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Blocking response surface designs
Get PDFThe design of experiments involving more than one blocking factor and quantitative explanatory variables is discussed, the focus being on two key aspects of blocked response surface designs: optimality and orthogonality. First, conditions for orthogonally blocked experiments are derived. Next, an algorithmic approach to compute D-optimal designs is presented. Finally, the relationships between design optimality and orthogonality in the context of response surface experiments are discussed in detail
Symmetric Linearizations for Matrix Polynomials
No full textA standard way of treating the polynomial eigenvalue problem
P(\l)x = 0 is to convert it
into an equivalent matrix pencil---a process known as linearization.
Two vector spaces of pencils
\Ell_1(P) and \Ell_2(P), and their intersection \DL(P),
have recently been defined and studied by
Mackey, Mackey, Mehl, and Mehrmann.
The aim of our work is to gain new insight into these spaces
and the extent to which their constituent pencils inherit
structure from \@.
For arbitrary polynomials we show that every pencil in \DL(P)
is block symmetric
and we obtain a convenient basis for \DL(P) built from block Hankel matrices.
This basis is then exploited to
prove that
the first pencils in a sequence constructed by
Lancaster in the 1960s generate \DL(P).
When is symmetric, we show that
the symmetric pencils in \Ell_1(P) comprise
\DL(P),
while for Hermitian the Hermitian pencils in \Ell_1(P)
form a proper subset of \DL(P) that we explicitly characterize.
Almost all pencils in
each of these subsets are shown to be linearizations.
In addition to obtaining new results, this work provides a self-contained
treatment of some of the key properties of \DL(P)
together with some new, more concise proofs
An Introduction to the Quality of Computed Solutions
No full textThis report is concerned with the quality of the computed numerical solutions of mathematical problems. We give an introduction to ideas that are important in understanding and measuring the quality of computed solutions. In particular we
review the ideas of condition, stability and error analysis, and their realisation in numerical software. A number of illustrative examples are given
Efficient Algorithms for the Matrix Cosine and Sine
No full textSeveral improvements are made to an algorithm of Higham and Smith for
computing the matrix cosine.
The original algorithm scales the matrix by a power of 2 to bring the
-norm to 1 or less,
evaluates the [8/8] Pad\'e approximant,
then uses the double-angle formula to recover the
cosine of the original matrix.
The first improvement is to phrase truncation error bounds in terms of
\norm{A^2}^{1/2} instead of the (no smaller and potentially much larger
quantity) \norm{A}.
The second is to choose the
degree of the Pad\'e approximant to minimize the computational cost
subject to achieving a desired truncation error.
A third improvement is to use an absolute, rather than relative,
error criterion in the choice of Pad\'e approximant;
this allows the use of higher degree approximants without
worsening an a priori error bound.
Our theory and experiments show that each of these modifications brings a
reduction in computational cost.
Moreover, because the modifications tend to reduce the number of double-angle
steps they usually result in a more accurate computed cosine
in floating point arithmetic.
We also derive an algorithm for computing
both and ,
by adapting the ideas developed for the
cosine and intertwining the cosine and sine double angle recurrences
Topics in Information Geometry
No full textWe introduce first some of the background ideas on information theory and its role in studying
analytic models for stochastic processes and the geometrization of families of measure functions.
This is then used to present the geometry of important examples of the Riemannian manifolds that
arise. Next, we obtain the proof of two theorems that characterise the metric neighbourhoods of the
two distinguished fundamental states: randomness and independence. These methods have had
applications in modelling cryptographic attacks, cosmological void distributions, porous media,
clustering of: galaxies, communications, and amino acids along protein chains in genomes
An Iterative Method for Electrostatic Object Reconstruction in a Half Space
No full textSensing electrodes arranged in or around a display can provide input function for interactive displays. Commercially this is interesting because the sensing electrodes and electronics can be made in the same manufacturing process as that of the display itself thus reducing cost. In engineering terms the electrodes measure capacitance changes resulting from the presence and movement of objects such as hands and fingers in front of the display. At the quasi static frequencies used (100kHz) the human body is conductive and the hands or fingers provide a screen between the capacitive electrodes. There is no need to touch the actual display and the overall system constitutes a touchless gesture input system. Determining the shape of the hand or fingers is a boundary condition reconstruction problem of finding the boundary of an earthed conductive object D from electrostatic measurements. This is the ill-posed problem of recovering the zero-surface of a solution to Laplace s equation from Cauchy data on part of the boundary of a domain. The problem has similarities with object reconstruction in EIT or inverse scattering but is complicated because only a partial Dirichelet-Neumannn map is available as experimental data. We suggest an algorithm where at each iteration we have an approximation on which we calculate approximate Cauchy data by solving a Tikhonov regularized linear system. This data is used to modify the approximation by extrapolation towards the zero-surface giving the next approximation. We implemented the algorithm in two and three space dimensions using the Boundary Element Method for discretization. Numerical results using simulated data with added noise show that simply connected but not necessarily convex objects can be reconstructed with reasonable positional accuracy and approximate shape, but as might be expected the shape is more accurately determined near the plane of measurements
A new orbital-based model for the analysis of experimental molecular change densities: an application to (Z)-N-methyl-C-phenylnitrone
No full textAn alternative to the usual atom-centred multipole expansion is presented for the analysis of high resolution, low-temperature X-ray scattering data. The molecular electron density is determined in a fixed basis of molecular orbitals with variable orbital occupation numbers, i.e. the same form which is used to represent the density in ab initio electron-correlated calculations. The advantages of such an approach include linear scaling (in the sense that the number of parameters to be determined by fitting varies linearly with system size) and ease of property calculation. The method is applied to experimental high-resolution structure factors for a phenylnitrone, and compared to the results of a multipole model of the same data. Finally, the model is critically compared with several related, published orbital-based models
The bredon cohomology of subgroup complexes
No full textWe develop the homological algebra of coefficient systems on a group, in particular from the point of view of calculating higher limits. We show how various sequences of modules associated to a class of subgroups of a given group can be analysed by methods from homological algebra. We are particularly interested in when these sequences are exact, or, if not, when their homology is equal to the higher limits of the coefficient system
Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics
Get PDFThe intended readership includes graduate students and researchers in engineering, numerical analysis, applied mathematics and interdisciplinary scientific computing. The publisher describes the book as follows:
* An excellent introduction to finite elements, iterative linear solvers and scientific computing
* Contains theoretical problems and practical exercises
* All methods and examples use freely available software
* Focuses on theory and computation, not theory for computation
* Describes approximation methods and numerical linear algebr
