1 Preliminaries 1
1.1 Review of Calculus 1
Exercise 7
1.2 Round-Off Errors and Computer Arithmetic 7
Exercise 17
2 The Solution of Nonlinear Equation f(x)=0 19
2.1 The Bisection Algorithm 20
Exercise 25
2.2 Fixed-Point Iteration 25
Exercise 33
2.3 The Newton-Raphson Method 34
Exercise 42
2.4 Error Analysis for Iterative Methods and Acceleration Techniques 42
Exercise 51
3 Interpolation and Polynomial Approximation 52
3.1 Interpolation and the Lagrange Polynomial 53
Exercise 61
3.2 Divided Differences 62
Exercise 70
3.3 Hermite Interpolation 72
Exercise 78
3.4 Cubic Spline Interpolation 79
4 Numerical Integration 88
4.1 Introduction to Quadrature 89
Exercise 97
4.2 Composite Trapezoidal and Simpson's Rule 98
Exercise 108
4.3 Recursive Rules and Romberg Integration 109
Exercise 120
5 Direct Methods for Solving Linear Systems 122
5.1 Linear Systems of Equations 122
Exercise 130
5.2 Pivoting Strategies 130
Exercise 137
5.3 Matrix Factorization 137
Exercise 145
5.4 Special Types of Matrices 145
Exercise 157
6 Iterative Techniques in Matrix Algebra 158
6.1 Norms of Vectors and Matrices 158
Exercise 166
6.2 Eigenvalues and Eigenvectors 167
Exercise 171
6.3 Iterative Techniques for Solving Linear Systems 172
Exercise 184
6.4 Error Estimates and Iterative Refinement 185
Exercise 193
7 Approximating Eigenvalues 194
7.1 Linear Algebra and Eigenvalues 194
Exercise 200
7.2 The Power Method 201
Exercise 214
7.3 Householder's Method 215
Exercise 222
7.4 The QR Algorithm 223
Exercise 233
8 Initial-Value Problems for Ordinary Differential Equations 235
8.1 The Elementary Theory of Initial-Value Problems 235
Exercise 240
8.2 Euler's Method 240
Exercise 247
8.3 Higher-Order Taylor Methods 248
Exercise 252
8.4 Runge-Kutta Methods 253
Exercise 260
8.5 Error Control and the Runge-Kutta-Fehlberg Method 261
Exercise 267
References 269