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This book is intended as a text for an introductory, one-year course in numerical
analysis for students in mathematics, engineering, and the physical sciences.
Historically, such a course was given by mathematics departments to mostly junior
and senior level students. In recent years a more elementary course has achieved
some popularity. This course is often taught in engineering or computer science
departments and has a younger audience of freshmen and sophomores. These two
groups have different skills and different needs, but there is a great deal
that is common. It is our view that these separate needs can be combined effectively
and that the subject matter can indeed profit from such a combination.
To serve the needs of the two different audiences, each topic is carefully
developed in a graduated manner. The first few sections of each chapter present
motivation and simple algorithms in an intuitive fashion. Later sections show
the underlying theory and introduce the more complicated, less common methods.
These later sections are starred to indicate their more advanced nature. The
unstarred sections are independent of the starred ones, so the elementary course
can be taught relying only on the simple material. For a more challenging course,
much of the material will come form the starred sections, with the elementary
part providing insight and motivation.
The material selected for the book is, for the most part, standard and traditional.
Only in the last three chapters some choices and compromises had to be made.
What to do about partial differential equations is not clear-cut for any author
of an introductory text, since any reasonable treatment presumes more theoretical
knowledge than the prospective audience can be expected to have. Additionally,
there are many practical complications that make it impossible to do justice
to the topic at the undergraduate level. But partial differential equations
are so important in practice that one cannot see what numerical analysis is
all about without some exposure to the main issues. Our way of dealing with
this dilemma is to present some simple prototype partial differential equations,
with a quick and intuitive overview of the difficulties of implementation and
the more theoretical question of stability. While this does not prepare students
to solve real-life partial differential equations, it presents them with the
flavor of the subject matter in preparation for more advanced courses.
One quite nontraditional topic is in the last chapter, the solution of inverse
and ill-posed problems. This inclusion not only reflects the author's special
interests, but also gives an introduction to an increasingly more important
topic. The matter is an advanced subject, so only the more intuitive aspects
are presented. It does show, though, that applying numerical methods naively
does not always work.
This book discusses the most important numerical algorithms, but it does not
attempt to be a reference work. Instead, it concentrates on the limited subject
matter that is most commonly presented to undergraduate students and stresses
pedagogical issues rather than completeness. The emphasis of the book is on
· providing insight and motivation for the construction of numerical methods,
· understanding the strengths and limitations of such methods,
· evaluating the effectiveness of available numerical software,
· modifying existing software for specific purposes,
· getting experience in choosing between alternative approaches,
· experimenting with numerical software in settings that mirror real-world situations,
· building a strong experiential base for continuing study and more specialized
courses.
These aims are greatly aided by the close connection between the discussion
of methods and algorithms and their implementation in MATLAB. While the MATLAB
library is much less extensive than many industrial libraries (such as IMSL),
it does give the student experience working with ready-made software whose internal
structure may not always entirely be clear. MATLAB's numerical analysis functions
are well constructed but are quite automatic and are used essentially as black
boxes. Since most libraries have similar characteristics, students learn how
to use numerical methods libraries effectively.
To complement the standard MATLAB functions, we provide an extension, NASOFT.
This set of functions does two things. First, it extends the MATLAB functionality
to problems such as two-point boundary value problems and some prototype partial
differential equations, allowing the student to experiment with fairly complex
algorithms. Secondly, NASOFT functions are coded in a straightforward manner
and the MATLAB source is available. This gives the student the opportunity to
critique the implementation and modify to improve it or to adapt it to different
purposes.
Many traditional numerical analysis courses are primarily lecture courses,
with perhaps a lab of secondary importance. This book envisions a different
emphasis in which lab work is at least as important as the lectures. For this
purpose, we have added a chapter called "Explorations" The problems in this
exploration section deal with the very practical issues of software evaluation,
selection, modification and the solution of not entirely specified, open-ended
problems. Students should conduct investigations using their own methodology,
and should be expected to write informative reports on their observations and
conclusions. This is the part of the course that most closely models real life
situations and should therefore be considered the heart of the course.
In summary, this book is designed for an undergraduate numerical analysis
course that stresses insight and hands-on experience over detailed knowledge
of a host of numerical methods and their mathematical justification.
Peter Linz
Richard L.C. Wang
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