本書介紹了科學(xué)計算中常用數(shù)值分析的基礎(chǔ)理論及計算機(jī)實現(xiàn)方法。主要內(nèi)容包括：誤差分析、插值、函數(shù)逼近、數(shù)值積分和數(shù)值微分、非線性方程的數(shù)值解法、線性方程組的直接解法、線性方程組的迭代解法、常微分方程的數(shù)值解法及相應(yīng)的上機(jī)實驗內(nèi)容等。各章都配有大量的習(xí)題及上機(jī)實驗題目，并附有部分習(xí)題的參考答案及數(shù)學(xué)專業(yè)軟件Mathematica和Matlab的簡介。
本書采用中、英兩種語言編寫，適合作為數(shù)學(xué)、計算機(jī)和其他理工類各專業(yè)本科“數(shù)值分析（計算方法）”雙語課程的教材或參考書，也可供從事科學(xué)計算的相關(guān)技術(shù)人員參考。

本書采用中、英兩種語言編寫，各章都配有大量的習(xí)題及上機(jī)實驗題目，并附有部分習(xí)題的參考答案及數(shù)學(xué)專業(yè)軟件Mathematica和Matlab的簡介。

本書第1 版出版以來，得到了許多專家、同仁及讀者的關(guān)心、支持和幫助，并提出了許多寶貴意見和建議。借再版之機(jī)，首先向關(guān)心本書的廣大讀者、專家、同行和本書的各位責(zé)任編輯表示由衷的謝意！
在修訂中，為了更適合當(dāng)前雙語教學(xué)的需求，我們保留了原教材的系統(tǒng)和編寫風(fēng)格（理論部分以中文為主，軟件實現(xiàn)部分以英文為主），注意吸收當(dāng)前國內(nèi)外教材改革中一些成功的經(jīng)驗，努力體現(xiàn)創(chuàng)新教學(xué)理念，以利于激發(fā)學(xué)生自主學(xué)習(xí)，提高實踐應(yīng)用能力，培養(yǎng)綜合素質(zhì)和創(chuàng)新能力。
本次再版修訂的內(nèi)容主要包括以下幾方面：
1．訂正了語言文字表達(dá)方面的不足之處，力求用詞規(guī)范，表達(dá)確切。
2．剔除了個別內(nèi)容重復(fù)和煩瑣之處，使理論部分更好地體現(xiàn)“夠用為度”的編寫原則。
3．恰當(dāng)?shù)靥幚碛嘘P(guān)定理的證明和有關(guān)例題的求解方法，使其更加通俗易懂。
4．增補(bǔ)了多重積分、有理逼近、Padé逼近等內(nèi)容，進(jìn)一步體現(xiàn)教材的先進(jìn)性。
5．結(jié)合增補(bǔ)內(nèi)容，對習(xí)題配置作了進(jìn)一步充實、完善。
6．在實驗部分，大量增加了算法的Matlab 實現(xiàn)程序及相應(yīng)的算例，以便于指導(dǎo)學(xué)生實踐應(yīng)用。
本書由浙江臺州學(xué)院蘇岐芳副教授主編，浙江臺州學(xué)院鄭學(xué)良教授、李希文副教授和應(yīng)瑋婷老師參與修訂。具體寫作分工為：第1 章、第2 章及附錄由李希文修訂；第3章由鄭學(xué)良修訂；第4章~第8章由蘇岐芳修訂；全書的計算機(jī)實驗由應(yīng)瑋婷修訂。
在本書修訂過程中，浙江師范大學(xué)徐秀斌教授為本書提出了許多寶貴意見，浙江海洋學(xué)院郝彥教授、朱玉輝老師及廈門理工學(xué)院陳淑萍老師，對本書的編寫都做了大量工作，在此一并表示衷心感謝！
編 者
2016年10月

蘇岐芳，副教授，臺州學(xué)院數(shù)學(xué)與信息工程學(xué)院副院長

1 Error Analysis ......1
1.1 Introduction ............ 1
1.2 Sources of Errors .... 2
1.3 Errors and Significant Digits .......... 4
1.4 Error Propagation ... 8
1.5 Qualitative Analysis and Control of Errors ............ 9
1.5.1 Ill-condition Problem and Condition Number....................... 9
1.5.2 The Stability of Algorithm .. 10
1.5.3 The Control of Errors .......... 11
1.6 Computer Experiments................. 14
1.6.1 Functions Needed in the Experiments by Mathematica ...... 14
1.6.2 Experiments by Mathematica...................... 14
1.6.3 Functions Needed in the Experiments by Matlab................ 16
1.6.4 Experiments by Matlab ....... 16
Exercises 1..................... 17
2 Interpolating.......19
2.1 Introduction .......... 20
2.2 Basic Concepts ..... 21
2.3 Lagrange Interpolation ................. 22
2.3.1 Linear and Parabolic Interpolation .............. 22
2.3.2 Lagrange Interpolation Polynomial............. 24
2.3.3 Interpolation Remainder and Error Estimate....................... 25
2.4 Divided-differences and Newton Interpolation .... 29
2.5 Differences and Newton Difference Formulae..... 33
2.5.1 Differences .. 33
2.5.2 Newton Difference Formulae ...................... 35
2.6 Hermite Interpolation ................... 38
2.7 Piecewise Low Degree Interpolation.................... 42
2.7.1 Ill-posed Properties of High Degree Interpolation .............. 42
2.7.2 Piecewise Linear Interpolation .................... 43
2.7.3 Piecewise Cubic Hermite Interpolation....... 44
2.8 Cubic Spline Interpolation............ 45
2.8.1 Definition of Cubic Spline... 45
2.8.2 The Construction of Cubic Spline ............... 46
2.9 Computer Experiments................. 49
2.9.1 Functions Needed in the Experiments by Mathematica ...... 49
2.9.2 Experiments by Mathematica...................... 50
2.9.3 Experiments by Matlab ....... 56
Exercises 2................... 64
3 Best Approximation ...................68
3.1 Introduction .......... 68
3.2 Norms ................... 69
3.2.1 Vector Norms ...................... 69
3.2.2 Matrix Norms ...................... 74
3.3 Spectral Radius..... 76
3.4 Best Linear Approximation .......... 79
3.4.1 Basic Concepts and Theories....................... 79
3.4.2 Best Linear Approximation . 81
3.5 Discrete Least Squares Approximation ................ 82
3.6 Least Squares Approximation and Orthogonal Polynomials........ 87
3.7 Rational Function Approximation 94
3.7.1 Continued Fractions ............ 94
3.7.2 Padé Approximation............ 97
3.8 Computer Experiments................. 99
3.8.1 Functions Needed in The Experiments by Mathematica..... 99
3.8.2 Experiments by Mathematica.................... 100
3.8.3 Functions Needed in The Experiments by Matlab ............ 106
3.8.4 Experiments by Matlab ..... 106
Exercises 3................. 111
4 Numerical Integration and Differentiation ........114
4.1 Introduction ........ 115
4.2 Interpolatory Quadratures........... 116
4.2.1 Interpolatory Quadratures.. 116
4.2.2 Degree of Accuracy........... 117
4.3 Newton-Cotes Quadrature Formula.................... 118
4.4 Composite Quadrature Formula . 123
4.4.1 Composite Trapezoidal Rule ..................... 123
4.4.2 Composite Simpson’s Rule ....................... 124
4.5 Romberg Integration................... 125
4.5.1 Recursive Trapezoidal Rule ...................... 125
4.5.2 Romberg Algorithm .......... 126
4.5.3 Richardson’s Extrapolation ....................... 128
4.6 Gaussian Quadrature Formula .... 129
4.7 Multiple Integrals ....................... 134
4.8 Numerical Differentiation........... 135
4.8.1 Numerical Differentiation . 135
4.8.2 Differentiation Polynomial Interpolation .. 137
4.8.3 Richardson’s Extrapolation ....................... 141
4.9 Computer Experiments............... 144
4.9.1 Functions Needed in the Experiments by Mathematica .... 144
4.9.2 Experiments by Mathematica.................... 144
4.9.3 Experiments by Matlab ..... 149
Exercises 4................... 153
5 Solution of Nonlinear Equations ......................156
5.1 Introduction ........ 156
5.2 Basic Theories .... 158
5.3 Bisection Method 159
5.4 Iterative Method and Its Convergence................ 162
5.4.1 Fixed Point and Iteration ... 162
5.4.2 Global Convergence.......... 163
5.4.3 Local Convergence............ 165
5.4.4 Order of Convergence ....... 167
5.5 Accelerating Convergence.......... 168
5.6 Newton’s Method ....................... 170
5.6.1 Newton’s Method and Its Convergence .... 170
5.6.2 Reduced Newton Method and Newton’s Descent Method ....................... 172
5.6.3 The Case of Multiple Roots....................... 173
5.7 Secant Method and Muller Method .................... 174
5.7.1 Secant Method................... 174
5.7.2 Muller Method................... 175
5.8 Systems of Nonlinear Equations. 176
5.9 Computer Experiments............... 179
5.9.1 Functions Needed in the Experiments by Mathematica .... 179
5.9.2 Experiments by Mathematica.................... 180
5.9.3 Experiments by Matlab ..... 185
Exercises 5................. 188
6 Direct Methods for Solving Linear Systems ....191
6.1 Introduction ........ 192
6.2 Gaussian Elimination.................. 193
6.2.1 Basic Gaussian Elimination....................... 193
6.2.2 Triangular Decomposition. 197
6.3 Gaussian Elimination with Column Pivoting ..... 200
6.4 Methods of the Triangular Decomposition......... 202
6.4.1 The Direct Methods of The Triangular Decomposition .... 202
6.4.2 The Square Root Method .. 203
6.4.3 The Speedup Method......... 206
6.5 Analysis of Round-off Errors ..... 210
6.5.1 Condition Number............. 210
6.5.2 Iterative Refinement .......... 214
6.6 Computer Experiments............... 215
6.6.1 Functions Needed in the Experiments by Mathematica .... 215
6.6.2 Experiments by Mathematica.................... 215
6.6.3 Functions Needed in the Experiments by Matlab.............. 222
6.6.4 Experiments by Matlab ..... 222
Exercises 6................... 227
7 Iterative Techniques for Solving Linear Systems ....................230
7.1 Introduction ........ 231
7.2 Basic Iterative Methods .............. 233
7.2.1 Jacobi Method ................... 234
7.2.2 Gauss-Seidel Method ........ 236
7.2.3 SOR Method...................... 237
7.3 Iterative Method Convergence ... 238
7.3.1 Basic Theorems ................. 238
7.3.2 Some Special Systems of Equations.......... 243
7.4 Computer Experiments............... 247
7.4.1 Functions Needed in The Experiments by Mathematica... 247
7.4.2 Experiments by Mathematica.................... 247
7.4.3 Experiments by Matlab ..... 251
Exercises 7................... 255
8 Numerical Solution of Ordinary Differential Equations ............258
8.1 Introduction ........ 258
8.2 The Existence and Uniqueness of Solutions....... 260
8.3 Taylor-Series Method................. 262
8.4 Euler’s Method ... 263
8.5 Single-step Methods ................... 267
8.5.1 Single-step Methods.......... 267
8.5.2 Local Truncation Error ...... 267
8.6 Runge-Kutta Methods ................ 268
8.6.1 Second-Order Runge-Kutta Method.......... 268
8.6.2 Fourth-Order Runge-Kutta Method........... 270
8.7 Multistep Methods...................... 271
8.7.1 General Formulas of Multistep Methods... 272
8.7.2 Adams Explicit and Implicit Formulas...... 273
8.8 Systems and Higher-Order Differential Equations..................... 275
8.8.1 Vector Notation ................. 276
8.8.2 Taylor-Series Method for Systems............ 278
8.8.3 Fourth-Order Runge-Kutta Formula for Systems.............. 279
8.9 Computer Experiments............... 281
8.9.1 Functions Needed in the Experiments by Mathematica .... 281
8.9.2 Experiments by Mathematica.................... 281
8.9.3 Experiments by Matlab ..... 286
Exercises 8................... 290
Appendix ...............293
Appendix A Mathematica Basic Operations ............ 293
Appendix B Matlab Basic Operations ...................... 309
Appendix C Answers to Selected Question.............. 327
Reference..............332