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Row-wise backward stable elimination methods for the equality constrained least squares problem
Get PDFIt is well known that the solution of the equality constrained least squares (LSE) problem min Bx=d ||b-Ax||2 is the limit of the solution of the unconstrained weighted least squares problem
\min_x\left\| \bmatrix{ \mu d \cr b } - \bmatrix{\mu B \cr A } x \right\|_2
as the weight tends to infinity, assuming that \bmatrix{B^T & A^T \cr}^T has full rank. We derive a method for the LSE problem by applying Householder QR factorization with column pivoting to this weighted problem and taking the limit analytically, with an appropriate rescaling of rows. The method obtained is a type of direct elimination method. We adapt existing error analysis for the unconstrained problem to obtain a row-wise backward error bound for the method. The bound shows that, provided row pivoting or row sorting is used, the method is well-suited to problems in which the rows of A and B vary widely in norm. As a by-product of our analysis, we derive a row-wise backward error bound of precisely the same form for the standard elimination method for solving the LSE problem. We illustrate our results with numerical tests
Relative Equilibria of Molecules
Get PDFWe describe a method for finding the families of relative equilibria of molecules that bifurcate from an equilibrium point as the angular momentum is increased from 0. Relative equilibria are steady rotations about a stationary axis during which the shape of the molecule remains constant.We show that the bifurcating families correspond bijectively to the critical points of a function h on the two-sphere which is invariant under an action of the symmetry group of the equilibrium point. From this it follows that for each rotation axis of the equilibrium configuration there is a bifurcating family of relative equilibria for which the molecule rotates about that axis. In addition, for each reflection plane there is a family of relative equilibria for which the molecule rotates about an axis perpendicular to the plane.
We also show that if the equilibrium is nondegenerate and stable, then the minima, maxima, and saddle points of h correspond respectively to relative equilibria which are (orbitally) Liapounov stable, linearly stable, and linearly unstable. The stabilities of the bifurcating branches of relative equilibria are computed explicitly for XY2, X3, and XY4 molecules.
These existence and stability results are corollaries of more general theorems on relative equilibria of G-invariant Hamiltonian systems that bifurcate from equilibria with finite isotropy subgroups as the momentum is varied. In the general case, the function h is defined on the Lie algebra dual g* and the bifurcating relative equilibria correspond to critical points of the restrictions of h to the coadjoint orbits in g*
Codes for spread spectrum applications generated using chaotic dynamical systems
No full textAn approach to finding codes for use in direct sequence
spread spectrum communications systems is described. It
is based upon an analogy between codes having auto- and
cross-correlation properties desirable for spread
spectrum systems, and certain dynamical systems
encountered in ergodic theory called systems with Lebesgue spectrum. Such systems are associated with
collections of orthogonal functions and these functions
can be used to generate collections of time series with
zero cross-correlation functions. To generate codewords
we must use truncated versions of these time series,
for which the cross-correlations are no longer
precisely zero: these truncated sequences correspond to
periodic orbits of the dynamical system. The method for
finding a code from a suitable periodic orbit is
described, and an example, using a simple dynamical
system known as the doubling map, is worked through in
some detail
The nearest definite pair for the Hermitian generalized eigenvalue problem
No full textThe generalized eigenvalue problem has special properties
when is a Hermitian and definite pair.
Given a general Hermitian pair it is of interest to find the nearest
definite pair having a specified Crawford number .
We solve the problem in terms of the inner numerical radius associated with the
field of values of .
We show that once the problem has been solved it is trivial to rotate the
perturbed pair
(A+\dA,B+\dB) to a pair
for which achieves its maximum value
,
which is a numerically desirable property when solving the
eigenvalue problem by methods that convert to a standard eigenvalue problem by
``inverting ''.
Numerical examples are given to illustrate the analysis
Spin-up of stratified rotating flows at large Schmidt number: experiment and theory
No full textWe consider the nonlinear spin-up/down of a rotating stratified fluid in a coni-
cal container. An analysis of axisymmetric similarity-type solutions to the relevant
boundary-layer problem, Duck, Foster & Hewitt (1997), has revealed three types of
behaviour for this geometry. In general, the boundary layer evolves to either a steady
state, or a gradually thickening boundary layer, or a finite-time singularity depending
on the Schmidt number, the ratio of initial to final rotation rates, and the relative
importance of rotation and stratification.
In this paper we emphasize the experimental aspects of an investigation into the
initial readjustment process. We make comparisons with the previously presented
boundary-layer theory, showing good quantitative agreement for positive changes
in the rotation rate of the container (relative to the initial rotation sense). The
boundary-layer analysis is shown to be less successful in predicting the flow evolution
for nonlinear decelerations of the container. We discuss the qualitative features of
the spin-down experiments, which, in general, are dominated by non-axisymmetric
effects. The experiments are conducted using salt-stratified solutions, which have a
Schmidt number of approximately 700.
The latter sections of the paper present some stability results for the steady
boundary-layer states. A high degree of non-uniqueness is possible for the system
of steady governing equations; however the experimental results are repeatable and
stability calculations suggest that �higher branch� solutions are, in general, unstable.
The eigenvalue spectrum arising from the linear stability analysis is shown to have
both continuous and discrete components. Some analytical results concerning the
continuous spectrum are presented in an appendix.
A brief appendix completes the previous analysis of Duck, Foster & Hewitt (1997),
presenting numerical evidence of a different form of finite-time singularity available
for a more general boundary-layer problem
The Ariel Project: A prospective cohort study of maternal-child transmission of human immunodeficiency virus type 1 in the era of maternal antiretroviral therapy
Get PDFIn a prospective cohort study, clinical and biologic factors that contribute to maternal-child transmission of human immunodeficiency virus type 1 (HIV-1) were studied. HIV-infected pregnant women and their infants were evaluated prospectively according to a standardized protocol. Of 204 evaluable women, 81% received zidovudine during their pregnancy. The infection rate among the 209 evaluable infants was 9.1%. By univariate analysis, histologic chorioamnionitis, prolonged rupture of membranes, and a history of genital warts were significantly associated with transmission. Additional factors associated with transmission that approached significance included a higher maternal virus load at delivery and the presence of cocaine in the urine. In a logistic regression model, histologic chorioamnionitis was the only independent predictor of transmission. Despite a significantly higher transmission rate at one site, no unique viral genotype was found at any site. Thus, chorioamnionitis was found to be the major risk factor for transmission among women receiving zidovudine
Backward error bounds for constrained least squares problems
No full textWe derive an upper bound on the
normwise backward error of an approximate solution to the
equality constrained least squares problem .
Instead of minimizing over the four perturbations
to , , and ,
we fix those to and and minimize over the remaining two;
we obtain an explicit solution of this simplified minimization problem.
Our experiments show that backward error bounds of practical use are obtained
when and are chosen as the optimal normwise relative backward
perturbations to the constraint system,
and we find that when the bounds are weak they can be improved by
direct search optimization.
We also derive upper and lower backward error bounds for the
problem of least squares minimization over a sphere:
Magnetohydrodynamic convection in molten gallium
Get PDFWe present the results of an experimental and numerical study of the effects of a steady magnetic field on sidewall convection in molten gallium. The magnetic field is applied in a direction which is orthogonal to the main flow which reduces the convection and good agreement is found for the scaling of this effect with the relevant parameters. Moreover, qualitatively similar changes in the structure of the bulk of the flow are observed in the experiment and the numerical simulations. In particular, the flow is restricted to two dimensions by the magnetic field, but it remains different to that found in two-dimensional free convection calculations. We also show that oscillations found at even greater temperature gradients can be suppressed by the magnetic field
Accuracy and stability of the null space method for solving the equality constrained least squares problem
No full textThe null space method is a standard method for solving the linear least squares problem subject to equality constraints (the LSE problem). We show that three variants of the method, including one used in LAPACK that is based on the generalized QR factorization, are numerically stable. We derive two perturbation bounds for the LSE problem: one of standard form that is not attainable, and a bound that yields the condition number of the LSE problem to within a small constant factor. By combining the backward error analysis and perturbation bounds we derive an approximate forward error bound suitable for practical computation. Numerical experiments are given to illustrate the sharpness of this bound