Preface

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Preface

Thank you for giving Tea Time Numerical Analysis a read. This textbook was born out of a desire to contribute a viable, complete, and completely free, Numerical Analysis textbook for instructors and students of mathematics. When this project began (summer 2012), there were traditionally published (very expensive hardcover) textbooks, notably the excellent Numerical Analysis by Burden and Faires, which was in its ninth edition. As you might guess by the number of editions, this text is a classic. It is one of very few numerical analysis texts geared for the mathematician, not the scientist or engineer. In fact, I studied from an early edition in the mid 1990's! And there were a couple of freely available websites, notably the popular http://nm.mathforcollege.com/, complete with video lectures. However, no resource I could find included a complete, single-pdf downloadable textbook designed for mathematics classes. To be just that is the ultimate goal of Tea Time Numerical Analysis.

I should apologize up front for the lack of conformance to the typical format of a college textbook, however. The phrase “tea time” is meant to do more than give the book a catchy title. It is meant to describe the general nature of the discourse within. Much of the material will be presented as if it were being told to a student during tea time at University, but with the benefit of careful planning. There will be no big blue boxes highlighting the main points, no stream of examples after a short introduction to a topic, and no theorem-proof theorem-proof structure. Instead, the necessary terms and definitions and theorems and examples will be woven into a more conversational style. My hope is that this blend of formal and informal mathematics will be easier to digest, and dare I say, students will be more invited to do their reading in this format. The content is meant to help guide the student's thoughts through the mathematics.

In an effort to avoid completely alienating those who enjoy a more typical presentation, there will be a summary of the key concepts at the end of each conversation, and a number of the exercises will be solved in complete detail in the appendix. So, one can get a closer-to-typical presentation by reading the key concepts and then skipping to the exercises with solutions, bypassing the discourse altogether! I hope most readers won't choose to do so, but it is an option. I should add, though, that this may be the preferred means of use by instructors. In any case, the exercises with solutions will be critical reading for most, despite their location in the back of the book. Learning by example is often the most effective means, so readers are strongly encouraged to read the exercises with solutions, contemplate their solutions, and then turn to the back of the book for full disclosure of the solutions. The hope is that readers will be more apt to consider solving the exercises on their own before looking at the solutions. Their placement in the appendix is only for this purpose---not to imply that they are of secondary importance.

The topical coverage in Tea Time Numerical Analysis will be fairly typical. The book starts with an introductory chapter, followed by root finding methods, followed by interpolation, and finally numerical calculus. This covers what, at SCSU, constitutes a first semester course on numerical analysis. Future efforts may add chapters on linear algebra and differential equations, making it a full year course. As this book is intended for use as a free download or an inexpensive print-on-demand volume, no effort has been made to keep the page count low or to spare copious diagrams and colors. In fact, I have taken the inexpensive mode of delivery as liberty to do quite the opposite. I have added many passages and diagrams that are not strictly necessary for the study of numerical analysis, but are at least peripherally related, and may be of interest to some readers. Most of these passages will be presented as digressions, so they will be easy to identify. For example, Taylor's Theorem plays such a central role in the subject that not only its statement is presented. Its proof and a bit of history are added as “gory details”. Of course they can be skipped, but are included to provide a more complete understanding of this fundamental theorem of numerical analysis. For another example, as a fan of dynamical systems, I found it impossible to refrain from including a section on visualizing Newton's Method. The powerful and beautiful pictures of Newton's Method as a dynamical system should be eyebrow-raising and question-provoking even if only tangentially important. There are, of course, other examples of somewhat atypical content, but each is there to enhance the reader's understanding or appreciation of the subject, even if the material is not strictly necessary for an introductory study of numerical analysis.

Along the way, implementation of the numerical methods in the form of computer code will also be discussed. While one could simply ignore the programming sections and exercises and still get something out of this text, it is my firm belief that full appreciation for the content can not be achieved without getting your hands dirty and doing some programming. It would be nice if readers have had at least some minimal exposure to programming whether it be Java, or C, or even web programming. But I have made every effort to give a complete introduction to Octave programming, so even those who have never written even a one-line program will be able to participate in this part of the study.

GNU Octave was chosen as the programming language for two reasons. First, MATLAB is perhaps the most widely used and best known language for numerical computation. GNU Octave (Octave for short) is a MATLAB clone. In this case, the clone has an extreme advantage over the original. MATLAB is far from free in any sense of the word. Octave is offered freely to anyone and everyone (http://www.gnu.org/software/octave/). It is free to download and use. Its source code is free to download and study. And anyone is welcome to modify or add to the code if so inclined.

Though it is easy to produce GNU Octave programs that will not run in MATLAB, by design nearly any program written in MATLAB will run in Octave without modification. In order to maintain both a MATLAB-compatible presentation and a free computing experience, I have made considerable effort to ensure that every line of Octave in this book will run verbatim under MATLAB. Even with this earnest effort, it is possible that some of the code will not run under MATLAB. I'd be happy to hear if you find such an instance! Please let me know.

As it is expected that most readers will use Octave for all programming in the course, there are subsections dedicated to programming and, in particular, Octave. There are also numerous exercises where Octave is required or highly recommended, each marked with the Octave logo, .

I hope you enjoy your reading of Tea Time Numerical Analysis. It was my pleasure to write it. Feedback is always welcome.