Preface Of Fundamental Numerical Method

The origins of this book can be found years ago when I was a doctoral candidate working on my thesis and finding that I needed numerical tools that I should have been taught years before. In the intervening decades, little has changed except for the worse. All fields of science have undergone an information explosion while the computer revolution has steadily and irrevocability been changing our lives. Although the crystal ball of the future is at best "seen through a glass darkly", most would declare that the advent of the digital electronic computer will change civilization to an extent not seen since the coming of the steam engine. Computers with the power that could be offered only by large institutions a decade ago now sit on the desks of individuals. Methods of analysis that were only dreamed of three decades ago are now used by students to do homework exercises. Entirely new methods of analysis have appeared that take advantage of computers to perform logical and arithmetic operations at great speed. Perhaps students of the future may regard the multiplication of two two-digit numbers without the aid of a calculator in the same vein that we regard the formal extraction of a square root. The whole approach to scientific analysis may change with the advent of machines that communicate orally. However, I hope the day never arrives when the investigator no longer understands the nature of the analysis done by the machine.

Unfortunately instruction in the uses and applicability of new methods of analysis rarely appears in the curriculum. This is no surprise as such courses in any discipline always are the last to be developed. In rapidly changing disciplines this means that active students must fend for themselves. With numerical analysis this has meant that many simply take the tools developed by others and apply them to problems with little knowledge as to the applicability or accuracy of the methods. Numerical algorithms appear as neatly packaged computer programs that are regarded by the user as "black boxes" into which they feed their data and from which come the publishable results. The complexity of many of the problems dealt with in this manner makes determining the validity of the results nearly impossible. This book is an attempt to correct some of these problems.

Some may regard this effort as a survey and to that I would plead guilty. But I do not regard the word survey as pejorative for to survey, condense, and collate, the knowledge of man is one of the responsibilities of the scholar. There is an implication inherent in this responsibility that the information be made more comprehensible so that it may more readily be assimilated. The extent to which I have succeeded in this goal I will leave to the reader. The discussion of so many topics may be regarded by some to be an impossible task. However, the subjects I have selected have all been required of me during my professional career and I suspect most research scientists would make a similar claim. xi

Unfortunately few of these subjects were ever covered in even the introductory level of treatment given here during my formal education and certainly they were never placed within a coherent context of numerical analysis.

The basic format of the first chapter is a very wide ranging view of some concepts of mathematics based loosely on axiomatic set theory and linear algebra. The intent here is not so much to provide the specific mathematical foundation for what follows, which is done as needed throughout the text, but rather to establish, what I call for lack of a better term, "mathematical sophistication". There is a general acquaintance with mathematics that a student should have before embarking on the study of numerical methods. The student should realize that there is a subject called mathematics which is artificially broken into sub-disciplines such a linear algebra, arithmetic, calculus, topology, set theory, etc. All of these disciplines are related and the sooner the student realizes that and becomes aware of the relations, the sooner mathematics will become a convenient and useful language of scientific expression. The ability to use mathematics in such a fashion is largely what I mean by "mathematical sophistication". However, this book is primarily intended for scientists and engineers so while there is a certain familiarity with mathematics that is assumed, the rigor that one expects with a formal mathematical presentation is lacking. Very little is proved in the traditional mathematical sense of the word. Indeed, derivations are resorted to mainly to emphasize the assumptions that underlie the results. However, when derivations are called for, I will often write several forms of the same expression on the same line. This is done simply to guide the reader in the direction of a mathematical development. I will often give "rules of thumb" for which there is no formal proof. However, experience has shown that these "rules of thumb" almost always apply. This is done in the spirit of providing the researcher with practical ways to evaluate the validity of his or her results.

The basic premise of this book is that it can serve as the basis for a wide range of courses that discuss numerical methods used in science. It is meant to support a series of lectures, not replace them. To reflect this, the subject matter is wide ranging and perhaps too broad for a single course. It is expected that the instructor will neglect some sections and expand on others. For example, the social scientist may choose to emphasize the chapters on interpolation, curve-fitting and statistics, while the physical scientist would stress those chapters dealing with numerical quadrature and the solution of differential and integral equations. Others might choose to spend a large amount of time on the principle of least squares and its ramifications. All these approaches are valid and I hope all will be served by this book. While it is customary to direct a book of this sort at a specific pedagogic audience, I find that task somewhat difficult. Certainly advanced undergraduate science and engineering students will have no difficulty dealing with the concepts and level of this book. However, it is not at all obvious that second year students couldn't cope with the material. Some might suggest that they have not yet had a formal course in differential equations at that point in their career and are therefore not adequately prepared. However, it is far from obvious to me that a student’s first encounter with differential equations should be in a formal mathematics course. Indeed, since most equations they are liable to encounter will require a numerical solution, I feel the case can be made that it is more practical for them to be introduced to the subject from a graphical and numerical point of view. Thus, if the

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