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 |  موضوع: كتاب Applied Numerical Methods Using Matlab الأربعاء 21 يونيو 2023, 10:07 am | |
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أخواني في الله
أحضرت لكم كتاب
Applied Numerical Methods Using Matlab
R. V. DukkipAti, PhD
و المحتوى كما يلي :
Preface xiii
CHAPTER 1: NUMERICAL COMPUTATIONS 1
1.1 Taylor’s Theorem 1
1.2 Number Representation 4
1.3 Error Considerations 11
1.3.1 Absolute and Relative Errors 12
1.3.2 Inherent Errors 14
1.3.3 Round-off Errors 14
1.3.4 Truncation Errors 19
1.3.5 Machine Epsilon 21
1.3.6 Error Propagation 22
1.4 Error Estimation 22
1.5 General Error Formula 24
1.5.1 Function Approximation 24
1.5.2 Stability and Condition 25
1.5.3 Uncertainty in Data or Noise 27
1.6 Sequences 28
1.6.1 Linear Convergence 28
1.6.2 Quadratic Convergence 29
1.6.3 Aitken’s Acceleration Formula 30
1.7 Summary 32
Exercises 33
Contentsvi • Contents
CHAPTER 2: LINEAR SYSTEM OF EQUATIONS 39
2.1 Introduction 39
2.2 Methods of Solution 40
2.3 The Inverse of a Matrix 41
2.4 Matrix Inversion Method 44
2.4.1 Augmented Matrix 50
2.5 Gauss Elimination Method 50
2.5.1 MATLAB Program for the Gauss Elimination Method 52
2.6 Gauss-Jordan Method 58
2.6.1 MATLAB Program for the Gauss Jordan Method 59
2.7 Cholesky’s Triangularization Method 68
2.8 Crout’s Method 79
2.8.1 MATLAB Program for Crout’s Method 81
2.9 Thomas Algorithm for Tridiagonal System 88
2.9.1 MATLAB Program for the Thomas Method for
Tridiagonal Systems 89
2.10 Jacobi’s Iteration Method 94
2.10.1 MATLAB Program for the Jacobi Iteration Method 95
2.11 Gauss-Seidel Iteration Method 104
2.11.1 MATLAB Program for the Gauss Seidel Method 106
2.12 Symmetric Matrix Eigenvalue Problems 113
2.12.1 The Jacobi Method 114
2.12.2 MATLAB Function for the Jacobi Method 120
2.12.3 Householder Reduction to Tridiagonal Form 133
2.12.4 Gerschgorin’s Circle Theorem 137
2.12.5 Sturm Sequence 139
2.12.6 QR Method 140
2.12.7 Power Method 157
2.12.8 Inverse Power Method 161
2.12 Summary 163
Exercises 164Contents • vii
CHAPTER 3: SOLUTION OF ALGEBRAIC AND
TRANSCENDENTAL EQUATIONS 177
3.1 Introduction 177
3.2 Bisection Method 178
3.2.1 Error Bounds 180
3.3 Method of False Position 186
3.3.1 MATLAB Program for the False Position Method 188
3.4 Newton-Raphson Method 194
3.4.1 Convergence of the Newton-Raphson Method 196
3.4.2 Rate of Convergence of the Newton-Raphson Method 197
3.4.3 MATLAB Program for the Newton Raphson Method 198
3.4.4 Modified Newton-Raphson Method 203
3.4.5 Rate of Convergence of Modified Newton-Raphson Method 204
3.5 Successive Approximation Method 206
3.5.1 Error Estimate in the Successive Approximation Method 207
3.6 Secant Method 211
3.6.1 Convergence of the Secant Method 212
3.6.2 MATLAB Program to Search for a Root of the
Function f(x) in the Interval (a,b) 214
3.6.3 MATLAB Program for Secant Method 214
3.7 Muller’s Method 219
3.7.1 MATLAB Program for Muller’s Method 222
3.8 Chebyshev Method 225
3.9 Aitken’s Δ2 Method 226
3.10 Brent’s Method 229
3.10.1 MATLAB Program for Brent’s Method 231
3.11 Newton Method for a System of Nonlinear Equations 236
3.12 Comparison of Iterative Methods 238
3.13 MATLAB Built-in Function: fzero 239
3.14 Summary 243
Exercises 244viii • Contents
CHAPTER 4: NUMERICAL DIFFERENTIATION 251
4.1 Introduction 251
4.2 Derivatives Based on Newton’s Forward Integration Formula 252
4.2.1 MATLAB Program for Derivatives Based on Newton’s
Forward Integration Formula—Equally Spaced Points 253
4.3 Derivatives Based on Newton’s Backward Interpolation Formula 260
4.4 Derivatives Based on Stirling’s Interpolation Formula 263
4.5 Maxima and Minima of a Tabulated Function 268
4.6 Cubic Spline Method 270
4.7 Richardson Extrapolation 271
4.8 Differentiation of Unequally Spaced Data 273
4.9 MATLAB Built-in Functions: diff and gradient 274
4.10 Summary 278
Exercises 278
CHAPTER 5: FINITE DIFFERENCES AND INTERPOLATION 285
5.1 Introduction 285
5.2 Finite Difference Operators 286
5.2.1 Forward Differences 287
5.2.2 Backward Differences 288
5.2.3 Central Differences 289
5.2.4 Error Propagation in a Difference Table 293
5.2.5 Properties of the Operator Δ 296
5.2.6 Difference Operators 298
5.2.7 Relation Among the Operators 299
5.2.8 Representation of a Polynomial using Factorial Notation 304
5.3 Interpolation with Equal Intervals 310
5.3.1 Missing Values 310
5.3.2 Newton’s Binomial Expansion Formula 310
5.3.3 Newton’s Forward Interpolation Formula 313
5.3.4 MATLAB M-file: Newtonint 314Contents • ix
5.3.5 Newton’s Backward Interpolation Formula 322
5.3.6 Error in the Interpolation Formula 328
5.4 Interpolation with Unequal Intervals 331
5.4.1 Lagrange’s Interpolating Polynomial for Equal Intervals 331
5.4.2 function yint = Lagrangeint (x,y,xx) 333
5.4.3 Lagrange’s Formula for Unequal Intervals 334
5.4.4 Hermite’s Interpolation Formula 338
5.4.5 Inverse Interpolation 340
5.4.6 Lagrange’s Formula for Inverse Interpolation 340
5.5 Central Difference Interpolation Formulae 342
5.5.1 Gauss’s Forward Interpolation Formula 343
5.5.2 Gauss Backward Interpolation Formula 346
5.5.3 Bessel’s Formula 348
5.5.4 Stirling’s Formula 351
5.5.5 Laplace-Everett’s Formula 352
5.5.6 Selection of an Interpolation Formula 355
5.6 Divided Differences 355
5.6.1 Newton’s Divided Difference Interpolation Formula 357
5.7 Cubic Spline Interpolation 360
5.8 Generalized Spline Method 366
5.8.1 Splines 366
5.8.2 Linear Splines 367
5.8.3 Quadratic Splines 368
5.8.4 Cubic Splines 371
5.8.5 End Conditions 376
5.8.6 MATLAB Built-in Function: spline 376
5.8.7 Multidimensional Interpolation 378
5.8.8 MATLAB Built-in Function: interpl 381
5.9 Summary 390
Exercises 391x • Contents
CHAPTER 6: CURVE FITTING, REGRESSION,
AND CORRELATION 405
Approximating Curves 406
Linear Regression 407
6.1 Linear Equation 407
6.2 Curve Fitting With a Linear Equation 408
6.3 Criteria for a Best Fit 410
6.4 Linear Least-Squares Regression 412
6.5 Linear Regression Analysis 414
6.5.1 MATLAB built-in function: polyfit 418
6.5.2 MATLAB built-in function: polyval 419
6.6 Interpretation of a and b 420
Assumptions in the Regression Model 421
6.7 Standard Deviation of Random Errors 421
6.8 Coefficient of Determination 422
6.9 Linear Correlation 425
Properties of the Linear Correlation Coefficient r 427
Explained and Unexplained Variation 428
6.10 Linearization of Nonlinear Relationships 429
6.11 Polynomial Regression 435
6.11.1 Polynomial Fit 436
6.11.2 MATLAB Built-in Functions for Polynomial Fit 438
6.12 Quantification of Error of Linear Regression 443
6.13 Multiple Linear Regression 445
6.14 Weighted Least-Squares Method 448
6.15 Orthogonal Polynomials and Least-Squares Approximation 449
6.16 Least-Squares Method for Continuous Data 449
6.17 Approximation Using Orthogonal Polynomials 451
6.18 Gram-Schmidt Orthogonalization Process 453
6.19 Fitting a Function Having a Specified Power 455
6.20 Fitting a Cubic Spring Model 455
6.21 Additional Example Problems and Solutions 456
6.22 Summary 468
Exercises 468Contents • xi
CHAPTER 7: NUMERICAL INTEGRATION 483
7.1 Introduction 483
7.1.1 Relative Error 484
7.2 Newton-Cotes Closed Quadrature Formula 485
7.3 Trapezoidal Rule 486
7.3.1 Error Estimate in Trapezoidal Rule 489
7.3.2 MATLAB Functions: trapz and cumtrapz 490
7.4 Simpson’s 1/3 Rule 493
7.4.1 Error Estimate in Simpson’s 1/3 Rule 495
7.4.2 MATLAB Program for Simpson’s Integration: simpsonint 496
7.4.3 MATLAB Built-in Functions: quad and quad1 497
7.5 Simpson’s 3/8 Rule 502
7.6 Boole’s and Weddle’s Rules 506
7.6.1 Boole’s Rule 506
7.6.2 Weddle’s Rule 507
7.7 Romberg’s Integration 512
7.7.1 Richardson’s Extrapolation 512
7.7.2 Romberg Integration Formula 513
7.7.3 MATLAB Program for Romberg Integration: Romberg 516
7.8 Gaussian Quadrature 523
7.8.1 Gaussian Integration Formulas 523
7.8.2 Orthogonal Polynomials 525
7.8.3 Gauss-Lagendre Quadrature 527
7.8.4 Gauss-Chebyshev Quadrature Method 533
7.8.5 Gauss-Laguerre Quadrature 535
7.8.6 Gauss-Hermite Quadrature 537
7.8.7 MATLAB Programs for Gaussian Quadrature:
gaussnodes and gaussquad 539
7.9 Double Integration 544
7.9.1 Trapezoidal Method 544
7.9.2 Simpson’s 1/3 Rule 548
7.9.3 MATLAB Built-in Function for Double Integration: dblquad 549
7.10 Summary 549
Exercises 550xii • Contents
CHAPTER 8: NUMERICAL SOLUTION OF ORDINARY
DIFFERENTIAL EQUATIONS 557
8.1 Introduction 557
8.2 One-Step Methods or Single-Step Methods 560
8.2.1 Picard’s Method of Successive Approximation 561
8.2.2 The Taylor’s Series Method 564
8.3 Step-by-Step Methods or Marching Methods 569
8.3.1 Euler’s Method 569
8.3.2 MATLAB Program for Euler’s Method: euler 571
8.3.3 Modified Euler’s Method 579
8.3.4 MATLAB Program for the Modified Euler’s
Method: modeuler 581
8.3.5 Runge-Kutta Methods 586
8.3.4 Predictor-Corrector Methods 604
8.4 MATLAB Functions for Ordinary Differential Equations:
ode45, ode23, ode113, ode15s, ode23s, ode23t, ode23tb 614
8.5 System of First-order Ordinary Differential Equations 626
8.6 Initial Value Problems 627
8.6.1 The Taylor Series Method 628
8.6.2 Picard’s Method 630
8.6.3 Second-Order Runge-Kutta Method 631
8.6.4 Fourth-Order Runge-Kutta Method 632
8.6.5 Euler’s Formula 634
8.6.6 Modified Euler’s Formula 635
8.6.7 Burlirsch-Stoer Method (Mid-Point Method) 635
8.6.8 The Runge-Kutta-Fehlberg Method 636
8.6.9 The Runge-Kutta-Butcher Method 637
8.7 Two-Point Boundary Value Problems 637
8.7.1 Finite Difference Method 638
8.7.2 Second-Order Differential Equations 639
8.7.3 The Shooting Method 641
8.8 Second-Order Initial Value Problem (IVP) 646Contents • xiii
8.9 Second-Order Boundary Value Problem (BVP) 648
8.10 MATLAB Built-in Functions 649
8.11 Summary 650
Exercises 650
CHAPTER 9: DIRECT NUMERICAL INTEGRATION METHODS 661
9.1 Introduction 661
9.2 Single Degree of Freedom System 661
9.2.1 Finite Difference Method 662
9.2.2 Central Difference Method 663
9.2.3 The Runge-Kutta Method 664
9.3 Multi-degree of Freedom Systems 666
9.4 Explicit Schemes 667
9.4.1 Central Difference Method 667
9.4.2 Two-Cycle Iteration with Trapezoidal Rule 670
9.4.3 Fourth-Order Runge-Kutta Method 671
9.5 Implicit Schemes 673
9.5.1 The Houbolt Method 673
9.5.2 Wilson Theta Method 676
9.5.3 The Newmark Beta Method 679
9.5.4 Park Stiffly Stable Method 682
9.6 Example Problems and Solutions Using MATLAB 684
9.7 Summary 747
Exercises 748
Additional Exercises 753
CHAPTER 10: MATLAB BASICS 757
10.1 Introduction 757
10.1.1 Starting and Quitting MATLAB 758
10.1.2 Display Windows 758
10.1.3 Entering Commands 758
10.1.4 MATLAB Expo 759
10.1.5 Abort 759
10.1.6 The Semicolon (;) 759xiv • Contents
10.1.7 Typing % 759
10.1.8 The clc Command 759
10.1.9 Help 759
10.1.10 Statements and Variables 759
10.2 Arithmetic Operations 760
10.3 Display Formats 760
10.4 Elementary Math Built-In Functions 761
10.5 Variable Names 764
10.6 Predefined Variables 764
10.7 Commands For Managing Variables 764
10.8 General Commands 765
10.9 Arrays 766
10.9.1 Row Vector 766
10.9.2 Column Vector 767
10.9.3 Matrix 767
10.9.4 Addressing Arrays 767
10.9.5 Adding Elements to a Vector or a Matrix 768
10.9.6 Deleting Elements 768
10.9.7 Built-in Functions 768
10.10 Operations with Arrays 770
10.10.1 Addition and Subtraction of Matrices 770
10.10.2 Dot Product 770
10.10.3 Array Multiplication 770
10.10.4 Array Division 771
10.10.5 Identity Matrix 771
10.10.6 Inverse of a Matrix 771
10.10.7 Transpose 771
10.10.8 Determinant 771
10.10.9 Array Division 771
10.10.10 Left Division 772
10.10.11 Right Division 772
10.11 Element-By-Element Operations 772
10.11.1 Built-In Functions For Arrays 773Contents • xv
10.12 Random Numbers Generation 774
10.12.1 The Random Command 775
10.13 Polynomials 775
10.14 System of Linear Equations 778
10.14.1 Matrix Division 778
10.14.2 Matrix Inverse 778
10.15 Script Files 784
10.15.1 Creating and Saving a Script File 784
10.15.2 Running a Script File 785
10.15.3 Input to a Script File 785
10.15.4 Output Commands 786
10.16 Programming in Matlab 787
10.16.1 Relational and Logical Operators 787
10.16.2 Order of Precedence 788
10.16.3 Built-in Logical Functions 788
10.16.4 Conditional Statements 789
10.16.5 Nested if Statements 790
10.16.6 else AND elseif Clauses 790
10.16.7 MATLAB while Structures 790
10.17 Graphics 792
10.17.1 Basic 2-D Plots 792
10.17.2 Specialized 2-D Plots 793
10.17.3 3-D Plots 795
10.17.4 Saving and Printing Graphs 802
10.18 Input/Output In Matlab 802
10.18.1 The fopen Statement 802
10.19 Symbolic Mathematics 803
10.19.1 Symbolic Expressions 804
10.19.2 Solution to Differential Equations 806
10.19.3 Calculus 807
10.23 Summary 843
References 844
Exercises 845xvi • Contents
CHAPTER 11: OPTIMIZATION 857
11.1 Introduction 857
11.2 Unconstrained Minimization of Functions 859
11.3 Minimization with Constraints Using Lagrange Multipliers 860
11.4 Numerical Optimization 864
11.4.1 Optimization Involving Single Variables 865
11.4.2 Local and Global Optima 873
11.4.3 Bracketing 875
11.4.4 Golden-Section Search 875
11.4.5 MATLAB Program for Bracketing Method 878
11.4.6 MATLAB Program for Golden-Section Search Method 879
11.5 Multidimensional Optimization 882
11.6 Gradient Methods 885
11.7 Newton’s Method 887
11.7.1 MATLAB Program for Newton’s Method 889
11.8 Methods Based on the Concept of Quadratic Convergence 893
11.8.1 Conjugate Directions for a Quadratic Function 893
11.9 Powell’s Method 901
11.9.1 MATLAB Program for Powell’s Optimization Method 901
11.10 Fletcher-Reeves Method 906
11.10.1 MATLAB Program for Fletcher-Reeves
Optimization Method 909
11.11 The Hooks and Jeeves Method 917
11.12 Method of Successive Linear Approximation 924
11.13 Interior Penalty Function Method 925
11.14 MATLAB Built-in Functions 931
11.14.1 MATLAB Function: fminbnd 931
11.14.2 MATLAB Function: fminsearch 935
11.15 Additional Example Problems and Solutions 940
11.16 Summary 949
References 950
Exercises 950Contents • xvii
CHAPTER 12: PARTIAL DIFFERENTIAL EQUATIONS 959
12.1 Introduction 960
12.2 Classification of Linear Second-Order Partial Differential Equation 961
12.3 Types of Problems 965
12.4 Finite-Difference Approximation to Partial Derivatives 967
12.5 Physical Phenomena 968
12.5.1 Laplace’s Equation 968
12.5.2 Heat Equation 969
12.5.3 Wave Equation 970
12.5.4 Equation Classification 971
12.6 Elliptic Equations 972
12.6.1 Central Difference Method 972
12.6.2 Boundary Conditions 975
12.6.3 Iterative Solution Methods 977
12.6.4 The Jacobi Method 985
12.6.5 Gauss-Seidel Method 986
12.6.6 Successive Over-Relaxation or S.O.R. Method 986
12.7 One-Dimensional Parabolic Equations 988
12.7.1 Explicit Forward Euler Method 989
12.7.2 Implicit Backward Euler Method 992
12.7.3 The Crank-Nicolson Implicit Method 993
12.7.4 function [t,x,U] =Heatone(T,a,m,n,beta,c,f,g) 996
12.7.5 function [x,y,U] = Heattwo(T,a,b,m,n,p,beta,f,g) 999
12.7.6 function [t,x,U] = Waveone(T,a,m,n,beta,f,g) 1005
12.7.7 function [x,y,U] = Wavetwo (T,a,b,m,n,p,beta,f,g) 1008
12.7.8 function [alpha,r,x,y,U] = Poisson (a,b,m,n,q,tol,f,g) 1013
12.8 Two-Dimensional Parabolic Equations 1030
12.9 One-Dimensional Hyperbolic Equations 1037
12.9.1 D’Alembert’s Solution 1038
12.9.2 Explicit Central Difference Method 1043
12.10 Two-Dimensional Hyperbolic Equations 1050
12.10.1 Explicit Central Difference Method 1051xviii • Contents
12.11 MATLAB Built-in Function: pdepe 1060
12.12 Summary 1079
Exercises 1081
APPENDIX A: PARTIAL FRACTION EXPANSIONS 1093
Case-I 1094
Partial Fraction Expansion when Q(s) has Distinct Roots 1094
Case-II 1096
Partial Fraction Expansion when Q(s) has Complex
Conjugate Roots 1096
Case-III 1097
Partial Fraction Expansion when Q(s) has Repeated Roots 1097
Exercises 1098
APPENDIX B: BASIC ENGINEERING MATHEMATICS 1101
B.1 Algebra 1101
B.1.1 Basic Laws 1101
B.1.2 Sums of Numbers 1101
B.1.3 Progressions 1102
B.1.4 Powers and Roots 1103
B.1.5 Binomial Theorem 1103
B.1.6 Absolute Values 1104
B.1.7 Logarithms 1104
B.2 Trigonometry 1105
B.2.1 Trigonometric Identities 1105
B.2.2 Cosine Law (Law of Cosines) 1107
B.2.3 Sine Law (Law of Sines) 1108
B.3 Differential Calculus 1108
B.3.1 List of Derivatives 1108
B.3.2 Expansion in Series 1111
B.4 Integral Calculus 1114
B.4.1 List of Most Common Integrals 1114Contents • xix
APPENDIX C: CRAMER’S RULE 1119
Exercises 1123
APPENDIX D: MATLAB BUILT-IN M-FILE FUNCTIONS 1125
APPENDIX E: MATLAB PROGRAMS 1129
APPENDIX F: ANSWERS TO ODD NUMBERED EXERCISES 1135
BIBLIOGRAPHY 1175
INDEX 1187A
Absolute and relative errors, 12–14
Adams-Bashforth formula, 606
Adams-Moulton corrector formula, 607
Adams-Moulton method, 605–611
Aitken’s acceleration formula, 30–32
Algebraic and transcendental equations
Aitken’s Δ2 method, 226–229
Brent’s method, 229–236
Chebyshev’s method, 225–226
direct methods, 177
indirect or iterative methods
bracketing methods, 177–178
comparison, 238–239
open methods, 178
MATLAB built-in function, 239–243
Muller’s method, 219–225
secant method, 212–219
Alternating direction implicit (ADI)
method, 1032
Approximating curve, 405
Arithmetic errors, 11
B
Backward error analysis, 23
Bisection method
advantages, 181
algorithm, 179–180
error bounds, 180–181
MATLAB program, 182–186
method of finding solution, 179
Blunders and mistakes, 11
Boole’s rule, 506–507
Bracketing method
algebraic and transcendental
equations, 177–178
numerical optimization methods, 875
Brent’s method, 229–236
Burlirsch-Stoer method, 635–636
C
Central difference method, 974
Chebyshev’s formula, 226
Cholesky’s triangularization
method, 68–78
Coefficient of determination
computational formula, 422
error sum of squares, 425
regression sum of squares, 423, 425
standard deviation of errors, 424
total sum of squares, 422, 425
value of, 423
Index1188 • Index
Compound quadratures, 523
Crank-Nicolson implicit method,
993–996
Crout’s triangularization method. see
Cholesky’s triangularization method
Curve fitting, 405
approximating curves, 406
cubic spring function, 455
power function with a specified
power, 455
D
Direct numerical integration methods
MATLAB program, 684–747
multi degree of freedom system
(see Explicit and Implicit direct
integration schemes)
single degree of freedom system
central difference method,
663–664
finite difference method,
662–663
Runge-Kutta method, 664–666
Double precision method, 22
E
Error propagation, 22
Expansion method, 5
Explicit central difference
method, 1044
Explicit direct integration schemes
central difference method, 667–669
fourth-order Runge-Kutta
method, 671–673
two-cycle iteration with trapezoidal
rule, 670–671
Explicit forward Euler method,
989–992
Extrapolation, 285
F
Finite difference operators
Δ operator properties, 296–297
average operator μ, 298
backward differences, 288–289
central differences, 289–291
differential operator, D, 299
error propagation in difference
table, 293–296
factorial notation, 304–310
forward differences, 287–288
relation among operators, 299–304
shift operator, E, 298
Fletcher-Reeves method, 906–917
Formula truncation error, 19
Forward error analysis, 23
Forward Euler method, 569
Fundamental theorem of calculus, 483
G
Gauss elimination method, 50–58
Gaussian quadrature methods, 524
Gauss-Chebyshev quadrature
method, 533–535
Gauss-Hermite quadrature, 537–538
Gaussian integration formula,
523–525
Gauss-Lagendre quadrature,
527–533
Gauss-Laguerre quadrature,
535–536
MATLAB programs, 539–543
orthogonal polynomials, 525–527
Gram-Schmidt orthogonalization
process, 453–455
H
Heun’s method, 579
Hexadecimal (base-16) system, 9–10Index • 1189
Hooke and Jeeves method, 917–924
Houbolt method, 673–676
I
Implicit direct integration schemes
Houbolt method, 673–676
Newmark beta integration
method, 679–682
Park Stiffly stable method, 682–684
Wilson theta method, 676–679
Inherent errors, 14
Initial-value problems, ODE, 558
Burlirsch-Stoer method, 635–636
Euler’s formula, 634
fourth-order Runge-Kutta
method, 632–634
modified Euler’s formula, 635
Picard’s method, 630–631
Runge-Kutta-Butcher method, 637
Runge-Kutta-Fehlberg method,
636–637
second-order Runge-Kutta
method, 631–632
Taylor series method, 628–630
Interior penalty function
method, 925–931
Interpolating quadratures, 523
Interpolation
central difference formulae
Bessel’s formula, 348–351
difference table, 342
Gauss’s backward interpolation
formula, 346–348
Gauss’s forward interpolation
formula, 343–345
Laplace-Everett’s formula,
352–355
selection of interpolation
formula, 355
Sheppard’s operator δ, 343
Stirling’s formula, 351–352
cubic spline interpolation method
advantages, 360
clamped boundary condition, 363
Lagrange’s two-point
interpolation, 361
natural boundary condition, 363
piecewise polynomial
approximation, 360
divided differences, 355–360
with equal intervals
error in the interpolation formula,
328–331
missing values, 310
Newton’s backward interpolation
formula, 322–328
Newton’s binomial expansion
formula, 310–312
Newton’s forward interpolation
formula, 313–322
formulae, 286
generalized spline method
cubic splines, 371–375
end conditions, 376
linear splines, 367–368
MATLAB interp1 function,
381–390
MATLAB spline function,
376–378
multidimensional interpolation,
378–381
quadratic splines, 368–371
spline functions, 366
Harper’s definition, 285
Hiral’s definition, 285
Theile’s definition, 2851190 • Index
with unequal intervals
Hermite’s interpolation formula,
338–340
inverse interpolation, 340
Lagrange’s formula for inverse
interpolation, 340–342
Lagrange’s formula for unequal
intervals, 334–338
linear Lagrange interpolating
polynomial for equal intervals,
331–332
Interval arithmetic method, 22–23
Interval halving method, 178. see also
Bisection method
L
Lagrange’s interpolation formula, 336
Least significant digit (LSD), 5
Least-squares approximation, 449
Least-squares method for continuous
data, 449–451
Linear convergence theorem, 29
Linear correlation coefficient, 425
explained and unexplained
variation, 428–429
perfect positive and negative, 425
properties, 427–428
strong and weak positive/negative,
426
Linear equation
criteria for best fit, 410–412
curve fitting, 408–410
with one independent variable, 407
Linear least-squares regression,
412–414
Linear regression, 407
quantification of error, 443–445
Linear regression analysis
assumptions, 421
estimated regression model, 416
MATLAB built-in function
polyfit, 418–419
polyval, 419–420
population regression line, 416
random error term, 415
relationship between two variables,
414–415
scatter diagram, 416
Linear regression model, 414
Linear systems of equations
algebraic equations, 39–40
Cholesky’s triangularization
method, 68–78
Crout’s method, 79–88
direct and iterative methods,
40–41
Gauss elimination method, 50–58
Gauss-Jordan method
LU decomposition, 67–68
MATLAB program, 59–67
Gauss-Seidel method, 104–113
inverse of a matrix, 41–44
Jacobi iteration method, 94–104
matrix eigenvalue problems
accumulated transformation
matrix, 136–137
Gerschgorin’s circle theorem,
137–139
Householder’s method, 133–136
inverse power method,
161–163
Jacobi method, 114–133
power method, 157–161
QR method, 146–157
Sturm sequence, 139–140
matrix inversion method, 44–50
Thomas method for tridiagonal
systems, 88–94Index • 1191
M
Machine epsilon, 21–22
Mathematical approximation errors, 11
MATrix LABoratory (MATLAB) basics
abort command, 759
arithmetic operations, 760
with complex numbers, 763
arrays, 766–768
built-in functions, 768–769
operations, 770–772
clc command, 759
commands, 758–759
common math functions, 761
complex number functions, 763
display formats, 760–761
display windows, 758
element-by-element operations,
772–774
Expo, 759
exponential functions, 762
functions, 791–792
general commands
directory information, 766
general information, 766
online help, 765
termination, 766
workspace information, 765
graphics
2-D plots, 792–793
3-D plots, 795–796
overlay plots, 794–795
save and print, 802
specialized 2-D plots, 793–794
help, 759
input/output in, 802–803
managing variables, commands
for, 765
percent symbol (%), 759
polynomials, 775–778
predefined variables, 764
programming in
built-in functions, 788–789
conditional statements, 789–790
else and elseif clauses, 790
nested if statements, 790
order of precedence, 788
relational and logical operators, 787
while loop, 790–791
random numbers, 774–775
round-off functions, 763
script files, 784–786
semicolon (;), 759
start and quit, 758
statements and variables, 759–760
symbolic mathematics, 803–809
system of linear equations, 778–784
trigonometric and hyperbolic
functions, 762
variable names, 764
Maximum likelihood principle, 443
Method of factorization. see Cholesky’s
triangularization method
Method of false position
algorithm, 188
description, 186–187
MATLAB program, 188–194
Milne’s predictor-corrector method,
611–614
Modeling errors, 11
Most and least significant bit, 5
Most significant digit (MSD), 5
Muller’s method, 219–225
Multiple linear regression, 445–448
N
Neumann boundary conditions, 976
Newmark beta integration method,
679–6821192 • Index
Newton-Cotes closed quadrature
formula, 484, 486
Newton-Raphson method
algorithm, 195
convergence of, 196–197
disadvantages, 211
drawbacks, 194
MATLAB program, 198–203
modified, 203–205
rate of convergence, 197
Newton’s forward interpolation
formula, 313–314
Newton’s method, 194
system of nonlinear equations,
236–238
Nonlinear regression model, 414
for curve fitting, 429–435
Numerical computations
error considerations
absolute and relative errors, 12–14
accuracy and precision, 11–12
error propagation, 22
inherent errors, 14
machine epsilon, 21–22
round-off errors, 14–19
sources of errors, 11
truncation errors, 19–21
error estimates
backward error analysis, 23
double precision method, 22
forward error analysis, 23
interval arithmetic method, 22–23
significant digit arithmetic
method, 23
statistical approach, 23
error formula
function approximation, 24–25
stability and condition, 25–27
uncertainty in data or
noise, 27–28
number representation
binary, decimal, and hexadecimal
conversion, 6–7
in computer languages, 5
in decimal and binary forms, 5
expansion method, 5
hexadecimal (base-16) system, 9–10
octal (base-8) numbering system, 8
sequences
Aitken’s acceleration process, 30–32
linear convergence, 28–29
quadratic convergence, 29–30
Taylor’s theorem, 1–4
Numerical differentiation
cubic spline method, 270–271
diff and gradient function, 274–277
maxima and minima of tabulated
function, 268–269
Newton’s backward interpolation
formula, 260–262
Newton’s forward interpolation
formula, 252–259
Richardson extrapolation, 271–273
Stirling’s interpolation formula,
263–267
of unequally spaced data, 273–274
Numerical integration
Boole’s rule, 506–507
closed and open integration, 483–484
double integral, 544–549
error of approximation, 485
Gaussian quadrature methods
Gauss-Chebyshev quadrature
method, 533–535
Gauss-Hermite quadrature,
537–538
Gaussian integration formula,
523–525
Gauss-Lagendre quadrature,
527–533Index • 1193
Gauss-Laguerre quadrature,
535–536
MATLAB programs, 539–543
orthogonal polynomials, 525–527
Newton-Cote’s method forward
interpolation formula, 485–486
relative error of approximation, 485
Romberg’s integration
MATLAB program, 516–523
Richardson extrapolation formula,
512–513
Simpson’s 1/3 rule
error estimate in, 495–496
MATLAB built-in functions,
497–502
MATLAB program, 496–497
Simpson’s 3/8 rule, 502–506
trapezoidal rule, 486–488
error estimate in, 489–490
MATLAB built-in function,
490–493
Weddle’s rule, 507–511
O
Octal (base-8) numbering system, 8
Optimization
conjugate directions
quadratic convergence, 893
(i - 1)th search step, 895
in two dimensions, 896–897
Fletcher-Reeves method, 906–917
gradient methods
concept of, 885
direction of step in, 886
using line searches, 887
Hooke and Jeeves method, 917–924
interior penalty function method,
925–931
MATLAB built-in functions
fminbnd, 931–935
fminsearch, 935–940
maxima and minima of a function,
857
minimization with constraints using
Lagrange multipliers, 860–864
multidimensional unconstrained
optimization techniques, 882–885
Newton’s method, 887–893
numerical methods
bracketing method, 875
golden-section search method,
875–878
local and global optima, 873–874
optimization involving single
variables, 865–873
Powell’s method, 901–906
successive linear approximation
method, 924–925
unconstrained minimization of
functions, 859
Ordinary differential equation (ODE)
boundary-value problem, 558
explicit and implicit methods,
559–560
first-order equation, 626–627
general form of, 557
initial-value problems, 558
Burlirsch-Stoer method, 635–636
Euler’s formula, 634
fourth-order Runge-Kutta
method, 632–634
modified Euler’s formula, 635
Picard’s method, 630–631
Runge-Kutta-Butcher method, 637
Runge-Kutta-Fehlberg method,
636–637
second-order Runge-Kutta
method, 631–632
Taylor series method, 628–6301194 • Index
MATLAB
built-in functions, 649
ODE solvers, 614–626
mesh points, 559
one-step or single-step methods,
558, 559
Picard’s method, 561–564
Taylor’s series method, 564–569
particular solution, 558
second-order boundary value
problem, 648–649
second-order linear initial value
problem, 646–647
single first-order, 558
step-by-step or marching methods,
558, 559
Euler’s explicit method, 569–579
modified Euler’s method,
579–586
predictor-corrector methods,
604–614
Runge-Kutta methods, 586–604
two-point boundary value problems
finite difference method, 638–639
second-order differential
equation, 639–641
shooting method, 641–646
Orthogonal polynomials, 449
Gram-Schmidt orthogonalization
process, 453–455
least-square approximation using,
451–453
P
Park Stiffly stable method, 682–684
Partial differential equations (PDE)
Cauchy’s problem, 966
classification, 971–972
Dirichlet’s problem, 965–966
elliptic equations
boundary conditions, 975–977
central difference method,
972–975
Gauss-Seidel method, 986
iterative solution methods,
977–985
Jacobi’s method, 985
successive over-relaxation method,
986–988
finite-difference approximation, 967
heat equation, 969–970
Laplace’s equation, 968–969
linear second-order partial
differential equation
elliptic equation, 963, 964
hyperbolic equation, 963, 965
parabolic equation, 962, 965
MATLAB built-in functions,
1060–1079
one-dimensional hyperbolic
equations
D’Alembert’s solution, 1038–1043
explicit central difference method,
1043–1050
one-dimensional parabolic equations
Crank-Nicolson implicit method,
993–996
explicit forward Euler method,
989–992
function [alpha,r,x,y,U] = Poisson
(a,b,m,n,q,tol,f,g), 1013–1019
function [t,x,U] =Heatone(T,a,m,
n,beta,c,f,g), 996–999
function [t,x,U] =
Waveone(T,a,m,n,beta,f,g),
1005–1008
function [x,y,U] = Heattwo(T,a,
b,m,n,p,beta,f,g), 999–1005
function [x,y,U] = Wavetwo (T,a,b,
m,n,p,beta,f,g), 1008–1013Index • 1195
implicit backward Euler method,
992–993
two-dimensional hyperbolic
equations, 1050–1059
two-dimensional parabolic equations,
1030–1037
wave equation, 970–971
Pearson product moment correlation
coefficient, 425
Picard’s method, 561–564
Poisson’s equation, 972
Polynomial regression
MATLAB built-in functions, 438–443
normal equations, 437
second-order polynomial, 435
Positional numbering system, 4
Powell’s method, 901–906
Q
Quadratic convergence theorem, 30
R
Recurrence formula, 664
Regula falsi method, 186. see also
Method of false position
Round-off errors, 14–19
Runge-Kutta-Butcher method, 637
Runge-Kutta-Fehlberg method, 636–637
Runge-Kutta method, 664–666
fourth-order, 671–673
second-order, 631–632
step-by-step or marching methods,
ODE, 586–604
S
Scatter diagram, 405
Secant method
convergence of, 212–213
formula, 212
MATLAB program, 214–219
rate of convergence, 213
Significant digit arithmetic method, 23
Significant digits, 13
Simply truncation error, 19
Simpson’s 1/3 rule
error estimate in, 495–496
MATLAB
built-in functions, 497–502
programs, 496–497
Simpson’s 3/8 rule, 502–506
Spline Toolbox , 387
Step-by-step or marching methods,
ODE, 558, 559
Euler’s explicit method, 569–579
modified Euler’s method, 579–586
predictor-corrector methods
Adams-Moulton method,
605–611
advantages, 604
Milne’s predictor-corrector
method, 611–614
Runge-Kutta methods of order four
vs. Euler’s method, 600
local truncation error, 592
MATLAB program, 594
Runge-Kutta methods of order two
local truncation error, 588
Taylor series expansion, 587
Successive approximation method
convergence criteria, 206–207
error estimate, 207–211
Successive linear approximation
method, 924–925
T
Taylor’s series expansion, 2
Taylor’s series method, 564–569
Taylor’s theorem, 1–4
Total numerical error, 201196 • Index
Trapezoidal rule, 486–488
error estimate in, 489–490
MATLAB built-in functions, 490–493
Truncation errors, 19–21. see also
Mathematical approximation errors
W
Weddle’s rule, 507–511
Weighted least-squares
method, 448–449
Wilson theta method, 676–679
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