## Handbook of Numerical Analysis: Solution of Equations in Rn (Prt 4), Techniques of Scientific Computer (Part 4), Numerical Methods for Fluids, Part 2Philippe G. Ciarlet, Jacques Louis Lions This series of volumes covers all the major aspects of numerical analysis, serving as the basic reference work on the subject. Each volume concentrates on one to three particular topics. Each article, written by an expert, is an in-depth survey, reflecting up-to-date trends in the field, and is essentially self-contained. The handbook will cover the basic methods of numerical analysis, under the following general headings: solution of equations in Rn; finite difference methods; finite element methods; techniques of scientific computing; optimization theory; and systems science. It will also cover the numerical solution of actual problems of contemporary interest in applied mathematics, under the following headings: numerical methods for fluids; numerical methods for solids; and specific applications - including meteorology, seismology, petroleum mechanics and celestial mechanics. |

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### Contents

Introduction | 9 |

Basic Theory | 21 |

Jacobis Method | 31 |

Methods for the Complete Eigenproblem | 37 |

Krylov Subspaces | 53 |

The Lanczos Method | 61 |

Lanczos in finite precision | 74 |

Alternative approximations | 75 |

The JacobiDavidson iteration method | 122 |

The iterative solution of the correction equation | 125 |

A numerical example | 127 |

A simple JacobiDavidson algorithm | 128 |

Restart and deflation strategies | 130 |

Preconditioning | 133 |

A complete algorithm | 134 |

Harmonic Ritz values | 137 |

Accuracy of harmonic Ritz values g I | 81 |

Refined eigenvector approximations | 82 |

Amoldis Method | 85 |

Numerical examples X6 30 Shiftandinvert m | 86 |

Implicitly Restarted Amoldi | 92 |

Harmonic Ritz values for Araoldis process | 96 |

Numerical examples | 98 |

The TwoSided Lanczos Method | 101 |

Numerical example | 106 |

Harmonic Petrov values for BiLanczos | 108 |

Breakdown and Lookahead 11 | 112 |

JacobiDavidson Methods | 115 |

Numerical example I jg 40 Davidsons method and preconditioning | 119 |

The methods of Jacobi | 120 |

Generalized Eigenproblems | 139 |

Reduction to standard form | 140 |

Rational Krylov methods | 142 |

The JacobiDavidson QZ method | 145 |

An example of a generalized eigenproblem | 149 |

Polynomial and Constrained Eigenproblems | 153 |

Quadratic eigenvalue problems | 154 |

Higher order polynomial eigenvalue problems | 157 |

Constrained eigenproblems | 160 |

Related Issues | 163 |

REFERENCES | 169 |

177 | |