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| موضوع: كتاب Applied Numerical Methods Using Matlab الأحد 11 أكتوبر 2020, 10:26 pm | |
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أخوانى فى الله أحضرت لكم كتاب 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
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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