CONTENTS 3
PART ONE MODELING,COMPUTERS,AND ERROR ANALYSIS 3
PT 1.1 Motivation 3
PT 1.2 Mathematical Background 5
PT 1.3 Orientation 8
CHAPTER 1 Mathematical Modeling and Engineering Problem-Solving 11
1.1 A Simple Mathematical Model 11
1.2 Conservation Laws and Engineering 18
Problems 21
CHAPTER 2 Computers and Software 24
2.1 Computing Environments 25
2.2 The Software Development Process 26
2.3 Algorithm Design 30
2.4 Program Composition 40
2.5 Quality Control 43
2.6 Documentation and Maintenance 47
2.7 Software Strategy 48
Problems 53
CHAPTER 3 Approximations and Round-Off Errors 56
3.1 Significant Figures 57
3.2 Accuracy and Precision 59
3.3 Error Definitions 60
3.4 Round-Off Errors 63
Problems 78
CHAPTER 4 Truncation Errors and the Taylor Series 79
4.1 The Taylor Series 79
4.2 Error Propagation 95
4.3 Total Numerical Error 99
4.4 Blunders,Formulation Errors,and Data Uncertainty 101
Problems 103
EpILOGUE:PART ONE 105
PT 1.4 Trade-Offs 105
PT 1.6 Advanced Methods and Additional References 108
PT 1.5 Important Relationships and Formulas 108
PART TWO ROOTS OF EQUATIONS 111
PT 2.1 Motivation 111
PT 2.2 Mathematical Background 113
PT 2.3 Orientation 114
CHAPTER 5 Bracketing Methods 118
5.1 Graphical Methods 118
5.2 The Bisection Method 122
5.3 The False-Position Method 132
5.4 Incremental Searches and Determining Initial Guesses 137
Problems 138
CHAPTER 6 Open Methods 141
6.1 Simple Fixed-Point Iteration 142
6.2 The Newton-Raphson Method 147
6.3 The Secant Method 153
6.4 Multiple Roots 158
6.5 Systems of Nonlinear Equations 161
Problems 165
CHAPTER 7 Roots of Polynomials 167
7.1 Polynomials in Engineering and Science 167
7.2 Computing with Polynomials 170
7.3 Conventional Methods 173
7.4 Müeller's Method 174
7.5 Bairstow's Method 178
7.6 Other Methods 183
7.7 Root Location with Libraries and Packages 183
Problems 193
CHAPTER 8 Engineering Applications:Roots of Equations 194
8.1 Ideal and Nonideal Gas Laws(Chemicol/Petroleum Engineering) 194
8.2 Open-Channel Flow(Civil/Environmental Engineering) 197
8.3 Design of an Electric Circuit(Electrical Engineering) 201
8.4 Vibration Analysis(Mechanical/Aerospace Engineering) 203
Problems 210
PT 2.4 Trade-Offs 215
EPILOGUE:PART TWO 215
PT 2.5 Important Relationships and Formulas 216
PT 2.6 Advanced Methods and Additional References 216
PART THREE LINEAR ALGEBRAIC EQUATIONS 219
PT 3.1 Motivation 219
PT 3.2 Mathematical Background 221
PT 3.3 Orientation 229
CHAPTER 9 Gauss Elimination 233
9.1 Solving Small Numbers of Equations 233
9.2 Naive Gauss Elimination 240
9.3 Pitfalls of Elimination Methods 246
9.4 Techniques for Improving Solutions 252
9.5 Complex Systems 259
9.6 Nonlinear Systems of Equations 259
9.7 Gauss-Jordan 261
Problems 263
9.8 Summary 263
CHAPTER 10 LU Decomposition and Matrix Inversion 266
10.1 LU Decomposition 266
10.2 The Matrix Inverse 275
10.3 Error Analysis and System Condition 279
Problems 287
CHAPTER 11 Special Matrices and Gauss-Seidel 288
11.1 Special Matrices 288
11.2 Gauss-Seidel 292
11.3 Linear Algebraic Equations with Libraries and Packages 299
Problems 307
CHAPTER 12 Engineering Applications:Linear Algebraic Equations 309
12.1 Steady-State Analysis of a System of Reactors (Chemical/Petroleum Engineering) 309
12.2 Analysis of a Statically Determinate Truss(Civil/Environmental Engineering) 312
12.3 Currents and Voltages in Resistor Circuits(Electrical Engineering) 316
12.4 Spring-Mass Systems (Mechanical/Aerospace Engineering) 318
Problems 321
PT 3.4 Trade-Offs 327
EPlLOGUE:PART THREE 327
PT 3.5 Important Relationships and Formulas 328
PT 3.6 Advanced Methods and Additional References 328
PART FOUR OPTIMIZATION 331
PT 4.1 Motivation 331
PT 4.2 Mathematical Background 336
PT 4.3 Orientation 337
CHAPTER 13 One-dimensional Unconstrained Optimization 341
13.1 Golden-Section Search 342
13.2 Quadratic Interpolation 349
13.3 Newton's Method 351
Problems 353
CHAPTER 14 Multidimensional Unconstrained Optimization 355
14.1 Direct Methods 356
14.2 Gradient Methods 360
Problems 373
15.1 Linear Programming 375
CHAPTER 15 Constrained Optimization 375
15.2 Nonlinear Constrained Optimization 386
15.3 Optimization with Packages 387
Problems 397
CHAPTER 16 Engineering Applications:Optimization 399
16.1 Least-Cost Design of a Tank(Chemical/Petroleum Engineering) 399
16.2 Least-Cost Treatment of Wastewater(Civil/Environmental Engineering) 403
16.3 Maximum Power Transfer for a Circuit(Electrical Engineering) 408
16.4 Mountain Bike Design (Mechanical/Aerospace Engineering) 412
Problems 414
EPILOGUE:PART FOUR 420
PT 4.4 Trade-Offs 420
PT 4.5 Additional References 421
PART FIVE CURVE FITTING 423
PT 5.1 Motivation 423
PT 5.2 Mathematical Background 425
PT 5.3 Orientation 434
17.1 Linear Regression 438
CHAPTER 17 Least-Squares Regression 438
17.2 Polynomial Regression 454
17.3 Multiple Linear Regression 460
17.4 General Linear Least-Squares 463
17.5 Nonlinear Regression 468
Problems 471
CHAPTER 18 Interpolation 473
18.1 Newton's Divided-Difference Interpolating Polynomials 474
18.2 Logrange Interpolating Polynomials 485
18.3 Coefficients of an Interpolating Polynomial 490
18.4 Inverse Interpolation 490
18.5 Additional Comments 491
18.6 Spline Interpolation 494
Problems 505
CHAPTER 19 Fourier Approximation 507
19.1 Curve Fitting with Sinusoidal Functions 508
19.2 Continuous Fourier Series 514
19.3 Frequency and Time Domains 517
19.4 Fourier Integral and Transform 521
19.5 Discrete Fourier Transform (DFT) 523
19.6 Fast Fourier Transform (FFT) 525
19.7 The Power Spectrum 532
19.8 Curve Fitting with Libraries and Packages 533
Problems 544
CHAPTER 20 Engineering Applications:Curve Fitting 546
20.1 Linear Regression and Population Models(Chemical/Petroleum Engineering) 546
20.2 Use of Splines to Estimate Heat Transfer (Civil/Environmental Engineering) 550
20.3 Fourier Analysis(Electrical Engineering) 552
20.4 Analysis of Experimental Data(Mechonical/Aerospace Engineering) 553
Problems 555
EPILOGUE:PART FIVE 561
PT 5.4 Trade-Offs 561
PT 5.5 Important Relationships and Formulas 562
PT 5.6 Advanced Methods and Additional References 564
PT 6.1 Motivation 567
PART SIX NUMERICAL DIFFERENTIATION AND INTEGRATION 567
PT 6.2 Mathematical Background 576
PT 6.3 Orientation 579
CHAPTER 21 Newton-Cotes Integration of Equations 582
21.1 The Trapezoidal Rule 584
21.2 Simpson's Rules 595
21.3 Integration with Unequal Segments 604
21.4 Open Integration Formulas 607
Problems 607
CHAPTER 22 Integration of Equations 610
22.1 Newton-Cotes Algorithms for Equations 610
22.2 Romberg Integration 612
22.3 Gauss Quadrature 617
22.4 Improper Integrals 624
Problems 628
23.1 High-Accuracy Differentiation Formulas 629
CHAPTER 23 Numerical Differentiation 629
23.2 Richardson Extrapolation 632
23.3 Derivatives of Unequally Spaced Data 634
23.4 Derivatives and Integrals for Data with Errors 635
23.5 Numerical Integration/Differentiation Formulas with Libraries and Packages 636
Problems 641
CHAPTER 24 Engineering Applications:Numerical Integration and Differentiation 643
24.1 Integration to Determine the Total Quantity of Heat (Chemical/Petroleum Engineering) 643
24.2 Effective Force on the Mast of a Racing Sailboat (Civil/Environmental Engineering) 645
24.3 Root-Mean-Square Current by Numerical Integration(Electrical Engineering) 647
24.4 Numerical Integration to Compute Work (Mechanical/Aerospace Engineering) 650
Problems 654
EPILOGUE:PART SIX 661
PT 6.4 Trade-Offs 661
PT 6.5 Important Relationships and Formulas 662
PT 6.6 Advanced Methods and Additional References 662
PT 7.1 Motivation 665
PART SEVEN ORDINARY DIFFERENTIAL EQUATIONS 665
PT 7.2 Mathematical Background 669
PT 7.3 Orientation 671
CHAPTER 25 Runge-Kutta Methods 675
25.1 Euler's Method 676
25.2 Improvements of Euler's Method 687
25.3 Runge-Kutta Method 695
25.4 Systems of Equations 705
25.5 Adaptive Runge-Kutta Method 710
Problems 718
CHAPTER 26 Stiffness and Multistep Methods 719
26.1 Stiffness 719
26.2 Multistep Methods 723
Problems 743
CHAPTER 27 Boundary-Value and Eigenvalue Problems 745
27.1 General Methods of Boundary-Value Problems 746
27.2 Eigenvalue Problems 752
27.3 ODEs and Eigenvalues with Libraries and Packages 768
Problems 775
CHAPTER 28 Engineering Applications:Ordinary Differential Equations 777
28.1 Using ODEs to Analyze the Transient Response of a Reactor(Chemical/Petroleum Engineering) 777
28.2 Predator-Prey Models and Chaos(Civil/Environmental Engineering) 784
28.3 Simulating Transient Current for an Electronic Circuit(Electrical Engineering) 788
28.4 The Swinging Pendulum(Mechanicol/Aerospace Engineering) 793
Problems 796
EPILOGUE:PART SEVEN 800
PT 7.4 Trade-Offs 800
PT 7.5 Important Relationships and Formulas 801
PT 7.6 Advanced Methods and Additional References 801
PART EIGHT PARTIAL DIFFERENTIAL EQUATIONS 805
PT 8.1 Motivation 805
PT 8.2 Orientation 808
29.1 The Laplace Equation 812
CHAPTER 29 Finite Difference:Elliptic Equations 812
29.2 Solution Techniques 814
29.3 Boundary Conditions 820
29.4 The Control Volume Approach 826
29.5 Software to Solve Elliptic Equations 829
Problems 830
CHAPTER 30 Finite Difference:Parabolic Equations 832
30.1 The Heat Conduction Equation 832
30.2 Explicit Methods 833
30.3 A Simple Implicit Method 837
30.4 The Crank-Nicholson Method 841
30.5 Parabolic Equations in Two Spatial Dimenstons 844
Problems 848
CHAPTER 31 Finite Element Method 849
31.1 The General Approach 850
31.2 Finite-Element Application in One Dimension 854
31.3 Two-Dimensional Problems 863
31.4 PDEs with Libraries and Packages 867
Problems 874
CHAPTER 32 Engineering Applications:Partial Differential Equations 876
32.1 One-Dimensional Mass Balance of a Reactor (Chemical/Petroleum Engineering) 876
32.2 Deflections of a Plate(Civil/Environmental Engineering) 880
32.3 Two-Dimensional Electrostatic Field Problems(Electrical Engineering) 882
32.4 Finite-Element Solution of a Series of Springs(Mechanical/Aerospace Engineering) 885
Problems 889
PT 8.4 Important Relationships and Formulas 891
EPILOGUE:PART EIGHT 891
PT 8.3 Trade-Offs 891
PT 8.5 Advanced Methods and Additional References 892
APPENDIX A:THE FOURIER SERIES 893
APPENDIX B:GETTING STARTED WITH MATHCAD 895
APPENDIX C:GETTING STARTED WITH MATLAB 905
BIBLIOGRAPHY 913
INDEX 916