Numerical Methods in Engineering with Python

Numerical Methods in Engineering with Python, 2nd Edition is a text for engineering students and a reference for practicing engineers, especially those who wish to explore Python. This new edition features 18 additional exercises and the addition of rational function interpolation. Brent's method of root finding was replaced by Ridder's method, and the Fletcher-Reeves method of optimization was dropped in favor of the downhill simplex method. Each numerical method is explained in detail, and its shortcomings are pointed out. The examples that follow individual topics fall into two categories: hand computations that illustrate the inner workings of the method and small programs that show how the computer code is utilized in solving a problem. This second edition also includes more robust computer code with each method, which is available on the book Web site. This code is made simple and easy to understand by avoiding complex bookkeeping schemes, while maintaining the essential features of the method. Contents......Page 6 Preface......Page 8 1.1 General Information......Page 10 1.2 Core Python......Page 13 1.3 Functions and Modules......Page 25 1.4 Mathematics Modules......Page 26 1.5 numarray Module......Page 28 1.6 Scoping of Variables......Page 32 1.7 Writing and Running Programs......Page 34 2.1 Introduction......Page 36 2.2 Gauss Elimination Method......Page 43 2.3 LU Decomposition Methods......Page 50 2.4 Symmetric and Banded Coefficient Matrices......Page 65 2.5 Pivoting......Page 76 ∗2.6 Matrix Inversion......Page 91 ∗2.7 Iterative Methods......Page 94 ∗2.8 Other Methods......Page 110 3.1 Introduction......Page 112 3.2 Polynomial Interpolation......Page 113 3.3 Interpolation with Cubic Spline......Page 124 3.4 Least-Squares Fit......Page 134 3.5 Other Methods......Page 150 4.1 Introduction......Page 151 4.2 Incremental Search Method......Page 152 4.3 Method of Bisection......Page 154 4.4 Brent’s Method......Page 157 4.5 Newton–Raphson Method......Page 163 4.6 Systems of Equations......Page 167 ∗4.7 Zeroes of Polynomials......Page 179 4.8 Other Methods......Page 188 5.1 Introduction......Page 190 5.2 Finite Difference Approximations......Page 191 5.3 Richardson Extrapolation......Page 196 5.4 Derivatives by Interpolation......Page 199 6.1 Introduction......Page 207 6.2 Newton–Cotes Formulas......Page 208 6.3 Romberg Integration......Page 216 6.4 Gaussian Integration......Page 225 ∗6.5 Multiple Integrals......Page 242 7.1 Introduction......Page 257 7.2 Taylor Series Method......Page 258 7.3 Runge–Kutta Methods......Page 264 7.4 Stability and Stiffness......Page 281 7.5 Adaptive Runge–Kutta Method......Page 284 7.6 Bulirsch–Stoer Method......Page 292 7.7 Other Methods......Page 303 8.1 Introduction......Page 304 8.2 Shooting Method......Page 305 8.3 Finite Difference Method......Page 319 9.1 Introduction......Page 333 9.2 Jacobi Method......Page 335 9.3 Inverse Power and Power Methods......Page 352 9.4 Householder Reduction to Tridiagonal Form......Page 367 9.5 Eigenvalues of Symmetric Tridiagonal Matrices......Page 374 9.6 Other Methods......Page 389 10.1 Introduction......Page 390 10.2 Minimization Along a Line......Page 392 10.3 Conjugate Gradient Methods......Page 398 10.4 Other Methods......Page 416 Appendices......Page 418 Index......Page 428 Numerical Methods in Engineering with Python is a text for engineering students and a reference for practicing engineers, especially those who wish to explore the power and efficiency of Python. Examples and applications were chosen for their relevance to real world problems, and where numerical solutions are most efficient. Numerical methods are discussed thoroughly and illustrated with problems involving both hand computation and programming. Computer code accompanies each method and is available on the book web site. This code is made simple and easy to understand by avoiding complex bookkeeping schemes, while maintaining the essential features of the method. Python was chosen as the example language because it is elegant, easy to learn and debug, and its facilities for handling arrays are unsurpassed. Moreover, it is an open-source software package; free and available to all students and engineers. Explore numerical methods with Python, a great language for teaching scientific computation. This text is for engineering students and a reference for practising engineers, especially those who wish to explore Python. This new edition features 18 additional exercises and the addition of rational function interpolation. Brent's method of root finding was replaced by Ridder's method, and the Fletcher-Reeves method of optimization was dropped in favor of the downhill simplex method. Each numerical method is explained in detail, and its shortcomings are pointed out. The examples that follow individual topics fall into two categories: hand computations that illustrate the inner workings of the method and small programs that show how the computer code is utilized in solving a problem. This second edition also includes more robust computer code with each method, which is available on the book website. This code is made simple and easy to understand by avoiding complex bookkeeping schemes, while maintaining the essential features of the method.