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.
Download or read book Numerical Algebra Matrix Theory Differential Algebraic Equations and Control Theory written by Peter Benner and published by Springer. This book was released on 2015-05-09 with total page 635 pages. Available in PDF, EPUB and Kindle. Book excerpt: This edited volume highlights the scientific contributions of Volker Mehrmann, a leading expert in the area of numerical (linear) algebra, matrix theory, differential-algebraic equations and control theory. These mathematical research areas are strongly related and often occur in the same real-world applications. The main areas where such applications emerge are computational engineering and sciences, but increasingly also social sciences and economics. This book also reflects some of Volker Mehrmann's major career stages. Starting out working in the areas of numerical linear algebra (his first full professorship at TU Chemnitz was in "Numerical Algebra," hence the title of the book) and matrix theory, Volker Mehrmann has made significant contributions to these areas ever since. The highlights of these are discussed in Parts I and II of the present book. Often the development of new algorithms in numerical linear algebra is motivated by problems in system and control theory. These and his later major work on differential-algebraic equations, to which he together with Peter Kunkel made many groundbreaking contributions, are the topic of the chapters in Part III. Besides providing a scientific discussion of Volker Mehrmann's work and its impact on the development of several areas of applied mathematics, the individual chapters stand on their own as reference works for selected topics in the fields of numerical (linear) algebra, matrix theory, differential-algebraic equations and control theory.
Download or read book The Matrix Eigenvalue Problem written by David S. Watkins and published by SIAM. This book was released on 2007-01-01 with total page 452 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first in-depth, complete, and unified theoretical discussion of the two most important classes of algorithms for solving matrix eigenvalue problems: QR-like algorithms for dense problems and Krylov subspace methods for sparse problems. The author discusses the theory of the generic GR algorithm, including special cases (for example, QR, SR, HR), and the development of Krylov subspace methods. This book also addresses a generic Krylov process and the Arnoldi and various Lanczos algorithms, which are obtained as special cases. Theoretical and computational exercises guide students, step by step, to the results. Downloadable MATLAB programs, compiled by the author, are available on a supplementary Web site. Readers of this book are expected to be familiar with the basic ideas of linear algebra and to have had some experience with matrix computations. Ideal for graduate students, or as a reference book for researchers and users of eigenvalue codes.
Download or read book Duality System in Applied Mechanics and Optimal Control written by Wan-Xie Zhong and published by Springer Science & Business Media. This book was released on 2006-04-11 with total page 467 pages. Available in PDF, EPUB and Kindle. Book excerpt: A unified approach is proposed for applied mechanics and optimal control theory. The Hamilton system methodology in analytical mechanics is used for eigenvalue problems, vibration theory, gyroscopic systems, structural mechanics, wave-guide, LQ control, Kalman filter, robust control etc. All aspects are described in the same unified methodology. Numerical methods for all these problems are provided and given in meta-language, which can be implemented easily on the computer. Precise integration methods both for initial value problems and for two-point boundary value problems are proposed, which result in the numerical solutions of computer precision. Key Features of the text include: -Unified approach based on Hamilton duality system theory and symplectic mathematics. -Gyroscopic system vibration, eigenvalue problems. -Canonical transformation applied to non-linear systems. -Pseudo-excitation method for structural random vibrations. -Precise integration of two-point boundary value problems. -Wave propagation along wave-guides, scattering. -Precise solution of Riccati differential equations. -Kalman filtering. -HINFINITY theory of control and filter.
Download or read book Error Analysis of the Symplectic Lanczos Method for the Symplectic Eigenvalue Problem written by Heike Faßbender and published by . This book was released on 1999 with total page 21 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Handbook of Linear Algebra written by Leslie Hogben and published by CRC Press. This book was released on 2013-11-26 with total page 1838 pages. Available in PDF, EPUB and Kindle. Book excerpt: With a substantial amount of new material, the Handbook of Linear Algebra, Second Edition provides comprehensive coverage of linear algebra concepts, applications, and computational software packages in an easy-to-use format. It guides you from the very elementary aspects of the subject to the frontiers of current research. Along with revisions and
Download or read book Structured Matrices in Mathematics Computer Science and Engineering I written by Vadim Olshevsky and published by American Mathematical Soc.. This book was released on 2001 with total page 346 pages. Available in PDF, EPUB and Kindle. Book excerpt: "The collection of the contributions to these volumes offers a flavor of the plethora of different approaches to attack structured matrix problems. The reader will find that the theory of structured matrices is positioned to bridge diverse applications in the sciences and engineering, deep mathematical theories, as well as computational and numberical issues. The presentation fully illustrates the fact that the technicques of engineers, mathematicisn, and numerical analysts nicely complement each other, and they all contribute to one unified theory of structured matrices"--Back cover.
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.
Download or read book An Implicitly Restarted Symplectic Lanczos Method for the Symplectic Eigenvalue Problem written by Peter Benner and published by . This book was released on 1998 with total page 31 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Inverse Eigenvalue Problems written by Moody Chu and published by Oxford University Press. This book was released on 2005-06-16 with total page 408 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Eigenvalue Algorithms for Symmetric Hierarchical Matrices written by Thomas Mach and published by Thomas Mach. This book was released on 2012 with total page 173 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis is on the numerical computation of eigenvalues of symmetric hierarchical matrices. The numerical algorithms used for this computation are derivations of the LR Cholesky algorithm, the preconditioned inverse iteration, and a bisection method based on LDL factorizations. The investigation of QR decompositions for H-matrices leads to a new QR decomposition. It has some properties that are superior to the existing ones, which is shown by experiments using the HQR decompositions to build a QR (eigenvalue) algorithm for H-matrices does not progress to a more efficient algorithm than the LR Cholesky algorithm. The implementation of the LR Cholesky algorithm for hierarchical matrices together with deflation and shift strategies yields an algorithm that require O(n) iterations to find all eigenvalues. Unfortunately, the local ranks of the iterates show a strong growth in the first steps. These H-fill-ins makes the computation expensive, so that O(n³) flops and O(n²) storage are required. Theorem 4.3.1 explains this behavior and shows that the LR Cholesky algorithm is efficient for the simple structured Hl-matrices. There is an exact LDLT factorization for Hl-matrices and an approximate LDLT factorization for H-matrices in linear-polylogarithmic complexity. This factorizations can be used to compute the inertia of an H-matrix. With the knowledge of the inertia for arbitrary shifts, one can compute an eigenvalue by bisectioning. The slicing the spectrum algorithm can compute all eigenvalues of an Hl-matrix in linear-polylogarithmic complexity. A single eigenvalue can be computed in O(k²n log^4 n). Since the LDLT factorization for general H-matrices is only approximative, the accuracy of the LDLT slicing algorithm is limited. The local ranks of the LDLT factorization for indefinite matrices are generally unknown, so that there is no statement on the complexity of the algorithm besides the numerical results in Table 5.7. The preconditioned inverse iteration computes the smallest eigenvalue and the corresponding eigenvector. This method is efficient, since the number of iterations is independent of the matrix dimension. If other eigenvalues than the smallest are searched, then preconditioned inverse iteration can not be simply applied to the shifted matrix, since positive definiteness is necessary. The squared and shifted matrix (M-mu I)² is positive definite. Inner eigenvalues can be computed by the combination of folded spectrum method and PINVIT. Numerical experiments show that the approximate inversion of (M-mu I)² is more expensive than the approximate inversion of M, so that the computation of the inner eigenvalues is more expensive. We compare the different eigenvalue algorithms. The preconditioned inverse iteration for hierarchical matrices is better than the LDLT slicing algorithm for the computation of the smallest eigenvalues, especially if the inverse is already available. The computation of inner eigenvalues with the folded spectrum method and preconditioned inverse iteration is more expensive. The LDLT slicing algorithm is competitive to H-PINVIT for the computation of inner eigenvalues. In the case of large, sparse matrices, specially tailored algorithms for sparse matrices, like the MATLAB function eigs, are more efficient. If one wants to compute all eigenvalues, then the LDLT slicing algorithm seems to be better than the LR Cholesky algorithm. If the matrix is small enough to be handled in dense arithmetic (and is not an Hl(1)-matrix), then dense eigensolvers, like the LAPACK function dsyev, are superior. The H-PINVIT and the LDLT slicing algorithm require only an almost linear amount of storage. They can handle larger matrices than eigenvalue algorithms for dense matrices. For Hl-matrices of local rank 1, the LDLT slicing algorithm and the LR Cholesky algorithm need almost the same time for the computation of all eigenvalues. For large matrices, both algorithms are faster than the dense LAPACK function dsyev.
Download or read book Foundations of Computational Mathematics Minneapolis 2002 written by Felipe Cucker and published by Cambridge University Press. This book was released on 2004-03-25 with total page 218 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume, first published in 2004, contains the plenary invited talks given at main conference in the subject.
Download or read book Numerical Methods for Linear Control Systems written by Biswa Datta and published by Academic Press. This book was released on 2004 with total page 737 pages. Available in PDF, EPUB and Kindle. Book excerpt: Numerical Methods for Linear Control Systems Design and Analysis is an interdisciplinary textbook aimed at systematic descriptions and implementations of numerically-viable algorithms based on well-established, efficient and stable modern numerical linear techniques for mathematical problems arising in the design and analysis of linear control systems both for the first- and second-order models. Unique coverage of modern mathematical concepts such as parallel computations, second-order systems, and large-scale solutions Background material in linear algebra, numerical linear algebra, and control theory included in text Step-by-step explanations of the algorithms and examples
Download or read book Templates for the Solution of Algebraic Eigenvalue Problems written by Zhaojun Bai and published by SIAM. This book was released on 2000-01-01 with total page 430 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematics of Computing -- Numerical Analysis.
Download or read book Differential Geometry The Interface between Pure and Applied Mathematics written by Mladen Luksic and published by American Mathematical Soc.. This book was released on 1987 with total page 286 pages. Available in PDF, EPUB and Kindle. Book excerpt: Contains papers that represent the proceedings of a conference entitled 'Differential Geometry: The Interface Between Pure and Applied Mathematics', which was held in San Antonio, Texas, in April 1986. This work covers a range of applications and techniques in such areas as ordinary differential equations, Lie groups, algebra and control theory.
Download or read book On different concepts for the linearization of matrix polynomials and canonical decompositions of structured matrices with respect to indefinite sesquilinear forms written by Philip Saltenberger and published by Logos Verlag Berlin GmbH. This book was released on 2019-05-30 with total page 191 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis, a novel framework for the construction and analysis of strong linearizations for matrix polynomials is presented. Strong linearizations provide the standard means to transform polynomial eigenvalue problems into equivalent generalized eigenvalue problems while preserving the complete finite and infinite eigenstructure of the problem. After the transformation, the QZ algorithm or special methods appropriate for structured linearizations can be applied for finding the eigenvalues efficiently. The block Kronecker ansatz spaces proposed here establish an innovative and flexible approach for the construction of strong linearizations in the class of strong block minimal bases pencils. Moreover, they represent a new vector-space-setting for linearizations of matrix polynomials that additionally provides a common basis for various existing techniques on this task (such as Fiedler-linearizations). New insights on their relations, similarities and differences are revealed. The generalized eigenvalue problems obtained often allow for an efficient numerical solution. This is discussed with special attention to structured polynomial eigenvalue problems whose linearizations are structured as well. Structured generalized eigenvalue problems may also lead to equivalent structured (standard) eigenvalue problems. Thereby, the transformation produces matrices that can often be regarded as selfadjoint or skewadjoint with respect to some indefinite inner product. Based on this observation, normal matrices in indefinite inner product spaces and their spectral properties are studied and analyzed. Multiplicative and additive canonical decompositions respecting the matrix structure induced by the inner product are established.
