Fundamentals of Matrix Computations
This book was written for advanced undergraduates, graduate students, and
mature
scientists in mathematics,computer science,engineering,and all disciplines in
which
numerical methods are used . At the heart of most scientific computer codes lie
matrix
computations,soitis important to understand how to perform such computation
seffi
ciently and accurately. This book meets that need by providing a detailed
introduction
to the fundamental ideas of numerical linear algebra .
The prerequisites are a first course in linear algebra and some experience with
computer programming. For the understanding of some of the examples, especially
in the second half of the book,the student will find it helpful to have had a
first course
in differential equations .
There are several other excellent books on this subject,including those by
Demmel
[15],GolubandVanLoan[33],andTrefethenandBau[71]. Students who are new to
this material often find those books quite difficult to read. The purpose of
this book
is to provide a gentler, more gradual introduction to the subject that is
nevertheless
mathematically solid . The strong positive student response to the first edition
has
assured me that my first attempt was successful and encouraged me to produce
this
updated and extended edition.
The first edition was aimed mainly at the undergraduate level. As it turned out,
the book also found a great deal of use as a graduate text. I have therefore
added
new material to make the book more attractive at the graduate level. These
additions
are detailed below. However, the text remains suitable for undergraduate use, as
the elementary material has been kept largely intact, and more elementary
exercises
have been added. The instructor can control the level of difficulty by deciding
which
sections to cover and how far to push into each section. Numerous advanced
topics
are developed in exercises at the ends of the sections.
The book contains many exercises, ranging from easy to moderately
difficult.
Some are interspersed with the textual material and others are collected at the
end
of each section. Those that are interspersed with the text are meant to be
worked
immediately by the reader. This is my way of getting students actively involved
in
the learning process. In order to get something out, you have to put something
in.
Many of the exercises at the ends of sections are lengthy and may appear
intimidating
at first. However,the persistent student will find that s/he can make it
through them
with the help of the ample hints and advice that are given. I encourage every
student
to work as many of the exercises as possible.
Nearly all numbered items in this book, including theorems, lemmas, numbered
equations,examples,and exercises,share a single numbering scheme. For example,
the first numbered item in Section 1.3 is Theorem 1.3.1. The next two numbered
items are displayed equations,which are numbered(1.3.2) and
(1.3.3),respectively.
These are followed by the first exercise of the section,which bears the number
1.3.4.
Thus each item has a unique number: the only item in the book that has the number
1.3.4 is Exercise1.3.4. Although this schemeis unusual,I believe that most
readers
will find it perfectly natural ,once they have gotten used to it. Its big
advantage is that
it makes things easy to find: The reader who has located Exercises 1.4.15 and
1.4.25
but is looking for Example 1.4.20,knows for sure that this example lies some
where
between the two exercises.
There are a couple of exceptions to the scheme. For technical reasons related
to the type setting, tables and figures (the socalled floating bodies) are
numbered
separately by chapter. For example,the third figure of Chapter1 is Figure1.3.
New Features of the Second Edition
Use of MATLAB
By now MATLAB is firmly established as the most widely used vehicle for teaching
matrix computations. MATLAB is an easy to use, very highlevel language that
allows the student to perform much more elaborate computational experiments than
before. MATLAB is also widely used in industry. I have therefore added many
examples and exercises that make use of MATLAB. This book is not, however, an
introduction to MATLAB,nor is it a MATLAB manual. For those purposes there are
other book savailable,for example,the MATLAB Guide by Higham and Higham[40].
However, MATLAB’s extensive help facilities are good enough that the reader
may
feel noneed for a supplementary text. In an effort to make it easier for the
student to
use MATLAB with this book,I have included an index of MATLAB terms ,separate
from the ordinary index.
I used to make my students write and debug their own Fortran programs. I have
left the Fortran exercises from the first edition largely intact. I hope a few
students
will choose to work through some of these worth while projects.
More Applications
In order to help the student better understand the importance of the subject
matter of
this book,I have included more examples and exercises on applications ( solved
using
MATLAB ),mostly at the beginnings of chapters. I have chosen very simple applica
tions: electrical circuits, massspring systems, simple partial differential
equations.
In my opinion the simplest examples are the ones from which we can learn the
most.
Earlier Introduction of the Singular Value Decomposition (SVD)
The SVD is one of the most important tools in numerical linear algebra. In the
first
edition it was placed in the final chapter of the book, because it is impossible
to
discuss methods for computing the SVD until after eigenvalue problems have been
discussed. I have since decided that the SVD needs to be introduced sooner, so
that the student can find out earlier about its properties and uses . With the
help
of MATLAB, the student can experiment with the SVD without knowing anything
about how it is computed. ThereforeI have addeda brief chapteron the SVD in the
middle of the book.
New Material on Iterative Methods
The biggest addition to the book is a chapter on iterative methods for solving
large,
sparse systems of linear equations . The main focus of the chapter is the
powerful
conjugategradient method for solving symmetric, positive definite systems. How
ever, the classical iterations are also discussed, and so are preconditioners.
Krylov
subspace methods for solving indefinite and non symmetric problems are surveyed
briefly.
There are also two new sections on methods for solving large, sparse eigenvalue
problems. The discussion includes the popular implicitlyrestarted Arnoldi and
JacobiDavid son methods.
I hope that these additions in particular will make the book more attractive as
a
graduate text.
Other New Features
To make the book more versatile,a number of other topics have been
added,including:
•a backward error analysis of Gaussian elimination , including a discussion of
the modern component wise error analysis.
•a discussion of reorthogonalization,apractical means of obtaining numerically
or thonormalvectors.
•a discussion of how to update the decomposition when arow or columnis
added to or deleted from the data matrix, as happens in signal processing and
data analysis applications.
•a section introducing new methods for the symmetric eigen value problem that
have been developed since the first edition was published.
A few topics have been deleted on the grounds that they are either obsolete or
too
specialized. I have also taken the opportunity to correct several vexing errors
from
the first edition. I can only hope that I have not introduced too many new ones.
Acknowledgments
I am greatly indebted to the authors of some of the early works in this
field. These
includeA.S.Householder[43],J.H.Wilkinson[81],G.E.ForsytheandC.B.Moler
[24], G. W. Stewart [67], C. L. Lawson and R . J. Hanson[48], B. N. Parlett [54],
A.
GeorgeandJ.W.Liu[30],aswellastheauthorsoftheHandbook[83],theEISPACK
Guide [64], and the LINPACK Users’ Guide [18]. All of them influenced me
profoundly. Bytheway,everyone of these books is still worth reading today.
Special
thanks goto CleveMoler for inventing MATLAB,which has changed everything .
Most of the first edition was written while I was on leave, at the University of
Bielefeld, Germany. I am pleased to thank once again my host and longtime
friend,
Ludwig Elsner. During that stay I received financial support from the Fulbright
commission. A big chunk of the second edition was also written in Germany,at the
Technical University of Chemnitz. I thank my host (and another longtime friend),
VolkerMehrmann. On that visit I received financial support from Sonder for
schungs
bereich 393, TU Chemnitz. I am also indebted to my home institution, Washington
State University,forits support of my work on both editions.
I thank once again professors Dale Olesky, Kemble Yates, and Tjalling Ypma,
who classtested a preliminary version of the first edition. Since publication
of the
first edition, numerous people have sent me corrections, feedback, and comments.
These include A. Cline, L. Dieci, E. Jessup, D. Koya, D. D. Olesky, B. N.
Parlett,
A. C. Raines, A. Witt, and K. Wright. Finally, I thank the many students who
have
helped me learn how to teachthis material over the years.
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