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Improved Inverse Scaling and Squaring Algorithms for the Matrix Logarithm
Get PDFA popular method for computing the matrix logarithm is the
inverse scaling and squaring method,
which essentially carries out the steps of the scaling and squaring
method for the matrix exponential in reverse order.
Here we make several improvements to the method,
putting its development on a par with our recent version
[\emph{SIAM J. Matrix Anal.\ Appl.}, 31 (2009), pp.\ 970--989]
of the scaling and squaring method for the exponential.
In particular,
we introduce backward error analysis to replace the previous forward
error analysis;
obtain backward error bounds in terms of the quantities
, for several small integer , instead of ;
and use special techniques to compute the argument of the
Pad\'e approximant more accurately.
We derive one algorithm
that employs a Schur decomposition,
and thereby works with triangular matrices,
and another that requires only matrix multiplications and the solution
of multiple right-hand side linear systems.
Numerical experiments show the new algorithms to be generally faster and more
accurate than their existing counterparts
and suggest that the Schur-based method is the method of choice for computing
the matrix logarithm
Loop Spaces and Choreographies in Dynamical Systems
Get PDFWe consider a subset of the set of solutions to the n-body problem, termed choreographies, which involve a motion of particles where each follows the same path in space with a fixed time delay. Focusing on planar choreographies, we use the action of symmetry groups on the spatial and temporal motion of such systems to restrict a space of loops and study the topology of the resulting manifolds.
As well as providing a framework of notation and terminology for the study of such systems, we prove various useful properties which allow us to classify the possible groups of symmetries, and discuss which are likely to be realisable as that of a motion of bodies
An Algorithm for the Complete Solution of Quadratic Eigenvalue Problems
No full textWe develop a new algorithm for the computation of all the eigenvalues and optionally the right and left eigenvectors of dense quadratic matrix polynomials.
It incorporates scaling of
the problem parameters prior to the computation of eigenvalues, a choice of linearization with favorable
conditioning and backward stability properties,
and a preprocessing step that
reveals and deflates the zero and infinite eigenvalues contributed by singular
leading and trailing matrix coefficients.
The algorithm is backward stable for quadratics that are not too heavily damped.
Numerical experiments show that
our MATLAB implementation of the algorithm,
quadeig, outperforms the MATLAB
function polyeig in terms of both stability and efficiency
Invariant measures of the border collision normal form
Get PDFThe border collision normal form is a two dimensional continuous, piecewise affine map which
arises naturally in models where the dynamics is defined by different systems of equations in different regions of phase space, but which are continuous across the boundaries between regions. There are theorems which establish the existence of invariant measures for chaotic attractors of these systems, but the conditions are hard to establish analytically. By verifying these conditions numerically it is possible to describe regions of parameter space for which invariant measures do exist (up to numerical confidence) and compare this with what is known about the dynamics in these regions
Kazhdan-Lusztig parameters and extended quotients
No full textThe Kazhdan-Lusztig parameters are important parameters in the representation theory of -adic groups and affine Hecke algebras. We show that the Kazhdan-Lusztig parameters have a definite geometric structure, namely that of the extended quotient of a complex torus by a finite Weyl group . More generally, we show that the corresponding parameters, in the principal series of a reductive -adic group with connected centre, admit such a geometric structure. This confirms, in a special case, our recently formulated geometric conjecture.
In the course of this study, we provide a unified framework for Kazhdan-Lusztig parameters on the one hand, and Springer parameters on the other hand. Our framework contains a complex parameter , and allows us to interpolate between and . When , we recover the parameters which occur in the Springer correspondence; when , we recover the Kazhdan-Lusztig parameters
Point Vortices on the Sphere: Stability of Symmetric Relative Equilibria
No full textWe describe the linear and nonlinear stability and instability of certain symmetric configurations of point vortices on the sphere forming relative equilibria. These configurations consist of one or two rings, and a ring with one or two polar vortices. Such configurations have dihedral symmetry, and the symmetry is used to block diagonalize the relevant matrices, to distinguish the subspaces on which their eigenvalues need to be calculated, and also to describe the bifurcations that occur as eigenvalues pass through zero
Micro-chaos in Relay Feedback Systems with Bang-Bang Control and Digital Sampling
No full textWe investigate a class of linear relay feedback systems with bang-bang control and
with the control input applied at discrete time instances. Using a third order system as a
representative example we show that stable oscillations with so-called sliding motion, with
sliding present in continuous time system, loose the sliding segment of evolution, but do not
loose their stability if the open loop system is stable. We then carry on our investigations
and consider a situation when stable self-sustained oscillations are generated with the unstable
open loop system. In the latter case a transition from a stable limit cycle to micro-chaotic
oscillations occurs. The presence of micro-chaotic oscillations is shown by considering a linearised
map that maps a small neighbourhood of initial conditions back to itself. Using this map the
presence of the positive Lyapunov exponent is shown. The largest Lyapunov exponent is then
calculated numerically for an open set of sampling times, and it is shown that it is positive. The
boundedness of the attractor is ensured for sufficiently small sampling times; with the sampling
time tending to zero these switchings become faster and they turn into sliding motion. It is
the presence of the underlying sliding evolution that ensures the boundedness of the chaotic
attractor. Our finding implies that what may be considered as noise in systems with digital
control should actually be termed as micro-chaotic behaviour. This information may be helpful
in designing digital control systems where any element contributing to what appears as noise
should be suppressed
Product Scheduling for Thermal Energy Reduction in Papermakig Industries
No full textPapermaking is considered as an energy-intensive industry partly due to the fact
that the machinery and procedures have been designed at the time when energy was both cheap
and plentiful. A typical paper machine manufactures a variety of di�erent products (grades)
which impose variable per-unit raw material and energy costs to the mill. It is known that
during a grade change operation the products are not market-worthy. Therefore, two di�erent
production regimes, i.e. steady state and grade transition can be recognised in papermaking
practice. Among the costs associated with paper manufacture, the energy cost is �more variable�
due to (usually) day-to-day variations of the energy prices. Moreover, the production of a grade
is often constrained by customer delivery time requirements. Given the above constraints and
production modes, the product scheduling technique proposed in this paper aims at optimising
the sequence of orders in a single machine so that the cost of production (mainly determined
by the energy) is minimised. Simulation results obtained from a commercial board machine in
the UK con�rm the e�ectiveness of the proposed method
A probabilistic approach to identify putative drug targets in biochemical networks
No full textNetwork-based drug design holds great promise in clinical research as a way to overcome the limitations of traditional approaches in the development of drugs with high efficacy and low toxicity. This novel strategy aims to study how a biochemical network as a whole, rather than its individual components, responds to specific perturbations in different physiological conditions. Proteins exerting little control over normal cells and larger control over altered cells may be considered as good candidates for drug targets. The application of network-based drug design would greatly benefit from using an explicit computational model describing the dynamics of the system under investigation. However, creating a fully characterized kinetic model is not an easy task, even for relatively small networks, as it is still significantly hampered by the lack of data about kinetic mechanisms and parameters values. Here, we propose a Monte Carlo approach to identify the differences between flux control profiles of a metabolic network in different physiological states, when information about the kinetics of the system is partially or totally missing. Based on experimentally accessible information on metabolic phenotypes, we develop a novel method to determine probabilistic differences in the flux control coefficients between the two observable phenotypes. Knowledge of how differences in flux control are distributed among the different enzymatic steps is exploited to identify points of fragility in one of the phenotypes. Using a prototypical cancerous phenotype as an example, we demonstrate how our approach can assist researchers in developing compounds with high efficacy and low toxicity
Distributional and local limit laws for a class of iterated maps that contract on average
No full textWe consider iterated function schemes that contract on average
with place-dependent probabilities. We are interested in generalisations
of the central limit theorem, particularly to observations with
infinite variance. By studying the spectral properties of an associated
one-parameter family of transfer operators acting on an appropriate
function space, we prove both a distributional and local limit law with
convergence to a stable distribution