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.
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.
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 Strangs Circulant Preconditioner
2.1 Introduction
2.2 Convergence rate
3 T. Chans 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
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