Chapter 1 Mathematical Preliminaries
　　(数学基础知识）
　　本章内容（Contents)
　　1.1 Mathematics English (数学英语）
　　1.2 Review of calculus (微积分回顾）
　　1.3 Errors and significant digits (误差和有效数字）
　　学习目标（Learning Objectives)
　　？熟悉常见数学公式及数学表达式的读法 ?掌握微积分学中的一些基本定义与引理 ?掌握极限与连续的定义 ?熟悉微分的定义并熟练运用其推广的重要定理 ?熟悉积分的定义并熟练运用积分中值定理 ?掌握泰勒多项式与级数的定义
　　？了解误差的来源
　　？熟悉绝对误差与相对误差的计算
　　？了解有效数字的概念
　　This chapter contains a short review of those topics from single-variable calculus that will be needed in later chapters. A solid knowledge of calculus is essential for the understanding of numerical techniques and their analysis， and more thorough review might be needed if you have been away from this subject for a while. In addition， there is an introduction to convergence， error analysis， the machine representation of numbers， and some techniques for categorizing and minimizing computational error.
　　1.1 Mathematics English (数学英语）
　　1/2 is read one half
　　1/3 is read one third
　　2/3 is read two thirds
　　1/4 is read one quarter
　　3/4 is read three quarters
　　5/14 is read five over fourteen
　　25 i is read twenty five and seven ninths
　　35% is read thirty five percent
　　-3.14 is read minus three point one four
　　a is read plus or minus a
　　is read minus or plus a
　　is read the absolute value of a
　　[a] is read the greatest integer not exceeding a
　　n! is read factorial n
　　Re(z) is read the real part of (complex number) z
　　Im(z) is read the imaginary part of (complex number) z
　　a + b = c is read a plus b equals (is) c
　　a-b = c is read a minus b equals (is) c
　　a×b = c is read a times b equals (is) c
　　a+b = c is read a divided by b equals (is) c
　　a 2 is read a square
　　a 3 is read a cube
　　a n is read a to the power of n
　　a- n is read a to the power of minus n
　　4a is read the square root of a
　　is read the cube root of a
　　is read the nth root of a
　　is read the nth root of a to the power of m
　　is read the exponential of x (to the base e )
　　is read the exponential of x (to the base a)
　　is read the natural logarithm of x (to the base e )
　　is read the (common) logarithm of x (to the base 10 )
　　is read the logarithm of x (to the base a)
　　is read a is equal to b
　　is read a is not equal to b
　　is read a is identically equal to b
　　is read a is approximately equal to b
　　a > b is read a is greater / larger than b
　　a < b is read a is less than b
　　a > b is read a is greater than or equal to b
　　a < b is read a is less than or equal to b
　　is read a is much greater than b
　　is read a is much less than b
　　is read a is parallel to b
　　is read a is perpendicular to b
　　is read open interval
　　[a，b] is read closed interval
　　(a，b] is read half-open interval open at the left
　　[a，b) is read half-open interval open at the right
　　is read the intersection of Aand B
　　is read the union of A and B
　　is read A is a subset ofB
　　is read A is a proper subset ofB
　　is read A is not a proper subset of B
　　is read the difference of A and B
　　is read the complement of A
　　is read the empty set
　　is read a belongs to A
　　is read a is not in A
　　is read the scalar product of vectors a and b
　　is read the vector product of vectors a and b
　　is read imply
　　is read be equivalent to
　　is read the limit of the sequence (an) is a as n tends to infinity
　　is read the limit of f of x is A as x tends to x〇
　　is read the limit of f of x is A as x tends to x〇 from the left
　　is read the limit of f of x is A as x tends to x— from the right
　　A is read delta
　　is read the first derivative of y with respect to x
　　is read the second derivative of y with respect to x
　　is read the n th derivative of y with respect to x
　　is read f prime of x ， f double prime of x
　　is read the first partial derivative of z with respect to x
　　is read the second partial derivative of z with respect to x
　　is read the nth partial derivative of y with respect to x
　　is read the (indefinite) integral of f of x ，dx
　　is read the (definite) integral from a to b of f of x ，dx
　　is read sigma of q as i runs from 1to n
　　is read sigma of a” as n runs from 1 to infinity
　　is read the determinant of (the matrix) A
　　AT is read the transpose of (the matrix) A
　　A-1 is read the inverse (matrix) of (the matrix) A
　　A* is read the adjoint matrix of (the matrix) A
　　A is read the conjugate matrix of (the matrix) A
　　rank A is read the rank of (the matrix) A
　　is read the trace of (the matrix) A
　　1.2 Review of calculus (微积分回顾）
　　？Limits and continuity (极限和连续性)
　　？Differentiability (可微性）
　　？Integration (积分)
　　？Taylor polynomials and series (泰勒多项式和级数）
　　？Examples (例题）
　　1.2.1 Limits and continuity (极限和连续性）
　　The concepts of limit (极限) and continuity (连续性) of a function are fundamental to thestudy of cal

目录
前言
Chapter 1 Mathematical Preliminaries (数学基础知识) 1
1.1 Mathematics English (数学英语) 1
1.2 Review of calculus (微积分回顾) 4
1.2.1 Limits and continuity (极限和连续性 ) 4
1.2.2 Differentiability (可微性) 6
1.2.3 Integration (积分) 6
1.2.4 Taylor polynomials and series (泰勒多项式和级数) 7
1.2.5 Examples (例题) 8
1.3 Errors and significant digits (误差和有效数字 ) 9
1.3.1 Source of errors (误差的来源) 9
1.3.2 Absolute error and relative error (绝对误差和相对误差) 11
1.3.3 Significant digit (or figure) (有效数字) 11
1.3.4 How to avoid the loss of accuracy (如何避免精度的丢失) 12
1.3.5 Examples (例题) 12
1.4本章要点 (Highlights) 14
1.5问题讨论 (Questions for discussion) 14
1.6关键术语 (Key terms) 14
1.7延伸阅读 (Extending reading) 15
1.7.1 背景知识 15
1.7.2 数学家传记：泰勒 (Taylor) 16
1.7.3 数学家传记：黎曼 (Riemann) 16
1.8习题 (Exercises) 18 Chapter 2 Direct Methods for Solving Linear Systems (解线性方程组的直接法) 21
2.1 Gauss elimination method (Gauss消元法 ) 21
2.1.1 Some preliminaries (预备知识) 21
2.1.2 Gauss elimination with backward-substitution process(可回代的 Gauss 消元法) 23
2.2 Pivoting strategies (选主元策略) 27
2.2.1 Partial pivoting (maximal column pivoting) (*大列主元) 28
2.2.2 Scaled partial pivoting (scaled-column pivoting) (按比例列主元) 29
2.3 Matrix factorization (矩阵分解法) 31
2.3.1 Doolittle factorization (Doolittle分解) 32
2.3.2 Crout factorization (Crout分解) 38
2.3.3 Permutation matrix (置换矩阵) 38
2.4 Special types of matrices (特殊形式矩阵的三角分解) 39
2.4.1 Strictly diagonally dominant matrix (严格对角占优矩阵 ) 39
2.4.2 Positive definite matrix (正定矩阵) 41
2.4.3 Strictly diagonally dominant tridiagonal matrix (严格对角占优三对角矩阵) 42
2.5本章算法程序及实例 (Algorithms and examples) 45
2.5.1 Gauss消元法 (Gauss elimination method) 45
2.5.2 选主元策略 (Pivoting strategies) 46
2.5.3 LU分解法 (LU decomposition) 48
2.6本章要点 (Hightlights) 49
2.7问题讨论 (Questions for discussion) 49
2.8 关键术语 (Key terms) 50
2.9 延伸阅读 (Extending reading) 51
2.10习题 (Exercises) 54 Chapter 3 Iterative Techniques in Matrix Algebra (矩阵代数迭代技术) 57
3.1 Norms of vectors and matrices (向量范数与矩阵范数) 58
3.1.1 Vector norm (向量范数 ) 58
3.1.2 Distance between vectors (向量之间的距离) 59
3.1.3 Matrix norm and distance (矩阵范数和距离) 60
3.1.4 Examples (例题) 61
3.2 Eigenvalues and eigenvectors (特征值和特征向量 ) 62
3.2.1 Eigenvalues and eigenvectors (特征值和特征向量) 63
3.2.2 Spectral radius (谱半径) 63
3.2.3 Convergent matrices (收敛矩阵 ) 64
3.2.4 Examples (例题) 64
3.3 Iterative techniques for solving linear systems (解线性方程组的迭代法 ) 66
3.3.1 Jacobi iterative method (Jacobi迭代法) 67
3.3.2 Gauss-Seidel iterative method (Gauss-Seidel迭代法) 68
3.3.3 General iteration method (一般迭代法) 69
3.3.4 Examples (例题) 70
3.4 Convergence analysis and SOR iterative method (收敛性分析与 SOR迭代法) 72
3.4.1 Convergence analysis (收敛性分析) 72
3.4.2 SOR iterative method (SOR迭代法) 73
3.4.3 SOR iterative method in matrix form (矩阵形式的 SOR迭代法) 74
3.4.4 Examples (例题) 75
3.5 Condition number and iterative refinement (条件数和迭代优化 ) 77
3.5.1 Condition number (条件数) 77
3.5.2 Iterative refinement (迭代优化) 79
3.5.3 Examples (例题) 80
3.6本章算法程序及实例 (Algorithms and examples) 82
3.6.1 雅可比迭代法 (Jacobi iterative method) 82
3.6.2 高斯-赛德尔迭代法 (Gauss-Seidel iterative method) 83
3.6.3 SOR迭代法 (SOR iterative method) 84
3.7本章要点 (Highlights) 85
3.8问题讨论 (Questions for discussion) 86
3.9关键术语 (Key terms) 87
3.10延伸阅读 (Extending reading) 87
3.10.1 背景知识 87
3.10.2 数学家传记：高斯 (Gauss) 88
3.10.3 数学家传记：雅可比 (Jacobi) 89
3.11习题 (Exercises) 90 Chapter 4 Solutions of Equations in One Variable (一元方程求根) 98
4.1 Bisection method (二分法 ) 99
4.2 Fixed-point iteration and error analysis (不动点迭代及误差分析 ) 101
4.2.1 Fixed-point iteration (不动点迭代法) 101
4.2.2 Convergence analysis and error estimation (收敛性分析和误差估计) 101
4.2.3 The order of convergence (收敛阶) 104
4.3 Newton’s method (牛顿法) 105
4.3.1 Newton’s method and convergence analysis (牛顿法及其收敛性分析) 105
4.3.2 How to handle multiple roots using Newton’s method (如何采用牛顿法处理重根问题) 107
4.4 The secant method (弦截法 ) 111
4.5本章算法程序及实例 (Algorithms and examples) 113
4.5.1 二分法求方程的根 (Root finding by bisection method) 113
4.5.2 不动点迭代法求方程的根 (Root finding by fix point iteration) 113
4.5.3 牛顿法求方程的根 (Root finding by Newton’s method) 114
4.5.4 牛顿法求一元方程重根(未知重数) (Multiple root finding by Newton’s method) 115
4.5.5 割线法求方程的根 (Root finding by secant method) 116
4.6本章要点 (Highlights) 117
4.7问题讨论 (Questions for discussion) 118
4.8关键术语 (Key terms) 118
4.9延伸阅读 (Extending reading) 119
4.10习题 (Exercises) 127 Chapter 5 Interpolation by Polynomials (多项式插值) 129
5.1 Lagrange interpolation (Lagrange插值) 130
5.1.1 Linear interpolation (线性插值) 130
5.1.2 Quadratic interpolation(二次插值) 131
5.1.3 nth-order polynomial interpolation ( n次多项式插值) 132
5.1.4 Uniqueness of interpolation (插值的唯一性) 133
5.1.5 Lagrange error formula (Lagrange误差公式) 134
5.1.6 Ex