Numerical Methods for Engineers, 6th Edition

Numerical methods are techniques by which mathematical problems are formulated so thatthey can be solved with arithmetic operations. Although there are many kinds of numericalmethods, they have one common characteristic: they invariably involve large numbers oftedious arithmetic calculations. It is little wonder that with the development of fast, efficientdigital computers, the role of numerical methods in engineering problem solving hasincreased dramatically in recent years. Cover Page ......Page 1 Title Page......Page 2 Copyright Page......Page 3 Dedication ......Page 4 Contents ......Page 5 Preface ......Page 15 Guided Tour ......Page 17 About the Authors ......Page 19 PT1.1 Motivation......Page 22 PT1.2 Mathematical Background......Page 24 PT1.3 Orientation......Page 27 1.1 A Simple Mathematical Model......Page 30 1.2 Conservation Laws and Engineering......Page 37 Problems......Page 40 2.1 Packages and Programming......Page 44 2.2 Structured Programming......Page 45 2.3 Modular Programming......Page 54 2.4 Excel......Page 56 2.5 MATLAB......Page 60 2.6 Mathcad......Page 64 2.7 Other Languages and Libraries......Page 65 Problems......Page 66 CHAPTER 3: Approximations and Round-Off Errors......Page 71 3.1 Significant Figures......Page 72 3.2 Accuracy and Precision......Page 74 3.3 Error Definitions......Page 75 3.4 Round-Off Errors......Page 81 Problems......Page 94 4.1 The Taylor Series......Page 96 4.2 Error Propagation......Page 113 4.3 Total Numerical Error......Page 117 4.4 Blunders, Formulation Errors, and Data Uncertainty......Page 122 Problems......Page 124 PT1.4 Trade-Offs......Page 126 PT1.6 Advanced Methods and Additional References......Page 129 PT2.1 Motivation......Page 132 PT2.2 Mathematical Background......Page 134 PT2.3 Orientation......Page 135 5.1 Graphical Methods......Page 139 5.2 The Bisection Method......Page 143 5.3 The False-Position Method......Page 151 5.4 Incremental Searches and Determining Initial Guesses......Page 157 Problems......Page 158 CHAPTER 6: Open Methods......Page 161 6.1 Simple Fixed-Point Iteration......Page 162 6.2 The Newton-Raphson Method......Page 167 6.3 The Secant Method......Page 173 6.4 Brent’s Method......Page 178 6.5 Multiple Roots......Page 183 6.6 Systems of Nonlinear Equations......Page 186 7.1 Polynomials in Engineering and Science......Page 193 7.2 Computing with Polynomials......Page 196 7.3 Conventional Methods......Page 199 7.4 Müller’s Method......Page 200 7.5 Bairstow’s Method......Page 204 7.7 Root Location with Software Packages......Page 209 Problems......Page 219 8.1 Ideal and Nonideal Gas Laws (Chemical/Bio Engineering)......Page 221 8.2 Greenhouse Gases and Rainwater (Civil/Environmental Engineering)......Page 224 8.3 Design of an Electric Circuit (Electrical Engineering)......Page 226 8.4 Pipe Friction (Mechanical/Aerospace Engineering)......Page 228 Problems......Page 232 PT2.4 Trade-Offs......Page 242 PT2.6 Advanced Methods and Additional References......Page 243 PT3.1 Motivation......Page 246 PT3.2 Mathematical Background......Page 248 PT3.3 Orientation......Page 256 9.1 Solving Small Numbers of Equations......Page 260 9.2 Naive Gauss Elimination......Page 267 9.3 Pitfalls of Elimination Methods......Page 273 9.4 Techniques for Improving Solutions......Page 279 9.6 Nonlinear Systems of Equations......Page 286 9.7 Gauss-Jordan......Page 288 Problems......Page 290 10.1 LU Decomposition......Page 293 10.2 The Matrix Inverse......Page 302 10.3 Error Analysis and System Condition......Page 306 Problems......Page 312 11.1 Special Matrices......Page 315 11.2 Gauss-Seidel......Page 319 11.3 Linear Algebraic Equations with Software Packages......Page 326 Problems......Page 331 12.1 Steady-State Analysis of a System of Reactors (Chemical/Bio Engineering)......Page 334 12.2 Analysis of a Statically Determinate Truss (Civil/Environmental Engineering)......Page 337 12.3 Currents and Voltages in Resistor Circuits (Electrical Engineering)......Page 341 12.4 Spring-Mass Systems (Mechanical/Aerospace Engineering)......Page 343 Problems......Page 346 PT3.4 Trade-Offs......Page 356 PT3.6 Advanced Methods and Additional References......Page 357 PT4.1 Motivation......Page 360 PT4.2 Mathematical Background......Page 365 PT4.3 Orientation......Page 366 CHAPTER 13: One-Dimensional Unconstrained Optimization......Page 370 13.1 Golden-Section Search......Page 371 13.2 Parabolic Interpolation......Page 378 13.3 Newton’s Method......Page 380 Problems......Page 383 CHAPTER 14: Multidimensional Unconstrained Optimization......Page 386 14.1 Direct Methods......Page 387 14.2 Gradient Methods......Page 391 Problems......Page 404 15.1 Linear Programming......Page 406 15.2 Nonlinear Constrained Optimization......Page 417 15.3 Optimization with Software Packages......Page 418 Problems......Page 429 16.1 Least-Cost Design of a Tank (Chemical/Bio Engineering)......Page 432 16.2 Least-Cost Treatment of Wastewater (Civil/Environmental Engineering)......Page 437 16.3 Maximum Power Transfer for a Circuit (Electrical Engineering)......Page 441 16.4 Equilibrium and Minimum Potential Energy (Mechanical/Aerospace Engineering)......Page 445 Problems......Page 447 PT4.4 Trade-Offs......Page 455 PT4.5 Additional References......Page 456 PT5.1 Motivation......Page 458 PT5.2 Mathematical Background......Page 460 PT5.3 Orientation......Page 469 17.1 Linear Regression......Page 473 17.2 Polynomial Regression......Page 489 17.3 Multiple Linear Regression......Page 493 17.4 General Linear Least Squares......Page 496 17.5 Nonlinear Regression......Page 500 Problems......Page 503 CHAPTER 18: Interpolation......Page 507 18.1 Newton’s Divided-Difference Interpolating Polynomials......Page 508 18.2 Lagrange Interpolating Polynomials......Page 519 18.4 Inverse Interpolation......Page 524 18.5 Additional Comments......Page 525 18.6 Spline Interpolation......Page 528 18.7 Multidimensional Interpolation......Page 538 Problems......Page 541 CHAPTER 19: Fourier Approximation......Page 543 19.1 Curve Fitting with Sinusoidal Functions......Page 544 19.2 Continuous Fourier Series......Page 550 19.3 Frequency and Time Domains......Page 553 19.4 Fourier Integral and Transform......Page 557 19.5 Discrete Fourier Transform (DFT)......Page 559 19.6 Fast Fourier Transform (FFT)......Page 561 19.7 The Power Spectrum......Page 568 19.8 Curve Fitting with Software Packages......Page 569 Problems......Page 578 20.1 Linear Regression and Population Models (Chemical/Bio Engineering)......Page 580 20.2 Use of Splines to Estimate Heat Transfer (Civil/Environmental Engineering)......Page 584 20.3 Fourier Analysis (Electrical Engineering)......Page 586 20.4 Analysis of Experimental Data (Mechanical/Aerospace Engineering)......Page 587 Problems......Page 589 PT5.4 Trade-Offs......Page 599 PT5.5 Important Relationships and Formulas......Page 600 PT5.6 Advanced Methods and Additional References......Page 602 PT6.1 Motivation......Page 604 PT6.2 Mathematical Background......Page 614 PT6.3 Orientation......Page 616 CHAPTER 21: Newton-Cotes Integration Formulas......Page 620 21.1 The Trapezoidal Rule......Page 622 21.2 Simpson’s Rules......Page 632 21.3 Integration with Unequal Segments......Page 641 21.5 Multiple Integrals......Page 644 Problems......Page 646 22.1 Newton-Cotes Algorithms for Equations......Page 650 22.2 Romberg Integration......Page 651 22.3 Adaptive Quadrature......Page 657 22.4 Gauss Quadrature......Page 659 22.5 Improper Integrals......Page 667 Problems......Page 670 23.1 High-Accuracy Differentiation Formulas......Page 672 23.2 Richardson Extrapolation......Page 675 23.3 Derivatives of Unequally Spaced Data......Page 677 23.4 Derivatives and Integrals for Data with Errors......Page 678 23.5 Partial Derivatives......Page 679 23.6 Numerical Integration/Differentiation with Software Packages......Page 680 Problems......Page 687 24.1 Integration to Determine the Total Quantity of Heat (Chemical/Bio Engineering)......Page 690 24.2 Effective Force on the Mast of a Racing Sailboat (Civil/Environmental Engineering)......Page 692 24.3 Root-Mean-Square Current by Numerical Integration (Electrical Engineering)......Page 694 24.4 Numerical Integration to Compute Work (Mechanical/Aerospace Engineering)......Page 697 Problems......Page 701 PT6.4 Trade-Offs......Page 711 PT6.6 Advanced Methods and Additional References......Page 712 PT7.1 Motivation......Page 715 PT7.2 Mathematical Background......Page 719 PT7.3 Orientation......Page 721 CHAPTER 25: Runge-Kutta Methods......Page 726 25.1 Euler’s Method......Page 727 25.2 Improvements of Euler’s Method......Page 738 25.3 Runge-Kutta Methods......Page 746 25.4 Systems of Equations......Page 756 25.5 Adaptive Runge-Kutta Methods......Page 761 Problems......Page 769 26.1 Stiffness......Page 771 26.2 Multistep Methods......Page 775 Problems......Page 795 CHAPTER 27: Boundary-Value and Eigenvalue Problems......Page 797 27.1 General Methods for Boundary-Value Problems......Page 798 27.2 Eigenvalue Problems......Page 805 27.3 Odes and Eigenvalues with Software Packages......Page 817 Problems......Page 824 28.1 Using ODEs to Analyze the Transient Response of a Reactor (Chemical/Bio Engineering)......Page 827 28.2 Predator-Prey Models and Chaos (Civil/Environmental Engineering)......Page 834 28.3 Simulating Transient Current for an Electric Circuit (Electrical Engineering)......Page 838 28.4 The Swinging Pendulum (Mechanical/Aerospace Engineering)......Page 843 Problems......Page 847 PT7.4 Trade-Offs......Page 857 PT7.6 Advanced Methods and Additional References......Page 858 PT8.1 Motivation......Page 862 PT8.2 Orientation......Page 865 29.1 The Laplace Equation......Page 869 29.2 Solution Technique......Page 871 29.3 Boundary Conditions......Page 877 29.4 The Control-Volume Approach......Page 883 29.5 Software to Solve Elliptic Equations......Page 886 Problems......Page 887 30.1 The Heat-Conduction Equation......Page 890 30.2 Explicit Methods......Page 891 30.3 A Simple Implicit Method......Page 895 30.4 The Crank-Nicolson Method......Page 899 30.5 Parabolic Equations in Two Spatial Dimensions......Page 902 Problems......Page 905 CHAPTER 31: Finite-Element Method......Page 907 31.1 The General Approach......Page 908 31.2 Finite-Element Application in One Dimension......Page 912 31.3 Two-Dimensional Problems......Page 921 31.4 Solving PDEs with Software Packages......Page 925 Problems......Page 929 32.1 One-Dimensional Mass Balance of a Reactor (Chemical/Bio Engineering)......Page 932 32.2 Deflections of a Plate (Civil/Environmental Engineering)......Page 936 32.3 Two-Dimensional Electrostatic Field Problems (Electrical Engineering)......Page 938 32.4 Finite-Element Solution of a Series of Springs (Mechanical/Aerospace Engineering)......Page 941 Problems......Page 945 PT8.4 Important Relationships and Formulas......Page 948 PT8.5 Advanced Methods and Additional References......Page 949 APPENDIX A: THE FOURIER SERIES......Page 950 APPENDIX B: GETTING STARTED WITH MATLAB......Page 952 APPENDIX C: GETTING STARTED WITH MATHCAD......Page 960 BIBLIOGRAPHY......Page 971 INDEX......Page 974

instructors Love numerical Methods For Engineers Because It Makes Teaching Easy! Students Love It Because It Is Written For Them—with Clear Explanations And Examples Throughout. The Text Features A Broad Array Of Applications That Span All Engineering Disciplines.

the Sixth Edition Retains The Successful Instructional Techniques Of Earlier Editions. Chapra And Canale's Unique Approach Opens Each Part Of The Text With Sections Called Motivation, Mathematical Background, And Orientation. This Prepares The Student For Upcoming Problems In A Motivating And Engaging Manner. Each Part Closes With An Epilogue Containing Trade-offs, Important Relationships And Formulas, And Advanced Methods And Additional References. Much More Than A Summary, The Epilogue Deepens Understanding Of What Has Been Learned And Provides A Peek Into More Advanced Methods. Helpful Separate Appendices. Getting Started With Matlab Abd Getting Started With Mathcad Which Make Excellent References.

numerous New Or Revised Problems Drawn From Actual Engineering Practice, Many Of Which Are Based On Exciting New Areas Such As Bioengineering. The Expanded Breadth Of Engineering Disciplines Covered Is Especially Evident In The Problems, Which Now Cover Such Areas As Biotechnology And Biomedical Engineering. Excellent New Examples And Case Studies Span Asll Areas Of Engineering Disciplines; The Students Using This Text Will Be Able To Apply Their New Skills To Their Chosen Field.

users Will Find Use Of Software Packages, Specifically Matlab®, Excel® With Vba And Mathcad®. This Includes Material On Developing Matlab® M-files And Vba Macros.

An introduction to numerical methods that covers the treatment of optimization and differential equations, Brent's methods for root location and optimization, and other topics, and includes problems based on actual engineering practice, examples, case studies, and charts