Introduction to Numerical Methods and Optimization Techniques | Date: 21 April 2011, 16:57
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The availability of computers has revolutionized every field which depends
on numerical calculations. The scientist or engineer can be greatly aided in
his work if he knows how to enlist the aid of his new ally. The purpose of
this book is to help the undergraduate student learn how to apply a
computer to many different types of practical problems. The book is
written in a manner that should instill enthusiasm in the reader and help
convince him that the computer is one of the greatest labor saving devices
ever to come along.
This book's philosophy differs from most others written on numerical
methods or numerical analysis. In a typical numerical-analysis text, much
time is spent on error analysis. However, in practical applications, usually
little time is devoted to rigorous error analysis.
Instead of relying solely on error analysis to estimate the accuracy of
answers, other methods are emphasized in this text. Observing how a
process converges can give insight into its accuracy. Also, solving a
problem two different ways can verify a solution. Although error analysis
is not very practical as a tool for estimating accuracy, it does have other
uses. It can be used to compare different numerical methods and demonstrate
that, on the average, one is superior to another. Or a knowledge of
error properties can be used to improve a numerical method.
Avoiding a lengthy investigation of error analysis allows time for the
reader to become acquainted with optimization techniques. Numerical
methods and optimization techniques are intimately related, but unfortunately
they are not generally taught in the same course. However, since
both numerical methods and optimization techniques are usually iterative
procedures, they have a common philosophy and application. In fact, as
demonstrated in this book, an optimization technique can be viewed as a
collection of numerical methods which have been linked together in a
specific way. Thus, once a student has become familiar with numerical
methods the extension to optimization techniques is very natural.
This text does not attempt to be a complete catalog of numerical
methods or optimization techniques-volumes would be needed for this.
For a specific problem, the specialist can probably consult the literature
and obtain a more efficient solution than presented in this book. If he uses
his sophisticated program very often, his time is well spent. In fact, this
text is not written for the specialist, but for the reader or student who will
probably not have the luxury of spending days on research to save
milliseconds of computer time.
Instead of overwhelming the reader with numerous methods for solving
the same problem, attention is focused on one or two. If a person is
familiar and confident with a specific method, he is much more likely to
apply it than if he only has a nodding acquaintance. Just because a
particular method has been included in this text, it need not be the best
one available. The choices were influenced by the desire to have an
introductory text which links numerical methods to optimization techniques.
At the end of most chapters is a section entitled "Suggested
Reading in Related Topics", which enables the enthusiastic reader to do
additional research on topics not essential to an introductory course.
The typical student using this book for a formal course will be a junior
or senior-a sophomore could understand the material, but might not
appreciate the range of applications. A knowledge of differentiation and
integration is essential for the course; the ability to solve differential
equations would be helpful, but is not essential.
Because the application of many of the algorithms in this text requires
the use of a computer, numerous programs are included. These programs
are written in a version of time-sharing FORTRAN that is similar to FORTRAN
iv. The programs were all run on a Control Data Cyber 70 computer
system.
The programs that have been included in the text are written with the
emphasis on clarity. Their purpose is to implement the methods described
in the text and provide a means for the student to apply the algorithms.
The programs have not been included in the hope that they will become
widely used in various computation centers; they are ill suited for that
purpose. The literature contains ample selections of programs that have
been written with the emphasis on speed and accuracy. However, even
though the programs in this book are relatively simple, they should be
adequate for the problems encountered by students who are at this
introductory level; and they provide a good basis for understanding the
more sophisticated programs that abound in the literature.
The problems at the end of the chapters serve various purposes. Some
help to extend the material presented in the text or to check the reader's
knowledge of subtle points. Others illustrate the application of the programs
or equations. Finally, many of the problems help the student
become familiar with the computer so that it will become an ally in future
endeavors.
The idea of combining numerical methods and optimization techniques
into one text occurred to the author while teaching at Tennessee State
University. My appreciation goes to those students who helped shape the
early direction of the book. Also, I have the good fortune to have had
many different experts review the text and offer numerous suggestions for
its improvement. To two of my colleagues go special thanks: R. P. Snicer
for his thorough review and detailed suggestions, and P. H. McDonald for
her helpful hints on structured programming.
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