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Loading...Numerical Methods for Engineers
Eighth Edition
Steven C. Chapra
Berger Chair in Computing and Engineering
Tufts University
Raymond P. Canale
Professor Emeritus of Civil Engineering
University of Michigan
CONTENTS
ABOUT THE AUTHORS iv
PREFACE xv
PART ONE
MODELING, PT1.1 Motivation 2
COMPUTERS, AND PT1.2 Mathematical Background 4
ERROR ANALYSIS 2 PT1.3 Orientation 7
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
Programming and Software 28
2.1 Packages and Programming 28
2.2 Structured Programming 29
2.3 Modular Programming 38
2.4 Excel 40
2.5 MATLAB 44
2.6 Mathcad 48
2.7 Other Languages and Libraries 49
Problems 50
CHAPTER 3
Approximations and Round-Off Errors 57
3.1 Significant Figures 58
3.2 Accuracy and Precision 60
3.3 Error Definitions 61
3.4 Round-Off Errors 67
Problems 81vi CONTENTS
CHAPTER 4
Truncation Errors and the Taylor Series 83
4.1 The Taylor Series 83
4.2 Error Propagation 99
4.3 Total Numerical Error 104
4.4 Blunders, Formulation Errors, and Data Uncertainty 108
Problems 110
EPILOGUE: PART ONE 112
PT1.4 Trade-Offs 112
PT1.5 Important Relationships and Formulas 115
PT1.6 Advanced Methods and Additional References 115
PART TWO
ROOTS OF PT2.1 Motivation 117
EQUATIONS 117 PT2.2 Mathematical Background 119
PT2.3 Orientation 120
CHAPTER 5
Bracketing Methods 124
5.1 Graphical Methods 124
5.2 The Bisection Method 128
5.3 The False-Position Method 136
5.4 Incremental Searches and Determining Initial Guesses 142
Problems 143
CHAPTER 6
Open Methods 146
6.1 Simple Fixed-Point Iteration 147
6.2 The Newton-Raphson Method 152
6.3 The Secant Method 158
6.4 Brent’s Method 163
6.5 Multiple Roots 167
6.6 Systems of Nonlinear Equations 170
Problems 174
CHAPTER 7
Roots of Polynomials 177
7.1 Polynomials in Engineering and Science 177
7.2 Computing with Polynomials 180
7.3 Conventional Methods 183CONTENTS vii
7.4 Müller’s Method 184
7.5 Bairstow’s Method 188
7.6 Other Methods 193
7.7 Root Location with Software Packages 193
Problems 204
CHAPTER 8
Case Studies: Roots of Equations 206
8.1 Ideal and Nonideal Gas Laws (Chemical/Bio Engineering) 206
8.2 Greenhouse Gases and Rainwater (Civil/Environmental Engineering) 209
8.3 Design of an Electric Circuit (Electrical Engineering) 211
8.4 Pipe Friction (Mechanical/Aerospace Engineering) 214
Problems 218
EPILOGUE: PART TWO 231
PT2.4 Trade-Offs 231
PT2.5 Important Relationships and Formulas 232
PT2.6 Advanced Methods and Additional References 232
PART THREE
LINEAR ALGEBRAIC PT3.1 Motivation 235
EQUATIONS 235 PT3.2 Mathematical Background 237
PT3.3 Orientation 245
CHAPTER 9
Gauss Elimination 249
9.1 Solving Small Numbers of Equations 249
9.2 Naive Gauss Elimination 256
9.3 Pitfalls of Elimination Methods 262
9.4 Techniques for Improving Solutions 268
9.5 Complex Systems 275
9.6 Nonlinear Systems of Equations 275
9.7 Gauss-Jordan 277
9.8 Summary 279
Problems 280
CHAPTER 10
LU Decomposition and Matrix Inversion 283
10.1 LU Decomposition 283
10.2 The Matrix Inverse 292
10.3 Error Analysis and System Condition 296
Problems 302viii CONTENTS
CHAPTER 11
Special Matrices and Gauss-Seidel 305
11.1 Special Matrices 305
11.2 Gauss-Seidel 309
11.3 Linear Algebraic Equations with Software Packages 316
Problems 321
CHAPTER 12
Case Studies: Linear Algebraic Equations 325
12.1 Steady-State Analysis of a System of Reactors (Chemical/Bio
Engineering) 325
12.2 Analysis of a Statically Determinate Truss (Civil/Environmental
Engineering) 328
12.3 Currents and Voltages in Resistor Circuits (Electrical
Engineering) 332
12.4 Spring-Mass Systems (Mechanical/Aerospace Engineering) 334
Problems 337
EPILOGUE: PART THREE 347
PT3.4 Trade-Offs 347
PT3.5 Important Relationships and Formulas 348
PT3.6 Advanced Methods and Additional References 348
PART FOUR
OPTIMIZATION 350 PT4.1 Motivation 350
PT4.2 Mathematical Background 355
PT4.3 Orientation 357
CHAPTER 13
One-Dimensional Unconstrained Optimization 360
13.1 Golden-Section Search 361
13.2 Parabolic Interpolation 368
13.3 Newton’s Method 370
13.4 Brent’s Method 371
Problems 373
CHAPTER 14
Multidimensional Unconstrained Optimization 375
14.1 Direct Methods 376
14.2 Gradient Methods 380
Problems 393CONTENTS ix
CHAPTER 15
Constrained Optimization 395
15.1 Linear Programming 395
15.2 Nonlinear Constrained Optimization 406
15.3 Optimization with Software Packages 407
Problems 418
CHAPTER 16
Case Studies: Optimization 421
16.1 Least-Cost Design of a Tank (Chemical/Bio Engineering) 421
16.2 Least-Cost Treatment of Wastewater (Civil/Environmental Engineering) 426
16.3 Maximum Power Transfer for a Circuit (Electrical Engineering) 430
16.4 Equilibrium and Minimum Potential Energy (Mechanical/Aerospace
Engineering) 434
Problems 436
EPILOGUE: PART FOUR 445
PT4.4 Trade-Offs 445
PT4.5 Additional References 446
PART FIVE
CURVE FITTING 447 PT5.1 Motivation 447
PT5.2 Mathematical Background 449
PT5.3 Orientation 458
CHAPTER 17
Least-Squares Regression 462
17.1 Linear Regression 462
17.2 Polynomial Regression 478
17.3 Multiple Linear Regression 482
17.4 General Linear Least Squares 485
17.5 Nonlinear Regression 489
Problems 493
CHAPTER 18
Interpolation 496
18.1 Newton’s Divided-Difference Interpolating Polynomials 497
18.2 Lagrange Interpolating Polynomials 508
18.3 Coefficients of an Interpolating Polynomial 513
18.4 Inverse Interpolation 513
18.5 Additional Comments 514
18.6 Spline Interpolation 517
18.7 Multidimensional Interpolation 529
Problems 531x CONTENTS
CHAPTER 19
Fourier Approximation 535
19.1 Curve Fitting with Sinusoidal Functions 536
19.2 Continuous Fourier Series 542
19.3 Frequency and Time Domains 545
19.4 Fourier Integral and Transform 549
19.5 Discrete Fourier Transform (DFT) 551
19.6 Fast Fourier Transform (FFT) 554
19.7 The Power Spectrum 560
19.8 Curve Fitting with Software Packages 561
Problems 570
CHAPTER 20
Case Studies: Curve Fitting 572
20.1 Fitting Enzyme Kinetics (Chemical/Bio Engineering) 572
20.2 Use of Splines to Estimate Heat Transfer (Civil/Environmental
Engineering) 576
20.3 Fourier Analysis (Electrical Engineering) 578
20.4 Analysis of Experimental Data (Mechanical/Aerospace
Engineering) 579
Problems 581
EPILOGUE: PART FIVE 592
PT5.4 Trade-Offs 592
PT5.5 Important Relationships and Formulas 593
PT5.6 Advanced Methods and Additional References 594
PART SIX
NUMERICAL PT6.1 Motivation 596
DIFFERENTIATION PT6.2 Mathematical Background 606
AND PT6.3 Orientation 608
INTEGRATION 596
CHAPTER 21
Newton-Cotes Integration Formulas 612
21.1 The Trapezoidal Rule 614
21.2 Simpson’s Rules 624
21.3 Integration with Unequal Segments 633
21.4 Open Integration Formulas 636
21.5 Multiple Integrals 636
Problems 638CONTENTS xi
CHAPTER 22
Integration of Equations 642
22.1 Newton-Cotes Algorithms for Equations 642
22.2 Romberg Integration 643
22.3 Adaptive Quadrature 649
22.4 Gauss Quadrature 651
22.5 Improper Integrals 659
22.6 Monte Carlo Integration 662
Problems 664
CHAPTER 23
Numerical Differentiation 667
23.1 High-Accuracy Differentiation Formulas 667
23.2 Richardson Extrapolation 670
23.3 Derivatives of Unequally Spaced Data 672
23.4 Derivatives and Integrals for Data with Errors 673
23.5 Partial Derivatives 674
23.6 Numerical Integration/Differentiation with Software Packages 675
Problems 682
CHAPTER 24
Case Studies: Numerical Integration and Differentiation 685
24.1 Integration to Determine the Total Quantity of Heat (Chemical/Bio
Engineering) 685
24.2 Effective Force on the Mast of a Racing Sailboat (Civil/Environmental
Engineering) 687
24.3 Root-Mean-Square Current by Numerical Integration (Electrical
Engineering) 689
24.4 Numerical Integration to Compute Work (Mechanical/Aerospace
Engineering) 692
Problems 696
EPILOGUE: PART SIX 708
PT6.4 Trade-Offs 708
PT6.5 Important Relationships and Formulas 709
PT6.6 Advanced Methods and Additional References 709
PART SEVEN
ORDINARY PT7.1 Motivation 711
DIFFERENTIAL PT7.2 Mathematical Background 715
EQUATIONS 711 PT7.3 Orientation 717xii CONTENTS
CHAPTER 25
Runge-Kutta Methods 721
25.1 Euler’s Method 722
25.2 Improvements of Euler’s Method 733
25.3 Runge-Kutta Methods 741
25.4 Systems of Equations 751
25.5 Adaptive Runge-Kutta Methods 756
Problems 764
CHAPTER 26
Stiffness and Multistep Methods 767
26.1 Stiffness 767
26.2 Multistep Methods 771
Problems 791
CHAPTER 27
Boundary-Value and Eigenvalue Problems 793
27.1 General Methods for Boundary-Value Problems 794
27.2 Eigenvalue Problems 801
27.3 ODEs and Eigenvalues with Software Packages 813
Problems 820
CHAPTER 28
Case Studies: Ordinary Differential Equations 823
28.1 Using ODEs to Analyze the Transient Response of a Reactor
(Chemical/Bio Engineering) 823
28.2 Predator-Prey Models and Chaos (Civil/Environmental Engineering) 830
28.3 Simulating Transient Current for an Electric Circuit (Electrical Engineering) 834
28.4 The Swinging Pendulum (Mechanical/Aerospace Engineering) 839
Problems 843
EPILOGUE: PART SEVEN 855
PT7.4 Trade-Offs 855
PT7.5 Important Relationships and Formulas 856
PT7.6 Advanced Methods and Additional References 856
PART EIGHT
PARTIAL PT8.1 Motivation 858
DIFFERENTIAL PT8.2 Orientation 862
EQUATIONS 858CONTENTS xiii
CHAPTER 29
Finite Difference: Elliptic Equations 865
29.1 The Laplace Equation 865
29.2 Solution Technique 867
29.3 Boundary Conditions 873
29.4 The Control-Volume Approach 879
29.5 Software to Solve Elliptic Equations 882
Problems 883
CHAPTER 30
Finite Difference: Parabolic Equations 886
30.1 The Heat-Conduction Equation 886
30.2 Explicit Methods 887
30.3 A Simple Implicit Method 891
30.4 The Crank-Nicolson Method 895
30.5 Parabolic Equations in Two Spatial Dimensions 898
Problems 901
CHAPTER 31
Finite-Element Method 903
31.1 The General Approach 904
31.2 Finite-Element Application in One Dimension 908
31.3 Two-Dimensional Problems 917
31.4 Solving PDEs with Software Packages 921
Problems 925
CHAPTER 32
Case Studies: Partial Differential Equations 928
32.1 One-Dimensional Mass Balance of a Reactor (Chemical/Bio
Engineering) 928
32.2 Deflections of a Plate (Civil/Environmental Engineering) 932
32.3 Two-Dimensional Electrostatic Field Problems (Electrical
Engineering) 934
32.4 Finite-Element Solution of a Series of Springs
(Mechanical/Aerospace Engineering) 937
Problems 941
EPILOGUE: PART EIGHT 944
PT8.3 Trade-Offs 944
PT8.4 Important Relationships and Formulas 944
PT8.5 Advanced Methods and Additional References 945xiv CONTENTS
APPENDIX A: THE FOURIER SERIES 946
APPENDIX B: GETTING STARTED WITH MATLAB 948
APPENDIX C: GETTING STARTED WITH MATHCAD 956
BIBLIOGRAPHY 967
INDEX 970
INDEX
A
Accuracy, 60–61, 114
Adams-Bashforth formula, 783–785, 787
Adams-Moulton formula, 771, 785–787
Adaptive integration, 642
Adaptive quadrature, 610, 649–651
Adaptive Runge-Kutta (RK) methods, 719, 756–763, 855
Adaptive step-size control, 757–758, 760–761
Addition, 75
estimated error bounds, 103
large and small number, 77–78
matrix operations, 240
smearing, 79–81
Advanced methods/additional references, 115–116
curve fitting, 594–595
linear algebraic equations, 348–349
numerical integration, 709
ordinary differential equations (ODEs), 856–857
partial differential equations (PDEs), 945
roots of equations, 232–234
Advection-diffusion equation, 929
Air resistance
falling parachutist problem, 13–18
formulation, 14
Allosteric enzymes, 572–576
Alternating-direction implicit (ADI) method, 862, 891–895, 898–901,
944, 945
Amplitude, 537–538
Analytical/direct approach
curve fitting, 447–448
falling parachutist problem, 13–18. See also Falling parachutist
problem
finite-element methods, 910–914
linear algebraic equations, 235–236
nature of, 14, 15
numerical differentiation, 599–600, 601–602
numerical integration, 600–602
optimization, 357, 375, 376–380
partial differential equations (PDEs), 861–862
roots of equations, 117–118, 231
Angular frequency, 538
Antidifferentiation, 608
Antoine’s equation, 437
Approximations, 57–66. See also Estimation
accuracy/inaccuracy, 60–61, 114
algorithm for iterative calculations, 64–66
approximate percent relative error, 62, 64, 116
continuous Fourier series, 543–545
error calculation, 61–64, 76
error definitions, 61–66
finite-element methods, 904–907
functional, 595
polynomial, 85–87
precision/imprecision, 60–61, 114
significant figures/digits, 58–59, 268
Taylor series, 83–99, 667–670
Archimedes’ principle, 25–26
Areal integrals, 605
Arithmetic mean, 450
Arithmetic operations, 75–76, 950–953, 957–958
Assemblage property matrix, 907
Associative property, matrix operations, 240
Augmentation, matrix operations, 243–244
Auxiliary conditions, 716
B
Background information
blunders, 108–109
computer programming and software, 28–56
conservation laws and engineering, 18–21
curve fitting, 449–458
data uncertainty, 60–61, 109, 674
eigenvalue problems, 801
error propagation, 99–103, 116
Excel, 40–44. See also Excel
formulation errors, 109
linear algebraic equations, 237–245
Mathcad, 48–49, 956–966. See also Mathcad
MATLAB, 44–48, 948–955. See also MATLAB
modular programming, 38–40
numerical differentiation, 95–99, 606–608
numerical integration, 606–608
optimization, 355–357
ordinary differential equations (ODEs), 715–717
root equation, 119–120, 180–183
roots of polynomials, 177–180
round-off errors, 67–81, 105–107INDEX 971
simple mathematical model, 11–18
structured programming, 29–38
Taylor series, 83–99
total numerical error, 104–108
truncation errors, 83, 91–99, 105–107
Back substitution, 256, 258–260, 261–262
LU decomposition, 288, 306
Backward deflation, 183
Backward difference approximation, 95–98, 669
Bairstow’s method, 122, 188–193, 231
Banded matrices, 305–306
Banded matrix, 239
Base-2 (binary) number system, 67, 72–73
Base-8 (octal) number system, 67
Base-10 (decimal) number system, 67, 68–69, 75–76
Basic feasible solution, 402
Basic variables, 402
Bernoulli’s equation, 229–230
BFGS algorithm, 393, 406
Bias/inaccuracy, 60–61
Bilinear interpolation, 529–531
Binary chopping. See Bisection method
Binary (base-2) number system, 67, 72–73
Binding constraints, 399
Bisection method, 120–121, 128–136, 231, 361
bisection algorithm, 134, 135, 233
computer methods, 133–134, 135
defined, 128
error estimates, 130–134
false-position method vs., 138–140
graphical method, 129–130, 131, 132, 233
incremental search methods vs., 128
minimizing function evaluations, 135–136
termination criteria, 130
Blasius formula, 220
Blunders, 108–109
Bolzano’s method. See Bisection method
Boole’s rule, 631, 632, 650
Boundary conditions, 873–879
derivative, 799–800, 873–876, 890
finite-element methods, 907, 916–917, 919–921, 940
irregular boundaries, 876–879
Laplace equation, 862, 868–871, 873–879
Boundary-value problems, 717, 794–801, 856
eigenvalue, 804–807
finite-difference method, 719
initial-value problems vs., 793
shooting method, 719, 795–798
Bracketing methods, 124–145, 371
bisection method, 120–121, 128–136, 231, 361
computer methods, 126–128
defined, 124
false-position method, 120–121, 136–142, 231
graphical method, 124–128
incremental searches/determining initial guesses, 142
Break command, 47
Break loops, 33, 34–35
Brent’s root-location method, 122, 163–167, 231, 232
algorithm, 165–167, 372–373
graphical method, 163, 164
inverse quadratic interpolation, 163–165
optimization, 357, 361, 371–373, 445
roots of polynomials, 201
Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithms, 393, 406
B splines, 595
Butcher’s fifth-order Runge-Kutta method, 749–751
Butterfly network, 557, 558
C
C++, 49
Cartesian coordinates, 605
CASE structure, 32, 33, 42, 46
Cash-Karp RK method, 759–760, 761–762
Centered finite divided-difference approximation, 96, 97,
98, 669
Central Limit Theorem, 455
Chaotic solutions, 834
Characteristic, 69–70
Characteristic equation, 178–179
Charge, conservation of, 20
Chebyshev economization, 595
Chemical/biological engineering
analyzing transient response of reactor, 823–830
conservation of mass, 20
curve fitting, 572–576
determining total quantity of heat, 685–687
fitting enzyme kinetics, 572–576
ideal gas law, 206–209
least-cost design of a tank, 421–425
linear algebraic equations, 325–328
numerical integration, 685–687
one-dimensional mass balance of reactor, 928–932
optimization, 421–425
ordinary differential equations (ODEs), 823–830
partial differential equations (PDEs), 928–932
roots of equations, 206–209
steady-state analysis of system of reactors, 325–328
Cholesky decomposition, 307–309
Chopping, 72–73, 76
Civil/environmental engineering
analysis of statically determinate truss, 328–332
conservation of momentum, 20
curve fitting, 576–577
deflections of a plate, 932–934972 INDEX
Civil/environmental engineering—Cont.
effective force on mast of racing sailboat, 687–689
greenhouse gases and rainwater, 209–211
least-cost treatment of wastewater, 426–430
linear algebraic equations, 328–332
numerical integration, 687–689
optimization, 421, 426–430
ordinary differential equations (ODEs), 830–834
partial differential equations (PDEs), 932–934
predator-prey models and chaos, 830–834
roots of equations, 209–211
splines to estimate heat transfer, 576–577
Clamped end condition, 528–529
Classical fourth-order Runge-Kutta method, 747–749, 857
Coefficient, method of undetermined, 653–654
Coefficient of determination, 469
Coefficient of interpolating polynomial, 513
Coefficient of thermal conductivity, 866
Coefficient of variation, 451
Colebrook equation, 214, 216, 227
Column, defined, 238
Column-sum norms, 298
Column vectors, 238
Commutative property, matrix operations, 240, 242
Complete pivoting, 268
Complex systems, linear algebraic equations, 275
Composite, integration formulas, 618–621
Computational error, 61–64, 76
Computer programming and software, 28–56. See also Pseudocode
algorithms
bisection method, 133–134, 135
bracketing methods, 126–128
computer programs, defined, 29
cost comparison, 112–115
curve fitting, 455–456, 460, 470–474, 561–569
eigenvalues, 813–820
Excel. See Excel
linear algebraic equations, 247–248, 272–274, 316–321
linear programming, 407–409
linear regression, 470–474
Mathcad. See Mathcad
MATLAB. See MATLAB
modular programming, 38–40
numerical integration/differentiation, 675–682
optimization, 357, 407–418, 424–425
ordinary differential equations (ODEs), 719, 813–820
other languages and libraries, 49
partial differential equations (PDEs), 864, 882–883, 921–925
roots of equations, 126–128, 193–203, 214–215
software user types, 28–29
step-size control, 780
structured programming, 29–38
Condition numbers, 102–103
matrix, 245, 299–301
Confidence intervals, 452–458, 487–488
Conjugate directions, 379–380
Conjugate gradient, 357, 391, 406
Conservation laws, 18–21
by field of engineering, 20
simple models in specific fields, 20
stimulus-response computations, 295–296
Conservation of charge, 20
Conservation of energy, 20
Conservation of mass, 20, 325–328
Conservation of momentum, 20
Constant of integration, 715–716
Constant step size, 780
Constitutive equation, 866–867
Constrained optimization, 356, 357, 395–420
linear programming, 356, 357, 395–406
nonlinear, 357, 406, 409–413, 418
Constraints
binding/nonbinding, 399
optimization, 353, 355
Continuous Fourier series, 542–545
approximation, 543–545
determination of coefficients, 543
Control-volume approach, 879–882
Convergences
defined, 890
fixed-point iteration, 148–151
Gauss-Seidel (Liebmann) method, 311–314
linear, 148–151
nature of, 151
Newton-Raphson method, 155–156
of numerical methods of problem solving, 113
Cooley-Tukey algorithm, 554, 558–560
Corrector equation, 735, 771–772
Corrector modifier, 777–779
Correlation coefficient, 469
Count-controlled loops, 34–35, 42, 46, 47, 64
Cramer’s rule, 252, 253–254, 347
Crank-Nicolson technique, 862, 895–898, 944, 945
Critically damped case, 179
Crout decomposition, 290–292
Cubic splines, 517, 523–528, 592–594, 709
derivation, 524–528
interpolation with Mathcad, 568–569
natural, 527
Cumulative normal distribution, 660–662
Current balance, 20
Curvature, 596
Curve fitting, 447–595
advanced methods and additional references, 594–595INDEX 973
Dependent variables, 11–12, 118, 711
Derivative
defined, 596
first, 596, 668–669
second, 596, 668–669
Derivative boundary conditions, 799–800, 873–876, 890
Derivative mean-value theorem, 90
Descriptive statistics, 109, 449–452
Design, 21
Design variables, 355
Determinants, in Gauss elimination, 252–253, 265–267
Determination, coefficient of, 469
DFP algorithm, 393, 446
Diagonally dominant systems, 313–314
Diagonal matrix, 239
Differential calculus. See Numerical differentiation
Differential equations, 14–16, 28, 39, 711
Dilatant (“shear thickening”) fluids, 589
Direct approach. See Analytical/direct approach
Directional derivative, 381
Dirichlet boundary condition, 799, 868–871, 873, 922
Discrete Fourier transform (DFT), 551–554
Discretization, finite-element methods, 904, 909, 917, 937–938
Discriminant, 179
DISPLAY statements, 39
Distributed-parameter system, 929
Distributed-variable systems, 236, 237, 316
Distributive property, matrix operations, 242
Division, 76
estimated error bounds, 103
synthetic, 181–182
by zero, 262
DOEXIT loops, 33, 34, 36, 42, 46, 47
DOFOR loops, 34–35
Double integrals, 637–638
Double roots, 167
Drag coefficient, 14
Dynamic instability, 932
E
Eigenvalue problems, 801–820, 856
boundary-value problem, 804–807
computer methods, 813–820
eigenvalue, defined, 801
eigenvalue analysis of axially loaded column, 806–807
eigenvectors, 801, 803–804
mass-spring system, 803–804
mathematical background, 801
other methods, 812–813
physical background, 802–804
polynomial method, 178–179, 719, 807–809
power method, 719, 809–812
case studies, 572–591
coefficients of an interpolating polynomial, 513
comparisons of alternative methods, 592–593
computer methods, 455–456, 460, 470–474, 561–569
defined, 447
engineering applications, 448–449, 572–591
estimation of confidence intervals, 452–458
extrapolation, 515
fast Fourier transform (FFT), 460
Fourier approximation, 485, 535–571
frequency domains, 545–549
general linear least squares model, 459, 485–489
goals/objectives, 460–461
important relationships and formulas, 593–594
interpolation, 447, 460, 496–534
inverse interpolation, 513–514
Lagrange interpolating polynomial, 460, 496, 508–513, 515,
592–594
least-squares regression, 447, 458, 462–495
linear regression, 458, 462–478
mathematical background, 449–458
multidimensional interpolation, 529–531
multiple linear regression, 458, 482–485, 592–594
Newton’s divided-difference interpolating polynomials, 497–508
Newton’s interpolating polynomial, 460, 496, 497–509, 515, 516,
592–594
noncomputer methods, 447–448
nonlinear regression, 460, 475–476, 489–492, 564, 592
normal distribution, 452
polynomial regression, 458, 478–482, 594
power spectrum, 560–561
scope/preview, 458–460
simple statistics, 449–452
with sinusoidal functions, 536–542
spline interpolation, 460, 517–529
time domains, 545–549
D
Dartboard Monte Carlo integration, 662–664
Data distribution, 452
Data uncertainty, 60–61, 109, 674
Davidon-Fletcher-Powell (DFP) method of optimization, 393, 446
Decimal (base-10) number system, 67, 68–69, 75–76
Decimation-in-frequency, 554–555
Decimation-in-time, 554–555, 559
Decision loops, 33, 64
Definite integration, 598n
Deflation, 812
backward, 183
forward, 183
polynomial, 181–183
Degrees of freedom, 450974 INDEX
numerical integration, 603–606, 675–682, 685–707
optimization, 351–355, 357, 421–444
ordinary differential equations (ODEs), 713–715, 719, 823–854
parameters, 11–12, 118, 828
partial differential equations (PDEs), 859–861, 862–863, 928–943
practical issues, 21
roots of equations, 118–119, 122, 177–180, 206–230
roots of polynomials, 177–180
two-pronged approach, 11, 13–18
Entering variables, 403–404
Epilimnion, 576
Equal-area graphical differentiation, 600
Equality constraint optimization, 356
Error(s)
approximations. See Approximations
bisection method, 130–134
blunders, 108–109
calculation, 61–64, 76
data uncertainty, 60–61, 109, 674
defined, 57, 60–61
estimates for iterative methods, 63–64
estimates in multistep method, 774–777
estimation, 470, 503–505, 625
estimation for Euler’s method, 724–729
falling parachutist problem, 17, 57
formulation, 109
Gauss quadrature, 658–659
linear algebraic equations, 296–302
Newton-Raphson estimation method, 153–155
Newton’s divided-difference interpolating polynomial estimation,
503–505
numerical differentiation, 105–108, 673–674
numerical integration, 673–674
predictor-corrector approach, 734–736, 771–779
quantizing, 72–74, 77
relative, 102
residual, 463, 467–470
round-off. See Round-off errors
Simpson’s 1/3 rule estimation, 625
total numerical, 104–108
trapezoidal rule, 616–617, 645–646, 771–772
true, 61, 116
true fractional relative error, 61–62
truncation. See Truncation errors
Error definitions, 61–66
approximate percent relative error, 62, 64, 116
stopping criterion, 64–65, 116
true error, 61, 116
true fractional relative error, 61–62
true percent relative error, 61–62, 66, 116
Error propagation, 99–103, 116
condition, 102–103
functions of more than one variable, 101–102
Eigenvectors, 801, 803–804
Electrical engineering
conservation of charge, 20
conservation of energy, 20
currents and voltages in resistor circuits, 332–334
curve fitting, 578–579
design of electric circuit, 211–214
Fourier analysis, 578–579
linear algebraic equations, 332–334
maximum power transfer for a circuit, 430–434
numerical integration, 689–692
optimization, 421, 430–434
ordinary differential equations (ODEs), 834–839
partial differential equations (PDEs), 934–937
root-mean-square current, 689–692
roots of equations, 211–214
simulating transient current for electric circuit, 834–839
two-dimensional electrostatic field problems, 934–937
Element properties, finite-element methods, 907
Element stiffness matrix, 907, 939
Elimination of unknowns, 254–260
back substitution, 256, 258–260, 261–262
forward, 256–258, 259
Elliptic partial differential equations (PDEs), 859–860, 865–885, 944, 945
boundary conditions, 862, 873–879
computer software solutions, 882–883
control-volume approach, 879–882
Gauss-Seidel (Liebmann) method, 862, 869–871, 894–895
Laplace equation, 859–860, 862, 865–867, 935–937
Laplacian difference equation, 868–869
solution technique, 867–873
Embedded Runge-Kutta (RK) method, 759–760
ENDDO statement, 34–35
End statement, 47
Energy
conservation of, 20
equilibrium and minimum potential, 434–435
Energy balance, 118
Engineering problem solving
chemical engineering. See Chemical/biological engineering
civil engineering. See Civil/environmental engineering
conservation laws, 18–21
curve fitting, 448–449, 572–591
dependent variables, 11–12, 118, 711
electrical engineering. See Electrical engineering
falling parachutist problem. See Falling parachutist problem
forcing functions, 11–12
fundamental principles, 118
independent variables, 11–12, 118, 711
linear algebraic equations, 236–237, 325–346
mechanical engineering. See Mechanical/aerospace engineering
Newton’s laws of motion, 11–18, 57, 118, 334, 714, 839–840
numerical differentiation, 602–603, 675–682INDEX 975
functions of single variable, 99–100
stability, 102–103
Estimated mean, 455
Estimation. See also Approximations
confidence interval, 452–458, 487–488
defined, 453
errors, 470, 503–505, 625, 724–729
Newton-Raphson estimation method, 153–155
parameter, 828
standard error of the estimate, 468
standard normal estimate, 454–455
Euclidean norms, 297–299
Euler-Cauchy method. See Euler’s method
Euler’s method, 16–17, 28, 39, 49, 179, 722–741
algorithm, 730–733
backward/implicit, 768–771
effect of reduced step size, 727–729
error analysis, 724–729
Euler’s formula, 806
improvements, 733–741
ordinary differential equations (ODEs), 719, 722–741, 835–836,
841–842, 855–857
as predictor, 771–772
systems of equations, 752
Excel, 28–29, 34–35, 40–44
computer implementation of iterative calculation, 65–66
curve fitting, 561–564
Data Analysis Toolpack, 562–564
described, 40
double precision to represent numerical quantities, 75
Goal Seek, 194
infinite series evaluation, 80
linear algebraic equations, 316–317
linear programming, 407–409
linear regression, 470
nonlinear constrained optimization, 409–413
optimization, 357, 407–413, 424–425, 428–430, 431–434
ordinary differential equations (ODEs), 813, 828–830
partial differential equations (PDEs), 921–923
pseudocode vs., 42
roots of equations, 78–79, 193–197, 214–215
Solver, 195–197, 407–413, 428–430, 431–434, 829–830
standard use, 40–41
Trendline command, 561–562
VBA macros, 40–44
Explicit solution technique, 119
ordinary differential equations (ODEs), 769–771
parabolic partial differential equations (PDEs), 887–892, 898, 944
Exponent, 69–70
Exponential model of linear regression, 474–475
Extended midpoint rule, 660
Extended precision, round-off error, 74–75
Extrapolation, 515
Extreme points, 400
Extremum, 361–364
F
Factors, polynomial, 181
Falling parachutist problem, 13–18, 119, 714, 717, 721–722
algorithm, 272–274
error, 17, 57
Gauss elimination, 272–274
Gauss quadrature application, 658
optimization of parachute drop cost, 351–355, 409–413
schematic diagram, 13
velocity of the parachutist, 471–473, 721–722
False-position method, 120–121, 136–142, 231
bisection method vs., 138–140
false-position formula, 137–139, 233
graphical method, 136, 140
modified false position, 141–142, 231
pitfalls, 139–141
secant method vs., 159–161
Fanning friction factor, 225
Faraday’s law, 714
Fast Fourier transform (FFT), 460, 554–560, 569
Cooley-Tukey algorithm, 554, 558–560
Sande-Tukey algorithm, 554–558
Feasible extreme points, 400
Feasible solution space, 397–400
Fibonacci numbers, 362–363
Fick’s law of diffusion, 696, 714, 929
Finish, 34–35
Finite-difference methods, 16–17, 83, 95–99
elliptic partial differential equations (PDEs), 859–860, 862, 865–885,
944, 945
high-accuracy differentiation formulas, 610, 667–670
optimization, 385–386
ordinary differential equations (ODEs), 719, 798–801
parabolic partial differential equations (PDEs), 860–861, 862,
886–902, 944, 945
Finite-divided-difference approximations of derivatives, 16–17, 95–99,
667–670
Finite-element methods, 903–927
assembly, 907, 914–916, 939–940
boundary conditions, 907, 916–917, 919–921, 940
defined, 903–904
discretization, 904, 909, 917, 937–938
element equations, 904–907, 910–914, 917–919, 938–939
general approach, 904–908
partial differential equations (PDEs), 862, 903–927, 944
single dimension, 908–917
solution and postprocessing, 907–908, 921, 940
two dimensions, 917–921
First backward difference, 95, 96
First derivative, 596, 668–669976 INDEX
Fourth-order methods
Adams, 785, 786, 789, 790–791, 855, 856, 857
Runge-Kutta, 747–749, 753–754, 755–756, 758, 762–763, 835,
841–842, 855–857
Fractional parts, 69–71
Frequency domain, 545–549
Frequency plane, 546–547
Friction factor, 214–217
Frobenius norms, 55, 299
Fully augmented version, 401
FUNCTION, 39
Function(s)
error propagation, 99–103
forcing, 11–12
interpolation, 905–906
mathematical behavior, 114
modular programming, 38–39
penalty, 406
sinusoidal, 536–542
spline, 517, 595
Functional approximation, 595
Fundamental frequency, 542
Fundamental theorem of integral calculus, 607
G
Gauss elimination, 249–282, 349
algorithm, 272–274
Cramer’s rule, 252, 253–254, 347
determinants, 252–253, 265–267
elimination of unknowns, 254–260
Gauss-Jordan method, 277–279, 405–406
graphical method, 249–251
improving solutions, 268–274
LU decomposition version, 285–290
more significant figures, 268
naive approach, 245, 256–262
operation counting, 260–262
pitfalls of elimination methods, 262–268
pivoting, 245, 257–258, 262, 268–272, 347
solving small numbers of equations, 249–255
Gauss-Jordan method, 277–279, 405–406
Gauss-Legendre formulas, 652, 654–658, 691, 709
higher-point, 657–658
two-point, 654–657
Gauss-Newton method, 489–492, 595
Gauss quadrature, 610, 642, 651–659, 692, 708–710
error analysis, 658–659
Gauss-Legendre formulas, 652, 654–658, 691, 709
method of undetermined coefficients, 653–654
Gauss-Seidel (Liebmann) method, 245–247, 305, 309–316, 347–348,
349, 944
algorithm, 314–315
First finite divided difference, 95
First forward difference, 95, 96, 116
First forward finite divided difference, 116
First-order approximation, 84, 86, 88, 95–98
First-order equations, 711–712, 727
First-order splines, 519–520
Fixed (Dirichlet) boundary condition, 799, 868–871, 873, 922
Fixed-point iteration, 147–152, 231
algorithm, 151–152
convergences, 148–151
graphical method, 148–151
nonlinear equations, 171–172
Fletcher-Reeves conjugate gradient algorithm, 391, 446
Floating-point operations/flops, 260–262
Floating-point representation
chopping, 72–73, 76
fractional part/mantissa/significand, 69–71
integer part/exponent/characteristic, 69–71
machine epsilon, 73–74
quantizing errors, 72–74, 77
Flowcharts, 30–35
defined, 30
sequence structure, 31
simple selection constructs, 32
symbols, 30
Force balance, 20, 118
Forcing functions, 11–12
Formulation errors, 109
Fortran 90, 49, 76–77
Forward deflation, 183
Forward difference approximation, 95, 96, 97, 668
Forward elimination of unknowns, 256–258, 259
Forward substitution, LU decomposition, 287–289, 306
Fourier approximation, 485, 535–571
continuous Fourier series, 542–545
curve fitting with sinusoidal functions, 536–542
defined, 535–536
discrete Fourier transform (DFT), 551–554
engineering applications, 578–579
fast Fourier transform (FFT), 460, 554–560, 569
Fourier integral and transform, 549–551
frequency domain, 545–549
power spectrum, 560–561
time domain, 545–549
Fourier integral, 549–551
Fourier series, 542–545, 546, 946–947
Fourier’s law of heat conduction, 603, 714, 866–867, 929
Fourier transform, 549–551
discrete Fourier transform (DFT), 551–554
fast Fourier transform (FFT), 460, 554–560, 569
Fourier transform pair, 549
Fourth derivative, 668–669INDEX 977
convergence criterion, 311–314
elliptic partial differential equations (PDEs), 862, 869–871, 894–895
graphical method, 312, 313–314
iteration cobwebs, 313–314
problem contexts, 315–316
relaxation, 314
Generalized reduced gradient (GRG), 406, 446
General linear least-squares model, 459, 485–489
confidence intervals for linear regression, 487–488
general matrix formulation, 485–486
statistical aspects of least-squares theory, 486–489
General solution, 178, 180
Genetic algorithm, 378
Given’s method, 813
Global truncation error, 725
Golden ratio, 362–364
Golden-section search optimization, 357, 361–368, 432–434, 445
extremum, 361–364
golden ratio, 362–364
single-variable optimization, 361
unimodal, 361–362
Gradient, defined, 603
Gradient methods of optimization, 375, 380–393
conjugate gradient method (Fletcher-Reeves), 357, 391, 446
finite-difference approximation, 385–386
gradients, 381–383
Hessian, 357, 383–386, 446
Marquardt’s method, 357, 392–393, 445, 595
Newton’s method, 357, 370–371, 385, 391–392, 445–446
path of steepest ascent/descent, 357, 382–383, 386–391, 595
quasi-Newton methods, 357, 393, 406, 446
Greenhouse gases, 209–211
H
Hagen-Poiseulle law, 438
Half-saturation constant, 220
Hamming’s method, 791
Harmonics, 542
Hazen-Williams equation, 580
Heat balance, 118
Heat-conduction equation, 860–861, 862, 886–887. See also Parabolic
partial differential equations (PDEs)
Hessenberg form, 813
Hessian, 357, 383–386, 446
Heun’s method, 719, 734–738, 740, 744–746, 855, 857
non-self-starting, 719, 771–779, 855, 857
High-accuracy differentiation formulas, 610, 667–670
Hilbert matrix, 300–301, 320–321
Histograms, 453–454
Hooke’s law, 334, 434–435
Hotelling’s method, 812
Householder’s method, 813
Humps function, 765
Hyperbolic partial differential equations (PDEs), 861
Hypolimnion, 576
Hypothesis testing, 449
I
Ideal gas law, 56, 206–209
Identity matrix, 239
IEEE format, 74
IF/THEN structure, 31–32, 38, 42, 46, 270
IF/THEN/ELSE structure, 31–32, 36, 42, 46
IF/THEN/ELSE/IF structure, 32, 33, 42, 46
Ill-conditioned systems, 103, 263–267
effect of scale on determinant, 265–267
elements of matrix inverse as measure of, 297
singular systems, 251, 267–268
Implicit solution technique, 119
ordinary differential equations (ODEs), 719, 767–771
parabolic partial differential equations (PDEs), 862, 891–895,
898–901, 944, 945
Imprecision, 60–61, 114
Improper integrals, 610, 642, 659–662
cumulative normal distribution, 660–662
extended midpoint rule, 660
normalized standard deviate, 660–662
Improved polygon (midpoint) method, 719, 739–740, 744–746,
775–776, 855
Inaccuracy, 60–61
Incremental search methods
bisection method vs., 128
defined, 128
determining initial guesses, 142
Increment function, 741–742
Indefinite integral, 712
Indefinite integration, 598n
Independent variables, 11–12, 118, 711
Indexes, 34–35
Inequality constraint optimization, 356
Inferential statistics, 453, 455
Infinite series
computation, 78, 80
smearing, 79–81
Initial value, 716–717
Initial-value problems, 717
boundary-value problems vs., 793
defined, 793
Inner products, 81
In place implementation, 270
INPUT statements, 39
Integer part, 69–71
Integer representation, 67–69
Integral calculus. See Numerical integration978 INDEX
Gauss-Seidel (Liebmann) method, 862, 869–871, 944
secondary variables, 872–873
solution technique, 867–873
Laplacian difference equation, 868–869
Large computations, interdependent computations, 76–77
Large versus small systems, 21
Law of mass action, 846
LC networks/circuits, 835–837
Least-squares fit of a sinusoid, 539–542
Least-squares regression, 447, 458, 462–495
general linear least-squares model, 459, 485–489
least-squares fit of a straight line, 465–467
linear regression, 458, 462–478, 592–593
multiple linear regression, 458, 482–485, 592–594
nonlinear, 460, 475–476, 489–492, 564, 592
polynomial regression, 458, 478–482, 594
Leaving variables, 403–404
Levenberg-Marquardt method, 417
Liebmann method. See Gauss-Seidel (Liebmann) method
Linear algebraic equations, 235–349
advanced methods and additional references, 348–349
case studies, 325–346
comparisons of methods, 347–348
complex systems, 275
computer methods, 247–248, 272–274, 316–321
Cramer’s rule, 252, 253–254, 347
determinants, 252–253
distributed-variable systems, 236, 237
division by zero, 262
elimination of unknowns, 254–260
engineering applications, 236–237, 325–346
error analysis, 296–302
Gauss elimination. See Gauss elimination
Gauss-Jordan method, 277–279, 405–406
Gauss-Seidel (Liebmann) method. See Gauss-Seidel (Liebmann)
method
general form, 235
goals/objectives, 247–248
graphical method, 249–251, 326, 327, 329, 331, 332–334, 347
ill-conditioned systems, 103, 251, 263–267
important relationships and formulas, 348, 349
Liebmann method. See Gauss-Seidel (Liebmann) method
LU decomposition methods, 245, 283–292, 336, 347, 349
lumped-variable systems, 236, 237, 316
mathematical background, 237–245
matrix inverse, 242–243, 245, 292–296
matrix notation, 238–239
matrix operating rules, 240–244
more significant figures, 268
noncomputer methods, 235–236
nonlinear systems of equations, 275–277
pivoting, 245, 257–258, 262, 268–272
representing in matrix form, 244–245
Integral form, 84
Integrand, 597, 676
Interdependent computations, 76–77
Interpolation, 496–534
coefficients of interpolating polynomial, 513
computers in, 511–513, 568–569
curve fitting, 447, 460
with equally spaced data, 516
finite-element methods, 905–906
interpolation functions, 905–906
inverse, 513–514
inverse quadratic interpolation method, 163–165
Lagrange interpolating polynomials, 460, 496, 508–513, 515, 592–594
linear interpolation method, 163, 497–498
multidimensional, 529–531
Newton’s divided-difference interpolating polynomials, 460, 496,
497–509, 515, 516, 592–594
polynomial, 496–534
quadratic, 499–501
spline, 460, 517–529
Interval estimator, 453
Interval halving. See Bisection method
Inverse Fourier transform, 549–550
Inverse interpolation, 513–514
Inverse quadratic interpolation, 163–165
Irregular boundaries, 876–879
Iterative approach to computation
algorithms, 64–66
defined, 62–63
error estimates, 63–64
Gauss-Seidel (Liebmann) method, 245–247, 305, 309–316, 347–348,
349, 944
iterative refinement, 301–302
J
Jacobian, 173, 174
Jacobi iteration, 311
Jacobi’s method, 812–813
Jenkins-Traub method, 193, 234
K
Kirchhoff’s laws, 118, 211–214, 332–333, 431, 834–837
L
Lagging phase angle, 538
Lagrange interpolating polynomials, 460, 496, 508–513, 515, 592–594
Lagrange multiplier, 351, 429
Lagrange polynomial, 164
Laguerre’s method, 193, 201, 234
Laplace equation, 859–860, 935–937
boundary conditions, 862, 868–871, 873–879
described, 865–867
flux distribution of heated plate, 872–873INDEX 979
round-off errors, 263
scaling, 265–267, 270–272
scope/preview, 245–247
singular systems, 251, 267–268
special matrices, 305–309
system condition, 296–302
Linear convergences, 148–151
Linear interpolation method. See also False-position method; Secant
method
defined, 163, 497–498
linear-interpolation formula, 497–498
Linearization, 713
Linear programming (LP)
computer solutions, 407–409
defined, 395
feasible solution space, 397–400
graphical solution, 397–400
optimization, 351, 356, 357, 395–406
possible outcomes, 399–400
setting up LP problem, 396–397
simplex method, 351, 357, 401–406
standard form, 395–397
Linear regression, 462–478
computer programs, 470–474
confidence intervals, 487–488
criteria for “best” fit, 464–465
curve fitting, 458
engineering applications, 572–576
estimation errors, 470
exponential model, 474–475
general comments, 478
general linear least-squares model, 459, 485–489
least-squares fit of straight line, 465–467
linearization of nonlinear relationships, 474–478
linearization of power equation, 476–478
minimax criterion, 465, 595
multiple, 458, 482–485, 592–594
quantification of error, 467–470
residual error, 463, 467–470
spread around the regression line, 468
standard error of the estimate, 468
Linear splines, 517–521
Linear trend, 84–85
Line spectra, 547–549
Local truncation error, 725
Logical loops, 34–35
Logical representation, 31–38
algorithm for roots of a quadratic equation, 35–38
repetition, 32–35
selection, 31–32
sequence, 31
Loops, 32–35, 64
Lorenz equations, 831–834
Lotka-Volterra equations, 830–834
Lower triangular matrix, 239
LR method (Rutishauser), 813
LU decomposition methods, 245, 283–292, 336, 347, 349
algorithm, 287, 288–290, 291–292
Crout decomposition, 290–292
defined, 283
LU decomposition step, 284, 285, 287–290, 305, 306
overview, 284–285
substitution step, 284, 285, 287–289
version of Gauss elimination, 285–290
Lumped-parameter systems, 928
Lumped-variable systems, 236, 237, 316
M
MacCormack’s method, 890–891
Machine epsilon, 73–74
Maclaurin series expansion, 63–64, 65–66
Manning equation, 440
Mantissa, storage, 74n
Maple V, 49
Marquardt’s method, 357, 392–393, 445, 595
Mass, conservation of, 20, 325–328
Mass balance, 20, 118
Mathcad, 48–49, 956–966
basics, 956–957
curve fitting, 567–569
double precision to represent numerical quantities, 75
entering text, 957–958
graphics, 962–964
linear algebraic equations, 319–321
mathematical functions and variables, 958–961
mathematical operations, 957–958
Minerr, 417
multigrid function, 924–925
multiline procedures/subprograms, 962
numerical integration/differentiation, 681–682
numerical methods function, 961
online help, 966
optimization, 357, 417–418
ordinary differential equations (ODEs), 818–820
partial differential equations (PDEs), 924–925
QuickSheets, 966
relax function, 924–925
resource center, 966
roots of equations, 200–203, 214–215
symbolic mathematics, 964–966
ToolTips, 966
Mathematical laws, 20
Mathematical models
defined, 11–12
overview of problem-solving process, 12
simple model, 11–18980 INDEX
conservation of momentum, 20
curve fitting, 579–580
equilibrium and minimum potential energy, 434–435
finite-element solution of series of springs, 937–940
linear algebraic equations, 334–336
numerical integration to compute work, 692–695
optimization, 421, 434–435
ordinary differential equations (ODEs), 839–842
partial differential equations (PDEs), 937–940
pipe friction, 214–217
roots of equations, 214–217
spring-mass systems, 334–336
swinging pendulum, 839–842
Method of false position. See False-position method
Method of lines, 890–891
Method of undetermined coefficients, 653–654
Method of weighted residuals (MWR), finite-element methods, 910–914
M-files (MATLAB), 44–48. See also MATLAB
Michaelis-Menten model, 220, 572–576, 846
Microsoft, Inc., 40
Midpoint (improved polygon) method, 719, 738–740, 744–746,
775–776, 855
Midtest loops, 34
Milne’s method, 788–789, 790–791, 855
Minimax criterion, 465, 595
MINPACK algorithms, 417
Mixed partial derivatives, 675
Model errors, 109
Modified Euler. See Midpoint (improved polygon) method
Modified false position, 141–142, 231
Modified fixed-point method, 231
Modified Newton-Raphson method, 168–170, 231, 232
Modified secant method, 162–163, 231
Modular programming, 38–40
advantages, 39
defined, 38
Momentum, conservation of, 20
Monte Carlo (MC) integration, 610, 642, 662–664, 708, 709
m surplus variables, 401–402
Müller’s method, 122, 184–188, 200–201, 231
Multidimensional interpolation, 529–531
Multidimensional unconstrained optimization, 356, 375–394
direct methods (nongradient), 357, 375, 376–380
gradient methods (descent/ascent), 357, 375, 380–393
MATLAB, 415–417
pattern searches/directions, 357, 379–380
Powell’s method, 379–380, 391, 445
random search method, 357, 376–378
univariate search method, 357, 378
Multimodal optimization, 360–361
Multiple-application trapezoidal rule, 618–621, 709, 710
Multiple integrals, 636–638
Mathematical programming. See Optimization
Mathsoft Inc., 48
MathWorks, Inc., The, 44
MATLAB, 28–29, 34–35, 948–955
assignment of values to variable names, 949–950
built-in functions, 952
computer implementation of iterative calculation, 65–66
curve fitting, 564–567
described, 44
double precision to represent numerical quantities, 75
Fourier analysis, 578–579
graphics, 952–953
linear algebraic equations, 317–319
linear regression, 470
mathematical operations, 950–953
M-files, 44–48, 105–107, 435
numerical differentiation errors, 105–107
numerical integration/differentiation, 675–681
optimization, 357, 371–373, 413–417, 435
ordinary differential equations (ODEs), 814–818, 827–829, 837–839
partial differential equations (PDEs), 923–924
polynomials, 953
roots of equations, 197–200, 214–215, 216–217
statistical analysis, 953–955
Matrix condition number, 245, 299–301
Matrix inverse, 242–243, 245, 292–296
calculating, 293–295
stimulus-response computations, 295–296
Matrix norms, 297–299
Matrix operations
banded matrices, 305–306
Cholesky decomposition, 307–309
components, 238–239
error analysis and system condition, 296–302
matrix, defined, 238
matrix condition number, 245, 299–301
matrix inverse, 242–243, 245, 292–296
matrix notation, 238–239
representing linear algebraic equations in matrix form, 244–245
rules, 240–244
special matrices, 305–309
symmetric matrices, 305
tridiagonal systems, 245, 306–307
Maximum likelihood principle, 467–468
Maximum-magnitude norms, 298, 299
Mean value, 450, 455, 537–538
confidence interval on the mean, 457–458
derivative mean-value theorem, 90
determining mean of discrete points, 603–604
spread around, 468
Mechanical/aerospace engineering
analysis of experimental data, 579–580INDEX 981
Multiple linear regression, 458, 482–485, 592–593, 594
Multiple roots, 126, 167–170
double roots, 167
modified Newton-Raphson method for multiple roots, 168–170, 231,
232
Newton-Raphson method, 167–168
secant method, 167–168
triple roots, 167
Multiplication, 76
estimated error bounds, 103
inner products, 81
matrix operations, 240–242
Multistep methods, 719, 767, 771–791, 855
defined, 771
higher-order methods, 787–791
integration formulas, 780–787
non-self-starting Heun, 719, 771–779, 855, 857
step-size control, 780
N
Naive Gauss elimination, 245, 256–262
back substitution, 256, 258–260, 261–262
forward elimination of unknowns, 256–258, 259
operation counting, 260–262
n-dimensional vector, 356
Nelder-Mead method, 415
Newmann boundary condition, 799–800, 873–876
Newton-Cotes integration formulas, 608–610, 612–641, 642–643, 740
Boole’s rule, 631, 632, 650
closed formulas, 783–785
comparisons, 708–709
defined, 612
higher-order, 631–633, 646–647, 708–709
open formulas, 782–785
ordinary differential equations (ODEs), 781, 782–787
Simpson’s 1/3 rule, 609–610, 624–629, 631–633, 634–636, 688–689,
693, 694, 708–709, 710
Simpson’s 3/8 rule, 609–610, 629–633, 634–636, 693, 694, 708–709,
710
trapezoidal rule, 609–610, 614–624, 633–636, 645–646, 688–689,
694, 708, 709, 710, 738
Newton-Gregory forward formula, 516
Newtonian fluid, 583
Newton-Raphson method, 122, 152–158, 208–209, 215–216, 370
algorithm, 156–158, 233
error estimates, 153–155
graphical method, 152, 157, 233
modified method for multiple roots, 168–170, 231, 232
multiple roots, 167–170
Newton-Raphson formula, 153
nonlinear equations, 172–174
pitfalls, 155–156
roots of polynomials, 181, 184, 189
slowly converging function, 155–156
Taylor series derivation, 153–154
Taylor series expansion, 276
termination criteria, 153–155
Newton’s divided-difference interpolating polynomials, 460, 496,
497–508, 515, 516, 592, 593, 594
algorithm, 505–508
defined, 502
derivation of Lagrange interpolating polynomial from, 508–509
error estimation, 503–505
general form, 501–503
quadratic interpolation, 499–501
Newton’s formula, 516
Newton’s law of cooling, 24, 704
Newton’s laws of motion, 118, 839–840
second law of motion, 11–18, 57, 119, 334, 714, 802
Newton’s method of optimization, 357, 370–371, 385, 391–392, 445–446
Nodal lines/planes, 904
Nonbasic variables, 402
Nonbinding constraints, 399
Nonideal versus idealized laws, 21
Nonlinear constrained optimization, 357, 406
Excel, 409–413
Mathcad, 418
Nonlinear equations
defined, 170
fixed-point iteration, 171–172
linear equations vs., 21, 170
Newton-Raphson method, 172–174
roots of equations, 122, 170–174
shooting method for boundary-value problems, 796–798
systems of equations, 122, 235–236, 275–277
Nonlinear programming optimization, 356
Nonlinear regression, 460, 475–476, 489–492, 564, 592
Non-self-starting Heun, 719, 771–779, 855, 857
Normal distribution, 452
Normalization, 70
Normalized standard deviate, 660–662
Norms
defined, 297
matrix, 297–299
vector, 297–299
“Not-a-knot” condition, 528–529
n structural variables, 401–402
nth finite divided difference, 501–502
Number systems, 67. See also specific number systems
Numerical differentiation, 95–99, 116, 667–684. See also Optimization
backward difference approximation, 95, 96, 97, 669
centered difference approximation, 96, 97, 98, 669
with computer software, 675–682
control of numerical errors, 107–108982 INDEX
Simpson’s 1/3 rule, 609–610, 624–633, 634–636, 688–689, 693, 694,
708–709, 710
Simpson’s 3/8 rule, 609–610, 629–633, 634–636, 693, 694, 708–709, 710
terminology, 597–598
trapezoidal rule, 609–610, 614–624, 632, 633–634, 645–646, 652,
688–689, 694, 708, 709, 710, 738
Numerical methods of problem solving, 112–115, 113–114
falling parachutist problem, 17–18
nature of, 15–16
Numerical Recipe library, 49
Numerical stability, 102–103
Nyquist frequency, 552
O
Objective function optimization, 353, 355, 356
Octal (base-8) number system, 67
ODEs. See Ordinary differential equations (ODEs)
Ohm’s law, 332
One-dimensional unconstrained optimization, 356, 357, 360–374
Brent’s root-location method, 357, 361, 371–373, 445
golden-section search, 357, 361–368, 432–434, 445
MATLAB, 414–415
multimodal, 360–361
Newton’s method, 357, 370–371, 385, 391–392, 445–446
parabolic interpolation, 357, 368–370, 445
One-point iteration, 122. See also Fixed-point iteration
One-sided interval, 453
One-step methods, 717–719, 721–766, 855
Open methods, 121–122, 146–176, 371
Brent’s root-location method, 122, 163–167, 231, 232
defined, 146–147
fixed-point iteration, 147–152, 231
graphical method, 146
multiple roots, 167–170
Newton-Raphson method, 122, 152–158, 208–209, 215–216, 232,
233, 370
secant method, 122, 157–163, 233
simple fixed-point iteration, 147–152
systems of nonlinear equations, 170–174
Optimal steepest ascent, 389–391, 595
Optimization, 350–446
additional references, 446
Brent’s root-location method, 357, 361, 371–373, 445
case studies, 421–444
computer methods, 357, 407–418, 424–425
defined, 350
engineering applications, 351–355, 357, 421–444
goals/objectives, 358–359
golden-section search, 357, 361–368, 432–434, 445
gradient methods. See Gradient methods of optimization
history, 351
linear programming, 351, 356, 357, 395–406
Numerical differentiation—Cont.
data with errors, 673–674
derivatives of unequally spaced data, 672–673
differentiate, defined, 596
engineering applications, 602–603, 675–682
error analysis, 105–108, 673–674
finite-divided-difference approximations, 16–17, 95–99
first derivative, 596, 668–669
forward difference approximations, 95, 96, 97, 668
goals/objectives, 610–611
high-accuracy differentiation formulas, 610, 667–670
mathematical background, 95–99, 606–608
noncomputer methods, 599–600, 601–602
ordinary differential equations. See Ordinary differential equations (ODEs)
partial derivatives, 596–597, 674–675
partial differential equations. See Partial differential equations (PDEs)
polynomials, 180–181
Richardson extrapolation, 610, 642, 644–647, 649–650, 670–672
round-off errors, 105–107
scope/preview, 608–610
second derivative, 596, 667–669
terminology, 596–597
Numerical integration, 642–666
Adams formula, 783–787, 789, 790–791, 855, 856, 857
adaptive integration, 642
adaptive quadrature, 610, 649–651
advanced methods and additional references, 709
Boole’s rule, 631, 632, 650
calculation of integrals, 603–606
case studies, 685–707
closed forms, 608–610, 613–614, 631–633, 783–785
comparisons, 708–709
with computer software, 675–682
data with errors, 673–674
engineering applications, 603–606, 675–682, 685–707
fundamental theorem, 607
Gauss quadrature, 610, 642, 651–659, 692, 708, 709, 710
goals/objectives, 610–611
important relationships and formulas, 709, 710
improper integrals, 610, 642, 659–662
integrate, defined, 597
integration with unequal segments, 610, 633–636
mathematical background, 606–608
Monte Carlo (MC) integration, 610, 642, 662–664, 708, 709
multiple integrals, 636–638
Newton-Cotes formulas, 608–610, 612–641, 642–643, 708–709, 740,
781, 782–787
noncomputer methods, 600–602
open forms, 610, 613–614, 636, 782–785
Richardson extrapolation, 610, 642, 644–647, 649–650, 670–672
Romberg integration, 610, 642, 643–649, 691, 708, 709, 710
scope/preview, 608–610INDEX 983
mathematical background, 355–357
multidimensional unconstrained, 356, 357, 375–394
Newton’s method, 357, 370–371, 385, 391–392, 445–446
noncomputer methods, 351
nonlinear constrained optimization, 357, 406, 409–413, 418
one-dimensional unconstrained, 356, 357, 360–374
parabolic interpolation, 357, 368–370, 445
problem classification, 355–357
random search method, 357, 376–378
scope/preview, 357–358
Order of polynomials, 120
Ordinary differential equations (ODEs), 177–180, 711–857
advanced methods and additional references, 856–857
algorithms, 730–733, 740, 741, 750–751, 754–756, 761–763
boundary-value problems, 717, 719, 794–801, 804–807, 856
case studies, 823–854
components, 711
computer programming and software, 719, 813–820
defined, 711
eigenvalue problems, 719, 801–820, 856
engineering applications, 713–715, 719, 823–854
Euler’s method, 719, 722–741, 835–836, 841–842, 855–857
explicit solution technique, 769–771
falling parachutist problem, 714, 717, 721–722
finite-difference methods, 719, 798–801
first-order equations, 711–712, 727
fourth-order Adams, 785, 786, 789, 790–791, 855–857
fourth-order RK, 747–749, 753–754, 755–756, 758, 762–763, 835,
841–842, 855–857
goals/objectives, 719–720
Heun’s method, 719, 734–738, 740, 744–746, 771–779, 855, 857
higher-order equations, 711–712, 733, 787–791
implicit solution technique, 719, 767–771
important relationships and formulas, 856, 857
initial-value problems, 717, 793
mathematical background, 715–717
midpoint (improved polygon) method, 719, 738–740, 744–746,
775–776, 855
Milne’s method, 788–789, 790–791, 855
multistep methods, 719, 767, 771–791, 855
Newton-Cotes integration formulas, 782
noncomputer methods, 712–713
one-step methods, 717–719, 721–766, 855
power methods, 719, 809–812
Ralston’s method, 744–746, 747, 855, 857
Runge-Kutta (RK) methods, 719, 741–751, 855, 856, 857
scope/preview, 717–719
second-order equations, 711–712, 742–746
shooting method, 719, 795–798
stiff systems, 719, 767–771, 816–817, 818, 856
systems of equations, 719, 751–756
third-order RK, 746–747
Orthogonal, 383
Orthogonal polynomials, 593, 594–595
Overconstrained optimization, 356
Overdamped case, 179
Overdetermined equations, 349
Overflow error, 71–72
Overrelaxation, 314
P
Parabolic interpolation optimization, 357, 368–370, 445
Parabolic partial differential equations (PDEs), 886–902
alternating-direction implicit (ADI) method, 862, 891–895, 898–901,
944, 945
Crank-Nicolson technique, 862, 895–898, 944, 945
explicit methods, 887–892, 898, 944
finite-difference methods, 860–861, 862, 886–902, 944, 945
heat-conduction equation, 860–861, 862, 886–887
implicit methods, 862, 891–895, 898–901, 944, 945
one-dimensional, 897–898, 944
two-dimensional, 898–901, 944
Parameter estimation, 828
Parameters, 11–12, 118
distributed-parameter system, 929
estimation, 828
lumped-parameter systems, 928
sinusoidal function, 537–539
Parametric Technology Corporation (PTC), 48
Partial derivatives, 596–597, 674–675
Partial differential equations (PDEs), 858–945
advanced methods and additional references, 945
boundary conditions, 868–871, 873–879
case studies, 928–943
characteristics, 858–859
computer solutions, 864, 882–883, 921–925
defined, 711, 858
elliptic equations, 859–860, 862, 865–885, 944, 945
engineering applications, 859–861, 862–863, 928–943
finite-difference methods, 859–861, 862, 886–902, 944, 945
finite-element methods, 862, 903–927, 944
goals/objectives, 863–864
higher-order temporal approximations, 890–891
hyperbolic equations, 861
important relationships and formulas, 944–945
order of, 858
parabolic equations, 860–861, 862, 886–902, 944, 945
precomputer methods, 861–862
scope/preview, 862–863
Partial pivoting, 245, 268–272, 273, 347
Pattern searches/directions, 357, 379–380
Penalty functions, 406
Period, sinusoidal function, 536–537
Phase-plane representation, 831–834984 INDEX
Phase shift, 538
Pivoting, 268–272, 347
complete, 268
division by zero, 262
effect of scaling, 270–272
partial, 245, 268–272, 273, 347
pivot coefficient/element, 257–258
Place value, 67
Point-slope method. See Euler’s method
Poisson equation, 867, 908–917
Polynomial regression, 458, 478–482, 594
algorithm, 481–482
fit of second-order polynomial, 479–481
Polynomials
computing with, 180–183
defined, 120
deflation, 181–183
eigenvalue problems, 178–179, 719, 807–809
engineering applications, 177–180
evaluation and differentiation, 180–181
factored form, 181
interpolation, 496–534
Lagrange, 164
Lagrange interpolating, 460, 496, 508–513, 515, 592–594
Newton’s divided-difference, 460, 496, 497–508, 508–509, 515, 516,
592, 593, 594
order, 120
ordinary differential equations (ODEs), 177–180, 719
orthogonal, 593, 594–595
polynomial approximation, 85–87
regression, 458, 478–482, 594
roots. See Roots of polynomials
synthetic division, 181–182
Polyroots, 201, 202
Populations, estimating properties of, 452–453
Positional notation, 67
Positive definite matrix, 309
Postprocessing, finite-element methods, 907–908, 921, 940
Posttest loops, 33–34
Potential energy, 434–435
Potentiometers, 430
Powell’s method of optimization, 379–380, 391, 445
Power equations, linear regression of, 476–478
Power methods, 719, 809–812
defined, 809
determining largest eigenvalue, 809–811
determining smallest eigenvalue, 811–812
Power spectrum, 560–561
Precision, 60–61, 114
Predator-prey models, 830–834
Predictor-corrector approach, 734–736, 771–779
Predictor equation, 734–735, 771
Predictor modifier, 777–779
Pretest loops, 33–34
Principal/main diagonal of matrix, 239
Product, matrix operations, 240
Programming and software. See Computer programming and software;
Pseudocode algorithms
Propagated truncation error, 725
Propagation problems, 860–861. See also Hyperbolic partial differential
equations (PDEs); Parabolic partial differential equations (PDEs)
Proportionality, 296
Pseudocode algorithms, 31–38
adaptive quadrature, 650–651
Bairstow’s method, 192–193
bisection, 134, 135
Brent’s root-location method, 165–167, 372–373
Cholesky decomposition, 309
computing with polynomials, 181–183
curve fitting, 460, 577
defined, 31
discrete Fourier transform (DFT), 551–554, 553–554
Euler’s method, 730–733
Excel VBA vs., 42
fast Fourier transform (FFT), 557–560
fixed-point iteration, 151–152
forward elimination, 258
function that solves differential equations, 39
Gauss-Jordan method, 279
Gauss-Seidel (Liebmann) method, 314–315
for generic iterative calculation, 64–66
golden-section-search optimization, 366, 367, 432–434
Lagrange interpolation, 511
linear regression, 471, 484
logical representation, 31–38
LU decomposition, 287, 289–290, 291–292
MATLAB vs., 46
matrix inverse, 294–295
modified false-position method, 141
Müller’s method, 187–188
multiple linear regression, 484
Newton’s divided-difference interpolating polynomials, 505–508
ordinary differential equations (ODEs), 730–733, 750–751, 754–756,
761–763
partial pivoting, 270, 273
polynomial regression, 482
Romberg integration, 647–649
roots of quadratic equation, 35–38, 78–79
Runge-Kutta (RK) method, 761–762
Simpson’s rules, 631–633, 635–636
Thomas algorithm, 306–307
trapezoidal rule, 621–624
Q
QR factorization, 488
QR method (Francis), 813
Quadratic equation, algorithm for roots, 35–38INDEX 985
open methods. See Open methods
optimization and, 350
polynomials. See Roots of polynomials
scope/preview, 120–122
secant method, 122, 157–163, 167–168, 231, 233
as zeros of equation, 117–118
Roots of polynomials, 177–205. See also Roots of equations
Bairstow’s method, 122, 188–193, 231
Brent’s method, 201
characteristic equation, 178–179
computer methods, 193–203
conventional methods, 183–184
critically damped case, 179
discriminant, 179
eigenvalue problems, 178–179
engineering applications, 177–180
general solution, 178, 180
Jenkins-Traub method, 193, 234
Laguerre’s method, 193, 201, 234
mathematical background, 177–180
Müller’s method, 122, 184–188, 200–201, 231
Newton-Raphson method, 181, 184, 189
other methods, 193
overdamped case, 179
Ridder method, 201
secant method, 200–201
underdamped case, 179
Rosin-Rommler-Bennet (RRB) equation, 706
Rounding, 73
Round-off errors, 67–81
adding a large and a small numbers, 77–78
arithmetic manipulation of computer numbers, 75–81
common arithmetic operations, 75–76
computer representation of numbers, 67–75
defined, 58, 61
Euler’s method, 725
extended precision, 74–75
Gauss elimination, 263
integer representation, 67–69
iterative refinement, 301–302
large computations, 76–77
linear algebraic equations, 263
number systems, 67
numerical differentiation, 105–107
polynomial deflation, 182–183
significant digits and, 58–59, 268
smearing, 79–81
subtractive cancellation, 78–79
total numerical error, 104–108
Row, defined, 238
Row-sum norms, 298, 299
Row vectors, 238
Runge-Kutta Fehlberg method, 759–760, 761–762
Quadratic interpolation, 499–501
Quadratic programming, 356
Quadratic splines, 521–523
Quadrature methods, 601
Quantizing errors, 72–74, 77
Quasi-Newton methods of optimization, 357, 393, 406, 446
Quotient difference (QD) algorithm, 233–234
R
Ralston’s method, 744–746, 747, 855, 857
Random search method of optimization, 357, 376–378
Rate equations, 711
Razdow, Allen, 48
Reaction kinetics, 828
Redlich-Kwong equation, 219
Regression. See Linear regression; Polynomial regression
Relative error, 102
Relaxation, 314, 924–925
Remainder, 116
Taylor series, 89–91, 116
Repetition, in logical representation, 32–35
Residual error, 463, 467–470
Respiratory quotient, 699
Response, 36
Richardson extrapolation, 610, 642, 644–647, 649–650, 670–672
Ridder method, root of polynomials, 201
Romberg integration, 610, 642, 643–649, 691, 708, 709, 710
Root-mean-square current, 689–692
Root polishing, 183
Roots of equations, 117–234
advanced methods and additional references, 232–234
analytical/direct method, 117, 231
bisection method, 120–121, 128–136, 231, 233, 361–362
bracketing methods. See Bracketing methods
Brent’s method, 163–167, 231, 232
case studies, 122, 206–230
computer methods, 126–128, 214–215
engineering applications, 118–119, 122, 177–180, 206–230
false-position method, 120–121, 136–142, 231, 233
fixed-point iteration, 147–152, 231
goals/objectives, 122
graphical methods, 117, 124–128, 146, 148–151, 158, 160–161, 231
important relationships and formulas, 232, 233
incremental searches/determined incremental guesses, 128, 142
Jenkins-Traub method, 234
Laguerre’s method, 234
mathematical background, 119–120, 180–183
multiple roots, 126, 167–170
nature of “roots,” 117
Newton-Raphson method, 122, 152–158, 167–170, 208–209, 231, 232,
233, 370
noncomputer methods, 117–118, 231
nonlinear equations, 122, 170–174986 INDEX
implementation, 403–406
slack variables, 401
Simpson’s 1/3 rule, 609–610, 624–633, 688–689, 693, 694, 708–709, 710
algorithms, 631–633
derivation and error estimate, 625
multiple-application, 627–629, 650
single-application, 626–627
with unequal segments, 634–636
Simpson’s 3/8 rule, 609–610, 629–633, 693, 694, 708–709, 710
algorithms, 631–633
with unequal segments, 634–636
Simultaneous overrelaxation, 314
Single-value decomposition, 595
Single-variable optimization, 361
Singular systems, 251, 267–268
Sinusoidal functions, 536–542
least-squares fit of sinusoid, 539–542
parameters, 537–539
Slack variables, 401
Smearing, 79–81
Software. See Computer programming and software
Special matrices, 305–309
Spectral norms, 298
Spline functions, 517, 595
Spline interpolation, 460, 517–529
cubic splines, 517, 523–528, 568–569, 592–594, 709
end conditions, 527, 528–529
engineering applications, 576–577
linear splines, 517–521
quadratic splines, 521–523
splines, defined, 517
Spread around the mean, 468
Spread around the regression line, 468
Spreadsheets. See Excel
Square matrices, 239
Stability
defined, 890
error propagation, 102–103
of multistep methods, 790–791, 856
of numerical methods of problem solving, 113–114
Standard atmosphere, 55
Standard deviation, 450
Standard error of the estimate, 468
Standard normal estimate, 454–455
Standard normal random variable, 455
Start, 34–35
Statistical inference, 453, 455
Statistics, 449–458
descriptive, 109, 449–452
estimation of confidence interval, 452–458, 487–488
inferential, 453, 455
least-squares theory, 486–489
Runge-Kutta (RK) methods, 719, 741–751, 855–857
adaptive, 719, 756–763, 855
adaptive step-size control, 757–758, 760–761
algorithms, 751, 761–763
Cash-Karp RK method, 759–760, 761–762
comparison, 749–750
embedded, 759–760
fourth-order, 747–749, 753–754, 755–756, 758, 762–763, 835,
841–842, 855–857
higher-order, 749–751
Runge-Kutta Fehlberg method, 759–760, 761–762
second-order, 742–746
systems of equations, 752–754
third-order, 746–747
S
Saddle, 384
Samples, estimating properties of, 452–453
Sande-Tukey algorithm, 554–558
Scaling
effect of scale on determinant in ill-conditioned systems, 265–267
effect on pivoting and round-off, 270–272
Secant method, 122, 157–163
algorithm, 162, 233
false-position method vs., 159–161
graphical method, 158, 160–161, 163, 233
modified, 162–163, 231
multiple roots, 167–168
root of polynomials, 200–201
Second Adams-Bashforth formula, 784–785
Second Adams-Moulton formula, 785–786
Second derivative, 596, 667–669
Second finite divided difference, 501
Second forward finite divided difference, 98–99
Second-order approximation, 84–85, 88
Second-order closed Adams formula, 785–786
Second-order equations, 711–712, 742–746
Second-order open Adams formula, 784–785
Selection, in logical representation, 31–32
Sensitivity analysis, 21, 43–44
Sentinel variables, 314–315
Sequence, in logical representation, 31
Shadow price, 429
Shooting method, 719, 795–798
Sigmoid (S-shaped), 572–573
Signed magnitude method, 67–69
Significance level, 454
Significand, 69–71
Significant figures/digits, 58–59, 268
Simple statistics, 449–452
Simplex method, 351, 357, 401–406
algebraic solution, 402–403INDEX 987
error propagation, 99–103
to estimate error for Euler’s method, 725–727, 733
to estimate truncation errors, 91–99, 725–727, 733
expansion of Newton-Raphson method, 276
expansion of Newton’s divided-difference interpolating polynomials,
503–504
expansions, 85–91, 116, 276, 503–504
finite-difference approximations, 95–99
finite-divided-difference approximations, 95–99, 667–670
first-order approximation, 84, 86, 88, 95–98
first theorem of mean for integrals, 84
forward difference approximations, 95, 96, 97, 668
infinite number of derivatives, 88–89
linear trend, 84–85
nonlinearity, 91–95, 489
numerical differentiation, 95–99
remainder, 89–91, 116
second-order approximation, 84–85, 88
second theorem of mean for integrals, 84
step size, 91–95
Taylor’s theorem/formula, 84
zero-order approximation, 83, 86, 88, 89
t distribution, 456
Terminal velocity, 15, 18
Thermal conductivity, 866
Thermal diffusivity, 866
Thermocline, 576
Third derivative, 668–669
Third-order methods, 746–747
Thomas algorithm, 306–307
Time domains, 545–549
Time plane, 545–546
Time-variable (transient) computation, 18
Topography, 357
Total numerical error, 104–108
Total sum of the squares, 468–469
Total truncation error, 725
Total value, 597–598
Trace, matrix operations, 243
Transcendental function, 120
Transient computation, 18
Transpose, matrix operations, 243
Trapezoidal rule, 609–610, 614–624, 632, 633–636, 688–689, 694, 708,
710, 738
algorithms, 621–624
defined, 614
error/error correction, 616–617, 645–646, 771–772
multiple-application, 618–621, 709, 710
single-application, 617–618
with unequal segments, 633–634
Trend analysis, 449
Tridiagonal systems, 245, 306–307
maximum likelihood principle, 467–468
normal distribution, 452
Steady-state computation, 19, 325–328. See also Elliptic partial
differential equations (PDEs)
Steepest ascent/descent optimization, 357, 386–391
optimal steepest ascent, 389–391, 595
using gradient to evaluate, 382–383
Stefan-Boltzmann law, 225
Step-halving method, 758
Stiffness matrix, 336
finite-element methods, 907, 939
Stiff ordinary differential equations (ODEs), 719, 767–771, 816–817,
818, 856
Euler’s method, 768–771
stiff system, defined, 767
Stimulus-response computations, 295–296
Stokes law, 229
Stopping criterion, 64–65, 116
Strange attractors, 834
Streeter-Phelps model, 439
Strip method, 601, 613
Structured programming, 29–38
defined, 30
flowcharts, 30–35
logical representation, 31–38
pseudocode, 31–38. See also Pseudocode algorithms
Subroutines, modular programming, 38–39
Subtraction, 76
estimated error bounds, 103
matrix operations, 240
subtractive cancellation, 78–79
Successive overrelaxation, 314
Successive substitution. See Fixed-point iteration
Summation, 597–598
Superposition, 296
Swamee-Jain equation, 214
Symmetric matrices, 305
Symmetric matrix, 239
Synthetic division, 181–182
Systems of equations
nonlinear equations, 122, 235–236, 275–277
ordinary differential equations (ODEs), 719, 751–756
T
Tableau, 403–406
Taylor series, 83–99. See also Finite-difference methods
approximation of polynomial, 85–87
backward difference approximation, 95, 96, 97, 98, 669
centered finite divided-difference approximation, 96, 97, 98, 669
defined, 83
derivation of Newton-Raphson method, 153–154
derivative mean-value theorem, 90988 INDEX
V
Van der Waals equation, 207–208
Variable metric methods of optimization, 357, 393
Variables
basic, 402
dependent, 11–12, 118, 711
design, 355
distributed-variable systems, 236, 237, 316
entering, 403–404
independent, 11–12, 118, 711
leaving, 403–404
lumped-variable systems, 236, 237, 316
single-variable optimization, 361
slack, 401
standard normal random, 455
Variable step size, 780
Variance, 450, 454–455
Vector norms, 297–299
Videoangiography, 698
Visual Basic Editor (VBE), 40–44
Voltage balance, 20
Volume-integral approach, 879–882
Volume integrals, 605
W
Waste minimization, 437
Wave equation, 859, 861
Well-conditioned systems, 103, 263
WHILE structure, 34
Wolf sunspot number, 578–579
Z
Zero, division by, 262
Zero-order approximation, 83, 86, 88, 89
Triple roots, 167
True derivative approximation, 96
True error, 61, 116
True fractional relative error, 61–62
True mean, 453–455
True percent relative error, 61–62, 66, 116
Truncation errors. See also Discretization, finite-element methods
defined, 58, 61, 83
Euler’s method, 724, 725–727, 733
numerical differentiation, 105–107
per-step, 776–777
significant digits and, 58–59, 268
Taylor series to estimate, 91–99, 725–727, 733. See also Taylor series
total numerical error, 104–108
types, 725
Twiddle factors, 557
Two-dimensional interpolation, 529–531
2’s complement, 69
Two-sided interval, 453–454
U
Uncertainty, 60–61, 109, 674
Unconditionally stable, 769
Unconstrained optimization, 356, 357
multidimensional. See Multidimensional unconstrained optimization
one-dimensional. See One-dimensional unconstrained optimization
Underdamped case, 179
Underdetermined equations, 348–349, 401
Underflow “hole,” 72
Underrelaxation, 314
Underspecified equations, 401
Uniform-matrix norms, 298, 299
Uniform-vector norms, 298, 299
Unimodal optimization, 361–362
Univariate search method, 357, 378
Upper triangular matrix, 239
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