## Matrix Preconditioning Techniques and ApplicationsPreconditioning techniques have emerged as an essential part of successful and efficient iterative solutions of matrices. Ke Chen's book offers a comprehensive introduction to these methods. A vast range of explicit and implicit sparse preconditioners are covered, including the conjugate gradient, multi-level and fast multi-pole methods, matrix and operator splitting, fast Fourier and wavelet transforms, incomplete LU and domain decomposition, Schur complements and approximate inverses. In addition, aspects of parallel realization using the MPI are discussed. Very much a users-guide, the book provides insight to the use of these techniques in areas such as acoustic wave scattering, image restoration and bifurcation problems in electrical power stations. Supporting MATLAB files are available from the Web to support and develop readers' understanding, and provide stimulus for further study. Pitched at graduate level, the book is intended to serve as a useful guide and reference for students, computational practitioners, engineers and researchers alike. |

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### Contents

Direct methods | 66 |

Iterative methods | 110 |

direct approximation | 165 |

direct | 191 |

coarse grid | 240 |

Multilevel recursive Schur complements | 289 |

approximation | 310 |

Wavelet Schur preconditioners T6 | 340 |

coupled matrix problems | 400 |

image restoration and inverse problems | 416 |

voltage stability in electrical power systems | 449 |

Parallel computing by examples | 475 |

a brief guide to linear algebra | 504 |

The Jordan decomposition | 507 |

list of supplied Mfiles and programs | 523 |

556 | |

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### Common terms and phrases

algebraic Algorithm applied approximate inverse preconditioner assume band matrix basis functions BCCB block block matrix boundary Chapter circulant matrix coarse level coarsest level coefficients column computing consider convergence decomposition defined denote dense discretization discuss domain eigenvalues eigenvectors entries example fast Figure finest level finite Fourier GMRES Hopf bifurcation idea Illustration implementation integral equations interpolation iterative methods iterative solver Lemma linear system LU decomposition MATLAB mesh minimization multigrid methods multilevel multiple neighbours nodes nonlinear nonzero notation Note obtain operator splitting orthogonal parallel partition permutation pivoting plot polynomial preconditioner problems processor Purcell method Remark residual correction result Schur complement Schur decomposition Section singular smoothing solution solving SPAI sparse approximate inverse sparse matrix sparsity pattern step subspace supplied Mfiles symmetric test function Theorem Toeplitz Toeplitz matrix triangular variables vector wavelet transform

### Popular passages

Page 539 - New methods for computing a closest saddle node bifurcation and worst case load power margin for voltage collapse", IEEE Trans, on Power Systems, vol.

Page 546 - YEREMIN, On a family of two-level preconditionings of the incomplete block factorization type, Soviet J.

Page 531 - A survey of preconditioned iterative methods for linear systems of algebraic equations".

Page 540 - Duff and J. Koster. On algorithms for permuting large entries to the diagonal of a sparse matrix.

Page 535 - VH Quintana. Comparison of Performance Indices for Detection of Proximity to Voltage Collapse.

Page 537 - On a class of preconditioning methods for dense linear systems from boundary elements, SIAM J. Sci.