accuracy the predicted answer of -1 as L
-> infinity. Our data are entirely consistent with the recent perturbative
analysis of Taylor and strongly support the idea that level-truncation is a
convergent approximation scheme. We also check systematically that our
numerical solution, which obeys the Siegel gauge condition, actually satisfies
the full gauge-invariant equations of motion. Finally we investigate the
presence of analytic patterns in the coefficients of the tachyon string field,
which we are able to reliably estimate in the L -> infinity limit.Comment: 37 pages, 6 figure
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