Applied Numerical Methods with MATLAB for Engineers and Scientists, 5th Edition

This book is designed to support a one-semester course in numerical methods. It has been written for students who want to learn and apply numerical methods in order to solve problems in engineering and science. As such, the methods are motivated by problems rather than by mathematics. That said, sufficient theory is provided so that students come away with insight into the techniques and their shortcomings. MATLAB provides a great environment for such a course. Although other environments (e.g., Excel/VBA, Mathcad) or languages (e.g., Fortran 90, C++, Python) could have been chosen, MATLAB presently offers a nice combination of handy programming features with powerful built-in numerical capabilities. On the one hand, its M-file programming environment allows students to implement moderately complicated algorithms in a structured and coherent fashion. On the other hand, its built-in, numerical capabilities empower students to solve more difficult problems without trying to “reinvent the wheel.” Algorithms Presented Using MATLAB M-files. Instead of using pseudocode, this book presents algorithms as well-structured MATLAB M-files. Aside from being useful computer programs, these provide students with models for their own M-files that they will develop as homework exercises. Cover Title Page Copyright Dedication About the Author Contents Preface viii Acknowledgments xiv Part One Modeling, Computers, and Error Analysis 1 1.1 Motivation 1 1.2 Part Organization 2 chapter 1: Mathematical Modeling, Numerical Methods, and Problem Solving 4 1.1 A Simple Mathematical Model 5 1.2 Conservation Laws in Engineering and Science 12 1.3 Numerical Methods Covered in This Book 13 1.4 Case Study: It’s a Real Drag 17 Problems 20 chapter 2: MATLAB Fundamentals 27 2.1 The Matlab Environment 28 2.2 Assignment 29 2.3 Mathematical Operations 36 2.4 Use of Built-In Functions 39 2.5 Graphics 42 2.6 Other Resources 46 2.7 Case Study: Exploratory Data Analysis 46 Problems 49 chapter 3: Programming with MATLAB 53 3.1 M-Files 54 3.2 Input-output 61 3.3 Structured Programming 65 3.4 Nesting and Indentation 79 3.5 Passing Functions to M-Files 82 3.6 Case Study: Bungee Jumper Velocity 88 Problems 92 chapter 4: Roundoff and Truncation Errors 100 4.1 Errors 101 4.2 Roundoff Errors 107 4.3 Truncation Errors 116 4.4 Total Numerical Error 127 4.5 Blunders, Model Errors, and Data Uncertainty 132 Problems 133 Part Two Roots and Optimization 137 2.1 Overview 137 2.2 Part Organization 138 chapter 5: Roots: Bracketing Methods 140 5.1 Roots in Engineering and Science 141 5.2 Graphical Methods 142 5.3 Bracketing Methods and Initial Guesses 143 5.4 Bisection 148 5.5 False Position 154 5.6 Case Study: Greenhouse Gases and Rainwater 158 Problems 161 chapter 6: Roots: Open Methods 166 6.1 Simple Fixed-Point Iteration 167 6.2 The Wegstein Method 173 6.3 Newton-Raphson 177 6.4 Secant Methods 182 6.5 Brent’s Method 184 6.6 Matlab Function: fzero 189 6.7 Polynomials 191 6.8 Case Study: Pipe Friction 194 Problems 199 chapter 7: Optimization 206 7.1 Introduction and Background 207 7.2 One-Dimensional Optimization 210 7.3 Multidimensional Optimization 219 7.4 Case Study: Equilibrium and Minimum Potential Energy 221 Problems 223 Part Three Linear Systems 231 3.1 Overview 231 3.2 Part Organization 233 chapter 8: Linear Algebraic Equations and Matrices 235 8.1 Matrix Algebra Overview 237 8.2 Solving Linear Algebraic Equations with Matlab 246 8.3 Case Study: Currents and Voltages in Circuits 248 Problems 252 chapter 9: Gauss Elimination 256 9.1 Solving Small Numbers of Equations 257 9.2 Naive Gauss Elimination 262 9.3 Pivoting 270 9.4 Tridiagonal Systems 273 9.5 Case Study: Model of a Heated Rod 276 Problems 279 chapter 10: LU Factorization 284 10.1 Overview of Lu Factorization 285 10.2 Gauss Elimination as Lu Factorization 286 10.3 Cholesky Factorization 293 10.4 Matlab Left Division 296 Problems 297 chapter 11: Matrix Inverse and Condition 298 11.1 The Matrix Inverse 298 11.2 Error Analysis and System Condition 302 11.3 Case Study: Indoor Air Pollution 307 Problems 310 chapter 12: Iterative Methods 315 12.1 Linear Systems: Gauss-Seidel 315 12.2 Nonlinear Systems 322 12.3 Case Study: Chemical Reactions 330 Problems 333 chapter 13: Eigenvalues 336 13.1 Eigenvalues and Eigenvectors—The Basics 338 13.2 Applications of Eigenvalues and Eigenvectors 341 13.3 Physical Settings—Mass-Spring Systems 347 13.4 The Power Method 350 13.5 MATLAB Function: eig 352 13.6 Case Study: Eigenvalues and Earthquakes 353 Problems 356 Part Four Curve Fitting 359 4.1 Overview 359 4.2 Part Organization 361 chapter 14: Linear Regression 362 14.1 Statistics Review 364 14.2 Random Numbers and Simulation 369 14.3 Linear Least-Squares Regression 374 14.4 Linearization of Nonlinear Relationships 382 14.5 Computer Applications 386 14.6 Case Study: Enzyme Kinetics 389 Problems 394 chapter 15: General Linear Least-Squares and Nonlinear Regression 401 15.1 Polynomial Regression 401 15.2 Multiple Linear Regression 405 15.3 General Linear Least Squares 407 15.4 Qr Factorization and the Backslash Operator 410 15.5 Nonlinear Regression 411 15.6 Case Study: Fitting Experimental Data 413 Problems 415 chapter 16: Fourier Analysis 420 16.1 Curve Fitting with Sinusoidal Functions 421 16.2 Continuous Fourier Series 427 16.3 Frequency and Time Domains 430 16.4 Fourier Integral and Transform 431 16.5 Discrete Fourier Transform (DFT) 434 16.6 The Power Spectrum 439 16.7 Case Study: Sunspots 441 Problems 442 chapter 17: Polynomial Interpolation 445 17.1 Introduction to Interpolation 446 17.2 Newton Interpolating Polynomial 449 17.3 Lagrange Interpolating Polynomial 457 17.4 Inverse Interpolation 460 17.5 Extrapolation and Oscillations 461 Problems 465 chapter 18: Splines and Piecewise Interpolation 469 18.1 Introduction to Splines 469 18.2 Linear Splines 471 18.3 Quadratic Splines 475 18.4 Cubic Splines 478 18.5 Piecewise Interpolation in MATLAB 484 18.6 Multidimensional Interpolation 489 18.7 Smoothing of Data Series 491 18.8 Case Study: Heat Transfer 501 Problems 505 Part Five Integration and Differentiation 511 5.1 Overview 511 5.2 Part Organization 512 chapter 19: Numerical Integration Formulas 514 19.1 Introduction and Background 515 19.2 Newton-Cotes Formulas 518 19.3 The Trapezoidal Rule 520 19.4 Simpson’s Rules 527 19.5 Higher-Order Newton-Cotes Formulas 533 19.6 Integration with Unequal Segments 534 19.7 Open Methods 538 19.8 Multiple Integrals 538 19.9 Case Study: Computing Work with Numerical Integration 541 Problems 544 chapter 20: Numerical Integration of Functions 550 20.1 Introduction 550 20.2 Romberg Integration 551 20.3 Gauss Quadrature 556 20.4 Adaptive Quadrature 563 20.5 Case Study: Root-Mean-Square Current 566 Problems 570 chapter 21: Numerical Differentiation 574 21.1 Introduction and Background 575 21.2 High-Accuracy Differentiation Formulas 579 21.3 Richardson Extrapolation 582 21.4 Tangent Line Differentiation of Functions 584 21.5 Derivatives of Unequally Spaced Data 587 21.6 Differentiation of Noisy Data 590 21.7 Partial Derivatives 596 21.8 Numerical Differentiation with Matlab 596 21.9 Case Study: Visualizing Fields 601 Problems 603 Part Six Ordinary Differential Equations 609 6.1 Overview 609 6.2 Part Organization 613 chapter 22: Initial-Value Problems 615 22.1 Overview 617 22.2 Euler’s Method 617 22.3 Improvements of Euler’s Method 623 22.4 Runge-Kutta Methods 629 22.5 Systems of Equations 634 22.6 Case Study: Predator-Prey Models and Chaos 640 Problems 645 chapter 23: Adaptive Methods and Stiff Systems 651 23.1 Adaptive Runge-Kutta Methods 651 23.2 Multistep Methods 660 23.3 Stiffness 664 23.4 Matlab Application: Bungee Jumper with Cord 670 23.5 Case Study: Pliny’s Intermittent Fountain 671 Problems 676 chapter 24: Boundary-Value Problems 682 24.1 Introduction and Background 683 24.2 The Shooting Method 687 24.3 Finite-Difference Methods 694 24.4 MATLAB Function: bvp4c 701 Problems 704 Appendix A: MATLAB Built-in Functions 710 Appendix B: MATLAB M-File functions 712 Appendix C: INTRODUCTION TO SIMULINK 713 Bibliography 721 Index 723 Applied Numerical Methods with MATLAB is designed to support a one-semester course in numerical methods. It has been written for students who want to learn and apply numerical methods in order to solve problems in engineering and science. As such, the methods are motivated by problems rather than by mathematics. That said, sufficient theory is provided so students come away with insight into the techniques and their shortcomings.This title will be available in Connect, featuring SmartBook, the MHeBook, and homework problems. Instructor Resources available for this title include: Image Library, Instructor Solutions Manual, Lecture PowerPoints, and MatLab Files.