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Advanced Algorithms For Polynomial Matrix Eigenvalue Decomposition
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Book Synopsis Advanced Algorithms for Polynomial Matrix Eigenvalue Decomposition by : Jamie Corr
Download or read book Advanced Algorithms for Polynomial Matrix Eigenvalue Decomposition written by Jamie Corr and published by . This book was released on 2017 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Matrix factorisations such as the eigen- (EVD) or singular value decomposition (SVD) offer optimality in often various senses to many narrowband signal processing algorithms.For broadband problems, where quantities such as MIMO transfer functions or cross spectral density matrices are conveniently described by polynomial matrices,such narrowband factorisations are suboptimal at best. To extend the utility of EVD and SVD to the broadband case, polynomial matrix factorisations have gained momentum over the past decade, and a number of iterative algorithms for particularly the polynomial matrix EVD (PEVD) have emerged. Existing iterative PEVD algorithms produce factorisations that are computationally costly (i) to calculate and (ii) to apply. For the former, iterative algorithms at every step eliminate off-diagonal energy, but this can be a slow process. For the latter,the polynomial order of the resulting factors, directly impacting on the implementation complexity, typically grows with every iteration of a PEVD algorithm. The work presented in this thesis helps to reduce both computational complexities. To address algorithmic complexity and convergence speed, this thesis firstly proposes a multiple shift approach, which can eliminate more off-diagonal energy at every iteration compared to existing methods. Equally applicable to the second order sequential best rotation (SBR2) algorithm, the idea here is applied to the family of sequential matrix diagonalisation (SMD) algorithms, for which a convergence proof is presented. Maximum energy transfer requires a very laborious parameter search; it is demonstrated that a very significant reduction of this search space can be gained while retaining almost all of the transfered energy. In addition a number of techniques have been developed which improve the efficiency of these PEVD algorithms. With each PEVD algorithm iteration the lengths of the polynomial eigenvectors and eigenvalues increase. To lower the order of the polynomial eigenvectors a novel truncation technique was developed which takes advantage of an ambiguity in the eigenvalues. A drawback of the multiple shift algorithms is the faster growth of the polynomial eigenvectors and eigenvalues; to mitigate this the search step has been analysed and modified to reduce the growth. The SMD algorithm suffers from a performance bottleneck in its EVD step. Here a cyclic-by-row Jacobi approximation is developed that significantly reduces the computational cost of the SMD algorithm with almost no impact on numerical performance. The impact of these algorithmic advances is studied in a number of applications. Firstly, based on a source model, it is investigated how the conditioning of the input data affects algorithm performance. Secondly, to highlight the benefit of the better converging multiple shift SMD algorithm it is compared to the SBR2 algorithm for broadband angle of arrival estimation, where the proposed multiple shift versions can achieve a more accurate subspace decomposition and hence a more accurate AoA estimation. Lastly, its demonstrated how PEVD algorithms can be utilised to extend other linear algebraic techniques to polynomial matrices. The example used here is the extension of the generalised eigenvalue decomposition (GEVD) to a polynomial matrix GEVD.
Book Synopsis Core-Chasing Algorithms for the Eigenvalue Problem by : Jared L. Aurentz
Download or read book Core-Chasing Algorithms for the Eigenvalue Problem written by Jared L. Aurentz and published by SIAM. This book was released on 2018-07-06 with total page 155 pages. Available in PDF, EPUB and Kindle. Book excerpt: Eigenvalue computations are ubiquitous in science and engineering. John Francis?s implicitly shifted QR algorithm has been the method of choice for small to medium sized eigenvalue problems since its invention in 1959. This book presents a new view of this classical algorithm. While Francis?s original procedure chases bulges, the new version chases core transformations, which allows the development of fast algorithms for eigenvalue problems with a variety of special structures. This also leads to a fast and backward stable algorithm for computing the roots of a polynomial by solving the companion matrix eigenvalue problem. The authors received a SIAM Outstanding Paper prize for this work. This book will be of interest to researchers in numerical linear algebra and their students.
Book Synopsis Numerical Methods for Large Eigenvalue Problems by : Yousef Saad
Download or read book Numerical Methods for Large Eigenvalue Problems written by Yousef Saad and published by SIAM. This book was released on 2011-01-01 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt: This revised edition discusses numerical methods for computing eigenvalues and eigenvectors of large sparse matrices. It provides an in-depth view of the numerical methods that are applicable for solving matrix eigenvalue problems that arise in various engineering and scientific applications. Each chapter was updated by shortening or deleting outdated topics, adding topics of more recent interest, and adapting the Notes and References section. Significant changes have been made to Chapters 6 through 8, which describe algorithms and their implementations and now include topics such as the implicit restart techniques, the Jacobi-Davidson method, and automatic multilevel substructuring.
Book Synopsis Polynomial Matrix Eigenvalue Decomposition Techniques for Multichannel Signal Processing by : Zeliang Wang
Download or read book Polynomial Matrix Eigenvalue Decomposition Techniques for Multichannel Signal Processing written by Zeliang Wang and published by . This book was released on 2017 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Symplectic Methods for the Symplectic Eigenproblem by : Heike Fassbender
Download or read book Symplectic Methods for the Symplectic Eigenproblem written by Heike Fassbender and published by Springer Science & Business Media. This book was released on 2007-05-08 with total page 277 pages. Available in PDF, EPUB and Kindle. Book excerpt: The solution of eigenvalue problems is an integral part of many scientific computations. For example, the numerical solution of problems in structural dynamics, electrical networks, macro-economics, quantum chemistry, and c- trol theory often requires solving eigenvalue problems. The coefficient matrix of the eigenvalue problem may be small to medium sized and dense, or large and sparse (containing many zeroelements). In the past tremendous advances have been achieved in the solution methods for symmetric eigenvalue pr- lems. The state of the art for nonsymmetric problems is not so advanced; nonsymmetric eigenvalue problems can be hopelessly difficult to solve in some situations due, for example, to poor conditioning. Good numerical algorithms for nonsymmetric eigenvalue problems also tend to be far more complex than their symmetric counterparts. This book deals with methods for solving a special nonsymmetric eig- value problem; the symplectic eigenvalue problem. The symplectic eigenvalue problem is helpful, e.g., in analyzing a number of different questions that arise in linear control theory for discrete-time systems. Certain quadratic eigenvalue problems arising, e.g., in finite element discretization in structural analysis, in acoustic simulation of poro-elastic materials, or in the elastic deformation of anisotropic materials can also lead to symplectic eigenvalue problems. The problem appears in other applications as well.
Book Synopsis Polynomial Zerofinding Matrix Algorithms by : Fadi Malek
Download or read book Polynomial Zerofinding Matrix Algorithms written by Fadi Malek and published by . This book was released on 1995 with total page 242 pages. Available in PDF, EPUB and Kindle. Book excerpt: In linear algebra, the eigenvalues of a matrix are equivalently defined as the zeros of its characteristic polynomial. Determining the zeros of polynomials by the computation of the eigenvalues of a corresponding companion matrix turns the table on the usual definition. In this dissertation, the work of Newbery has been expanded and a (complex) symmetric or nonsymmetric companion matrix associated with a given characteristic polynomial has been constructed. Schmeisser's technique for the construction of a tridiagonal companion matrix associated with a polynomial with real zeros has been generalized to polynomials with complex zeros. New matrix algorithms based on Schmeisser's and Fiedler's companion matrices are developed. The matrix algorithm which is based on Schmeisser's matrix uses no initial values and computes the simple and multiple zeros with high accuracy. The algorithms based on Fielder's matrices are applied recursively, and require initial values as approximations to the true zeros of the polynomial. A few techniques concerning the choice of the required initial values are also presented. An important part of this thesis is the design of a new composite three-stage matrix algorithm for finding the real and complex zeros of polynomials. The composite algorithm reduces a polynomial with multiple zeros to another polynomial with simple zeros which are then computed with high accuracy. The exact multiplicities of these zeros are then calculated by means of Lagouanelle's limiting formula. The QR algorithm has been used in all the algorithms to find the eigenvalues of the companion matrices. The effectiveness of these algorithms is illustrated by presenting numerical results based on polynomials taken from the literature and considered to be ill-conditioned, as well as random polynomials with randomly generated zeros in small and large clusters. Polynomials are represented and evaluated in quadruple precision; but it suffices to use a double precision QR algorithm in order to obtain almost double precision in the zeros of polynomials.
Book Synopsis Structural Dynamics Fundamentals and Advanced Applications, Volume I by : Alvar M. Kabe
Download or read book Structural Dynamics Fundamentals and Advanced Applications, Volume I written by Alvar M. Kabe and published by Academic Press. This book was released on 2020-06-28 with total page 930 pages. Available in PDF, EPUB and Kindle. Book excerpt: The two-volume work, Structural Dynamics Fundamentals and Advanced Applications, is a comprehensive work that encompasses the fundamentals of structural dynamics and vibration analysis, as well as advanced applications used on extremely large and complex systems. Volume I covers Newton’s Laws, single-degree-of-freedom systems, damping, transfer and frequency response functions, transient vibration analysis (frequency and time domain), multi-degree-of-freedom systems, forced vibration of single and multi-degree-of-freedom systems, numerical methods for solving for the responses of single and multi-degree-of-freedom systems, and symmetric and non-symmetric eigenvalue problems. In addition, a thorough discussion of real and complex modes, and the conditions that lead to each is included. Stochastic methods for single and multi-degree-of-freedom systems excited by random forces or base motion are also covered. Dr. Kabe’s training and expertise are in structural dynamics and Dr. Sako’s are in applied mathematics. Their collaboration has led to the development of first-of-a-kind methodologies and solutions to complex structural dynamics problems. Their experience and contributions encompass numerous past and currently operational launch and space systems. The two-volume work was written with both practicing engineers and students just learning structural dynamics in mind Derivations are rigorous and comprehensive, thus making understanding the material easier Presents analysis methodologies adopted by the aerospace community to solve extremely complex structural dynamics problems
Book Synopsis Structured Matrix Based Methods for Approximate Polynomial GCD by : Paola Boito
Download or read book Structured Matrix Based Methods for Approximate Polynomial GCD written by Paola Boito and published by Springer Science & Business Media. This book was released on 2012-03-13 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt: Defining and computing a greatest common divisor of two polynomials with inexact coefficients is a classical problem in symbolic-numeric computation. The first part of this book reviews the main results that have been proposed so far in the literature. As usual with polynomial computations, the polynomial GCD problem can be expressed in matrix form: the second part of the book focuses on this point of view and analyses the structure of the relevant matrices, such as Toeplitz, Toepliz-block and displacement structures. New algorithms for the computation of approximate polynomial GCD are presented, along with extensive numerical tests. The use of matrix structure allows, in particular, to lower the asymptotic computational cost from cubic to quadratic order with respect to polynomial degree.
Book Synopsis 2021 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA) by : IEEE Staff
Download or read book 2021 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA) written by IEEE Staff and published by . This book was released on 2021-10-17 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: WASPAA is sponsored by the Audio and Acoustic Signal Processing Technical Committee of the IEEE Signal Processing Society The objective of this workshop is to provide an informal environment for the discussion of problems in audio and acoustics and signal processing techniques leading to novel solutions Topic areas broadly include acoustic signal processing and music signal processing, together with relevant applications
Book Synopsis Algorithms and Perturbation Theory for Matrix Eigenvalue Problems and the Singular Value Decomposition by : Yuji Nakatsukasa
Download or read book Algorithms and Perturbation Theory for Matrix Eigenvalue Problems and the Singular Value Decomposition written by Yuji Nakatsukasa and published by . This book was released on 2011 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation is about algorithmic and theoretical developments for eigenvalue problems in numerical linear algebra. The first part of this dissertation proposes algorithms for two important matrix decompositions, the symmetric eigenvalue decomposition and the singular value decomposition. Recent advances and changes in computational architectures have made it necessary for basic linear algebra algorithms to be well-adapted for parallel computing. A few decades ago, an algorithm was considered faster if it required fewer arithmetic operations. This is not the case anymore, and now it is vital to minimize both arithmetic and communication when designing algorithms that are well-suited for high performance scientific computing. Unfortunately, for the above two matrix decompositions, no known algorithm minimizes communication without needing significantly more arithmetic. The development of such algorithms is the main theme of the first half of the dissertation. Our algorithms have great potential as the future approach to computing these matrix decompositions. The second part of this dissertation explores eigenvalue perturbation theory. Besides being of theoretical interest, perturbation theory is a useful tool that plays important roles in many applications. For example, it is frequently employed in the stability analysis of a numerical algorithm, for examining whether a given problem is well-conditioned under perturbation, or for bounding errors of a computed solution. However, there are a number of phenomena that still cannot be explained by existing theory. We make contributions by deriving refined eigenvalue perturbation bounds for Hermitian block tridiagonal matrices and generalized Hermitian eigenproblems, giving explanations for the perturbation behavior of a multiple generalized eigenvalue, presenting refined eigenvector perturbation bounds, and developing new Gerschgorin-type eigenvalue inclusion sets.
Book Synopsis Matrix Computations by : Gene H. Golub
Download or read book Matrix Computations written by Gene H. Golub and published by JHU Press. This book was released on 1996-10-15 with total page 734 pages. Available in PDF, EPUB and Kindle. Book excerpt: Revised and updated, the third edition of Golub and Van Loan's classic text in computer science provides essential information about the mathematical background and algorithmic skills required for the production of numerical software. This new edition includes thoroughly revised chapters on matrix multiplication problems and parallel matrix computations, expanded treatment of CS decomposition, an updated overview of floating point arithmetic, a more accurate rendition of the modified Gram-Schmidt process, and new material devoted to GMRES, QMR, and other methods designed to handle the sparse unsymmetric linear system problem.
Book Synopsis On the Error Analysis and Implementation of Some Eigenvalue Decomposition and Singular Value Decomposition Algorithms by : Huan Ren
Download or read book On the Error Analysis and Implementation of Some Eigenvalue Decomposition and Singular Value Decomposition Algorithms written by Huan Ren and published by . This book was released on 1996 with total page 342 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Numerical Methods for General and Structured Eigenvalue Problems by : Daniel Kressner
Download or read book Numerical Methods for General and Structured Eigenvalue Problems written by Daniel Kressner and published by Springer Science & Business Media. This book was released on 2006-01-20 with total page 272 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is about computing eigenvalues, eigenvectors, and invariant subspaces of matrices. Treatment includes generalized and structured eigenvalue problems and all vital aspects of eigenvalue computations. A unique feature is the detailed treatment of structured eigenvalue problems, providing insight on accuracy and efficiency gains to be expected from algorithms that take the structure of a matrix into account.
Book Synopsis Advanced Statistics for the Behavioral Sciences by : Jonathon D. Brown
Download or read book Advanced Statistics for the Behavioral Sciences written by Jonathon D. Brown and published by Springer. This book was released on 2019-04-30 with total page 526 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book demonstrates the importance of computer-generated statistical analyses in behavioral science research, particularly those using the R software environment. Statistical methods are being increasingly developed and refined by computer scientists, with expertise in writing efficient and elegant computer code. Unfortunately, many researchers lack this programming background, leaving them to accept on faith the black-box output that emerges from the sophisticated statistical models they frequently use. Building on the author’s previous volume, Linear Models in Matrix Form, this text bridges the gap between computer science and research application, providing easy-to-follow computer code for many statistical analyses using the R software environment. The text opens with a foundational section on linear algebra, then covers a variety of advanced topics, including robust regression, model selection based on bias and efficiency, nonlinear models and optimization routines, generalized linear models, and survival and time-series analysis. Each section concludes with a presentation of the computer code used to illuminate the analysis, as well as pointers to packages in R that can be used for similar analyses and nonstandard cases. The accessible code and breadth of topics make this book an ideal tool for graduate students or researchers in the behavioral sciences who are interested in performing advanced statistical analyses without having a sophisticated background in computer science and mathematics.
Book Synopsis Polynomial and Matrix Computations: Fundamental algorithms by : Dario Bini
Download or read book Polynomial and Matrix Computations: Fundamental algorithms written by Dario Bini and published by . This book was released on 1994 with total page 474 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Matrix Algorithms by : G. W. Stewart
Download or read book Matrix Algorithms written by G. W. Stewart and published by SIAM. This book was released on 1998-08-01 with total page 476 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is the first in a self-contained five-volume series devoted to matrix algorithms. It focuses on the computation of matrix decompositions--that is, the factorization of matrices into products of similar ones. The first two chapters provide the required background from mathematics and computer science needed to work effectively in matrix computations. The remaining chapters are devoted to the LU and QR decompositions--their computation and applications. The singular value decomposition is also treated, although algorithms for its computation will appear in the second volume of the series. The present volume contains 65 algorithms formally presented in pseudocode. Other volumes in the series will treat eigensystems, iterative methods, sparse matrices, and structured problems. The series is aimed at the nonspecialist who needs more than black-box proficiency with matrix computations. To give the series focus, the emphasis is on algorithms, their derivation, and their analysis. The reader is assumed to have a knowledge of elementary analysis and linear algebra and a reasonable amount of programming experience, typically that of the beginning graduate engineer or the undergraduate in an honors program. Strictly speaking, the individual volumes are not textbooks, although they are intended to teach, the guiding principle being that if something is worth explaining, it is worth explaining fully. This has necessarily restricted the scope of the series, but the selection of topics should give the reader a sound basis for further study.
Book Synopsis Advanced Probability and Statistics by : Harish Parthasarathy
Download or read book Advanced Probability and Statistics written by Harish Parthasarathy and published by Taylor & Francis. This book was released on 2022-11-17 with total page 427 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book surveys some of the important research work carried out by Indian scientists in the field of pure and applied probability, quantum probability, quantum scattering theory, group representation theory and general relativity. It reviews the axiomatic foundations of probability theory by A.N. Kolmogorov and how the Indian school of probabilists and statisticians used this theory effectively to study a host of applied probability and statistics problems like parameter estimation, convergence of a sequence of probability distributions, and martingale characterization of diffusions. It will be an important resource to students and researchers of Physics and Engineering, especially those working with Advanced Probability and Statistics.
