Thesis (Ph. D.)--University of Hong Kong, 2006. / Title proper from title frame. Also available in printed format.

published_or_final_version / Mathematics / Master / Master of Philosophy

published_or_final_version / Mathematics / Master / Master of Philosophy

Thesis (M. Phil.)--University of Hong Kong, 1994. / Includes bibliographical references (leaves 40-42).

Thesis (M. Phil.)--University of Hong Kong, 1993. / Photocopy typescript.

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The research contained in this thesis can be divided into two related, but distinct parts. The rst chapter deals with block Toeplitz operators de ned by rational matrix function symbols on discrete sequence spaces. Here we study sequences of operators that converge to the inverses of these Toeplitz operators via an invertibility result involving a special representation of the symbol of these block Toeplitz operators. The second part focuses on a special class of matrices generated by banded Toeplitz matrices, i.e., Toeplitz matrices with a nite amount of non-zero diagonals. The spectral theory of banded Toeplitz matrices is well developed, and applied to solve questions regarding the behaviour of the singular values of Toeplitz-generated matrices. In particular, we use the behaviour of the singular values to deduce bounds for the growth of the norm of the inverse of Toeplitz-generated matrices. In chapter 2, we use a special state-space representation of a rational matrix function on the unit circle to de ne a block Toeplitz operator on a discrete sequence space. A discrete Riccati equation can be associated with this representation which can be used to prove an invertibility theorem for these Toeplitz operators. Explicit formulas for the inverse of the Toeplitz operators are also derived that we use to de ne a sequence of operators that converge in norm to the inverse of the Toeplitz operator. The rate of this convergence, as well as that of a related Riccati di erence equation is also studied. We conclude with an algorithm for the inversion of the nite sections of block Toeplitz operators. Chapter 3 contains the main research contribution of this thesis. Here we derive sharp growth rates for the norms of the inverses of Toeplitz-generated matrices. These results are achieved by employing powerful theory related to the Avram-Parter theorem that describes the distribution of the singular values of banded Toeplitz matrices. The investigation is then extended to include the behaviour of the extreme and general singular values of Toeplitz-generated matrices. We conclude with Chapter 4, which sets out to answer a very speci c question regarding the singular vectors of a particular subclass of Toeplitz-generated matrices. The entries of each singular vector seems to be a permutation (up to sign) of the same set of real numbers. To arrive at an explanation for this phenomenon, explicit formulas are derived for the singular values of the banded Toeplitz matrices that serve as generators for the matrices in question. Some abstract algebra is also employed together with some results from the previous chapter to describe the permutation phenomenon. Explicit formulas are also shown to exist for the inverses of these particular Toeplitz-generated matrices as well as algorithms to calculate the norms and norms of the inverses. Finally, some additional results are compiled in an appendix. / PhD (Mathematics), North-West University, Potchefstroom Campus, 2015

The research contained in this thesis can be divided into two related, but distinct parts. The rst chapter deals with block Toeplitz operators de ned by rational matrix function symbols on discrete sequence spaces. Here we study sequences of operators that converge to the inverses of these Toeplitz operators via an invertibility result involving a special representation of the symbol of these block Toeplitz operators. The second part focuses on a special class of matrices generated by banded Toeplitz matrices, i.e., Toeplitz matrices with a nite amount of non-zero diagonals. The spectral theory of banded Toeplitz matrices is well developed, and applied to solve questions regarding the behaviour of the singular values of Toeplitz-generated matrices. In particular, we use the behaviour of the singular values to deduce bounds for the growth of the norm of the inverse of Toeplitz-generated matrices. In chapter 2, we use a special state-space representation of a rational matrix function on the unit circle to de ne a block Toeplitz operator on a discrete sequence space. A discrete Riccati equation can be associated with this representation which can be used to prove an invertibility theorem for these Toeplitz operators. Explicit formulas for the inverse of the Toeplitz operators are also derived that we use to de ne a sequence of operators that converge in norm to the inverse of the Toeplitz operator. The rate of this convergence, as well as that of a related Riccati di erence equation is also studied. We conclude with an algorithm for the inversion of the nite sections of block Toeplitz operators. Chapter 3 contains the main research contribution of this thesis. Here we derive sharp growth rates for the norms of the inverses of Toeplitz-generated matrices. These results are achieved by employing powerful theory related to the Avram-Parter theorem that describes the distribution of the singular values of banded Toeplitz matrices. The investigation is then extended to include the behaviour of the extreme and general singular values of Toeplitz-generated matrices. We conclude with Chapter 4, which sets out to answer a very speci c question regarding the singular vectors of a particular subclass of Toeplitz-generated matrices. The entries of each singular vector seems to be a permutation (up to sign) of the same set of real numbers. To arrive at an explanation for this phenomenon, explicit formulas are derived for the singular values of the banded Toeplitz matrices that serve as generators for the matrices in question. Some abstract algebra is also employed together with some results from the previous chapter to describe the permutation phenomenon. Explicit formulas are also shown to exist for the inverses of these particular Toeplitz-generated matrices as well as algorithms to calculate the norms and norms of the inverses. Finally, some additional results are compiled in an appendix. / PhD (Mathematics), North-West University, Potchefstroom Campus, 2015

Some recently discussed stationary processes like fractionally integrated processes cannot be described by low order autoregressive or moving average (ARMA) models rendering the common algorithms for generation estimation and prediction partly very misleading. We offer an unified approach based on the Cholesky decomposition of the covariance matrix which makes these problems exactly solvable in an efficient way. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

In this thesis, we mainly study preconditioning techniques for all-at-once linear systems arising from discretization of three types of time-dependent advection-diffusion equation: linear diffusion equation, constant-coefficients advection-diffusion equation, time-fractional sub-diffusion equation. The proposed preconditioners are used with Krylov subspace solvers. The preconditioner developed for linear diffusion equation is based on -circulant ap- proximation of temporal discretization. Diagonalizability, clustering of spectrum and identity-plus-low-rank decomposition are derived for the preconditioned matrix. We also show that generalized minimal residual (GMRES) solver for the preconditioned system has a linear convergence rate independent of matrix-size. The preconditioner for constant-coefficients advection-diffusion equation is based on approximating the discretization of advection term with a matrix diagonalizable by sine transform. Eigenvalues of the preconditioned matrix are proven to be lower and upper bounded by positive constants independent of discretization parameters. Moreover, as the preconditioner is based on spatial approximation, it is also applicable to steady-state problem. We show that GMRES for the preconditioned steady-state problem has a linear convergence rate independent of matrix size. The preconditioner for time-fractional sub-diffusion equation is based on approximat- ing the discretization of diffusion term with a matrix diagonalizable by sine transform. We show that the condition number of the preconditioned matrix is bounded by a constant independent of discretization parameters so that the normalized conjugate gradient (NCG) solver for the preconditioned system has a linear convergence rate independent of discretization parameters and matrix size. Fast implementations based on fast Fourier transform (FFT), fast sine transform (FST) or multigrid approximation are proposed for the developed preconditioners. Numerical results are reported to show the performance of the developed preconditioners