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- 原书名：Numerical linear algebra and its applications
内容简介
书籍
数学书籍
Numerical linear algebra, also called matrix computation, has been a center of scientific and engineering computing since 1946. Most of problems in science and engineering finally become problems in matrix computations. This book gives an elementary introduction to matrix computation and it also includes some new results obtained in recent years. .
This book consists of nine chapters. It includes the Gaussian elimination, classical iterative methods and Krylov subspace methods for solving linear systems; the perturbation analysis of linear systems; the rounding error analysis of elimination; the orthogonal decompositions for solving linear least squares problem; and some classical methods for eigen-problems. In the last chapter, a brief survey of the latest developments in using boundary value methods for solving initial value problems of ordinary differential equations is given. ..
This is texbook for the senior students majoring in scientific computing and information science. It will be also useful to all who teach or study the subject. ...
数学书籍
Numerical linear algebra, also called matrix computation, has been a center of scientific and engineering computing since 1946. Most of problems in science and engineering finally become problems in matrix computations. This book gives an elementary introduction to matrix computation and it also includes some new results obtained in recent years. .
This book consists of nine chapters. It includes the Gaussian elimination, classical iterative methods and Krylov subspace methods for solving linear systems; the perturbation analysis of linear systems; the rounding error analysis of elimination; the orthogonal decompositions for solving linear least squares problem; and some classical methods for eigen-problems. In the last chapter, a brief survey of the latest developments in using boundary value methods for solving initial value problems of ordinary differential equations is given. ..
This is texbook for the senior students majoring in scientific computing and information science. It will be also useful to all who teach or study the subject. ...
目录
Contents .
Preface
Chapter 1 Introduction
1.1 Basic symbols
1.2 Basic problems in NLA
1.3 Why shall we study numerical methods?
1.4 Matrix factorizations (decompositions)
1.5 Perturbation and error analysis
1.6 Operation cost and convergence rate
Exercises
Chapter 2 Direct Methods for Linear Systems
2.1 Triangular linear systems and LU factorization
2.2 LU factorization with pivoting
2.3 Cholesky factorization
Exercises
Chapter 3 Perturbation and Error Analysis
3.1 Vector and matrix norms
3.2 Perturbation analysis for linear systems
3.3 Error analysis on floating point arithmetic
3.4 Error analysis on partial pivoting
Preface
Chapter 1 Introduction
1.1 Basic symbols
1.2 Basic problems in NLA
1.3 Why shall we study numerical methods?
1.4 Matrix factorizations (decompositions)
1.5 Perturbation and error analysis
1.6 Operation cost and convergence rate
Exercises
Chapter 2 Direct Methods for Linear Systems
2.1 Triangular linear systems and LU factorization
2.2 LU factorization with pivoting
2.3 Cholesky factorization
Exercises
Chapter 3 Perturbation and Error Analysis
3.1 Vector and matrix norms
3.2 Perturbation analysis for linear systems
3.3 Error analysis on floating point arithmetic
3.4 Error analysis on partial pivoting
前言
Numerical linear algebra, also called matrix computation, has been a center of scientific and engineering computing since 1946, the first modern com- puter was born. Most of problems in science and engineering finally become problems in matrix computation. Therefore, it is important for us to study numerical linear algebra. This book gives an elementary introduction to matrix computation and it also includes some new results obtained in recent years. In the beginning of this book, we first give an outline of numerical linear algebra in Chapter 1. .
In Chapter 2, we introduce Gaussian elimination, a basic direct method, for solving general linear systems. Usually, Gaussian elimination is used for solving a dense linear system with median size and no special structure. The operation cost of Gaussian elimination is O(n3) where n is the size of the system. The pivoting technique is also studied.
In Chapter 3, in order to discuss effects of perturbation and error on numerical solutions, we introduce vector and matrix norms and study their prop- erties. The error analysis on floating point operations and on partial pivoting technique is also given.
In Chapter 4, linear least squares problems are studied. We will concentrate on the problem of finding the least squares solution of an over determined linear system Ax = b where A has more rows than columns. Some orthogonal transformations and the QR decomposition are used to design efficient algorithms for solving least squares problems.
We study classical iterative methods for the solution of Ax = b in Chapter
5. Iterative methods are quite different from direct methods such as Gaussian elimination. Direct methods based on an LU factorization of the matrix A are prohibitive in terms of computing time and computer storage if A is quite large. Usually, in most large problems, the matrices are sparse. The spar sity may be lost during the LU factorization procedure and then at the end of LU factorization, the storage becomes a crucial issue. For such kind of problem, we can use a class of methods called iterative methods. We only consider some classical iterative methods in this chapter.
In Chapter 6, we introduce another class of iterative methods called Krylov subspace methods proposed recently. We will only study two versions among those Krylov subspace methods: the conjugate gradient (CG) method and the generalized minimum residual (GMRES) method. The CG method proposed in 1952 is one of the best known iterative method for solving symmetric positive definite linear systems. The GMRES method was proposed in 1986 for solving nonsymmetric linear systems. The preconditioning technique is also studied. ..
Eigenvalue problems are particularly interesting in scientific computing. In Chapter 7, nonsymmetric eigenvalue problems are studied. We introduce some well-known methods such as the power method, the inverse power method and the QR method.
The symmetric eigenvalue problem with its nice properties and rich mathematical theory is one of the most interesting topics in numerical linear algebra. In Chapter 8, we will study this topic. -The symmetric QR iteration method, the Jacobi method, the bisection method and a divide-and-conquer technique will be discussed in this chapter.
In Chapter 9, we will briefly survey some of the latest developments in using boundary value methods for solving systems of ordinary differential equations with initial values. These methods require the solutions of one or more nonsymmetric, large and sparse linear systems. Therefore, we will use the GMRES method in Chapter 6 with some preconditioners for solving these linear systems. One of the main results is that if an Av1,v2-stable boundary value method is used for an m-by-m system of ODEs, then the preconditioned matrix can be decomposed as I + L where I is the identity matrix and the rank of L is at most 2rn(v1 + v2). It follows that when the GMRES method is applied to the preconditioned system, the method will converge in at most 2m(v1 +v2)+ 1 iterations. Applications to different delay differential equations are also given.
"If any other mathematical topic is as fundamental to the mathematical sciences as calculus and differential equations, it is numerical linear algebra. "— L. Trefethen and D. Bau III
Acknowledgments: We would like to thank Professor Raymond H. F. Chan of the Department of Mathematics, Chinese University of Hong Kong, for his constant encouragement, long-standing friendship, financial support; Professor Z. H. Cao of the Department of Mathematics, Fudan University, for his many helpful discussions and useful suggestions. We also would like to thank our friend Professor Z. C. Shi for his encouraging support and valuable comments. Of course, special appreciation goes to two important institutions in the authors' life: University of Macau and Fudan University for providing
a wonderful intellectual atmosphere for writing this book. Most of the writing was done during evenings, weekends and holidays. Finally, thanks are also due to our families for their endless love, understanding, encouragement and support essential to the completion of this book. The most heartfelt thanks to all of them!
The publication of the book is supported in part by the research grants No. RG024/01-02S/JXQ/FST, No. RG031/02-03S/JXQ/FST and No. RG064/0304S/JXQ/FST from University of Macau; the research grant No.10471027 from the National Natural Science Foundation of China and some financial support from Shanghai Education Committee and Fudan University.
Authors' words on the corrected and revised second printing: In its second printing, we corrected some minor mathematical and typographical mistakes in the first printing of the book. We would like to thank all those people who pointed these out to us. Additional comments and some revision have been made in Chapter 7. The references have been updated. More exercises are also to be found in the book. The second printing of the book is supported by the research grant No. RG081/04-05S/JXQ/FST. ...
In Chapter 2, we introduce Gaussian elimination, a basic direct method, for solving general linear systems. Usually, Gaussian elimination is used for solving a dense linear system with median size and no special structure. The operation cost of Gaussian elimination is O(n3) where n is the size of the system. The pivoting technique is also studied.
In Chapter 3, in order to discuss effects of perturbation and error on numerical solutions, we introduce vector and matrix norms and study their prop- erties. The error analysis on floating point operations and on partial pivoting technique is also given.
In Chapter 4, linear least squares problems are studied. We will concentrate on the problem of finding the least squares solution of an over determined linear system Ax = b where A has more rows than columns. Some orthogonal transformations and the QR decomposition are used to design efficient algorithms for solving least squares problems.
We study classical iterative methods for the solution of Ax = b in Chapter
5. Iterative methods are quite different from direct methods such as Gaussian elimination. Direct methods based on an LU factorization of the matrix A are prohibitive in terms of computing time and computer storage if A is quite large. Usually, in most large problems, the matrices are sparse. The spar sity may be lost during the LU factorization procedure and then at the end of LU factorization, the storage becomes a crucial issue. For such kind of problem, we can use a class of methods called iterative methods. We only consider some classical iterative methods in this chapter.
In Chapter 6, we introduce another class of iterative methods called Krylov subspace methods proposed recently. We will only study two versions among those Krylov subspace methods: the conjugate gradient (CG) method and the generalized minimum residual (GMRES) method. The CG method proposed in 1952 is one of the best known iterative method for solving symmetric positive definite linear systems. The GMRES method was proposed in 1986 for solving nonsymmetric linear systems. The preconditioning technique is also studied. ..
Eigenvalue problems are particularly interesting in scientific computing. In Chapter 7, nonsymmetric eigenvalue problems are studied. We introduce some well-known methods such as the power method, the inverse power method and the QR method.
The symmetric eigenvalue problem with its nice properties and rich mathematical theory is one of the most interesting topics in numerical linear algebra. In Chapter 8, we will study this topic. -The symmetric QR iteration method, the Jacobi method, the bisection method and a divide-and-conquer technique will be discussed in this chapter.
In Chapter 9, we will briefly survey some of the latest developments in using boundary value methods for solving systems of ordinary differential equations with initial values. These methods require the solutions of one or more nonsymmetric, large and sparse linear systems. Therefore, we will use the GMRES method in Chapter 6 with some preconditioners for solving these linear systems. One of the main results is that if an Av1,v2-stable boundary value method is used for an m-by-m system of ODEs, then the preconditioned matrix can be decomposed as I + L where I is the identity matrix and the rank of L is at most 2rn(v1 + v2). It follows that when the GMRES method is applied to the preconditioned system, the method will converge in at most 2m(v1 +v2)+ 1 iterations. Applications to different delay differential equations are also given.
"If any other mathematical topic is as fundamental to the mathematical sciences as calculus and differential equations, it is numerical linear algebra. "— L. Trefethen and D. Bau III
Acknowledgments: We would like to thank Professor Raymond H. F. Chan of the Department of Mathematics, Chinese University of Hong Kong, for his constant encouragement, long-standing friendship, financial support; Professor Z. H. Cao of the Department of Mathematics, Fudan University, for his many helpful discussions and useful suggestions. We also would like to thank our friend Professor Z. C. Shi for his encouraging support and valuable comments. Of course, special appreciation goes to two important institutions in the authors' life: University of Macau and Fudan University for providing
a wonderful intellectual atmosphere for writing this book. Most of the writing was done during evenings, weekends and holidays. Finally, thanks are also due to our families for their endless love, understanding, encouragement and support essential to the completion of this book. The most heartfelt thanks to all of them!
The publication of the book is supported in part by the research grants No. RG024/01-02S/JXQ/FST, No. RG031/02-03S/JXQ/FST and No. RG064/0304S/JXQ/FST from University of Macau; the research grant No.10471027 from the National Natural Science Foundation of China and some financial support from Shanghai Education Committee and Fudan University.
Authors' words on the corrected and revised second printing: In its second printing, we corrected some minor mathematical and typographical mistakes in the first printing of the book. We would like to thank all those people who pointed these out to us. Additional comments and some revision have been made in Chapter 7. The references have been updated. More exercises are also to be found in the book. The second printing of the book is supported by the research grant No. RG081/04-05S/JXQ/FST. ...
