Numerical Methods in Engineering with Python Book

Preface to the First Edition viii

Preface to the Second Edition x

1 Introduction to Python 1

1.1 General Information 1

1.2 Core Python 3

1.3 Functions and Modules 15

1.4 Mathematics Modules 17

1.5 numpy Module 18

1.6 Scoping of Variables 24

1.7 Writing and Running Programs 25

2 Systems of Linear Algebraic Equations 27

2.1 Introduction 27

2.2 Gauss Elimination Method 33

2.3 LU Decomposition Methods 40

Problem Set 2.1 51

2.4 Symmetric and Banded Coefficient Matrices 54

2.5 Pivoting 64

Problem Set 2.2 73

*2.6 Matrix Inversion 79

*2.7 Iterative Methods 82

Problem Set 2.3 93

*2.8 Other Methods 97

3 Interpolation and Curve Fitting 99

3.1 Introduction 99

3.2 Polynomial Interpolation 99

3.3 Interpolation with Cubic Spline 114

Problem Set 3.1 121

3.4 Least-Squares Fit 124

Problem Set 3.2 135

4 Roots of Equations 139

4.1 Introduction 139

4.2 Incremental Search Method 140

4.3 Method of Bisection 142

4.4 Methods Based on Linear Interpolation 145

4.5 Newton-Raphson Method 150

4.6 Systems of Equations 155

Problem Set 4.1 160

*4.7 Zeroes of Polynomials 166

Problem Set 4.2 174

5 Numerical Differentiation 177

5.1 Introduction 177

5.2 Finite Difference Approximations 177

5.3 Richardson Extrapolation 182

5.4 Derivatives by Interpolation 185

Problem Set 5.1 189

6 Numerical Integration 193

6.1 Introduction 193

6.2 Newton-Cotes Formulas 194

6.3 Romberg Integration 202

Problem Set 6.1 207

6.4 Gaussian Integration 211

Problem Set 6.2 225

*6.5 Multiple Integrals 227

Problem Set 6.3 239

7 Initial Value Problems 243

7.1 Introduction 243

7.2 Taylor Series Method 244

7.3 Runge-Kutta Methods 249

Problem Set 7.1 260

7.4 Stability and Stiffness 266

7.5 Adaptive Runge-Kutta Method 269

7.6 Bulirsch-Stoer Method 277

Problem Set 7.2 284

7.7 Other Methods 289

8 Two-Point Boundary Value Problems 290

8.1 Introduction 290

8.2 Shooting Method 291

Problem Set 8.1 301

8.3 Finite Difference Method 305

Problem Set 8.2 314

9 Symmetric Matrix Eigenvalue Problems 319

9.1 Introduction 319

9.2 Jacobi Method 321

9.3 Power and Inverse Power Methods 337

Problem Set 9.1 345

9.4 Householder Reduction to Tridiagonal Form 351

9.5 Eigenvalues of Symmetric Tridiagonal Matrices 358

Problem Set 9.2 367

9.6 Other Methods 373

10 Introduction to Optimization 374

10.1 Introduction 374

10.2 Minimization along a Line 376

10.3 Powell's Method 382

10.4 Downhill Simplex Method 392

Problem Set 10.1 399

10.5 Other Methods 406

A1 Taylor Series 407

A2 Matrix Algebra 410

List of Program Modules (by Chapter) 416

Index 419