Iterative Methods for Linear and Nonlinear Equations
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Authors(Editors):
        C. T. Kelley
                North Carolina State University
Publisher: Society for Industrial and Applied Mathematics
Pub Date: 1995
Pages: 172
ISBN:

Preface
This book on iterative methods for linear and nonlinear equations can be used
as a tutorial and a reference byan yone who needs to solve nonlinear systems
of equations or large linear systems. It may also be used as a textbook for
introductorycourses in nonlinear equations or iterative methods or as source
material for an introductorycourse in numerical analysis at the graduate level.
We assume that the reader is familiar with elementaryn umerical analysis,
linear algebra, and the central ideas of direct methods for the numerical
solution of dense linear systems as described in standard texts such as [7],
[105], or [184].
Our approach is to focus on a small number of methods and treat them
in depth. Though this book is written in a finite-dimensional setting, we
have selected for coverage mostlyalgorithms and methods of analysis which
extend directlyto the infinite-dimensional case and whose convergence can be
thoroughly analyzed. For example, the matrix-free formulation and analysis for
GMRES and conjugate gradient is almost unchanged in an infinite-dimensional
setting. The analysis of Broyden’s method presented in Chapter 7 and
the implementations presented in Chapters 7 and 8 are different from the
classical ones and also extend directlyto an infinite-dimensional setting. The
computational examples and exercises focus on discretizations of infinitedimensional
problems such as integral and differential equations.
We present a limited number of computational examples. These examples
are intended to provide results that can be used to validate the reader’s own
implementations and to give a sense of how the algorithms perform. The
examples are not designed to give a complete picture of performance or to be
a suite of test problems.
The computational examples in this book were done with MATLAB
(version 4.0a on various SUN SPARCstations and version 4.1 on an Apple
Macintosh Powerbook 180) and the MATLAB environment is an excellent one
for getting experience with the algorithms, for doing the exercises, and for
small-to-medium scale production work.1 MATLAB codes for manyof the
algorithms are available byanon ymous ftp. A good introduction to the latest
version (version 4.2) of MATLAB is the MATLAB Primer [178]; [43] is also
a useful resource. If the reader has no access to MATLAB or will be solving
verylarge problems, the general algorithmic descriptions or even the MATLAB
codes can easilyb e translated to another language.
Parts of this book are based upon work supported bythe National
Science Foundation and the Air Force Office of Scientific Research over
several years, most recently under National Science Foundation Grant Nos.
DMS-9024622 and DMS-9321938. Anyopinions, findings, and conclusions or
recommendations expressed in this material are those of the author and do not
necessarilyreflect the views of the National Science Foundation or of the Air
Force Office of Scientific Research.
Manyof mystuden ts and colleagues discussed various aspects of this
project with me and provided important corrections, ideas, suggestions, and
pointers to the literature. I am especiallyindebted to Jim Banoczi, Jeff Butera,
Steve Campbell, TonyChoi, MoodyCh u, Howard Elman, Jim Epperson,
Andreas Griewank, Laura Helfrich, Ilse Ipsen, Lea Jenkins, Vickie Kearn,
Belinda King, Debbie Lockhart, Carl Meyer, Casey Miller, Ekkehard Sachs,
Jeff Scroggs, Joseph Skudlarek, Mike Tocci, Gordon Wade, Homer Walker,
Steve Wright, Zhaqing Xue, Yue Zhang, and an anonymous reviewer for their
contributions and encouragement.
Most importantly, I thank Chung-Wei Ng and my parents for over one
hundred and ten years of patience and support.
C. T. Kelley
Raleigh, North Carolina
January, 1998


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