Numerical Methods Algorithms and Applications

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, is designed with student learning as the primary objective. A detailed algorithm is given for each method presented, so that students can write simple programs implementing the technique in the computer language of their choice. Numerous examples of the use of the methods are also included. The first chapter sets the stage for the material in the rest of the text, giving a brief introduction to the long history of numerical techniques and a "preview of coming attractions" for some of the recurring themes in the remainder of the text. It also presents a summary of the key components of a computer program for solving problems involving numerical techniques such as those given in the text. The trapezoid rule for numerical integration is used to illustrate the relationship between a numerical algorithm and a computer program implementing the algorithm. Sample programs are given in several different languages. 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, of suggestions for further reading, and an extensive set of exercises. The first group of problems provides fairly routine practice of the techniques; the second group includes applications adapted from a variety of fields, and the final group of problems encourages 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. The last section of each chapter gives a brief discussion of more advanced methods for solving the kinds of problems covered in the chapter, including methods used in MATLAB, Mathcad,Mathematica,and various software libraries. The chapters are arranged according to the following general areas: Chapter 2 deals with solving nonlinear equations of a single variable. Chapters 3-6 treat topics from numerical linear algebra. Chapter 7 considers nonlinear functions of several variables Chapters 8-10 cover numerical methods for data interpolation and approximation. Chapter 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 OR 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) assumes that the students have somewhat more mathematical maturity sinFausett, Laurene V. is the author of 'Numerical Methods Algorithms and Applications', published 2002 under ISBN 9780130314000 and ISBN 0130314005.