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Extensions of the Nyström method for the numerical solution of linear integral equations of the second kindAtkinson, Kendall E. January 1966 (has links)
Thesis (Ph. D.)--University of Wisconsin, 1966. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.
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Aspects of Toeplitz operators and matrices : asymptotics, norms, singular values / Hermann RabeRabe, Hermann January 2015 (has links)
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
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Aspects of Toeplitz operators and matrices : asymptotics, norms, singular values / Hermann RabeRabe, Hermann January 2015 (has links)
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
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Spectral theory of self-adjoint higher order differential operators with eigenvalue parameter dependent boundary conditionsZinsou, Bertin 05 September 2012 (has links)
We consider on the interval [0; a], rstly fourth-order di erential operators with eigenvalue
parameter dependent boundary conditions and secondly a sixth-order di erential operator
with eigenvalue parameter dependent boundary conditions. We associate to each of these
problems a quadratic operator pencil with self-adjoint operators. We investigate the spectral
proprieties of these problems, the location of the eigenvalues and we explicitly derive the rst
four terms of the eigenvalue asymptotics.
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A GL(3) Kuznetsov Trace Formula and the Distribution of Fourier Coefficients of Maass FormsGuerreiro, João Leitão January 2016 (has links)
We study the problem of the distribution of certain GL(3) Maass forms, namely, we obtain a Weyl’s law type result that characterizes the distribution of their eigenvalues, and an orthogonality relation for the Fourier coefficients of these Maass forms. The approach relies on a Kuznetsov trace formula on GL(3) and on the inversion formula for the Lebedev-Whittaker transform. The family of Maass forms being studied has zero density in the set of all GL(3) Maass forms and contains all self-dual forms. The self-dual forms on GL(3) can also be realised as symmetric square lifts of GL(2) Maass forms by the work of Gelbart-Jacquet. Furthermore, we also establish an explicit inversion formula for the Lebedev-Whittaker transform, in the nonarchimedean case, with a view to applications.
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On the Structured Eigenvalue Problem: Methods, Analysis, and ApplicationsJames P. Vogel (5930360) 17 January 2019 (has links)
<div>This PhD thesis is an important development in the theories, methods, and applications of eigenvalue algorithms for structured matrices. Though eigenvalue problems have been well-studied, the class of matrices that admit very fast (near-linear time) algorithms was quite small until very recently. We developed and implemented a generalization of the famous symmetric tridiagonal divide-and-conquer algorithm to a much larger class of rank structured matrices (symmetric hierarchically semisperable, or HSS) that appear frequently in applications. Altogether, this thesis makes valuable contributions to three different major areas of scientific computing: algorithmic development, numerical analysis, and applications. In addition to the previously stated divide-and-conquer algorithm, we generalize to larger classes of eigenvalue problems and provide several key new low-rank update algorithms. A major contribution the analysis of the structured eigenvalue problem. In addition to standard perturbation analysis, we elucidate some subtle and previously under-examined issues in structured matrix eigenvalue problems such as subspace contributions and secular equation conditioning. Finally, several applications are studied.</div>
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Improved estimation of the eigenvalues in a one-sample and two-sample problem.January 2001 (has links)
Chan Pui Man. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 103-105). / Abstracts in English and Chinese. / Chapter Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Main Problems --- p.1 / Chapter 1.2 --- Motivation --- p.5 / Chapter 1.3 --- Present Works --- p.7 / Chapter Chapter 2 --- Estimation of the Eigenvalues in a Wishart Distribution --- p.11 / Chapter 2.1 --- Review of Previous Works --- p.14 / Chapter 2.2 --- Some Useful Statistical and Mathematical Results --- p.17 / Chapter 2.3 --- Improved Estimation of A under Squared Error Loss L1 --- p.23 / Chapter 2.4 --- Simulation Study for the Wishart Distribution under Squared Error Loss --- p.28 / Chapter 2.5 --- Discussions on Wishart Distribution Under Squared Error Loss --- p.32 / Chapter 2.6 --- Improved Estimation of A under the Entropy Loss(det) L2 --- p.33 / Chapter 2.7 --- Simulation Study for the Wishart Distribution Under Entropy Loss L2 --- p.38 / Chapter 2.8 --- Discussions on Wishart Distribution Under Entropy Loss --- p.44 / Chapter Chapter 3 --- Estimation of the Eigenvalues in a Multivariate F Distribution --- p.46 / Chapter 3.1 --- Review of Previous Works --- p.49 / Chapter 3.2 --- Some Useful Statistical and Mathematical Results --- p.50 / Chapter 3.3 --- Improved Estimation of A under the Squared Loss L1 --- p.54 / Chapter 3.4 --- Simulation Study for F Distribution under Squared Error Loss L1 --- p.62 / Chapter 3.5 --- Discussions on F distribution under Squared Error Loss --- p.68 / Chapter 3.6 --- Improved Estimation of A under the Entropy Loss(det) L2 --- p.69 / Chapter 3.7 --- Simulation Study for Multivariate F Distribution under Entropy Loss(det) L2 --- p.76 / Chapter 3.8 --- Discussions on F distribution under Entropy Loss --- p.86 / Chapter Chapter 4 --- Inheritance of Dominance between Eigenvalues Loss Function and Matrix Function --- p.87 / Chapter 4.1 --- Significance of The Problem --- p.87 / Chapter 4.2 --- Inheritance of Dominance between Eigenvalues Estimator and Matrix Estimator under Squared Error Loss --- p.92 / Chapter 4.3 --- Inheritance of Dominance between Eigenvalues Estimator and Matrix Estimator under Entropy Loss --- p.97 / Chapter 4.4 --- Conclusion --- p.102 / BIBLIOGRAPHY --- p.103
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An Eigenanalysis and Synthesis of Unitary Operators used in Quantum Computing AlgorithmsHutsell, Steven Randall 01 January 2009 (has links)
In this work we tackle the challenge of designing quantum unitary operators which represent solutions to optimization problems. We start with a novel method which combines an evolutionary algorithm known as an Evolution Strategy (ES) with a method to randomly generate unitary operators. With this new method, a quantum operator is represented for the first time using real-valued vectors and can be "evolved" or designed to meet certain target criteria. This criteria could be the solution to an optimization problem. With the ability to evolve quantum operators, we attempt to evolve various known single and multi-qubit quantum gates as well as quantum oracles. We evolve quantum operators which solve instance problems of a known NP-Hard problem and even attempt to evolve a generalized solution operator. We evolve multiple operators with varying size and investigate their properties through eigenanalysis methods as well as by synthesizing them into quantum logic gates using the quantum compiler Qubiter. We also present a new quantum logic algebra which offers a new way to represent quantum circuits and demonstrate its immediate uses in quantum computing.
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Research and Tutorial ExpositionStrang, Gilbert 01 1900 (has links)
My research is concentrated on applications of linear algebra in engineering, including wavelet analysis and structured matrices and (currently) approximation of large dense matrices by a mosaic of low rank blocks. / Singapore-MIT Alliance (SMA)
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Iteration methods for approximating the lowest order energy eigenstate of a given symmetry for one- and two-dimensional systems /Junkermeier, Chad E. January 2003 (has links) (PDF)
Thesis (M.S.)--Brigham Young University. Dept. of Physics and Astronomy, 2003. / Includes bibliographical references (p. 73).
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