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**Fast reliable algorithms for matrices with structure.**
*(English)*
Zbl 0931.65018

Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). xvi, 342 p. (1999).

Recent contributions to the theory and the practice of fast numerical algorithms for large scale structured linear systems are described in this book. It has ten chapters and two appendices.

The first four chapters are on fast direct methods for the triangular factorization of structured matrices and the solution of the corresponding equations. This is followed by three chapters that deal with fast iterative methods for the solution of structured linear equations. The primary emphasis in these chapters is on preconditioned conjugate gradient methods and on Newton’s method. The next three chapters are on extensions of the notion of structure to the block cases, tensor case, and to the input-output framework. A variety of matrix results, pertinent to the book, are given in the appendices.

The editors have done an excellent work in blending the chapters, written by different authors, into a reasonably consistent presentation. Enough background material is given to put the work in a proper context. The notations are consistent throughout the edited book. All the references are together at the end rather than at the end of each chapter, as is the usual practice in edited volumes. The editors have succeeded in presenting in a coherent manner the state of the art algorithms in the subject.

The first four chapters are on fast direct methods for the triangular factorization of structured matrices and the solution of the corresponding equations. This is followed by three chapters that deal with fast iterative methods for the solution of structured linear equations. The primary emphasis in these chapters is on preconditioned conjugate gradient methods and on Newton’s method. The next three chapters are on extensions of the notion of structure to the block cases, tensor case, and to the input-output framework. A variety of matrix results, pertinent to the book, are given in the appendices.

The editors have done an excellent work in blending the chapters, written by different authors, into a reasonably consistent presentation. Enough background material is given to put the work in a proper context. The notations are consistent throughout the edited book. All the references are together at the end rather than at the end of each chapter, as is the usual practice in edited volumes. The editors have succeeded in presenting in a coherent manner the state of the art algorithms in the subject.

Reviewer: R.P.Tewarson (Stony Brook)

### MSC:

65F05 | Direct numerical methods for linear systems and matrix inversion |

65F10 | Iterative numerical methods for linear systems |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

15B57 | Hermitian, skew-Hermitian, and related matrices |