Toeplitz系统预处理方法

所属分类:数学  
出版时间:2010-3   出版时间:高等教育出版社   作者:金小庆   页数:94  

前言

  Toeplitz systems arise in a variety of applications, for instance, numerical differ-ential equations, numerical integral equations of convolution-type, stationary auto-regressive time series, system identification problems, and image restoration prob-lems [24, 28, 40, 47, 56, 61, 73], to mention just a few. In this book, we give anelementary introduction to preconditioning techniques developed recently for solv-ing Toeplitz systems.  The use of the preconditioned conjugate gradient (PCG) method with circu-lant preconditioners to solve Toeplitz systems was proposed in 1986 [74, 81]. Oneof the main results of this iterative method is that the complexity of solving a largeclass of n-by-n Toeplitz systems Tnx —— b is only required O(n log n) operations. Alot of different preconditioners have been proposed, used, and analyzed since then.Within limited space and time, we mainly study some well-known precondition-ers from a theoretical viewpoint. An application of preconditioners to systems ofordinary differential equations (ODEs) is also discussed.  The book is organized into seven chapters. In Chapter 1, we provide somebackground knowledge of numerical linear algebra that will be used throughoutthe book. Two important Krylov subspace methods, the conjugate gradient (CG)method and the generalized minimum residual (GMRES) method, are introduced.A basiC knowledge of the development in using the CG method with circulantpreconditioners for solving Toeplitz systems is also given.

内容概要

The use of the preconditioned conjugate gradient method with circulant preconditioners to solve Toeplitz systems was proposed in 1986. In this short book,the author mainly studies some well-known preconditioners from a theoretical viewpoint. An application of preconditioners to systems of ordinary differential equations is also discussed. The book contains several important research results on iterative Toeplitz solvers obtained in recent years. It could be accessible to senior undergraduate students who, in various scientific computing disciplines, have a basic linear algebra, calculus, numerical analysis, and computing knowledge.The book is also useful to researchers and computational' practitioners who are interested in fast iterative Toeplitz solvers.    Dr. Xiao-Qing Jin is a Professor at the Department of Mathematics, University of Macau. He is the author of 4 books and over 70 research papers. He is also a member of the editorial beards of Journal on Numerical Methods and Computer Applications, Numerical Mathematics: Theory, Methods and Applications, and East Asia Journal of Applied Mathematics.

书籍目录

1
Introduction
1.1
Background
in
numerical
linear
algebra
1.1.1
Basic
symbols,
notations,
and
definitions
1.1.2
Spectral
properties
of
Hermitian
matrix
1.1.3
Norms
and
condition
number
1.2
Toeplitz
systems
1.3
Conjugate
gradient
method
1.4
GMRES
method
1.5
Basic
knowledge
of
iterative
Toeplitz
solvers
1.5.1
Circulant
preconditioners
1.5.2
Generating
function
and
spectral
analysis
2
Strang's
Circulant
Preconditioner
2.1
Introduction
2.2
Convergence
rate3
T.
Chan's
Optimal
Preconditioner
3.1
Introduction
3.2
Convergence
rate
3.3
Non-circulant
optimal
preconditioners
3.3.1
Optimal
sine
transform
preconditioner
3.3.2
Optimal
cosine
transform
preconditioner
3.3.3
Optimal
Hartley
transform
preconditioner
3.3.4
Convergence
result
and
operation
cost
3.4
Linear
operator
cu
3.5
Stability4
Superoptimal
Preconditioner
4.1
Introduction
4.2
Convergence
rate
4.3
Spectral
relation
of
preconditioned
matrices
4.4
Numerical
results5
Ill-Conditioned
Toeplitz
Systems
5.1
Band-Toeplitz
preconditioner
5.2
{w}-circulant
preconditioner
5.2.1
Construction
of
preconditioner
5.2.2
Spectral
analysis6
Block
Preconditioner
6.1
Block
operator■
6.2
Complexity
of
preconditioned
system
6.3
Convergence
rate
6.4
Numerical
results7
Application
in
ODEs
7.1
Background
of
BVMs
7.1.1
Linear
multistep
formulas
7.1.2
Block-BVMs
and
their
matrix
forms
7.2
Construction
of
preconditioner
7.3
Convergence
rate
and
operation
cost
7.4
Numerical
resultsA
M-files
used
in
Chapter
7
A.1
A.2
A.3
A.4
A.5BibliographyIndex

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