这是一部非常成功的学术著作，它介绍了科学计算需要的各类数值分析。不但在严谨的数学科学背景下进行讨论，而且给出了数值分析方法的严格证明。
　　本书适合作为数学、工程、计算机科学和其他相关专业高年级本科生或研究生数值分析课程的教材本书特点：
　　·涵盛了科学计算中数值分桥的广泛主题，除数值分析的基础知识外，还涉及线性代数和非线性代数系统的求解、数值微分与数值积分．常微分方程和偏微分方程的数值解、函数逼近等方面的内容
　　·增加了优化方面的内容和查询相关信息的网络资源
　　·书中并不详细分析算法，而是着重讲解相关的理论基础
　　·算法以伪代码的形式给出，以便学生可以立即用标准语言和交互数据包来编写实现算法的计算机程序

David Kincaid是得克萨斯大学奥斯汀分校计算机科学系及数学系的高级讲师，他还是得克萨斯计算及应用 数学学会数值分析中心的代主任。
　　Ward Cheney是得克萨斯大学奥斯汀分校数学系教授，他的研究方向主要是逼近理论、数值分析和极大化问题。

This book has evolved over many years from lecture notes that accompany certain upper-division and graduate courses in mathematics and computer sciences at The University of Texas at Austin. These courses introduce students to the algorithms and methods that are commonly needed in scientific computing. The mathematical underpinnings of these methods are emphasized as much as their algorithmic aspects. The students have been diverse: mathematics, engineering, science, and computer science undergraduates, as well as graduate students from various disciplines. Portions of the book also have been used to lay the groundwork in several graduate courses devoted to special topics in numerical analysis, such as the numerical solution of differential equations, numerical linear algebra, and approximation theory. Our approach has always been to treat the subject from a mathematical point of view, with attention given to its rich offering of theorems, proofs, and interesting ideas. From these arise many computational procedures and intriguing questions of computer science. Of course, our motivation comes from the practical world of scientific computing, which dictates the choice of topics and the manner of treating each. For example, with some topics it is more instructive to discuss the theoretical
foundations of the subject and not attempt to analyze algorithms in detail. In other cases, the reverse is true, and the students learn much from programming simple algorithms themselves and experimenting with them-although we offer a blanket admonition to use well-tested software, such as from program libraries, on problems that arise from applications.
　　There is some overlap between this book and our more elementary text, Numerical Mathematics and Computing, Fourth Edition (Brooks/Cole). That book is addressed to students having more modest mathematical preparation (and sometimes less enthusiasm for the theoretical side of the subject). In that text, there is a different menu of topics, and no topic is pursued to any great depth. The present book, on the other hand, is intended for a course that offers a more scholarly treatment of the subject; many topics are dealt with at length. Occasionally we broach topics that heretofore have not found their way into standard textbooks at this level. In this category are the multigrid method, procedures for multivariate interpolation, homotopy (or continuation) methods, delay differential equations, and optimization.
　　The algorithms in the book are presented in a pseudocode that contains additional details beyond the mathematical formulas. The reader can easily write routines based on the pseudocode in any standard computer language. We believe that students learn and understand numerical methods best by seeing how algorithms are developed from the mathematical theory and then writing and testing computer implementations of them. Of course, such computer programs do not contain all the complicated procedures and sophisticated checks found in robust routines available in scientific libraries. Examples of general-purpose mathematical libraries are found in the appendix on ,An Overview of Mathematical Software. For most applications. such libraries are strongly preferred to code written oneself.
　　An important constituent of the book (and essential to its pedagogic purpose) is he abundance of problems for the student. These are of two types: analytic problems and computer problems. The computer problems are, in tum, of two types: those in which students write their own code, and those in which they employ existing software. We believe that both kinds of programming practice are necessary. Using someone else's software is not always a trivial exercise, "even when it is as well documented as with large program libraries or packages. On the other hand, students usually acquire much more insight into an algorithm after coding and testing it themselves, rather than simply using a software package. In most cases the computer problems require access to a computer that has at least a 32-bit word length.
　　Software, errata, and teaching aids are available via the Internet as discussed in the appendix. Also, the publisher has made available a Solution Manual for instructors who adopt the book for their classes.
　　The third edition contains new problems, re-ordering of some problems, and corrections to all known errors in the previous edition. Updating of the information about resources on the Internet has been done in the appendix on mathematical software. Also, the bibliography has been updated. Many references to problems and to other parts of the book are now given with page numbers to help the reader easily find them. Also, most theorems are displayed with names or titles
to help the reader remember them. The entire book has a new design style and it has been reformatted for improved appearance. Many improvements have been made throughout. For example, in this new edition, we have added a chapter on optimization with subtopics on methods of descent, quadratic fitting algorithms. Nelder-Meade algorithm, simulated annealing, genetic algorithms, Pareto optimization, and convex programming. A standard course of one semester can be based on selected sections from Chapters 1-4 and 6-8. A two-semester course could cover selected sections in Chapters l-9 plus other topics of interest. Chapters 4 and 5 could be taught independently from the previous chapters as a short course on numerical linear algebra. Because of the ambitious scope of this book, some sections make greater demands on the preparation of the reader. These sections usually occur late in any given chapter so that the reader is not unduly challenged at the start, and they may be. skipped at the reader's discretion. Such sections are marked with an asterisk. Page numbers are included with references to problems, computer problems, and items such as theorems and equations outside the section being read. Unless it says otherwise, references to equations, theorems, lemmas, corollaries, etc. are assumed to be in the current section and page numbers are not included.
　　
Acknowledgments
　　We are glad to express our indebtedness to many persons who have assisted us in the writing of this book.
　　First Edition
　　Administrative support was provided by Sheri Brice, Margaret Combs, Jan Duffy, Katherine Mueller, and Jenny Tsao of The University of Texas at Austin. Foremost among these is Margaret Combs of the Mathematics Department, who rendered into TEX innumerable versions of each section, patiently preparing new ones as they were needed for classroom notes in successive years. No technical problem of typesetting was too difficult for her as she mastered the arcane art of dissecting and reassembling TEX macros to serve unusual needs. It is appropriate at this point that we also express our public thanks to Donald Knuth for his magnificent contribution to the scientific community embodied in the TEX typesetting system. We appreciate the suggestions made by the following astute reviewers of preliminary versions of the manuscript: Thomas A. Atchison, Frederick J. Carter, Philip Crooke, Jim D'Archangelo, R. S. Falk, J. R. Hubbard, Patrick Lang, Giles Wilson Maloof, A. K. Rigler, F. Schumann, A. J. Worsey, and Charles Votaw. In addition, thanks are due the following persons for technical help and critical reading of the manuscript: Victoria Hunter, Carole Kincaid, Tad Liszka, Rio Hirowati Shariffudin, and Laurette Tuckerman. David Young was always generous with suggestions and advice. A number of advanced students who served as teaching assistants in our classes also helped; in particular. we thank David Bruce, Nai-ching Chen, Ashok Hattangady, Ru Huang, Wayne Joubert, Irina Mukherjee, Bill Nanry, Tom Oppe, Marcos Raydan, Malathi Ramdas, John Respess, Phien Tran, Linda Tweedy, and Baba Vemuri. The editors and technical staff at Brooks/Cole Publishing Company have been most cooperative and supportive during this project. In particular, we are pleased to thank Jeremy Hayhurst and Marlene Thom for their assistance. Stacey Sawyer of Sawyer and Williams was responsible for a careful copyediting of the manuscript, and Ralph Youngen of the American Mathematical Society provided technical assistance and supervision in turning our TEX files into final printed copy.
　　Second Edition
　　We would like to acknowledge the reviewers for their work: Dan Boley, University of Minnesota; Min Chen, Pennsylvania State University; John Harper, University of Rochester; Ramon Moore, Ohio State University; Yves Nievergelt, Eastern Washington University; and Elinor Velasquez, University of California-Berkeley. We especially thank Ron Boisvert for clarifying our understanding of the different categories of mathematical software and for the examples given in the appendix. Also, we want to thank those who took the trouble to contact us with suggestions and corrections in the first edition. Some of these are Victor M. Afram, Roger Alexander.A. Awwal, Carl de Boor, T. P. Brown, James Caveny, George J. Davis, Hakan Ekblom, Mariano Gasca, Bill Gearhart, Patrick Goetz, Gary L. Gray, Bob Katherine Hua Guo, Cecilia Jea, Liz Jessup, Grant Keady, Baker Kearfott. Junjiangegeany Lei, Teck C. Lim, Julio Lopez, C. Lu, Taketomo Mitsui, Irina Mukherjee, Teresa Perez, Robert Piche, Sherman Riemenschneider, Maria Teresa Rodriqaez, Ulf Roennow, Larry Schumaker, Wei-Chang Shann, Christopher J. van Wyk, Kang Zhao, and Mark Zhou.
　　Third Edition
　　We wish to thank those who have contacted as with suggestions and corrections in the second edition: Eyal Arian, Carl de Boor, Yung-Ming Chang, Antonella Cupillari Paul Eenigenburg, Leopoldo P. Franca, Henry Greenside, R.J. Gregorac, Scott A. King, Robert Piche, N. Shamsundar, Topi Urponen, and Yuan Xu. Also, we are grateful to those who have been helpfal in putting out the new edition: Pauick Goetz, Shashank Khandelwal, and especially Durene Ngo.
　　We are grateful to tbe Center for Numerical Analysis, the Texas Institute of Computational and Applied Mathematics, and ale Computer Sciences Department and the Mathematics Department of The University of Texas at Austin for technical support and for furnishing excellent computing facilities.
　　Finally, the editors and technical staff at Brooks/Cole-Thomson Learning have been most helpful and supportive during the revision of this book, particularly Bob Pirtle, Janet Hill, and Molly Nance. We also thank Don DeLand, Leslie Galen, Joe Albrecht, and others at Integre Technical Publishing Company for their fine work.
　　Software supporting this textbook is available online at the Web sites listed in the Appendix (p. 737).
　　We would appreciate any comments, suggestions, questions, criticisms, or corrections that readers may take the trouble of communicating to us.
　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　 David Kincaid
Ward Cheney

1 Mathematical Preliminaries
1.0 Introduction
1.1 Basic Concepts and Taylor's Theorem
1.2 Orders of Convergence and Additional Basic concepts
1.3 Difference Equations
2 Computer Arithmetic
2.0 Introduaion
2.1 Floating-Point Numbers and Roundoff Errors
2.2 Absolute and Relative Errors: Loss of Significance
2.3 Stable and Unstable Computations: Conditioning
3 Solution of Nonlinear Equations
3.0 Introduction
3.1 Bisection (Interval Halving Method
3.2 Newton's Method
3.3 Secant Method
3.4 Fixed Points. and Functional Iteration
3.5 Computing Roots of Polynomials
3.6 Homotopy and Continuation Methods
4 Solving Systems of Linear Equations
4.0 Introduction
4.1 Matrix Algebra
4.2 LU and Cholesky Factorizations
4.3 Pivoting and Constructing an Algorithm
4.4 Norms and the Analysis of Errors
4.5 Neumann Series and Iterative Refinement
4.6 Solution of Equations by Iterative Methods
4.7 Steepest Descent and Conjugate Gradient Methods
4.8 Analysis of Roundoff Error in the Gaussian Algorithm
5 Selected Topics in Numerical Linear Algebra
5.0 Review of Basic Concepts
5.1 Matrix Eigenvalue Problem: Power Method
5.2 Schur's and Cershgorin's Theorems
5.3 Orthogonal Factorizations and Least-Squares Problems
5.4 Singular-Value Decomposition and Pseudoinverses
5.5 QR-Algorithm of Francis for the Eigenvalue Problem
6 Approximating Functions
6.0 Introduction
6.1 Polynomial Interpolation
6.2 Divided Differences
6.3 Hermite Interpolation
6.4 Spline Interpolation
6.5 B-Splines: Basic Theory
6.6 B-Splines: Applications
6.7 Taylor Series
6.8 Best Approximation: Least-Squares Theory
6.9 Best Approximation: Chebyshev Theory
6.1O Interpolatlon In Higher Dimensions
6.11 Continued Fractions
6.12 Trigonometric Interpolation
6.13 Fast Fourier Transform
6.14 Adaptive Approximation
7 Numerical Differentiation and
7.1 Numerical Differentiation and
7.2 Numerical Integration Based on
7.3 Gaussian Quadrature
7.4 Romberg Integration
7.5 Adaptive Quadrature
7.6 Sard's Theory of Approximating Functionals
7.7 Bernoulli Polynomials and the Euler-Maclaurin Formula
8 Numeriral Solution ot Ordinary
Differential Equations
8.0 Introduction
8.1 The Existence and Uniqueness of Solutions
8.2 Taylor-Series Method
8.3 Runge-Kutta Methods
8.4 Multistep Methods
8.5 Local and Global Errors: Stability
8.6 Systems and Higher-Order Ordinary Differential Equations
8.7 Boundary-Value Problems
8.8 Boundary-Value Problems: Shooting Methods
8.9 Boundary-Value Problems: Finite-Differences
8.1 0 Boundary-Value Problems: Collocation
8.1 1 Linear Differential Equations
8.1 2 Stiff Equations
9 Numetical Selution of Partial Differential Equations
9.0 Introduction
9.1 Parabolic Equations: Explicit Methods
9.2 Parabolic Equations: Implicit Methods
9.3 Problems Without Without Time Dependence: Finite-Differences
9.4 Problems Without Without Time Dependence: Galerkin Methods
9.5 First-Order Partial Differential Equations: Characteristics
9.6 Quasilinear Second-Order Equations: Characteristics
9.7 Other Methods for Hyperbolic Problems
9.8 Multigrid Method
9.9 Fast Methods for Poisson's Equation
10 Linear Programming and Related Topics
10.1 Convexity and Linear Inequalities
10.2 Linear Inequalities
10.3 Linear Programming
10.4 The Simplex Algorithm
11 Oytimization
11.0 Introdudion
11.1 One-Variable Case
11.2 Descent Methods
11.3 Analysis of Quadratic Objective Functions
11.4 Quadratic-Fitting Algorithms
11.5 Nelder-Meade Algorithm
11.6 Simulated Annealing
11.7 Cenetic Algorithms
11.8 Convex Programming
11.9 Constrained Minimization
11.10 Pareto Optimization
Appendix A An Overview of Mathematical Software
Bibliography
Index