This book presents the fundamental numerical techniques used in engineering, applied mathematics, computer science, and the physical and life sciences in a way that is both interesting and understandable. Using a wide range of examples and problems, this book focuses on the use of MathCAD functions and worksheets to illustrate the methods used when discussing the following concepts: solving linear and nonlinear equations, numerical linear algebra, numerical methods for data interpolation and approximation, numerical differentiation and integration, and numerical techniques for solving differential equations. For professionals in the fields of engineering, mathematics, computer science, and physical or life sciences who want to learn MathCAD functions for all major numerical methods.
The purpose of this text is to present the fundamental numerical techniques used in engineering, applied mathematics, computer science, and the physical and life sciences in a manner that is both interesting and understandable to undergraduate and beginning graduate students in those fields. The organization of the chapters, and of the material within each chapter, the use of Mathcad worksheets and functions to illustrate the methods, and the exercises provided are all designed with student learning as the primary objective.
The first chapter sets the stage for the material in the rest of the text, by giving a brief introduction to the long history of numerical techniques, and a "preview of coming attractions" for some of the recurring themes of the remainder of the text. It also presents enough description of Mathcad to allow students to use the Mathcad functions presented for each of the numerical methods discussed in the other chapters. An algorithmic statement of each method is also included; the algorithm may be used as the basis for computations using a variety of types of technological support, ranging from paper and pencil, to calculators, Mathcad worksheets or developing computer programs.
Each of the subsequent chapters begins with a one-page overview of the subject matter, together with an indication as to how the topics presented in the chapter are related to those in previous and subsequent chapters. Introductory examples are presented to suggest a few of the types of problems for which the topics of the chapter may be used. Following the sections in which the methods are presented, each chapter concludes with a summary of the most important formulas, a selection of suggestions for further reading, and an extensive set of exercises. The first group of problems provide fairly routine practice of the techniques; the second group are applications adapted from a variety of fields, and the final group of problems encourage students to extend their understanding of either the theoretical or the computational aspects of the methods.
The presentation of each numerical technique is based on the successful teaching methodology of providing examples and geometric motivation for a method, and a concise statement of the steps to carry out the computation, before giving a mathematical derivation of the process or a discussion of the more theoretical issues that are relevant to the use and understanding of the topic. Each topic is illustrated by examples that range in complexity from very simple to moderate.
Geometrical or graphical illustrations are included whenever they are appropriate. A simple Mathcad function is presented for each method, which also serves as a clear step-by-step description of the process; discussion of theoretical considerations is placed at the conclusion of the section. The last section of each chapter gives a brief discussion of Mathcad's built-in functions for solving the kinds of problems covered in the chapter.
The chapters are arranged according to the following general areas:
* Chapters 2-5 deal with solving linear and nonlinear equations.
* Chapters 6 and 7 treat topics from numerical linear algebra.
* Chapters 8-10 cover numerical methods for data interpolation and approximation.
* Chapters 11 presents numerical differentiation and integration.
* Chapters 12-15 introduce numerical techniques for solving differential equations.
For much of the material, a calculus sequence that includes an introduction to differential equations and linear algebra provides adequate background. For more in depth coverage of the topics from linear algebra (especially the QR method for eigenvalues) a linear algebra course would be an appropriate prerequisite. The coverage of Fourier approximation and FFT (Chapter 10) and partial differential equations (Chapter 15) also assumes that the students have somewhat more mathematical maturity than the other chapters, since the material in intrinsically more challenging. The subject matter included is suitable for a two-semester sequence of classes, or for any of several different one-term courses, depending on the desired emphasis, student background, and selection of topics.
Many people have contributed to the development of this text. My colleagues at Florida Institute of Technology, the Naval Postgraduate School, the University of South Carolina Aiken, and Georgia Southern University have provided support, encouragement, and suggestions. I especially want to thank Jacalyn Huband for the development of the Mathcad functions and examples. I also wish to thank three other colleagues for their particular contributions: Jane Lybrand for the data from classroom experiments used in several examples and exercises in Ch 9. Jack Leifer for providing data, as well as helpful discussions on engineering applications and the use of Mathcad in engineering; Pierre Larochelle for the example of robot motion in Ch 13.1 also appreciate the many contributions my students have made to this text, which was after all written with them in mind. The comments made by the reviewers of the text have helped greatly in the fine-tuning of the final presentation. The editorial and production staff at Prentice Hall, as well as Patty Donovan, and the rest of the staff at Pinetree Composition, have my heartfelt gratitude for their efforts in insuring that the text is as accurate and as well designed as possible. And, saving the most important for last, I thank my husband and colleague, Don Fausett, for his patience and support. Laurene Fausett
