Toeplitz systems arise in a variety of applications, for instance, numerical differential equations, numerical integral equations of convolution-type, stationary autoregressive time series, system identification problems, and image restoration problems, to mention just a few. In this book, we give an elementary introduction to preconditioning techniques developed recently for solving Toeplitz systems.

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.

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 rate

3 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 Stability

4 Superoptimal Preconditioner

 4.1 Introduction

 4.2 Convergence rate

 4.3 Spectral relation of preconditioned matrices

 4.4 Numerical results

5 Ill-Conditioned Toeplitz Systems

 5.1 Band-Toeplitz preconditioner

 5.2 {w}-circulant preconditioner

 5.2.1 Construction of preconditioner

 5.2.2 Spectral analysis

6 Block Preconditioner

 6.1 Block operator■

 6.2 Complexity of preconditioned system

 6.3 Convergence rate

 6.4 Numerical results

7 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 results

A M-files used in Chapter 7

 A.1

 A.2

 A.3

 A.4

 A.5

Bibliography

Index