The Lanczos And Conjugate Gradient Algorithms
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The Lanczos And Conjugate Gradient Algorithms
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Author : Gerard Meurant
language : en
Publisher: SIAM
Release Date : 2006-01-01
The Lanczos And Conjugate Gradient Algorithms written by Gerard Meurant and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-01-01 with Computers categories.
The Lanczos and conjugate gradient (CG) algorithms are fascinating numerical algorithms. This book presents the most comprehensive discussion to date of the use of these methods for computing eigenvalues and solving linear systems in both exact and floating point arithmetic. The author synthesizes the research done over the past 30 years, describing and explaining the "average" behavior of these methods and providing new insight into their properties in finite precision. Many examples are given that show significant results obtained by researchers in the field. The author emphasizes how both algorithms can be used efficiently in finite precision arithmetic, regardless of the growth of rounding errors that occurs. He details the mathematical properties of both algorithms and demonstrates how the CG algorithm is derived from the Lanczos algorithm. Loss of orthogonality involved with using the Lanczos algorithm, ways to improve the maximum attainable accuracy of CG computations, and what modifications need to be made when the CG method is used with a preconditioner are addressed.
Conjugate Gradient Algorithms In Nonconvex Optimization
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Author : Radoslaw Pytlak
language : en
Publisher: Springer Science & Business Media
Release Date : 2008-11-18
Conjugate Gradient Algorithms In Nonconvex Optimization written by Radoslaw Pytlak 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 2008-11-18 with Mathematics categories.
This book details algorithms for large-scale unconstrained and bound constrained optimization. It shows optimization techniques from a conjugate gradient algorithm perspective as well as methods of shortest residuals, which have been developed by the author.
Conjugate Gradient Algorithms And Finite Element Methods
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Author : Michal Krizek
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Conjugate Gradient Algorithms And Finite Element Methods written by Michal Krizek 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 2012-12-06 with Science categories.
The position taken in this collection of pedagogically written essays is that conjugate gradient algorithms and finite element methods complement each other extremely well. Via their combinations practitioners have been able to solve complicated, direct and inverse, multidemensional problems modeled by ordinary or partial differential equations and inequalities, not necessarily linear, optimal control and optimal design being part of these problems. The aim of this book is to present both methods in the context of complicated problems modeled by linear and nonlinear partial differential equations, to provide an in-depth discussion on their implementation aspects. The authors show that conjugate gradient methods and finite element methods apply to the solution of real-life problems. They address graduate students as well as experts in scientific computing.
Fitting Linear Models
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Author : A. McIntosh
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Fitting Linear Models written by A. McIntosh 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 2012-12-06 with Mathematics categories.
The increasing power and decreasing price of smalI computers, especialIy "personal" computers, has made them increasingly popular in statistical analysis. The day may not be too far off when every statistician has on his or her desktop computing power on a par with the large mainframe computers of 15 or 20 years ago. These same factors make it relatively easy to acquire and manipulate large quantities of data, and statisticians can expect a corresponding increase in the size of the datasets that they must analyze. Unfortunately, because of constraints imposed by architecture, size or price, these smalI computers do not possess the main memory of their large cousins. Thus, there is a growing need for algorithms that are sufficiently economical of space to permit statistical analysis on smalI computers. One area of analysis where there is a need for algorithms that are economical of space is in the fitting of linear models.
Linear And Nonlinear Conjugate Gradient Related Methods
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Author : Loyce M. Adams
language : en
Publisher: SIAM
Release Date : 1996-01-01
Linear And Nonlinear Conjugate Gradient Related Methods written by Loyce M. Adams and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996-01-01 with Mathematics categories.
Proceedings of the AMS-IMS-SIAM Summer Research Conference held at the University of Washington, July 1995.
Predicting The Behavior Of Finite Precision Lanczos And Conjugate Gradient Computations Classic Reprint
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Author : Anne Greenbaum
language : en
Publisher: Forgotten Books
Release Date : 2017-11-20
Predicting The Behavior Of Finite Precision Lanczos And Conjugate Gradient Computations Classic Reprint written by Anne Greenbaum and has been published by Forgotten Books this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-11-20 with categories.
Excerpt from Predicting the Behavior of Finite Precision Lanczos and Conjugate Gradient Computations Finite precision CG computations for solving an n by n symmetric positive definite linear system Ar b sometimes fail to converge after n steps, especially when n is small. In such cases, it is demonstrated that exact CG applied to the corresponding large linear system A5: b also requires more than 11 iterations to converge. More commonly, finite precision CG computations converge in far fewer than n steps, and the same holds for the exact CG algorithm applied to any matrix A whose eigenvalues are clustered in tiny intervals about the eigenvalues of A. Frequently, finite precision CG computations go through several steps at which there is only a modest reduction in the error and then at the next step there is a very sharp decrease in the error. This same behavior is observed in the exact CG algorithm applied to matrices A whose eigenvalues are distributed in it tight clusters about the eigenvalues of A. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
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.
Hybrid Conjugate Gradient Algorithms
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Author : Dianne Prost O'Leary
language : en
Publisher:
Release Date : 1976
Hybrid Conjugate Gradient Algorithms written by Dianne Prost O'Leary and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1976 with Algorithms categories.
Error Norm Estimation In The Conjugate Gradient Algorithm
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Author : Gérard Meurant
language : en
Publisher: SIAM
Release Date : 2024-01-30
Error Norm Estimation In The Conjugate Gradient Algorithm written by Gérard Meurant and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-01-30 with Mathematics categories.
The conjugate gradient (CG) algorithm is almost always the iterative method of choice for solving linear systems with symmetric positive definite matrices. This book describes and analyzes techniques based on Gauss quadrature rules to cheaply compute bounds on norms of the error. The techniques can be used to derive reliable stopping criteria. How to compute estimates of the smallest and largest eigenvalues during CG iterations is also shown. The algorithms are illustrated by many numerical experiments, and they can be easily incorporated into existing CG codes. The book is intended for those in academia and industry who use the conjugate gradient algorithm, including the many branches of science and engineering in which symmetric linear systems have to be solved.
Lanczos Algorithms For Large Symmetric Eigenvalue Computations Vol Ii Programs
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Author : Cullum
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Lanczos Algorithms For Large Symmetric Eigenvalue Computations Vol Ii Programs written by Cullum 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 2012-12-06 with Mathematics categories.