Eigenvalue Algorithms For Symmetric Hierarchical Matrices
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Eigenvalue Algorithms For Symmetric Hierarchical Matrices
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Author : Thomas Mach
language : en
Publisher: Thomas Mach
Release Date : 2012
Eigenvalue Algorithms For Symmetric Hierarchical Matrices written by Thomas Mach and has been published by Thomas Mach this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with Mathematics categories.
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.
Eigenvalue Algorithms For Symmetric Hierarchical Matrices
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Author : Thomas Mach
language : en
Publisher:
Release Date : 2012
Eigenvalue Algorithms For Symmetric Hierarchical Matrices written by Thomas Mach and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with categories.
The Symmetric Eigenvalue Problem
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Author : Beresford N. Parlett
language : en
Publisher: SIAM
Release Date : 1980-01-01
The Symmetric Eigenvalue Problem written by Beresford N. Parlett and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 1980-01-01 with Mathematics categories.
Mathematics of Computing -- Numerical Analysis.
Lanczos Algorithms For Large Symmetric Eigenvalue Computations
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Author : Jane K. Cullum
language : en
Publisher: SIAM
Release Date : 1985-01-01
Lanczos Algorithms For Large Symmetric Eigenvalue Computations written by Jane K. Cullum and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 1985-01-01 with Mathematics categories.
First published in 1985, Lanczos Algorithms for Large Symmetric Eigenvalue Computations; Vol. 1: Theory presents background material, descriptions, and supporting theory relating to practical numerical algorithms for the solution of huge eigenvalue problems. This book deals with "symmetric" problems. However, in this book, "symmetric" also encompasses numerical procedures for computing singular values and vectors of real rectangular matrices and numerical procedures for computing eigenelements of nondefective complex symmetric matrices. Although preserving orthogonality has been the golden rule in linear algebra, most of the algorithms in this book conform to that rule only locally, resulting in markedly reduced memory requirements. Additionally, most of the algorithms discussed separate the eigenvalue (singular value) computations from the corresponding eigenvector (singular vector) computations. This separation prevents losses in accuracy that can occur in methods which, in order to be able to compute further into the spectrum, use successive implicit deflation by computed eigenvector or singular vector approximations.
Hierarchical Matrices
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Author : Mario Bebendorf
language : en
Publisher: Springer Science & Business Media
Release Date : 2008-06-25
Hierarchical Matrices written by Mario Bebendorf and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-06-25 with Mathematics categories.
Hierarchical matrices are an efficient framework for large-scale fully populated matrices arising, e.g., from the finite element discretization of solution operators of elliptic boundary value problems. In addition to storing such matrices, approximations of the usual matrix operations can be computed with logarithmic-linear complexity, which can be exploited to setup approximate preconditioners in an efficient and convenient way. Besides the algorithmic aspects of hierarchical matrices, the main aim of this book is to present their theoretical background. The book contains the existing approximation theory for elliptic problems including partial differential operators with nonsmooth coefficients. Furthermore, it presents in full detail the adaptive cross approximation method for the efficient treatment of integral operators with non-local kernel functions. The theory is supported by many numerical experiments from real applications.
Hierarchical Matrices Algorithms And Analysis
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Author : Wolfgang Hackbusch
language : en
Publisher: Springer
Release Date : 2015-12-21
Hierarchical Matrices Algorithms And Analysis written by Wolfgang Hackbusch and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-12-21 with Mathematics categories.
This self-contained monograph presents matrix algorithms and their analysis. The new technique enables not only the solution of linear systems but also the approximation of matrix functions, e.g., the matrix exponential. Other applications include the solution of matrix equations, e.g., the Lyapunov or Riccati equation. The required mathematical background can be found in the appendix. The numerical treatment of fully populated large-scale matrices is usually rather costly. However, the technique of hierarchical matrices makes it possible to store matrices and to perform matrix operations approximately with almost linear cost and a controllable degree of approximation error. For important classes of matrices, the computational cost increases only logarithmically with the approximation error. The operations provided include the matrix inversion and LU decomposition. Since large-scale linear algebra problems are standard in scientific computing, the subject of hierarchical matrices is of interest to scientists in computational mathematics, physics, chemistry and engineering.
Matrix Algorithms
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Author : G. W. Stewart
language : en
Publisher: SIAM
Release Date : 2001-08-30
Matrix Algorithms written by G. W. Stewart and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001-08-30 with Mathematics categories.
This is the second volume in a projected five-volume survey of numerical linear algebra and matrix algorithms. It treats the numerical solution of dense and large-scale eigenvalue problems with an emphasis on algorithms and the theoretical background required to understand them. The notes and reference sections contain pointers to other methods along with historical comments. The book is divided into two parts: dense eigenproblems and large eigenproblems. The first part gives a full treatment of the widely used QR algorithm, which is then applied to the solution of generalized eigenproblems and the computation of the singular value decomposition. The second part treats Krylov sequence methods such as the Lanczos and Arnoldi algorithms and presents a new treatment of the Jacobi-Davidson method. These volumes are not intended to be encyclopedic, but provide the reader with the theoretical and practical background to read the research literature and implement or modify new algorithms.
Lanczos Algorithms For Large Symmetric Eigenvalue Computations
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Author : Jane K. Cullum
language : en
Publisher: SIAM
Release Date : 2002-09-01
Lanczos Algorithms For Large Symmetric Eigenvalue Computations written by Jane K. Cullum and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002-09-01 with Mathematics categories.
First published in 1985, this book presents background material, descriptions, and supporting theory relating to practical numerical algorithms for the solution of huge eigenvalue problems. This book deals with 'symmetric' problems. However, in this book, 'symmetric' also encompasses numerical procedures for computing singular values and vectors of real rectangular matrices and numerical procedures for computing eigenelements of nondefective complex symmetric matrices. Although preserving orthogonality has been the golden rule in linear algebra, most of the algorithms in this book conform to that rule only locally, resulting in markedly reduced memory requirements. Additionally, most of the algorithms discussed separate the eigenvalue (singular value) computations from the corresponding eigenvector (singular vector) computations. This separation prevents losses in accuracy that can occur in methods which, in order to be able to compute further into the spectrum, use successive implicit deflation by computed eigenvector or singular vector approximations.
The Science Of High Performance Algorithms For Hierarchical Matrices
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Author : Chen-Han Yu (Ph. D.)
language : en
Publisher:
Release Date : 2018
The Science Of High Performance Algorithms For Hierarchical Matrices written by Chen-Han Yu (Ph. D.) and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018 with categories.
Many matrices in scientific computing, statistical inference, and machine learning exhibit sparse and low-rank structure. Typically, such structure is exposed by appropriate matrix permutation of rows and columns, and exploited by constructing an hierarchical approximation. That is, the matrix can be written as a summation of sparse and low-rank matrices and this structure repeats recursively. Matrices that admit such hierarchical approximation are known as hierarchical matrices (H-matrices in brief). H-matrix approximation methods are more general and scalable than solely using a sparse or low-rank matrix approximation. Classical numerical linear algebra operations on H-matrices-multiplication, factorization, and eigenvalue decomposition-can be accelerated by many orders of magnitude. Although the literature on H-matrices for problems in computational physics (low-dimensions) is vast, there is less work for generalization and problems appearing in machine learning. Also, there is limited work on high-performance computing algorithms for pure algebraic H-matrix methods. This dissertation tries to address these open problems on building hierarchical approximation for kernel matrices and generic symmetric positive definite (SPD) matrices. We propose a general tree-based framework (GOFMM) for appropriately permuting a matrix to expose its hierarchical structure. GOFMM supports both static and dynamic scheduling, shared memory and distributed memory architectures, and hardware accelerators. The supported algorithms include kernel methods, approximate matrix multiplication and factorization for large sparse and dense matrices.
Eigenvalue Problems Algorithms Software And Applications In Petascale Computing
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Author : Tetsuya Sakurai
language : en
Publisher: Springer
Release Date : 2018-01-03
Eigenvalue Problems Algorithms Software And Applications In Petascale Computing written by Tetsuya Sakurai and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-01-03 with Computers categories.
This book provides state-of-the-art and interdisciplinary topics on solving matrix eigenvalue problems, particularly by using recent petascale and upcoming post-petascale supercomputers. It gathers selected topics presented at the International Workshops on Eigenvalue Problems: Algorithms; Software and Applications, in Petascale Computing (EPASA2014 and EPASA2015), which brought together leading researchers working on the numerical solution of matrix eigenvalue problems to discuss and exchange ideas – and in so doing helped to create a community for researchers in eigenvalue problems. The topics presented in the book, including novel numerical algorithms, high-performance implementation techniques, software developments and sample applications, will contribute to various fields that involve solving large-scale eigenvalue problems.