Numerical Methods For Singular Multiparameter Eigenvalue Problems
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Numerical Methods For Singular Multiparameter Eigenvalue Problems
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Author : Andrej Muhič
language : en
Publisher:
Release Date : 2011
Numerical Methods For Singular Multiparameter Eigenvalue Problems written by Andrej Muhič and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with categories.
In the 1960s Atkinson introduced an abstract algebraic setting for multiparameter eigenvalue problems. He showed that a nonsingular multiparameter eigenvalue problem is equivalent to the associated system of generalized eigenvalue problems. Many theoretical results and numerical methods for nonsingular multiparameter eigenvalue problems are based on this relation. We extend the above relation to singular two-parameter eigenvalue problems and show that the simple finite regular eigenvalues of a two-parameter eigenvalue problem and the common regular eigenvalues of the coupled generalized eigenvalue problem agree. Using the theory on the pencils of matrix polynomials we furthermore generalize the theory to the nonregular singular two-parameter eigenvalue problems. This enables one to solve a singular two-parameter eigenvalue problem by computing the common regular eigenvalues of the associated system of two singular generalized eigenvalue problems. There are various numerical methods for twoparameter eigenvalue problems, but all of them can only be applied to nonsingular problems. We develop a numerical method that can be applied to the singular two-parameter eigenvalue problems. It is based on the staircase algorithm for the extraction of the common regular part of two singular matrix pencils. We introduce the quadratic two-parameter eigenvalue problem (QMEP) and show that we can linearize it as a regular singular two-parameter eigenvalue problem. We present several transformations that can be used to solve the QMEP, by formulating an associated linear multiparameter eigenvalue problem. We also generalize the linearization to the polynomial twoparameter eigenvalue problem(PMEP). As an alternative approach to the linearization, we propose the transformation of the QMEP into a nonsingular five-parameter eigenvalue problem. We also consider several special cases of the QMEP, where some matrix coefficients are zero, which allows us to solve such QMEP more efficiently. We propose a Jacobi-Davidson type method for regular singular problems. We modify the Jacobi-Davidson type method for nonsingular two-parameter eigenvalue problem so that it can be applied to regular singular problems. The obtained algorithm can then be used to solve the problem obtained by linearizing the PMEP. If the dimension of matrices is large, then we cannot use the approach with linearization. If order of polynomials is small enough, then we can apply a Jacobi-Davidson type method directly to the polynomial system. This method is a generalization of the method for polynomial eigenvalue problems. We give some numerical results that illustrate the convergence of the introduced Jacobi-Davidson type methods.
Multiparameter Eigenvalue Problems
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Author : Atkinson
language : en
Publisher: Academic Press
Release Date : 1972-06-16
Multiparameter Eigenvalue Problems written by Atkinson and has been published by Academic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1972-06-16 with Computers categories.
Multiparameter eigenvalue problems
Multiparameter Eigenvalue Problems
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Author : F. V. Atkinson
language : en
Publisher:
Release Date : 1972-01-01
Multiparameter Eigenvalue Problems written by F. V. Atkinson and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1972-01-01 with Eigenvalues categories.
Multiparameter Eigenvalue Problems
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Author : F.V. Atkinson
language : en
Publisher: CRC Press
Release Date : 2010-12-07
Multiparameter Eigenvalue Problems written by F.V. Atkinson and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-12-07 with Mathematics categories.
One of the masters in the differential equations community, the late F.V. Atkinson contributed seminal research to multiparameter spectral theory and Sturm-Liouville theory. His ideas and techniques have long inspired researchers and continue to stimulate discussion. With the help of co-author Angelo B. Mingarelli, Multiparameter Eigenvalue Problem
Numerical Methods For Large Eigenvalue Problems
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Author : Y. Saad
language : en
Publisher: Manchester University Press
Release Date : 1992
Numerical Methods For Large Eigenvalue Problems written by Y. Saad and has been published by Manchester University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1992 with Eigenvalues categories.
Fitted Numerical Methods For Singular Perturbation Problems Error Estimates In The Maximum Norm For Linear Problems In One And Two Dimensions Revised Edition
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Author : John J H Miller
language : en
Publisher: World Scientific
Release Date : 2012-02-29
Fitted Numerical Methods For Singular Perturbation Problems Error Estimates In The Maximum Norm For Linear Problems In One And Two Dimensions Revised Edition written by John J H Miller and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-02-29 with Mathematics categories.
Since the first edition of this book, the literature on fitted mesh methods for singularly perturbed problems has expanded significantly. Over the intervening years, fitted meshes have been shown to be effective for an extensive set of singularly perturbed partial differential equations. In the revised version of this book, the reader will find an introduction to the basic theory associated with fitted numerical methods for singularly perturbed differential equations. Fitted mesh methods focus on the appropriate distribution of the mesh points for singularly perturbed problems. The global errors in the numerical approximations are measured in the pointwise maximum norm. The fitted mesh algorithm is particularly simple to implement in practice, but the theory of why these numerical methods work is far from simple. This book can be used as an introductory text to the theory underpinning fitted mesh methods.
Multiparameter Eigenvalue Problems
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Author : F. V. Atkinson
language : en
Publisher:
Release Date : 1972
Multiparameter Eigenvalue Problems written by F. V. Atkinson and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1972 with Differential equations categories.
Spectral Methods For Non Standard Eigenvalue Problems
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Author : Călin-Ioan Gheorghiu
language : en
Publisher: Springer Science & Business
Release Date : 2014-04-22
Spectral Methods For Non Standard Eigenvalue Problems written by Călin-Ioan Gheorghiu and has been published by Springer Science & Business this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-04-22 with Mathematics categories.
This book focuses on the constructive and practical aspects of spectral methods. It rigorously examines the most important qualities as well as drawbacks of spectral methods in the context of numerical methods devoted to solve non-standard eigenvalue problems. In addition, the book also considers some nonlinear singularly perturbed boundary value problems along with eigenproblems obtained by their linearization around constant solutions. The book is mathematical, poising problems in their proper function spaces, but its emphasis is on algorithms and practical difficulties. The range of applications is quite large. High order eigenvalue problems are frequently beset with numerical ill conditioning problems. The book describes a wide variety of successful modifications to standard algorithms that greatly mitigate these problems. In addition, the book makes heavy use of the concept of pseudospectrum, which is highly relevant to understanding when disaster is imminent in solving eigenvalue problems. It also envisions two classes of applications, the stability of some elastic structures and the hydrodynamic stability of some parallel shear flows. This book is an ideal reference text for professionals (researchers) in applied mathematics, computational physics and engineering. It will be very useful to numerically sophisticated engineers, physicists and chemists. The book can also be used as a textbook in review courses such as numerical analysis, computational methods in various engineering branches or physics and computational methods in analysis.
Numerical Methods For Eigenvalue Problems
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Author : Steffen Börm
language : en
Publisher: De Gruyter Textbook
Release Date : 2012
Numerical Methods For Eigenvalue Problems written by Steffen Börm and has been published by De Gruyter Textbook this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with Eigenvalues categories.
This textbook presents a number of the most important numerical methods for finding eigenvalues and eigenvectors of matrices. The authors discuss the central ideas underlying the different algorithms and introduce the theoretical concepts required t
Inverse Eigenvalue Problems
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Author : Moody Chu
language : en
Publisher: Oxford University Press
Release Date : 2005-06-16
Inverse Eigenvalue Problems written by Moody Chu and has been published by Oxford University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-06-16 with Mathematics categories.
Inverse eigenvalue problems arise in a remarkable variety of applications and associated with any inverse eigenvalue problem are two fundamental questions--the theoretical issue of solvability and the practical issue of computability. Both questions are difficult and challenging. In this text, the authors discuss the fundamental questions, some known results, many applications, mathematical properties, a variety of numerical techniques, as well as several open problems.This is the first book in the authoritative Numerical Mathematics and Scientific Computation series to cover numerical linear algebra, a broad area of numerical analysis. Authored by two world-renowned researchers, the book is aimed at graduates and researchers in applied mathematics, engineering and computer science and makes an ideal graduate text.