The Lanczos and Conjugate Gradient Algorithms
Title | The Lanczos and Conjugate Gradient Algorithms PDF eBook |
Author | Gerard Meurant |
Publisher | SIAM |
Pages | 380 |
Release | 2006-01-01 |
Genre | Computers |
ISBN | 9780898718140 |
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.
The Equivalence of the Lanczos and the Conjugate
Title | The Equivalence of the Lanczos and the Conjugate PDF eBook |
Author | Jane Cullum |
Publisher | |
Pages | 25 |
Release | 1977 |
Genre | |
ISBN |
Conjugate Gradient Algorithms and Finite Element Methods
Title | Conjugate Gradient Algorithms and Finite Element Methods PDF eBook |
Author | Michal Krizek |
Publisher | Springer Science & Business Media |
Pages | 405 |
Release | 2012-12-06 |
Genre | Science |
ISBN | 3642185606 |
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.
The Symmetric Eigenvalue Problem
Title | The Symmetric Eigenvalue Problem PDF eBook |
Author | Beresford N. Parlett |
Publisher | SIAM |
Pages | 422 |
Release | 1998-01-01 |
Genre | Mathematics |
ISBN | 9781611971163 |
According to Parlett, "Vibrations are everywhere, and so too are the eigenvalues associated with them. As mathematical models invade more and more disciplines, we can anticipate a demand for eigenvalue calculations in an ever richer variety of contexts." Anyone who performs these calculations will welcome the reprinting of Parlett's book (originally published in 1980). In this unabridged, amended version, Parlett covers aspects of the problem that are not easily found elsewhere. The chapter titles convey the scope of the material succinctly. The aim of the book is to present mathematical knowledge that is needed in order to understand the art of computing eigenvalues of real symmetric matrices, either all of them or only a few. The author explains why the selected information really matters and he is not shy about making judgments. The commentary is lively but the proofs are terse. The first nine chapters are based on a matrix on which it is possible to make similarity transformations explicitly. The only source of error is inexact arithmetic. The last five chapters turn to large sparse matrices and the task of making approximations and judging them.
Preconditioning and the Conjugate Gradient Method in the Context of Solving PDEs
Title | Preconditioning and the Conjugate Gradient Method in the Context of Solving PDEs PDF eBook |
Author | Josef Malek |
Publisher | SIAM |
Pages | 106 |
Release | 2014-12-22 |
Genre | Mathematics |
ISBN | 161197383X |
Preconditioning and the Conjugate Gradient Method in the Context of Solving PDEs?is about the interplay between modeling, analysis, discretization, matrix computation, and model reduction. The authors link PDE analysis, functional analysis, and calculus of variations with matrix iterative computation using Krylov subspace methods and address the challenges that arise during formulation of the mathematical model through to efficient numerical solution of the algebraic problem. The book?s central concept, preconditioning of the conjugate gradient method, is traditionally developed algebraically using the preconditioned finite-dimensional algebraic system. In this text, however, preconditioning is connected to the PDE analysis, and the infinite-dimensional formulation of the conjugate gradient method and its discretization and preconditioning are linked together. This text challenges commonly held views, addresses widespread misunderstandings, and formulates thought-provoking open questions for further research.?
The Lanczos Tridiagonalization and the Conjugate Gradient
Title | The Lanczos Tridiagonalization and the Conjugate Gradient PDF eBook |
Author | Jane Cullum |
Publisher | |
Pages | 20 |
Release | 1978 |
Genre | |
ISBN |
The Lanczos Tridiagonalization and the Conjugate Gradient Algorithms with Local, $ \epsilon $ -orthogonality of the Lanczos Vector
Title | The Lanczos Tridiagonalization and the Conjugate Gradient Algorithms with Local, $ \epsilon $ -orthogonality of the Lanczos Vector PDF eBook |
Author | J. Cullum |
Publisher | |
Pages | 20 |
Release | 1978 |
Genre | |
ISBN |