Iterative Solution of Symmetric Quasi-Definite Linear Systems

Numerous applications, including computational optimization and fluid dynamics, give rise to block linear systems of equations said to have the quasi-definite structure. In practical situations, the size or density of those systems can preclude a factorization approach, leaving only iterative methods as the solution technique. Known iterative methods, however, are not specifically designed to take advantage of the quasi-definite structure.

This book discusses the connection between quasi-definite systems and linear least-squares problems, the most common and best understood problems in applied mathematics, and explains how quasi-definite systems can be solved using tailored iterative methods for linear least squares (with half as much work!). To encourage researchers and students to use the software, it is provided in MATLAB, Python, and Julia.

The authors provide a concise account of the most well-known methods for symmetric systems and least-squares problems, research-level advances in the solution of problems with specific illustrations in optimization and fluid dynamics, and a website that hosts software in three languages.

Audience: This book is intended for researchers and advanced graduate students in computational optimization, computational fluid dynamics, computational linear algebra, data assimilation, and virtually any computational field in which saddle-point systems occur. The software should appeal to all practitioners, even those not technically inclined.

Contents: List of Algorithms; List of Theorems; Preface; Chapter 1: Introduction; Chapter 2: Preliminaries; Chapter 3: Overview of Existing Direct and Iterative Methods; Chapter 4: Fundamental Processes; Chapter 5: Iterative Methods Based on Reduced Equations; Chapter 6: Full-Space Iterative Methods; Chapter 7: Software and Numerical Experiments; Chapter 8: Discussion and Open Questions; Bibliography; Index.