Werner C. Rheinboldt
CBMS-NSF Regional Conference Series in Applied Mathematics 70
This second edition provides much-needed updates to the original volume. Like the first edition, it emphasizes the ideas behind the algorithms as well as their theoretical foundations and properties, rather than focusing strictly on computational details; at the same time, this new version is now largely self-contained and includes essential proofs.
Additions have been made to almost every chapter, including an introduction to the theory of inexact Newton methods, a basic theory of continuation methods in the setting of differentiable manifolds, and an expanded discussion of minimization methods. New information on parametrized equations and continuation incorporates research since the first edition.
Applied scientists will find this new edition most useful.
Preface to the Second Edition; Preface to the First Edition; Chapter 1: Introduction. Problem Overview; Notation and Background; Chapter 2: Model Problems. Discretization of Operator Equations; Minimization; Discrete Problems; Chapter 3: Iterative Processes and Rates of Convergence. Characterization of Iterative Processes; Rates of Convergence; Evaluation of Convergence Rates; On Efficiency and Accuracy; Chapter 4: Methods of Newton Type. The Linearization Concept; Methods of Newton Form; Discretized Newton Methods; Attraction Basins; Chapter 5: Methods of Secant Type. General Secant Methods; Consistent Approximations; Update Methods; Chapter 6: Combinations of Processes. The Use of Classical Linear Methods; Nonlinear SOR Methods; Residual Convergence Controls; Inexact Newton Methods; Chapter 7: Parametrized Systems of Equations. Submanifolds of Rn; Continuation Using ODEs; Continuation with Local Parametrizations; Simplicial Approximations of Manifolds; Chapter 8: Unconstrained Minimization Methods. Admissible Step Length Algorithms; Gradient Related Methods; Collectively Gradient Related Directions; Trust Region Methods; Chapter 9: Nonlinear Generalizations of Several Matrix Classes. Basic Function Classes; Properties of the Function Classes; Convergence of Iterative Processes; Chapter 10: Outlook at Further Methods. Higher Order Methods; Piecewise-Linear Methods; Further Minimization Methods; Bibliography; Index.
1998 / x + 148 pages / Softcover / ISBN-13: 978-0-898714-15-9 / ISBN-10: 0-89871-415-X /
List Price $58.00 / SIAM/CBMS Member Price $40.60 / Order Code CB70