11 research outputs found
The numerical solution of sparse matrix equations by fast methods and associated computational techniques
The numerical solution of sparse matrix equations by fast methods and associated computational technique
The numerical solution of banded linear systems by generallized factorization procedures
The numerical solution of banded linear systems by generallized factorization procedure
An elementary algorithm for computing the determinant of pentadiagonal Toeplitz matrices
AbstractOver the last 25 years, various fast algorithms for computing the determinant of a pentadiagonal Toeplitz matrices were developed. In this paper, we give a new kind of elementary algorithm requiring 56⋅⌊n−4k⌋+30k+O(logn) operations, where k≥4 is an integer that needs to be chosen freely at the beginning of the algorithm. For example, we can compute det(Tn) in n+O(logn) and 82n+O(logn) operations if we choose k as 56 and ⌊2815(n−4)⌋, respectively. For various applications, it will be enough to test if the determinant of a pentadiagonal Toeplitz matrix is zero or not. As in another result of this paper, we used modular arithmetic to give a fast algorithm determining when determinants of such matrices are non-zero. This second algorithm works only for Toeplitz matrices with rational entries
Generalized preconditioning strategies
Over the past decade Professor David J. Evans [1968] has suggested the
use of ‘Preconditioning’ in iterative methods for solving large, sparse
systems of linear equations, which arise from the finite difference
approximations to the partial differential equations. Since then, certain
aspects on preconditioning have appeared in the literature and a whole new
theory constructed. The versatility of the preconditioning concept is shown
by the stimulating exploration of new numerical algorithms and methods of
their realization.
The aim of this thesis is to emphasise in the theory we use and
develop together with the practice we state. This study led to a new form
of preconditioning, which has not yet appeared in the literature.
Specifically, we consider the conditioning matrix factorized into two
rectangular matrices, so as to develop a new preconditioned iterative
method and its related properties as well. It requires the selection of
two parameters to be applied, a preconditioning parameter at its optimal
value and an acceleration parameter in such a fashion that a simultaneous displacement method is applicable. [Continues.
Numerical studies of fast electron transport in laser irradiated targets
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Parallel algorithms for the solution of elliptic and parabolic problems on transputer networks
This thesis is a study of the implementation of parallel algorithms for solving
elliptic and parabolic partial differential equations on a network of transputers.
The thesis commences with a general introduction to parallel processing. Here a
discussion of the various ways of introducing parallelism in computer systems and the
classification of parallel architectures is presented.
In chapter 2, the transputer architecture and the associated language OCCAM are
described. The transputer development system (TDS) is also described as well as a
short account of other transputer programming languages. Also, a brief description of
the methodologies for programming transputer networks is given. The chapter is
concluded by a detailed description of the hardware used for the research. [Continues.
Design and analysis of numerical algorithms for the solution of linear systems on parallel and distributed architectures
The increasing availability of parallel computers is having a very significant impact on
all aspects of scientific computation, including algorithm research and software
development in numerical linear algebra. In particular, the solution of linear systems,
which lies at the heart of most calculations in scientific computing is an important
computation found in many engineering and scientific applications.
In this thesis, well-known parallel algorithms for the solution of linear systems are
compared with implicit parallel algorithms or the Quadrant Interlocking (QI) class of
algorithms to solve linear systems. These implicit algorithms are (2x2) block
algorithms expressed in explicit point form notation. [Continues.
Analysis and design of parallel algorithms
The present state of electronic technology is such that factors
affecting computation speed have almost been minimised; switching for
instance is almost instantaneous. Electronic components are so good,
in fact, that the time taken for a logic signal to travel between two
points is now a significant factor of instruction times.
Clearly, with the actual physical size of components being very
small and the high circuit density, there is little scope for improving
computation speech significantly by such means as even denser circuitry
or still faster electronic components. Thus, development of faster
computers will require a new approach that depends on the imaginative
use of existing knowledge.
One such approach is to increase computation speed through
parallelism. Obviously, a parallel computer with p identical processors
is potentially p times as fast as a single computer, although this
limit can rarely be achieved
Preconditioned iterative methods for solving elliptic partial differential equations
Preconditioned iterative methods for solving elliptic partial differential equation
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The Foundations of Infinite-Dimensional Spectral Computations
Spectral computations in infinite dimensions are ubiquitous in the sciences. However, their many applications and theoretical studies depend on computations which are infamously difficult. This thesis, therefore, addresses the broad question,
“What is computationally possible within the field of spectral theory of separable Hilbert spaces?”
The boundaries of what computers can achieve in computational spectral theory and mathematical physics are unknown, leaving many open questions that have been unsolved for decades. This thesis provides solutions to several such long-standing problems.
To determine these boundaries, we use the Solvability Complexity Index (SCI) hierarchy, an idea which has its roots in Smale's comprehensive programme on the foundations of computational mathematics. The Smale programme led to a real-number counterpart of the Turing machine, yet left a substantial gap between theory and practice. The SCI hierarchy encompasses both these models and provides universal bounds on what is computationally possible. What makes spectral problems particularly delicate is that many of the problems can only be computed by using several limits, a phenomenon also shared in the foundations of polynomial root-finding as shown by McMullen. We develop and extend the SCI hierarchy to prove optimality of algorithms and construct a myriad of different methods for infinite-dimensional spectral problems, solving many computational spectral problems for the first time.
For arguably almost any operator of applicable interest, we solve the long-standing computational spectral problem and construct algorithms that compute spectra with error control. This is done for partial differential operators with coefficients of locally bounded total variation and also for discrete infinite matrix operators. We also show how to compute spectral measures of normal operators (when the spectrum is a subset of a regular enough Jordan curve), including spectral measures of classes of self-adjoint operators with error control and the construction of high-order rational kernel methods. We classify the problems of computing measures, measure decompositions, types of spectra (pure point, absolutely continuous, singular continuous), functional calculus, and Radon--Nikodym derivatives in the SCI hierarchy. We construct algorithms for and classify; fractal dimensions of spectra, Lebesgue measures of spectra, spectral gaps, discrete spectra, eigenvalue multiplicities, capacity, different spectral radii and the problem of detecting algorithmic failure of previous methods (finite section method). The infinite-dimensional QR algorithm is also analysed, recovering extremal parts of spectra, corresponding eigenvectors, and invariant subspaces, with convergence rates and error control. Finally, we analyse pseudospectra of pseudoergodic operators (a generalisation of random operators) on vector-valued spaces.
All of the algorithms developed in this thesis are sharp in the sense of the SCI hierarchy. In other words, we prove that they are optimal, realising the boundaries of what digital computers can achieve. They are also implementable and practical, and the majority are parallelisable. Extensive numerical examples are given throughout, demonstrating efficiency and tackling difficult problems taken from mathematics and also physical applications.
In summary, this thesis allows scientists to rigorously and efficiently compute many spectral properties for the first time. The framework provided by this thesis also encompasses a vast number of areas in computational mathematics, including the classical problem of polynomial root-finding, as well as optimisation, neural networks, PDEs and computer-assisted proofs. This framework will be explored in the future work of the author within these settings