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A Rational Krylov Toolbox for MATLAB
Get PDFThe Rational Krylov Toolbox contains MATLAB implementations of Ruhe's rational Krylov sequence method, algorithms for the implicit and explicit relocation of the poles of a rational Krylov space, and an implementation of RKFIT, a robust algorithm for rational least squares fitting
A new strain energy function for the hyperelastic modelling of ligaments and tendons based on fascicle microstructure
Get PDFA new strain energy function for the hyperelastic modelling of ligaments and tendons based on the geometrical arrangement of their fibrils is derived. The distribution of the crimp angles of the fibrils is used to determine the stress-strain response of a single fascicle, and this stress-strain response is used to determine the form of the strain energy function, the parameters of which can all potentially be directly measured via experiments - unlike those of commonly used strain energy functions such as the Holzapfel-Gasser-Ogden (HGO) model, whose parameters are phenomenological. We compare the new model with the HGO model and show that the new model gives a better match to existing stress-strain data for human patellar tendon than the HGO model, with the average relative error in matching this data when using the new model being 0.053 (compared with 0.57 when using the HGO model), and the average absolute error when using the new model being 0.12MPa (compared with 0.31MPa when using the HGO model)
A Rational Krylov Toolbox for MATLAB
Get PDFThe Rational Krylov Toolbox contains MATLAB implementations of Ruhe's rational Krylov sequence method, algorithms for the implicit and explicit relocation of the poles of a rational Krylov space, and an implementation of RKFIT, a robust algorithm for rational least squares fitting
A stochastic model for early placental development
Get PDFIn the human, placental structure is closely related to placental function and consequent pregnancy outcome. Studies have noted abnormal placental shape in small-for-gestational-age infants which extends to increased lifetime risk of cardiovascular disease. The origins and determinants of placental shape are incompletely understood and are difficult to study in vivo. In this paper, we model the early development of the human placenta, based on the hypothesis that this is driven by a chemoattractant effect emanating from proximal spiral arteries in the decidua. We derive and explore a two-dimensional stochastic model, and investigate the effects of loss of spiral arteries in regions near to the cord insertion on the shape of the placenta. This model demonstrates that disruption of spiral arteries can exert profound effects on placental shape, particularly if this is close to the cord insertion. Thus, placental shape reflects the underlying maternal vascular bed. Abnormal placental shape may reflect an abnormal uterine environment, predisposing to pregnancy complications. Through statistical analysis of model placentas, we are able to characterize the probability that a given placenta grew in a disrupted environment, and even able to distinguish between different disruptions
Error Analysis of Diffusion Approximation Methods for Multiscale Systems in Reaction Kinetics
Get PDFSeveral different methods exist for efficient approximation of paths
in multiscale stochastic chemical systems. Another approach is to
use bursts of stochastic simulation to estimate the parameters of a
stochastic differential equation approximation of the paths. In this
paper, multiscale methods for approximating paths are used to
formulate different strategies for estimating the dynamics by
diffusion processes. We then analyse how efficient and accurate
these methods are in a range of different scenarios, and compare
their respective advantages and disadvantages to other methods
proposed to analyse multiscale chemical networks
Invariant measures for the n-dimensional border collision normal form
Get PDFThe border collision normal form is a continuous piecewise affine map of \BR^n with applications in piecewise smooth bifurcation theory. We show that these maps have absolutely continuous invariant measures for an open set of parameter space and hence that the attractors have Hausdorff (fractal) dimension . If the attractors have topological dimension two, i.e. they contain open sets, and if then they have topological dimension generically
Functions of Matrices
No full textMatrix functions are used in many areas of linear algebra and arise
in numerous applications in science and engineering.
The most common matrix function is the matrix inverse;
it is not treated specifically in this chapter,
but is covered in Section~1.5 and Section~51.3.
This chapter is concerned with general matrix functions as
well as specific cases such as matrix square roots,
trigonometric functions,
and the exponential and logarithmic functions
Tropical roots as approximations to eigenvalues of matrix polynomials
No full textThe tropical roots of
are points at which the maximum is attained at least twice.
These roots, which can be computed in only operations, can be good approximations to the moduli of
the eigenvalues of the matrix polynomial , in particular when the norms of the matrices vary widely.
Our aim is to investigate this observation and its applications.
We start by providing annuli defined in terms of the tropical roots of that contain the eigenvalues of .
Our localization results yield conditions under which tropical roots offer order of magnitude approximations to the moduli of the eigenvalues of .
Our tropical localization of eigenvalues are less tight than
eigenvalue localization results derived from a generalized matrix version of Pellet's theorem but they are easier to interpret.
Tropical roots are already used to determine the starting points for matrix polynomial eigensolvers based on scalar polynomial root solvers such as the Ehrlich-Aberth method
and our results further justify this choice.
Our results provide the basis for analyzing the effect of Gaubert and Sharify's tropical scalings for
on (a) the conditioning of linearizations of tropically scaled and
(b) the backward stability of eigensolvers based on linearizations of tropically scaled .
We anticipate that the tropical roots of , on which
the tropical scalings are based, will help designing polynomial eigensolvers with better numerical properties than standard algorithms for polynomial eigenvalue problems
such as that implemented in the MATLAB function \texttt{polyeig}
Higher Order Frechet Derivatives of Matrix Functions and the Level-2 Condition Number
Get PDFThe Fr\'echet derivative of a matrix function
controls the sensitivity of the function to small perturbations
in the matrix.
While much is known about the properties of
and how to compute it,
little attention has been given to higher order Fr\'echet derivatives.
We derive sufficient conditions for the
th Fr\'echet derivative to exist and be continuous
in its arguments
and we develop algorithms for computing the th derivative and its
Kronecker form.
We analyze the level-2 absolute condition number of a
matrix function (``the condition number of the condition number'')
and bound it in terms of the second Fr\'echet derivative.
For normal matrices and the exponential we show that in the 2-norm
the level-1 and level-2 absolute condition numbers are equal
and that the relative condition numbers
are within a small constant factor of each other.
We also obtain an exact relationship between the level-1 and level-2
absolute condition numbers for
the matrix inverse and arbitrary nonsingular matrices,
as well as a weaker connection for Hermitian matrices
for a class of functions that includes the logarithm and square root.
Finally, the relation between the level-1 and level-2 condition numbers
is investigated more generally through numerical experiments