DSpace Collection:
https://dspace.lboro.ac.uk/2134/4642
2015-03-31T23:59:51ZThe numerical solution of quadratic matrix equations
https://dspace.lboro.ac.uk/2134/16687
Title: The numerical solution of quadratic matrix equations
Authors: Mahmood, Khalid
Abstract: Methods for computing an efficient and accurate numerical solution of the real monic
unilateral quadratic matrix equation,
are few. They are not guaranteed to work on all problems. One of the methods performs a
sequence of Newton iterations until convergence occurs whilst another is a matrix analogy
of the scalar polynomial algorithm. The former fails from a poor starting point and the
latter fails if no dominant solution exists. A recent approach, the Elimination method,
is analysed and shown to work on problems for which other methods fail. . The method
requires the coefficients of the characteristic polynomial of a matrix to be computed and
to this end a comparative numerical analysis of a number of methods for computing the
coefficients is performed. A new minimisation approach for solving the quadratic matrix
equation is proposed and shown to compare very favourably with existing methods .
. A special case of the quadratic matrix equation is the matrix square root problem,
where P = o. There have been a number of method proposed for it's solution, the more
successful ones being based upon Newton iterations or the Schur factorisation. The Elimination
method is used as a basis for generating three methods for solving the matrix square
root problem. By means of a numerical analysis and results it is shown that for small order
problems the Elimination methods compare favourably with the existing methods.
The algebraic Riccati equation of stochastic and optimal control is,
where the solution of interest is the symmetric non-negative definite one. The current
methods are based on Newton iterations or the determination of the invariant subspace of
the associated Hamiltonian matrix. A new method based on a reformulation of Newton's
method is presented. The method reduces the work involved at each iteration by introducing
a Schur factorisation and a sparse linear system solver. Numerical results suggest
that it may compare favourably with well-established methods.
Central to the numerical issues are the discussions on conditioning, stability and accuracy.
For a method to yield accurate results, the problem must be well-conditioned and the
method that solves the problem must be stable-consequently discussions on conditioning
and stability feature heavily in this thesis.
The units of measure we use to compare the speed of the methods are the operations
count and the Central Processor Unit (CPU) time. We show how the CPU time accurately
reflects the amount of work done by an algorithm and that the operations counts of the
algorithms correspond with the respective CPU times.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.1990-01-01T00:00:00ZA study of matrix equations
https://dspace.lboro.ac.uk/2134/16686
Title: A study of matrix equations
Authors: McDonald, Eileen M.
Abstract: Matrix equations have been studied by Mathematicians for many
years. Interest in them has grown due to the fact that these
equations arise in many different fields such as vibration analysis,
optimal control, stability theory etc.
This thesis is concerned with methods of solution of various
matrix equations with particular emphasis on quadratic matrix
equations. Large scale numerical techniques are not investigated
but algebraic aspects of matrix equations are considered.
Many established methods are described and the solution of a
matrix equation by consideration of an equivalent system of
multivariable polynomial equations is investigated. Matrix equations
are also solved by a method which combines the given equation with
the characteristic equation of the unknown matrix.
Several iterative processes used for the solution of scalar
equations are applied directly to the matrix equation. A new
iterative process based on elimination methods is also described
and examples given.
The solutions of the equation x2 = P are obtained by a method
which derives a set of polynomial equations connecting the
characteristic coefficients of X and P. It is also shown that
the equation X2 = P has an infinite number of solutions if P is a
derogatory matrix.
Acknowledgements
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.1987-01-01T00:00:00ZModelling of driven free surface liquid films
https://dspace.lboro.ac.uk/2134/16574
Title: Modelling of driven free surface liquid films
Authors: Galvagno, Mariano
Abstract: In several types of coating processes a solid substrate is removed at a controlled velocity U from a liquid bath. The shape of the liquid meniscus and the thickness of the coating layer depend on U. These dependencies have to be understood in detail for non-volatile liquids to control the deposition of such a liquid and to lay the basis for the control in more complicated cases (volatile pure liquid, solution with volatile solvent). We study the case of non-volatile liquids employing a precursor film model that describes partial wettability with a Derjaguin (or disjoining) pressure. In particular, we focus on the relation of the deposition of (i) an ultrathin precursor film at small velocities and (ii) a macroscopic film of thickness h ∝ U^(2/3) (corresponding to the classical Landau Levich film). Depending on the plate inclination, four regimes are found for the change from case (i) to (ii). The different regimes and the transitions between them are analysed employing numerical continuation of steady states and saddle-node bifurcations and simulations in time. We discuss the relation of our results to results obtained with a slip model.
In connection with evaporative processes, we will study the pinning of a droplet due to a sharp corner. The approach employs an evolution equation for the height profile of an evaporating thin film (small contact angle droplet) on a substrate with a rounded edge, and enables one to predict the dependence of the apparent contact angle on the position of the contact line. The calculations confirm experimental observations, namely that there exists a dynamically produced critical angle for depinning that increases with the evaporation rate. This suggests that one may introduce a simple modification of the Gibbs criterion for pinning that accounts for the non-equilibrium effect of evaporation.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2015-01-01T00:00:00ZHigher-order airy functions of the first kind and spectral properties of the massless relativistic quartic anharmonic oscillator
https://dspace.lboro.ac.uk/2134/16497
Title: Higher-order airy functions of the first kind and spectral properties of the massless relativistic quartic anharmonic oscillator
Authors: Durugo, Samuel
Abstract: This thesis consists of two parts. In the first part, we study a class of special functions Aik (y), k = 2, 4, 6, · · · generalising the classical Airy function Ai(y) to higher orders and in the second part, we apply expressions and properties of Ai4(y) to spectral problem of a specific operator. The first part is however motivated by latter part.
We establish regularity properties of Aik (y) and particularly show that Aik (y) is smooth, bounded, and extends to the complex plane as an entire function, and obtain pointwise bounds on Aik (y) for all k. Some analytic properties of Aik (y) are also derived allowing one to express Aik (y) as a finite sum of certain generalised hypergeometric functions. We further obtain full asymptotic expansions of Aik (y) and their first derivative Ai'(y) both for y > 0 and for y < 0. Using these expansions, we derive expressions for the negative real zeroes of Aik (y) and Ai'(y).
Using expressions and properties of Ai4(y), we extensively study spectral properties of a non-local operator H whose physical interpretation is the massless relativistic quartic anharmonic oscillator in one dimension. Various spectral results for H are derived including estimates of eigenvalues, spectral gaps and trace formula, and a Weyl-type asymptotic relation. We study asymptotic behaviour, analyticity, and uniform boundedness properties of the eigenfunctions ψn(x) of H. The Fourier transforms of these eigenfunctions are expressed in two terms, one involving Ai4(y) and another term derived from Ai4(y) denoted by AI4(y). By investigating the small effect generated by AI4(y) this work shows that eigenvalues λn of H are exponentially close, with increasing n ∈ N, to the negative real zeroes of Ai4(y) and those of its first derivative Ai'4(y) arranged in alternating and increasing order of magnitude. The eigenfunctions ψn(x) are also shown to be exponentially well-approximated by the inverse Fourier transform of Ai4(|y| − λn) in its normalised form.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2014-01-01T00:00:00Z