Numerical Methods for Engineers

Cover Page\n......Page 1
Title Page......Page 2
Copyright Page......Page 3
Dedication \n......Page 4
Contents\n......Page 5
Preface\n......Page 15
Guided Tour\n......Page 17
About the Authors\n......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