Elementary Numerical Analysis: An Algorithmic Approach | Date: 08 May 2011, 00:38
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PREFACE
This is the third edition of a book on elementary numerical analysis which is designed specifically for the needs of upper-division undergraduate students in engineering, mathematics, and science including, in particular, computer science. On the whole, the student who has had a solid college calculus sequence should have no difficulty following the material. Advanced mathematical concepts, such as norms and orthogonality, when they are used, are introduced carefully at a level suitable for undergraduate students and do not assume any previous knowledge. Some familiarity with matrices is assumed for the chapter on systems of equations and with differential equations for Chapters 8 and 9. This edition does contain some sections which require slightly more mathematical maturity than the previous edition. However, all such sections are marked with asterisks and all can be omitted by the instructor with no loss in continuity.
This new edition contains a great deal of new material and significant changes to some of the older material. The chapters have been rearranged in what we believe is a more natural order. Polynomial interpolation (Chapter 2) now precedes even the chapter on the solution of nonlinear systems (Chapter 3) and is used subsequently for some of the material in all chapters. The treatment of Gauss elimination (Chapter 4) has been simplified. In addition, Chapter 4 now makes extensive use of Wilkinson's backward error analysis, and contains a survey of many well-known methods for the eigenvalue-eigenvector problem. Chapter 5 is a new chapter on systems of equations and unconstrained optimization. It contains an introduction to steepest-descent methods, Newton's method for nonlinear systems of equations, and relaxation methods for solving large linear systems by iteration. The chapter on approximation (Chapter 6) has been enlarged. It now treats best approximation and good approximation by polynomials, also approximation by trigonometric functions, including the Fast Fourier Transforms, as well as least-squares data fitting, orthogonal polynomials, and curve fitting by splines. Differentiation and integration are now treated in Chapter 7, which contains a new section on
adaptive quadrature. Chapter 8 on ordinary differential equations contains considerable new material and some new sections. There is a new section on step-size control in Runge-Kutta methods and a new section on stiff differential equations as well as an extensively revised section on numerical instability. Chapter 9 contains a brief introduction to collocation as a method for solving boundary-value problems. This edition, as did the previous one, assumes that students have access to a computer and that they are familiar with programming in some procedure-oriented language. A large number of algorithms are presented in the text, and FORTRAN programs for many of these algorithms have been provided. There are somewhat fewer complete programs in this edition. All the programs have been rewritten in the FORTRAN 77 language which uses modern structured-programming concepts. All the programs have been tested on one or more computers, and in most cases machine results are presented. When numerical output is given, the text will indicate which machine (IBM, CDC, UNIVAC) was used to obtain the results. The book contains more material than can usually be covered in a typical one-semester undergraduate course for general science majors. This gives the instructor considerable leeway in designing the course. For this, it is important to point out that only the material on polynomial interpolation in Chapter 2, on linear systems in Chapter 4, and on differentiation and integration in Chapter 7, is required in an essential way in subsequent chapters. The material in the first seven chapters (exclusive of the starred sections) would make a reasonable first course.
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