Error Norm Estimation In The Conjugate Gradient Algorithm
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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.
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.
Advanced Numerical Methods For Complex Environmental Models Needs And Availability
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Author : István Faragó
language : en
Publisher: Bentham Science Publishers
Release Date : 2013-12-10
Advanced Numerical Methods For Complex Environmental Models Needs And Availability written by István Faragó and has been published by Bentham Science Publishers this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-12-10 with Nature categories.
High air pollution levels pose a significant threat to plants, animals and human beings. Efforts by researchers are directed towards keeping air pollution levels below well defined ‘critical‘ levels in order to maintain a sustainable atmosphere and environmental system. The application of advanced mathematical models is important for researchers to achieve this goal as efficiently as possible. Mathematical models can be used to predict answers to many important questions about the environment. This application comes with several complex theoretical and practical obstacles which need to be resolved. A successfully applicable mathematical model needs to enable researchers to • Mathematically describe all important physical and chemical processes. • Apply fast and sufficiently accurate numerical methods. • Ensure that the model runs efficiently on modern high speed computers. • Use high quality input data, both meteorological data and emission inventories, in the runs. • Verify the model results by comparing them with reliable measurements taken in different parts of the spatial domain of the model. • Carry out long series of sensitivity experiments to check the response of the model to changes of different key parameters. • Visualize and animate the output results in order to make them easily understandable even to non-specialists. This monograph thoroughly describes mathematical methods useful for various situations in environmental modeling - including finite difference methods, splitting methods, parallel computation, etc. - and provides a framework for resolving problems posed in relation to the points listed above. Chapters are written by well-known specialists making this book a handy reference for researchers, university teachers and students working and studying in the areas of air pollution, meteorology, applied mathematics and computer science.
Iterative Solution Methods
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Author : Owe Axelsson
language : en
Publisher: Cambridge University Press
Release Date : 1996-03-29
Iterative Solution Methods written by Owe Axelsson and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996-03-29 with Mathematics categories.
This book deals primarily with the numerical solution of linear systems of equations by iterative methods. The first part of the book is intended to serve as a textbook for a numerical linear algebra course. The material assumes the reader has a basic knowledge of linear algebra, such as set theory and matrix algebra, however it is demanding for students who are not afraid of theory. To assist the reader, the more difficult passages have been marked, the definitions for each chapter are collected at the beginning of the chapter, and numerous exercises are included throughout the text. The second part of the book serves as a monograph introducing recent results in the iterative solution of linear systems, mainly using preconditioned conjugate gradient methods. This book should be a valuable resource for students and researchers alike wishing to learn more about iterative methods.
Walter Gautschi Volume 3
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Author : Claude Brezinski
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-10-24
Walter Gautschi Volume 3 written by Claude Brezinski 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 2013-10-24 with Mathematics categories.
Walter Gautschi has written extensively on topics ranging from special functions, quadrature and orthogonal polynomials to difference and differential equations, software implementations, and the history of mathematics. He is world renowned for his pioneering work in numerical analysis and constructive orthogonal polynomials, including a definitive textbook in the former, and a monograph in the latter area. This three-volume set, Walter Gautschi: Selected Works with Commentaries, is a compilation of Gautschi’s most influential papers and includes commentaries by leading experts. The work begins with a detailed biographical section and ends with a section commemorating Walter’s prematurely deceased twin brother. This title will appeal to graduate students and researchers in numerical analysis, as well as to historians of science. Selected Works with Commentaries, Vol. 1 Numerical Conditioning Special Functions Interpolation and Approximation Selected Works with Commentaries, Vol. 2 Orthogonal Polynomials on the Real Line Orthogonal Polynomials on the Semicircle Chebyshev Quadrature Kronrod and Other Quadratures Gauss-type Quadrature Selected Works with Commentaries, Vol. 3 Linear Difference Equations Ordinary Differential Equations Software History and Biography Miscellanea Works of Werner Gautschi
Milestones In Matrix Computation
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Author : Raymond Chan
language : en
Publisher: OUP Oxford
Release Date : 2007-02-22
Milestones In Matrix Computation written by Raymond Chan and has been published by OUP Oxford this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-02-22 with Mathematics categories.
The text presents and discusses some of the most influential papers in Matrix Computation authored by Gene H. Golub, one of the founding fathers of the field. The collection of 21 papers is divided into five main areas: iterative methods for linear systems, solution of least squares problems, matrix factorizations and applications, orthogonal polynomials and quadrature, and eigenvalue problems. Commentaries for each area are provided by leading experts: Anne Greenbaum, Ake Bjorck, Nicholas Higham, Walter Gautschi, and G. W. (Pete) Stewart. Comments on each paper are also included by the original authors, providing the reader with historical information on how the paper came to be written and under what circumstances the collaboration was undertaken. Including a brief biography and facsimiles of the original papers, this text will be of great interest to students and researchers in numerical analysis and scientific computation.
Pole Swapping Algorithms For The Eigenvalue Problem
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Author : Daan Camps
language : en
Publisher: SIAM
Release Date : 2025-05-01
Pole Swapping Algorithms For The Eigenvalue Problem written by Daan Camps and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-05-01 with Mathematics categories.
Matrix eigenvalue problems arise in a wide variety of fields in science and engineering, so it is important to have reliable and efficient methods for solving them. Of the methods devised, bulge-chasing algorithms, such as the famous QR and QZ algorithms, are the most important. This book focuses on pole-swapping algorithms, a new class of methods that are generalizations of bulge-chasing algorithms and a bit faster and more accurate owing to their inherent flexibility. The pole-swapping theory developed by the authors sheds light on the functioning of the whole class of algorithms, including QR and QZ. Pole-Swapping Algorithms for the Eigenvalue Problem is the only book on the topic. It describes the state of the art on eigenvalue methods and provides an improved understanding and explanation of why these important algorithms work. This book is for researchers and students in the field of matrix computations, software developers, and anyone in academia or industry who needs to understand how to solve eigenvalue problems, which are ubiquitous in science and engineering.
Hessenberg And Tridiagonal Matrices
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Author : Gérard Meurant
language : en
Publisher: SIAM
Release Date : 2025-06-10
Hessenberg And Tridiagonal Matrices 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 2025-06-10 with Mathematics categories.
This is the only book devoted exclusively to Hessenberg and tridiagonal matrices. Hessenberg matrices are involved in Krylov methods for solving linear systems or computing eigenvalues and eigenvectors, in the QR algorithm for computing eigenvalues, and in many other areas of scientific computing (for instance, control theory). Matrices that are both upper and lower Hessenberg are tridiagonal. Their entries are zero except for the main diagonal and the subdiagonal and updiagonal next to it. Hessenberg and Tridiagonal Matrices: Theory and Examples presents known and new results; describes the theoretical properties of the matrices, their determinants, LU factorizations, inverses, and eigenvalues; illustrates the theoretical properties with applications and examples as well as numerical experiments; and considers unitary Hessenberg matrices, inverse eigenvalue problems, and Toeplitz tridiagonal matrices. This book is intended for applied mathematicians, especially those interested in numerical linear algebra, and it will also be of interest to physicists and engineers.
Background Modeling And Foreground Detection For Video Surveillance
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Author : Thierry Bouwmans
language : en
Publisher: CRC Press
Release Date : 2014-07-25
Background Modeling And Foreground Detection For Video Surveillance written by Thierry Bouwmans and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-07-25 with Computers categories.
Background modeling and foreground detection are important steps in video processing used to detect robustly moving objects in challenging environments. This requires effective methods for dealing with dynamic backgrounds and illumination changes as well as algorithms that must meet real-time and low memory requirements.Incorporating both establish
Kwic Index For Numerical Algebra
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Author : Alston Scott Householder
language : en
Publisher:
Release Date : 1972
Kwic Index For Numerical Algebra written by Alston Scott Householder and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1972 with Algebra categories.