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Author: Joohwan Chun
Publisher:
ISBN:
Category :
Languages : en
Pages : 298
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Book Description
Many engineering or mathematical problems require to factorize structured matrices (Toeplitz, Hankel, Vandermonde products of such matrices and their inverses. Schur complements, etc) either in explicit or in disguised form. Consequently there exist various analytic tools regarding structured matrices as well as several fast factorization algorithms. In this thesis, we show that many of these results and several significant generalizations can be obtained in a very constructive way. The genetic form is to use elementary circular and hyperbolic transformations to triangularize a certain array of numbers derived from the displacement representation of the given structured matrix; the desired results can then be read off from the resulting array. These fast array algorithms require O (mn) operations for LU and QR factorizations of m x n structured matrices, and O(mn) or even O(n log square n) operations for solving matrix equations. Also the array form suggests various alternative algorithms, depending upon the order in which the transformations are applied; these variations can have different numerical properties and lead to different implementations.
Author: Joohwan Chun
Publisher:
ISBN:
Category :
Languages : en
Pages : 298
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Book Description
Many engineering or mathematical problems require to factorize structured matrices (Toeplitz, Hankel, Vandermonde products of such matrices and their inverses. Schur complements, etc) either in explicit or in disguised form. Consequently there exist various analytic tools regarding structured matrices as well as several fast factorization algorithms. In this thesis, we show that many of these results and several significant generalizations can be obtained in a very constructive way. The genetic form is to use elementary circular and hyperbolic transformations to triangularize a certain array of numbers derived from the displacement representation of the given structured matrix; the desired results can then be read off from the resulting array. These fast array algorithms require O (mn) operations for LU and QR factorizations of m x n structured matrices, and O(mn) or even O(n log square n) operations for solving matrix equations. Also the array form suggests various alternative algorithms, depending upon the order in which the transformations are applied; these variations can have different numerical properties and lead to different implementations.
Author: Vadim Olshevsky
Publisher: American Mathematical Soc.
ISBN: 0821831771
Category : Mathematics
Languages : en
Pages : 448
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Book Description
One of the best known fast computational algorithms is the fast Fourier transform method. Its efficiency is based mainly on the special structure of the discrete Fourier transform matrix. Recently, many other algorithms of this type were discovered, and the theory of structured matrices emerged. This volume contains 22 survey and research papers devoted to a variety of theoretical and practical aspects of the design of fast algorithms for structured matrices and related issues. Included are several papers containing various affirmative and negative results in this direction. The theory of rational interpolation is one of the excellent sources providing intuition and methods to design fast algorithms. The volume contains several computational and theoretical papers on the topic. There are several papers on new applications of structured matrices, e.g., to the design of fast decoding algorithms, computing state-space realizations, relations to Lie algebras, unconstrained optimization, solving matrix equations, etc. The book is suitable for mathematicians, engineers, and numerical analysts who design, study, and use fast computational algorithms based on the theory of structured matrices.
Author: T. Kailath
Publisher: SIAM
ISBN: 0898714311
Category : Computers
Languages : en
Pages : 350
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Book Description
This book deals with the combined issues of speed and numerical reliability in algorithm development.
Author: T. Kailath
Publisher: SIAM
ISBN: 9781611971354
Category : Computers
Languages : en
Pages : 351
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Book Description
This book is the first to pay special attention to the combined issues of speed and numerical reliability in algorithm development. These two requirements have often been regarded as competitive, so much so that the design of fast and numerically reliable algorithms for large-scale structured systems of linear equations, in many cases, remains a significant open issue. Fast Reliable Algorithms for Matrices with Structure helps bridge this gap by providing the reader with recent contributions written by leading experts in the field. The authors deal with both the theory and the practice of fast numerical algorithms for large-scale structured linear systems. Each chapter covers in detail different aspects of the most recent trends in the theory of fast algorithms, with emphasis on implementation and application issues. Both direct and iterative methods are covered. This book is not merely a collection of articles. The editors have gone to considerable lengths to blend the individual papers into a consistent presentation. Each chapter exposes the reader to some of the most recent research while providing enough background material to put the work into proper context.
Author: Peter Arbenz
Publisher: Nova Publishers
ISBN: 9781560725947
Category : Business & Economics
Languages : en
Pages : 228
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Book Description
Comprises 10 contributions that summarize the state of the art in the areas of high performance solutions of structured linear systems and structured eigenvalue and singular-value problems. Topics covered range from parallel solvers for sparse or banded linear systems to parallel computation of eigenvalues and singular values of tridiagonal and bidiagonal matrices. Specific paper topics include: the stable parallel solution of general narrow banded linear systems; efficient algorithms for reducing banded matrices to bidiagonal and tridiagonal form; a numerical comparison of look-ahead Levinson and Schur algorithms for non-Hermitian Toeplitz systems; and parallel CG-methods automatically optimized for PC and workstation clusters. Annotation copyrighted by Book News, Inc., Portland, OR
Author: Victor Y. Pan
Publisher: Springer Science & Business Media
ISBN: 1461201292
Category : Mathematics
Languages : en
Pages : 299
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Book Description
This user-friendly, engaging textbook makes the material accessible to graduate students and new researchers who wish to study the rapidly exploding area of computations with structured matrices and polynomials. The book goes beyond research frontiers and, apart from very recent research articles, includes previously unpublished results.
Author: Debajyoti Pal
Publisher:
ISBN:
Category :
Languages : en
Pages : 422
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Book Description
Triangular factorization, solution to linear equations, inversion, computation of rank profile and inertia (in the Hermitian case) etc. of general n x n matrices require O(n cubed) operations. For certain structured matrices including Toeplitz and Hankel matrices the computational complexity is known to be O(n squared) or better. These structured matrices often arise in a wide variety of areas including Signal processing. Systems theory and Communications. Fast (i.e. O(n squared)) algorithms for these structured matrices have been actively studied for over twenty five years. However almost all the authors have assumed that the underlying matrices are strongly regular i.e. every principal submatrix is nonsingular. Although some fast algorithms have recently been developed for certain problems involving some of these structured matrices which may have one or more zero minors, several other problems is lacking. In this dissertation, we obtain several new results through a unified approach to the problems mentioned earlier.
Author: David Miller
Publisher:
ISBN:
Category : Electronic dissertations
Languages : en
Pages :
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Book Description
The Vandermonde matrix and Cauchy matrix are classical and are encountered in polynomial and rational interpolation computation respectively. The structure of these matrices lead to fast inversion algorithms and system solvers. We look to extend these properties to other structured matrices, including Cauchy-Vandermonde matrices and systems involving Laurent polynomials.
Author: Michele Benzi
Publisher: Springer
ISBN: 3319498878
Category : Mathematics
Languages : en
Pages : 413
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Book Description
Focusing on special matrices and matrices which are in some sense `near’ to structured matrices, this volume covers a broad range of topics of current interest in numerical linear algebra. Exploitation of these less obvious structural properties can be of great importance in the design of efficient numerical methods, for example algorithms for matrices with low-rank block structure, matrices with decay, and structured tensor computations. Applications range from quantum chemistry to queuing theory. Structured matrices arise frequently in applications. Examples include banded and sparse matrices, Toeplitz-type matrices, and matrices with semi-separable or quasi-separable structure, as well as Hamiltonian and symplectic matrices. The associated literature is enormous, and many efficient algorithms have been developed for solving problems involving such matrices. The text arose from a C.I.M.E. course held in Cetraro (Italy) in June 2015 which aimed to present this fast growing field to young researchers, exploiting the expertise of five leading lecturers with different theoretical and application perspectives.
Author: Yuli Eidelman
Publisher: Springer Science & Business Media
ISBN: 303480606X
Category : Mathematics
Languages : en
Pages : 404
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Book Description
This two-volume work presents a systematic theoretical and computational study of several types of generalizations of separable matrices. The main attention is paid to fast algorithms (many of linear complexity) for matrices in semiseparable, quasiseparable, band and companion form. The work is focused on algorithms of multiplication, inversion and description of eigenstructure and includes a large number of illustrative examples throughout the different chapters. The first volume consists of four parts. The first part is of a mainly theoretical character introducing and studying the quasiseparable and semiseparable representations of matrices and minimal rank completion problems. Three further completions are treated in the second part. The first applications of the quasiseparable and semiseparable structure are included in the third part where the interplay between the quasiseparable structure and discrete time varying linear systems with boundary conditions play an essential role. The fourth part contains factorization and inversion fast algorithms for matrices via quasiseparable and semiseparable structure. The work is based mostly on results obtained by the authors and their coauthors. Due to its many significant applications and the accessible style the text will be useful to engineers, scientists, numerical analysts, computer scientists and mathematicians alike.
