كتاب Applied Numerical Methods Using Matlab
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 كتاب Applied Numerical Methods Using Matlab

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Applied Numerical Methods Using Matlab
Second Edition
Won Y. Yang ,
Wenwu Cao ,
Jaekwon Kim ,
Kyung W. Park ,
Ho-Hyun Park ,
Jingon Joung ,
Jong-Suk Ro ,
Han L. Lee ,
Cheol-Ho Hong ,
Taeho Im  

كتاب Applied Numerical Methods Using Matlab  A_n_m_10
و المحتوى كما يلي :


CONTENTS
Preface xv
Acknowledgments xvii
About the Companion Website xix
1 MATLAB Usage and Computational Errors 1
1.1 Basic Operations of MATLAB / 2
1.1.1 Input/Output of Data from MATLAB Command
Window / 3
1.1.2 Input/Output of Data Through Files / 3
1.1.3 Input/Output of Data Using Keyboard / 5
1.1.4 Two-Dimensional (2D) Graphic Input/Output / 6
1.1.5 Three Dimensional (3D) Graphic Output / 12
1.1.6 Mathematical Functions / 13
1.1.7 Operations on Vectors and Matrices / 16
1.1.8 Random Number Generators / 25
1.1.9 Flow Control / 27
1.2 Computer Errors vs. Human Mistakes / 31
1.2.1 IEEE 64-bit Floating-Point Number Representation / 31
1.2.2 Various Kinds of Computing Errors / 35
1.2.3 Absolute/Relative Computing Errors / 37
1.2.4 Error Propagation / 38
1.2.5 Tips for Avoiding Large Errors / 39
1.3 Toward Good Program / 42
1.3.1 Nested Computing for Computational Efficiency / 42
1.3.2 Vector Operation vs. Loop Iteration / 43
1.3.3 Iterative Routine vs. Recursive Routine / 45
1.3.4 To Avoid Runtime Error / 45
viiviii CONTENTS
1.3.5 Parameter Sharing via GLOBAL Variables / 49
1.3.6 Parameter Passing Through VARARGIN / 50
1.3.7 Adaptive Input Argument List / 51
Problems / 52
2 System of Linear Equations 77
2.1 Solution for a System of Linear Equations / 78
2.1.1 The Nonsingular Case (M = N) / 78
2.1.2 The Underdetermined Case (M < N): Minimum-norm
Solution / 79
2.1.3 The Overdetermined Case (M > N): Least-squares Error
Solution / 82
2.1.4 Recursive Least-Squares Estimation (RLSE) / 83
2.2 Solving a System of Linear Equations / 86
2.2.1 Gauss(ian) Elimination / 86
2.2.2 Partial Pivoting / 88
2.2.3 Gauss-Jordan Elimination / 97
2.3 Inverse Matrix / 100
2.4 Decomposition (Factorization) / 100
2.4.1 LU Decomposition
(Factorization) – Triangularization / 100
2.4.2 Other Decomposition (Factorization) – Cholesky, QR and
SVD / 105
2.5 Iterative Methods to Solve Equations / 108
2.5.1 Jacobi Iteration / 108
2.5.2 Gauss-Seidel Iteration / 111
2.5.3 The Convergence of Jacobi and Gauss-Seidel
Iterations / 115
Problems / 117
3 Interpolation and Curve Fitting 129
3.1 Interpolation by Lagrange Polynomial / 130
3.2 Interpolation by Newton Polynomial / 132
3.3 Approximation by Chebyshev Polynomial / 137
3.4 Pade Approximation by Rational Function / 142
3.5 Interpolation by Cubic Spline / 146
3.6 Hermite Interpolating Polynomial / 153CONTENTS ix
3.7 Two-Dimensional Interpolation / 155
3.8 Curve Fitting / 158
3.8.1 Straight-Line Fit – A Polynomial Function of Degree
1 / 158
3.8.2 Polynomial Curve Fit – A Polynomial Function of
Higher Degree / 160
3.8.3 Exponential Curve Fit and Other Functions / 165
3.9 Fourier Transform / 166
3.9.1 FFT vs. DFT / 167
3.9.2 Physical Meaning of DFT / 169
3.9.3 Interpolation by Using DFS / 172
Problems / 175
4 Nonlinear Equations 197
4.1 Iterative Method toward Fixed Point / 197
4.2 Bisection Method / 201
4.3 False Position or Regula Falsi Method / 203
4.4 Newton(-Raphson) Method / 205
4.5 Secant Method / 208
4.6 Newton Method for a System of Nonlinear Equations / 209
4.7 Bairstow’s Method for a Polynomial Equation / 212
4.8 Symbolic Solution for Equations / 215
4.9 Real-World Problems / 216
Problems / 223
5 Numerical Differentiation/Integration 245
5.1 Difference Approximation for the First Derivative / 246
5.2 Approximation Error of the First Derivative / 248
5.3 Difference Approximation for Second and Higher
Derivative / 253
5.4 Interpolating Polynomial and Numerical Differential / 258
5.5 Numerical Integration and Quadrature / 259
5.6 Trapezoidal Method and Simpson Method / 263
5.7 Recursive Rule and Romberg Integration / 265
5.8 Adaptive Quadrature / 268
5.9 Gauss Quadrature / 272x CONTENTS
5.9.1 Gauss-Legendre Integration / 272
5.9.2 Gauss-Hermite Integration / 275
5.9.3 Gauss-Laguerre Integration / 277
5.9.4 Gauss-Chebyshev Integration / 277
5.10 Double Integral / 278
5.11 Integration Involving PWL Function / 281
Problems / 285
6 Ordinary Differential Equations 305
6.1 Euler’s Method / 306
6.2 Heun’s Method – Trapezoidal Method / 309
6.3 Runge-Kutta Method / 310
6.4 Predictor-Corrector Method / 312
6.4.1 Adams-Bashforth-Moulton Method / 312
6.4.2 Hamming Method / 316
6.4.3 Comparison of Methods / 317
6.5 Vector Differential Equations / 320
6.5.1 State Equation / 320
6.5.2 Discretization of LTI State Equation / 324
6.5.3 High-order Differential Equation to State Equation / 327
6.5.4 Stiff Equation / 328
6.6 Boundary Value Problem (BVP) / 333
6.6.1 Shooting Method / 333
6.6.2 Finite Difference Method / 336
Problems / 341
7 Optimization 375
7.1 Unconstrained Optimization / 376
7.1.1 Golden Search Method / 376
7.1.2 Quadratic Approximation Method / 378
7.1.3 Nelder-Mead Method / 380
7.1.4 Steepest Descent Method / 383
7.1.5 Newton Method / 385
7.1.6 Conjugate Gradient Method / 387
7.1.7 Simulated Annealing / 389
7.1.8 Genetic Algorithm / 393CONTENTS xi
7.2 Constrained Optimization / 399
7.2.1 Lagrange Multiplier Method / 399
7.2.2 Penalty Function Method / 406
7.3 MATLAB Built-In Functions for Optimization / 409
7.3.1 Unconstrained Optimization / 409
7.3.2 Constrained Optimization / 413
7.3.3 Linear Programming (LP) / 416
7.3.4 Mixed Integer Linear Programming (MILP) / 423
7.4 Neural Network[K-1] / 433
7.5 Adaptive Filter[Y-3] / 439
7.6 Recursive Least Square Estimation (RLSE)[Y-3] / 443
Problems / 448
8 Matrices and Eigenvalues 467
8.1 Eigenvalues and Eigenvectors / 468
8.2 Similarity Transformation and Diagonalization / 469
8.3 Power Method / 475
8.3.1 Scaled Power Method / 475
8.3.2 Inverse Power Method / 476
8.3.3 Shifted Inverse Power Method / 477
8.4 Jacobi Method / 478
8.5 Gram-Schmidt Orthonormalization and QR
Decomposition / 481
8.6 Physical Meaning of Eigenvalues/Eigenvectors / 485
8.7 Differential Equations with Eigenvectors / 489
8.8 DoA Estimation with Eigenvectors[Y-3] / 493
Problems / 499
9 Partial Differential Equations 509
9.1 Elliptic PDE / 510
9.2 Parabolic PDE / 515
9.2.1 The Explicit Forward Euler Method / 515
9.2.2 The Implicit Backward Euler Method / 516
9.2.3 The Crank-Nicholson Method / 518
9.2.4 Using the MATLAB function ‘pdepe()’ / 520
9.2.5 Two-Dimensional Parabolic PDEs / 523xii CONTENTS
9.3 Hyperbolic PDES / 526
9.3.1 The Explicit Central Difference Method / 526
9.3.2 Two-Dimensional Hyperbolic PDEs / 529
9.4 Finite Element Method (FEM) for Solving PDE / 532
9.5 GUI of MATLAB for Solving PDES – PDE tool / 543
9.5.1 Basic PDEs Solvable by PDEtool / 543
9.5.2 The Usage of PDEtool / 545
9.5.3 Examples of Using PDEtool to Solve PDEs / 549
Problems / 559
Appendix A Mean Value Theorem 575
Appendix B Matrix Operations/Properties 577
B.1 Addition and Subtraction / 578
B.2 Multiplication / 578
B.3 Determinant / 578
B.4 Eigenvalues and Eigenvectors of a Matrix / 579
B.5 Inverse Matrix / 580
B.6 Symmetric/Hermitian Matrix / 580
B.7 Orthogonal/Unitary Matrix / 581
B.8 Permutation Matrix / 581
B.9 Rank / 581
B.10 Row Space and Null Space / 581
B.11 Row Echelon Form / 582
B.12 Positive Definiteness / 582
B.13 Scalar (Dot) Product and Vector (Cross) Product / 583
B.14 Matrix Inversion Lemma / 584
Appendix C Differentiation W.R.T. A Vector 585
Appendix D Laplace Transform 587
Appendix E Fourier Transform 589
Appendix F Useful Formulas 591
Appendix G Symbolic Computation 595
G.1 How to Declare Symbolic Variables and Handle Symbolic
Expressions / 595
G.2 Calculus / 597
G.2.1 Symbolic Summation / 597CONTENTS xiii
G.2.2 Limits / 597
G.2.3 Differentiation / 598
G.2.4 Integration / 598
G.2.5 Taylor Series Expansion / 599
G.3 Linear Algebra / 600
G.4 Solving Algebraic Equations / 601
G.5 Solving Differential Equations / 601
Appendix H Sparse Matrices 603
Appendix I MATLAB 605
References 611
Index 613
Index for MATLAB Functions 619
Index for Tables 629
INDEX
A
absolute error, 37–38
acceleration of Aitken, 228
Adams-Bashforth-Moulton (ABM) method,
314–318
adaptive filter, 439–443
adaptive input argument, 51
adaptive quadrature, see quadrature
alternating direction implicit (ADI) method,
527
animation, 352, 555, 574
apostrophe, 17
B
back-propagation algorithm, 434
backslash, 21–22, 65–66, 82, 127
backward difference approximation, 247, 254
backward substitution, 87–90, 104–105
Bairstow’s method, 212–214
basic variable, 418–419, 432
basis function (of FEM), 535–540
beamforming, 496, 498
bilinear interpolation, 155–157
bisection method, 201–203
BJT circuit, 219–222
Boltzmann, 390
boundary condition, 333, 356–360, 510–512
∼ for a cubic spline, 147
Dirichlet ∼, 510, 544, 545
Neumann ∼, 514, 544
Boundary mode, 548
boundary node, 533
Applied Numerical Methods Using MATLAB®, Second Edition. Won Y. Yang, Jaekwon Kim, Kyung W. Park,
Donghyun Baek, Sungjoon Lim, Jingon Joung, Suhyun Park, Han L. Lee, Woo June Choi, and Taeho Im.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
Companion website: www.wiley.com/go/yang/appliednumericalmethods
boundary value problem (BVP), 306, 333–340
bracketing method, 205, 207
branch-and-bound (BB) method, 425–427
Bulirsch-Stoer, 178–179
C
cantilever beam, 346
case, 29
catastrophic cancellation, 37
central difference approximation, 247, 249,
254
cftool (curve fitting tool), 165–166
characteristic equation, 468, 579
characteristic value, see eigenvalue
characteristic vector, see eigenvector
Chebyshev coefficient polynomial, 139–140
Chebyshev node, 137–139, 177
Chebyshev polynomial, 140–141
chemical reactor, 345
Cholesky decomposition (factorization), 105
circuit, 219–222, 232–237
circulant matrix, 499
conjugate gradient method, 387–389
Fletcher-Reeves (FR) ∼, 388
Polak-Ribiere (PR) ∼, 388
constrained linear least squares, see least
squares
constrained optimization, 376, 399–409,
413–433
constructive solid geometry (CSG), 546
contour, 12, 410
corrector, 314, 315
613614 INDEX
correlation matrix, 495–496
covariance matrix, 486–488
Crank-Nicholson method, 518–520
cross entropy function, 462
CtFS (Continuous-time Fourier series), 270
CtFT (Continuous-time Fourier transform),
74–76, 589–590
cubic spline, 146–153
curve fitting, 130, 158–166, 182–185
cutting plane method, 428–432
D
damped Newton method, 211–212, 244
dat-file, 3–4
data file, 3–4
debugging, 625
decoupling, 471–474
deep neural network (DNN), see neural
network
departing (leaving) variable, 420
determinant, 578
DFT (Discrete Fourier Transform), 167–171
recursive computation of ∼, 195
diagonalization, 469–474
symmetric ∼ theorem, see symmetric
difference approximation
∼ of the first derivative, 246–248
∼ of the second or higher derivative,
253–257
differential equation, 306–340, 601
Dirichlet boundary condition, see boundary
condition
discrete Fourier transform, see DFT
discrete-time Fourier Transform, see DtFT
discretization of LTI state equation, see state
equation
distinct eigenvalues, 470
divided difference, 133–135
DoA (degree of arrival) estimation, see
eigenvector
double integral/integration, 278–281, 303
Draw mode, 546
DtFT (Discrete-time Fourier Transform), 590
E
eigenmode PDE, 544
eigenpair, 468
eigenvalue, 468–469, 579
physical meaning of ∼, 485–489
∼problem, 366–372
eigenvector, 468–469, 579
differential equation with ∼s, 489–493
DoA (degree of arrival) estimation with ∼s,
493–498
physical meaning of ∼, 485–489
electric potential, 543, 556
electronic circuit, 219–222, 232–237
element-by-element operation, 15, 18
elliptic PDE, 510–515, 532, 543
entering (incoming) variable, 419–420
eps, 15, 34
error, 35–36
absolute ∼, 37–38
∼ analysis of difference approximation, 246,
249–252
∼ analysis of midpoint rule, 261
∼ analysis of Simpson’s rule, 262
∼ analysis of trapezoidal rule, 262
∼ estimate, 263
∼ magnification, 36
∼ propagation, 38
relative ∼, 37–38
errorbar(), 163
Euler’ s method, 306–309
explicit central difference method, 526–528
explicit forward Euler method, 515–516
exponent field, 31–32
F
false position (regular falsi), 203–205
FFT (Fast Fourier Transform), 167–168
finite difference method (FDM), 336–340
finite element method (FEM), 532–543
fixed-point
∼ iteration, 198–201, 223–227
∼ theorem, 198
Fletcher-Reeves (FR) conjugate gradient
method, see conjugate gradient method
forward difference approximation, 246, 254
forward substitution, 104–105
Fourier transform
continuous-time ∼, see CtFT
discrete-time ∼, see DtFT
full pivoting, 95, 119
G
GA, see genetic algorithm
Gauss(ian) elimination, 86–95
Gauss quadrature, see quadrature
Gauss-Chebyshev, 277–278
Gauss-Hermite, 275–276, 294
Gauss-Jordan elimination, 97–99INDEX 615
Gauss-Laguerre integration, 277, 295
Gauss-Legendre integration, 272–275
Gauss-Seidel iteration, 111–116
Gaussian (normal) distribution, 26–27
genetic algorithm (GA), 393–399
Gerschgorin’s disk theorem, 477
golden search method, 376–378, 448
Gomory cut, 428–430
gradient, 383–384, 585
graphic, 6–13, 608
H
Hamming method, 316–317
heat equation, 515, 523, 525
heat flow equation, 521
Helmholtz’s equation, 510
Hermite interpolating polynomial, 153–154
Hermite polynomial, 72, 275
Hermitian, 580
Hessenberg form, 504–505
Hessian, 385, 386, 404, 586
Heun’s method, 309–310
hidden bit, 32–35, 69
Hilbert matrix, 96–97
histogram, 10–11, 26
Householder, 501–504
hyperbolic PDE, 526–532
I
IDFT, 167
IEEE 64-bit floating-point number, 31–35
ill-condition, 95–96
impedance matching, 238–242
implicit backward Euler method, 516–518,
522
improper integral, 289–294
inactive inequality, 400
inconsistency, 91, 93, 95
interpolation
∼ by Chebyshev polynomial, 137–142
∼ by Lagrange polynomial, 130–131
∼ by Newton polynomial, 132–136
two-dimensional (2D) ∼, 155–158
∼ using DFS, 172–174
interpolation function, see basis function
intersection of two circles, 243–244
inverse matrix, 100, 580
inverse power method, see power method
shifted ∼, see power method
IVP (initial value problem), 327
J
Jacobi iteration, 108–111
Jacobi method, 478–481
Jacobian, 210, 586, 598
K
keyboard input, 3, 6
L
Lagrange coefficient polynomial, 130–131
Lagrange multiplier method, 80, 399–405
Lagrange polynomial, 130–131
Laguerre polynomial, 277
Laplace transform, 322, 588
Laplace’s equation, 510, 513, 543, 549, 556
largest number in MATLAB, 15, 30–31, 34
learning factor, 439, 447
least mean square (LMS), 441
least square (LS), 82–83, 159
constrained linear ∼, 415
∼ error (LSE), 78
nonlinear ∼ (NLLS), 411–412
nonnegative ∼ (NLS), 416, 454
recursive ∼ estimation (RLSE), 83–86,
443–447
weighted ∼ (WLS), 159
Legendre polynomial, 273–274
length of arc/curve, 298
limit, 597–598
linear programming (LP), 416–433
mixed integer ∼, 423–433
logical operator, 28
loop, 29–30
∼ iteration, 29, 43–44
Lorenz equation, 345
loss of significance, 36–39
LP, see linear programming
∼ relaxation, 424
LSE, see least square
LU decomposition (factorization), 100–104
M
mantissa field, 31–32
mat-file, 3–4
mathematical functions, 13–14
matrix, 16–25, 578–584
matrix inversion lemma, 84, 445, 584
mean value theorem, 575
mesh, 12–13, 54, 608
Mesh mode, 548
midpoint rule, 259–262616 INDEX
minimum ratio rule, 419, 422
minimum-norm solution, 79–82
mixed boundary condition, 339, 357–360
mixed integer linear programming (MILP), see
linear programming
modal matrix, 470
mode, 328, 474–486
modification formula, 315, 317
mu (????)-law, 57–58
mu-inverse (????−1) law, 57–58, 391–393
N
negligible addition, 35
Nelder-Mead Algorithm, 380–382
nested
∼ computing, 42–43, 71–72, 134
Neumann boundary condition, see boundary
condition
neural network (NN), 433–438
deep ∼ (DNN), 463–465
Newton (-Raphson) method, 205–208,
385–386
Newton polynomial, 132–137
nonbasic variables, 419, 432
nonlinear BVP, 339, 363–365, 372
nonlinear least squares (NLLS), see least
squares
nonnegative least squares (NLS), see least
squares
normal (Gaussian) distribution, 26–27
normalized exponential function, 434
normalized range, 33–34
null space, 79–81, 125, 581
numerical differentiation, 246–258
numerical integration, 259–271
O
orthogonal, 581
orthogonalization, 483–484
orthonormal, 478, 479, 481–482
overdetermined, 82–83, 106–107
overflow, 35, 39, 70
P
Pade approximation, 142–146, 177
parabolic PDE, 510, 515–526
two-dimensional ∼, 423–526
parallelepiped, 489
parallelogram, 488–489
parameter passing through VARARGIN,
50–51
parameter sharing via GLOBAL, 49–50
partial differential equation (PDE), 510–543
partial pivoting, 88–95, 117–118
scaled ∼, 93
PDE mode, 548
PDEtool, 543–558
penalty function method, 406–409
permutation, 100, 581
persistent excitation, 186
plot, see graphic
Plot mode, 549
Poisson’s equation, 510
Polak-Ribiere (PR) method, See conjugate
gradient method
polynomial
∼ approximation, 137
∼ wiggle, 137
positive definite, 582
power method, 475–478
inverse ∼, 476
scaled ∼, 475–476
shifted inverse ∼, 477
predictor, 309, 312–317
product
cross (outer, vector) ∼, 584
dot (inner, scalar) ∼, 583
inner (dot, scalar) ∼, 583
outer (cross, vector) ∼, 584
scalar (dot, inner) ∼, 583
vector (cross, outer) ∼, 584
projection operator, 80, 481
pseudo (generalized) inverse, 80, 82,
124–125
PWL (piecewise linear) function, 281–282
Q
QR decomposition (factorization), 105–107,
481–483
quadratic approximation method, 378–380
quadratic interpolation, 175
quadratically convergent, 206
quadrature, 259
adaptive ∼, 268–271
Gauss ∼, 272–278
quantization error, 35, 69, 70, 249
quenching factor, 390, 391
R
rank, 581
rational interpolation, 178–179
rectified linear unit (ReLU), 463INDEX 617
recursive, 45, 132, 195, 227, 265
∼ (self-calling) routine, 45, 227
recursive least square estimation (RLSE), see
least squares
redundancy, 90, 93, 95
regula falsi, see false position
relational operators, 28
relaxation, 116
reserved constants/variables, 15
Richardson’s extrapolation, 248
RLSE, see least squares
robot path planning, 180–181
Romberg integration, 265–267
rotation, 488
∼ matrix, 478–481
round-off error, 35, 39
row echelon form, 582
row space, 581
row switching, 92
Runge phenomenon, 137
Runge-Kutta (RK4), 310–311
S
saddle point, 347
satellite orbit problem, 216–218
scaled partial pivoting, see partial pivoting
Schroder method, 230
secant method, 208–209
self-calling routine, see recursive routine
set path, 2
shape function, see basis function
shifted inverse power method, see power
method
shooting method, 333–336
shooting position, 356
similarity transformation, 469–474
simplex method, 417–422
simplex tableau, 417, 419
Simpson’s rule, 259–264
simulated annealing (SA), 389–393
singular value decomposition (SVD), 106–108,
124–127
slack variable, 400, 419, 421
smallest positive number in MATLAB, 15, 30,
33
softmax function, see normalized exponential
function,
Solve mode, 548
SOR (successive over-relaxation), 116
source row, 428
sparse, 603–604
stability, 515–516, 518–519, 522, 528, 531
state equation, 320–328
discretization of LTI, 324–327
high-order differential equation to ∼,
327–328
steepest descent, 383–384
∼ rule, 420
Steffensen method, 228–229
step-size, 211, 246, 249–252, 387
adaptation ∼, 439, 440
∼ dilemma, 250
stiff, 328–332, 347–349, 486
Sturm-Liouville (BVP) equation, 370–372
surface area of revolutionary object,
299–300
SVD, see singular value decomposition
symbolic, 257, 291, 595–602
∼ algebraic equation, 601
∼ computation, 595–602
∼ differential equation, 601
∼ differentiation, 598
∼ integration, 598
∼ linear algebra, 600
priority of∼ variables, 215
∼ solution, 215
∼ variable, 595–596
symmetric, 580
∼ diagonalization theorem, 402
T
Taylor series theorem, 576, 599
term-wise operation, see element-by-element
operation
Toeplitz matrix, 499
trapezoidal rule, 259–260, 263–265
tridiagonal, 120
truncation error, 35, 246, 248–250
U
unbounded case, 419
unconstrained optimization, 376, 376–399,
409–413
underdetermined, 79–82
underflow, 35, 70
uniform probabilistic distribution, 25
unitary, 581
un-normalized range, 33–34
V
Van der Pol equation, 331–333, 344
Van der Waals Isotherms, 242–243618 INDEX
vector
differentiation w.r.t. a ∼, 585–586
∼ operation, 43–44
volume, 279
∼ of revolutionary 3D object, 300
W
wave equation, 526, 528–531
2D ∼, 529–531
weight least square (WLS), see least squareINDEX FOR MATLAB
FUNCTIONS
(cf) A/C/E/P/R/S/T stand for Appendix/Chapter/Example/Problems/Remark/Section/
Table, respectively.
Name Place Description
ABMc() S6.4.1 Predictor/Corrector coefficients in
Adams-BashforthMoulton ODE solver
adapt_smpsn() S5.8 integration by the adaptive
Simpson method
adc1() P1.10 AD conversion
adc2() P1.10 AD conversion
axis() S1.1.4 specify axis limits or appearance
backslash(\) S1.1.7/R1.2 left matrix division
backsubst() S2.4.1 backward substitution for
lower-triangular matrix equation
bar()/barh() S1.1.4 a vertical/horizontal bar chart
bisct() S4.2 bisection method to solve a
nonlinear equation
bisct_r() P4.4 bisection method to solve a
nonlinear equation (self-calling)
BJT_DC_analysis_exp0() S4.9 DC analysis of a BJT circuit
BJT2_complementary0() S4.9 Analysis of a complementary
BJT circuit
branch_and_bound() S7.3.4 use the branch-and-bound method
to solve a MILP problem
break S1.1.9 terminate execution of a for
loop or while loop
bvp2_eig() P6.12 solve an eigenvalue BVP2
Applied Numerical Methods Using MATLAB®, Second Edition. Won Y. Yang, Jaekwon Kim, Kyung W. Park,
Donghyun Baek, Sungjoon Lim, Jingon Joung, Suhyun Park, Han L. Lee, Woo June Choi, and Taeho Im.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
Companion website: www.wiley.com/go/yang/appliednumericalmethods
619620 INDEX FOR MATLAB FUNCTIONS
bvp2_fdf() S6.6.2 FDM (Finite difference method)
for a BVP
bvp2_fdfp() P6.7 FDM for a BVP with initial
derivative fixed
bvp2_shoot() S6.6.1 Shooting method for a BVP
(boundary value problem)
bvp2_shootp() P6.7 Shooting method for a BVP
with initial derivative fixed
bvp2_fdf() S6.6.2 FDM (Finite difference method)
for a BVP
bvp2_fdfp() P6.7 FDM for a BVP with initial
derivative fixed
bvp2m_shoot() P6.8 Shooting method for BVP with
mixed boundary condition I
bvp2m_fdfp() P6.8 FDM for a BVP with mixed
boundary condition I
bvp2mm_shoot() P6.9 Shooting method for BVP with
mixed boundary condition II
bvp2mm_fdf() P6.9 FDM for a BVP with mixed
boundary condition II
bvp4c() S6.6.2,
P6.11∼13
Finite difference method for a BVP
with initial derivative fixed
c2d_steq() S6.5.2 continuous-time state equation
to discrete-time one
ceil() S1.1.5/T1.3 round toward infinity
Cheby() S3.3 Chebyshev polynomial approximation
chol() S2.4.2 Cholesky factorization
clear S1.1.2 remove items from workspace,
freeing up system memory
clf S1.1.4 clear current figure window
compare_DFT_FFT S3.9.1 compare DFT with FFT
cond() S2.2.2 condition number
contour() S1.1.5 2-D contour plot of a scalar-valued
function of 2-D variable
conv() S1.1.6 convolution of two sequences or
multiplication of two polynomials
cspline() S3.5 cubic spline interpolation
CTFS_exp() S5.11 exponential CtFS (Continuous-time
Fourier Series) coefs
CtFT1() P1.28 CtFT (Continuous-time Fourier
Transform)
curve_fit() P3.11, p157 weighted least square (WLS)
curve fitting
cutting_plane() S7.3.4 use the cutting plane method to
solve a MILP problem
dblquad() S5.10 2-D (double) integralINDEX FOR MATLAB FUNCTIONS 621
diag() S1.1.7/R1.3 construct a diagonal matrix or
get diagonals of a matrix
difapx() S5.3 difference approximation for
numerical derivatives
diff() AG2.3, S5.4 differences between neighboring
elements in an array
disp() S1.1.3 display text or array onto the
(monitor) screen
DNN_multiclass() P7.14 implement a deep neural
network (DNN)
do_Cheby S3.3 approximate by Chebyshev
polynomial
do_condition S2.2.2 condition numbers for
ill-conditioned matrices
do_csplines S3.5 interpolate by cubic splines
do_FFT S3.9.1 do FFT (Fast Fourier Transform)
do_Gauss S2.2.2 do Gauss elimination
do_interp2 S3.7 do 2D interpolation
do_Lagranp S3.1 do Lagrange polynomial interpolation
do_lagnewch S3.3 try Lagrange/Newton/Chebyshev
polynomial
do_lu_dcmp S2.4.1 do LU decomposition (factorization)
do_MBK P6.4 simulate a mass-damper-spring system
do_Newtonp S3.2 do Newton polynomial interpolation
do_Newtonp1 S3.2 do Newton polynomial interpolation
do_Pade S3.4 do Pade (rational polynomial)
approximation
do_polyfit S3.8.2 do polynomial curve fitting
do_RDFT P3.21 do recursive DFT
do_quiver P6.0 use quiver() to plot the gradient vectors
do_rlse S2.1.4 do recursive least-squares estimation
do_wlse S3.8.2 do weighted least-squares curve fitting
DoA_estimation_with_
eigenvector
S8.8 use eigenvectors to estimate the DoA
(degree of arrival)
double() AG.1 convert to double-precision
draw_MBK P6.4 simulate a mass-damper-spring system
dsolve() S6.5.1, AG.5 symbolic differential equation solver
eig() S8.1 eigenvalues and eigenvectors
of a matrix
eig_Jacobi() S8.4 find the eigenvalues/eigenvectors of a
symmetric matrix
eig_power() S8.3 find the largest eigenvalue & the
corresponding eigenvector
eig_QR() P8.8 find eigenvalues using QR factorization
eig_QR_Hs() P8.8 find eigenvalues using QR
factorization via Hessenberg622 INDEX FOR MATLAB FUNCTIONS
else S1.1.9 for conditional execution of statements
elseif S1.1.9 for conditional execution of statements
end S1.1.9 terminate for/while,/witch/try/if
statements or last index
error_DE_sol() P6.5 evaluate the error of solution of
differential eq.
error_DE2_sols() P6.10 evaluate the error of solution(s) of
2nd-order differential eq.
eval() S1.1.5/T1.3 evaluate a string containing a
literal/symbolic expression
eye() S1.1.7 identity matrix (having 1/0
on/off its diagonal)
ezplot() S1.3.6 easy plot
falsp() S4.3 false position method to solve a
nonlinear equation
fem_basis_ftn() S9.4 coefficients of each basis
function for subregions
fem_coef() S9.4 coefficients for subregions
feval() S1.1.6 evaluation of a function defined by inline()
or in an M-file
find() P1.10 find indices of nonzero (true) elements
find_CTFS_PWL() S5.11 find the CtFS coefficients of a
PWL function
findsym() S4.8 find symbolic variables in a
symbolic expression
fix() S1.1.6/T1.3 round towards zero
fixpt() S4.1 fixed-point iteration to solve a
nonlinear equation
fliplr() S1.1.7 flip the elements of a matrix left-right
flipud() S1.1.7 flip the elements of a matrix up-down
floor() S1.1.6/T1.3 round to –infinity
fminbnd() S7.1.2 unconstrained minimization of
one-variable function
fmincon() S7.3.2 constrained minimization
fminimax() S7.3.2 minimize the maximum of
vector/matrix-valued function
fminsearch() S7.2.2, S7.3.1 unconstrained nonlinear minimization
(Nelder-Mead)
fminunc() S7.2.2, S7.3.1 unconstrained nonlinear minimization
(gradient-based)
for S1.1.9 repeat statements a specific
number of times
format S1.1.3 control display format for numbers
forsubst() S2.4.1 forward substitution for lower-triangular
matrix equation
fprintf() S1.1.3, P1.2 write formatted data to screen or fileINDEX FOR MATLAB FUNCTIONS 623
fsolve() S4.6,4.9, E4.3 solve nonlinear equations by a least
squares (LS) method
Gauseid() S2.5.2 Gauss-Seidel method to solve a system of
linear equations
Gauss() S2.2.2 Gauss elimination to solve a system of
linear equations
Gauss_legendre() S5.9.1 Gauss-Legendre integration
Gausslp() S5.9.1 grid points of Gauss-Legendre
integration formula
Gausshp() S5.9.2 grid points of Gauss-Hermite
integration formula
genetic() S7.1.8 optimization by the genetic algorithm (GA)
ginput() S1.1.4 input the x- & y-coordinates of point(s)
clicked by mouse
global S1.3.5 declare global variables
gradient() P6.0 numerical gradient
grid on/off S1.1.4 grid lines for 2-D or 3-D graphs
gtext() S1.1.4 mouse placement of text in a 2-D graph
help S1.1 display help comments for
MATLAB routines
Hermit() S3.6 Hermite polynomial interpolation
Hermitp() S5.9.2 Hermite polynomial
Hermits() S3.6 multiple Hermite polynomial interpolations
Hessenberg() P8.6 transform a matrix into almost
upper-triangular one
hist() S1.1.4, 1.1.8 plot a histogram
hold on/off S1.1.4 hold on/off current graph in the figure
Housholder() P8.5 Householder matrix to zero-out
the tail part of a vector
ICtFT1() P1.28 ICtFT (Inverse Continuoustime Fourier Transform)
iD_NMOS_at_vDS_vGS() P4.12 drain current of an NMOS
if S1.1.9 for conditional execution of statements
ilaplace() inverse Laplace transform
imp_match_1stub0() P4.14 single-stub impedance matching
imp_match_1stub1() P4.14 single-stub impedance matching
inline() S1.1.6 define a function inside the program
inpolygon() S9.4 is the point inside an polygonal region?
input() S1.1.3 request and get user input
int() S5.8, AG2.4 numerical/symbolic integration
interp1() S3.5 1-D interpolation
interp2() S3.7 2-D interpolation
intlinprog() S7.3.4 solve an integer linear programming (ILP)
problem
intrp1() P3.10 1-D interpolation
intrp2() S3.7 2-D interpolation624 INDEX FOR MATLAB FUNCTIONS
interpolate_by_DFS S3.9.3 interpolation using DFS
int2s() S5.10, P5.15 2-D (double) integral
inv() S2.1.1 the inverse of a matrix
isempty() P1.1, P1.10 is it empty (no value)?
isnumeric() S1.3.7 has it a numeric value?
jacob() S4.6 Jacobian matrix of a given function
jacob1() P5.3 Jacobian matrix of a given function
jacobi() S2.5.1 Jacobi iteration to solve a equation
Jkb() P1.23 1st kind of k-th order Bessel function
Lagranp() S3.1 Lagrange polynomial interpolation
Lgndrp() S5.9.1 Legendre polynomial
length() S1.1.7 the length of a vector (sequence)
or a matrix
limit() AG2.2 limit of a symbolic expression
lin_eq() S2.1.3 solve linear equation(s)
linprog() S7.3.3 solve a linear programming (LP) problem
load S1.1.2,4 read variable(s) from file
loglog() S1.1.4 plot data as logarithmic scales for the
x-axis and y-axis
lookfor S1.1 search for string in the first comment line
in all M-files
lscov() S3.8.1 weighted least-squares with known (error)
covariance
lsqcurvefit() P3.11, S7.3.1 weighted nonlinear least-squares
curve fitting
lsqlin() S7.3.1 solve a linear least squares (LLS) problem
lsqnonlin() S7.3.1 solve a non-linear least squares (NLLS)
problem
lsqnonneg() S7.3.2 find a non-negative least squares (NNLS)
solution
lu() S2.4.1 LU decomposition (factorization)
lu_dcmp() S2.4.1 LU decomposition (factorization)
max() S1.1.7 find the maximum element(s) of an array
mesh() S1.1.5, S3.7 plot a mesh-type graph of f(x,y)
meshgrid() S1.1.5, S3.7 grid points for plotting a mesh-type graph
min() S1.1.7 find the minimum element(s) of an array
mkpp() P1.13 make a piece-wise polynomial
mod() S1.1.6/T1.3 remainder after division
mulaw() P1.9 ????-law
mu_inv() S7.1.7 ????-1-law
multiply_matrix() P1.14 matrix multiplication
Newton() S4.4 Newton method to solve a
nonlinear equation
Newtonp() S3.2 Newton polynomial interpolation
Newtons() S4.6 Newton method to solve a system of
nonlinear equationINDEX FOR MATLAB FUNCTIONS 625
nm07e07 S7.5 apply the LMS/steepest descent/Newton
methods for …
nm07e08 S7.6 apply the RLS/LMS methods for
parameter estimation
NN_multiclass() S7.4 perform a NN(neural network)
based multi-classification
norm() P1.15 norm of vector/matrix
numel() S1.1.7 number of elements in a given array
ode_ABM() S6.4.1 solve a state equation by
Adams-Bashforth-Moulton solver
ode_Euler() S6.1 solve a state equation by Euler’s method
ode_Ham() S6.4.2 solve a state equation by Hamming
ODE solver
ode_Heun() S6.2 solve a state equation by Heun’s method
ode_RK4() S6.3 solve a state equation by
Runge-Kutta method
ode23()/ode45()/ode113() S6.4.3 ODE solver
ode15s()/ode23s()/
ode23t()/ode23tb()
S6.5.4 solve (stiff) ODEs
ones() S1.1.7 constructs an array of ones
opt_conjg() S7.1.6 optimization by Conjugate gradient method
opt_gs() S7.1.1 optimization by Golden search
opt_quad() S7.1.2 optimization by quadratic approximation
opt_Nelder() S7.1.3 optimization by Nelder-Mead method
opt_steep() S7.1.4 optimization by steepest descent
Padeap() S3.4 Pade approximation
pdetool S9.5 start the PDE toolbox GUI
(graphical user interface)
pinv() S1.1.7/R1.1 pseudo-inverse (generalized inverse)
pivoting() S7.3.4 pivoting at a given pivot element
plot() S1.1.4 linear 2-D plot
plot3() S1.1.5 linear 3-D plot
pde_heat_exp() S9.2.1 explicit forward Euler method for
parabolic PDE (heat eq)
pde_heat_imp() S9.2.2 implicit backward Euler method for
parabolic PDE (heat eq)
pde_heat_CN() S9.2.3 Crank-Nicholson method for
parabolic PDE (heat eq)
pde_heat2_ADI() S9.2.5 ADI method for parabolic PDE (2-D heat
equation)
pde_poisson() S9.1 central difference method for
elliptic PDE (Poisson’s eq)
pde_wave() S9.3.1 central difference method for hyperbolic
PDE (wave eq)
pde_wave2() S9.3.2 central difference method for hyperbolic
PDE (2-D wave eq)626 INDEX FOR MATLAB FUNCTIONS
pdepe() S9.2.4 solve a 1D parabolic or elliptic PDE
polar() S1.1.4 plot polar coordinates in a
Cartesian plane with polar grid
poly_der() P1.13 derivative of polynomial
polyder() P1.13 derivative of polynomial
polyfit() P3.13 polynomial curve fitting
polyfits() S3.8.2 polynomial curve fitting
polyint() P1.13 integral of polynomial
polyval() S1.1.6 evaluate a polynomial
ppval() P1.13 evaluate a set of piece-wise polynomials
pretty() AG print symbolic expression like in type-set
form
prod() S1.1.7/T1.3 product of array elements
qr() S2.4.2 QR factorization
qr_Hessenberg() P8.7 QR factorization of Hessenberg form by
Givens rotation
quad() P1.8, S5.8 numerical integration
quadl() S5.8 numerical integration
quiver() P6.0 plot gradient vectors
quiver3() P6.0 plot normal vectors on a surface
rand() S1.1.8 uniform random number generator
randn() S1.1.8 Gaussian random number generator
rational_interpolation() P3.6 rational polynomial interpolation
repetition() P1.17 repetition of a matrix
repmat() S1.1.7 repetition of a matrix
reshape() S1.1.7 a matrix into one with given numbers of
row/columns
residue() P1.13 partial fraction expansion of
Laplace-transformed function
residuez() P1.13 partial fraction expansion of
z-transformed rational function
rlse_online() S2.1.4 on-line Recursive Least-Squares
Estimation
Rmbrg() S5.7 Integration by Romberg method
robot_path P3.9 determine a path of robot using
cubic splines
roots() P1.13 roots of a polynomial equation
roots_Bairstow() S4.7 roots of a polynomial equation
(using Bairstow’s method)
round() S1.1.6/T1.3 round to nearest integer
rot90() S1.1.7 rotate a matrix by 90 degrees
save S1.1.2 save variable(s) into a file
secant() S4.5 secant method to solve a
nonlinear equation
semilogx() S1.1.4 plot data as logarithmic scales
for the x-axisINDEX FOR MATLAB FUNCTIONS 627
semilogy() S1.1.4 plot data as logarithmic scales for the
y-axis
sigmoid() S7.4 sigmoid function defined (Eq. (7.4.4))
size() S1.1.7 the numbers of rows/columns/ …
of a 1-D/2-D/3-D array
sim_anl() S7.1.7 optimization by simulated annealing (SA)
simplex() S7.3.3 implement the simplex method for linear
programming (LP)
simplify() AG simplify a symbolic expression
Smpsns() S5.6 integration by Simpson rule
Smpsns_fxy() S5.10, P5.16 2D integration of a function f(x,y) along y
softmax() S7.4 softmax or ‘normalized exponential
function’ (Eq. (7.4.6))
solve() P3.1,S4.8,AG.4 solve a (set of) symbolic
algebraic equation(s)
sort() S1.1.4 arranges the elements of an array in
ascending order
spline() S3.5 cubic spline
sprintf() S1.1.4 make formatted data to a string
stairs() S1.1.4 stair-step plot of zero-hold signal
of sampled data systems
stem() S1.1.4 plot discrete sequence data
Stfns() P4.6 Steffensen Method to solve a
nonlinear equation
subplot() S1.1.4 divide the current figure into
rectangular panes
subs() AG.1 substitute
sum() S1.1.7/R1.3 sum of elements of an array
surf() P6.0 plot a surface-type graph of f(x,y)
surfnorm() P6.0 generate vectors normal to a surface
svd() S2.4.2 singular value decomposition
switch S1.1.9 switch among several cases
syms AG.1 declare symbolic variable(s)
sym2poly() S5.3, AG.2.5 extract the coefficients of symbolic
polynomial expression
taylor() S5.3, AG.2.5 Taylor series expansion
test_DNN_multiclass() P7.14 test ‘train_DNN_multiclass()’/‘
DNN_multiclass()’
test_NN_multiclass() S7.4 test ‘train_NN_multiclass()’/‘
NN_multiclass()’
text() S1.1.4 add a text at the specified location on the
graph
title() S1.1.4 add a title string to current axes
train_DNN_multiclass P7.14 train a (multi-layer) DNN for multi-class
classification628 INDEX FOR MATLAB FUNCTIONS
train_NN_multiclass() S7.4 train a (possibly multi-layer) NN for
multi-class classification
trid() S6.6.2 solve a tri-diagonal system of linear
equations
trimesh() S9.4 plot a triangular-mesh-type graph
trpzds() S5.6 Integration by trapezoidal rule
try_beamforming S8.8 beamforming toward estimated
DoA to get the arriving signal
varargin() S1.3.6 variable length input argument list
view() S1.1.5, P1.4 3-D graph viewpoint specification
vpa() AG.1 evaluate double array by
variable precision arithmetic
while S1.1.9 repeat statements an indefinite
number of times
windowing() P3.18 multiply a sequence by the
specified window sequence
xlabel()/ylabel() S1.1.4 label the x-axis/y-axis
zeros() S1.1.7 make an array of zeros
zeroing() P1.17 cross out every (kM-m)th element to zeroINDEX FOR TABLES
Table Number Place Description
Conversion type specifiers & special characters in fprintf()
Graphic line specifications used in the plot() command
Functions and variables inside MATLAB
Relational operators and logical operators
Residual error and the number of floating-point operations of
various solutions
Divided difference table
Divided differences
Chebyshev coefficient polynomials
Boundary conditions for a cubic spline
Linearization of nonlinear functions by parameter/data
transformation
The forward difference approximation (5.1.4) for the 1st
derivative and its error depending on the step-size
The central difference approximation (5.1.8) for the 1st
derivative and its error depending on the step-size
The difference approximation formulas for the 1st and 2nd
derivatives
Romberg table
A numerical solution of DE (6.1.1) obtained by the Euler’s
method
Results of applying several routines for solving a simple DE
Results of running several unconstrained optimization routines
with various initial values
Results of running several unconstrained optimization routines
with various initial values
The names of the MATLAB built-in minimization routines
(cf) A: Appendix, P: Problem, S: Section, T: Table, R: Remark
Applied Numerical Methods Using MATLAB®, Second Edition. Won Y. Yang, Jaekwon Kim, Kyung W. Park,
Donghyun Baek, Sungjoon Lim, Jingon Joung, Suhyun Park, Han L. Lee, Woo June Choi, and Taeho Im.


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