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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