Solving applied mathematical problems with MATLAB /

Contents
1 Computer Mathematics Languages ¿ An Overview 1
1.1 Computer Solutions to Mathematics Problems . . . . . . . . 1
1.1.1 Why should we study computer mathematics language? 1
1.1.2 Analytical solutions versus numerical solutions . . . . . 4
1.1.3 Mathematics software packages: an overview . . . . . . 5
1.2 Summary of Computer Mathematics Languages . . . . . . . . 6
1.2.1 A brief historic review of MATLAB . . . . . . . . . . . 6
1.2.2 Three widely used computer mathematics languages . 7
1.2.3 Introduction to Free Scientific Open-source Softwares . 7
1.3 Outline of the Book . . . . . . . . . . . . . . . . . . . . . . . 8
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Fundamentals of MATLAB Programming 11
2.1 Fundamentals of MATLAB Programming . . . . . . . . . . . 12
2.1.1 Variables and constants in MATLAB . . . . . . . . . . 12
2.1.2 Data structure . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.3 Basic structure of MATLAB . . . . . . . . . . . . . . . 14
2.1.4 Colon expressions and sub-matrices extraction . . . . . 15
2.2 Fundamental Mathematical Calculations . . . . . . . . . . . . 16
2.2.1 Algebraic operations of matrices . . . . . . . . . . . . . 16
2.2.2 Logic operations of matrices . . . . . . . . . . . . . . . 18
2.2.3 Relationship operations of matrices . . . . . . . . . . . 19
2.2.4 Simplifications and presentations of analytical results . 19
2.2.5 Basic number theory computations . . . . . . . . . . . 21
2.3 Flow Control Structures of MATLAB Language . . . . . . . 22
2.3.1 Loop control structures . . . . . . . . . . . . . . . . . . 22
2.3.2 Conditional control structures . . . . . . . . . . . . . . 24
2.3.3 Switch structure . . . . . . . . . . . . . . . . . . . . . . 25
2.3.4 Trial structure . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Writing and Debugging MATLAB Functions . . . . . . . . . 27
2.4.1 Basic structure of MATLAB functions . . . . . . . . . 27
2.4.2 Programming of functions with variable inputs/outputs 30
2.4.3 Inline functions and anonymous functions . . . . . . . 31
2.5 Two-dimensional Graphics . . . . . . . . . . . . . . . . . . . 31
2.5.1 Basic statements of two-dimensional plotting . . . . . . 31
2.5.2 Other two-dimensional plotting statements . . . . . . . 34
2.5.3 Implicit function plotting and applications . . . . . . . 35
2.5.4 Graphics decorations . . . . . . . . . . . . . . . . . . . 37
2.6 Three-dimensional Graphics . . . . . . . . . . . . . . . . . . . 39
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8 Solving Applied Mathematical Problems with MATLAB
2.6.1 Plotting of three-dimensional curves . . . . . . . . . . . 39
2.6.2 Plotting of three-dimensional surfaces . . . . . . . . . . 39
2.6.3 Viewpoint setting in 3D graphs . . . . . . . . . . . . . 42
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3 Calculus Problems 47
3.1 Analytical Solutions to Calculus Problems . . . . . . . . . . . 47
3.1.1 Analytical solutions to limit problems . . . . . . . . . . 48
3.1.2 Analytical solutions to derivative problems . . . . . . . 50
3.1.3 Analytical solutions to integral problems . . . . . . . . 55
3.2 Series Expansions and Series Evaluations . . . . . . . . . . . 59
3.2.1 Taylor series expansion . . . . . . . . . . . . . . . . . . 59
3.2.2 Fourier series expansion . . . . . . . . . . . . . . . . . 63
3.2.3 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.4 Sequence product . . . . . . . . . . . . . . . . . . . . . 67
3.3 Numerical Differentiation . . . . . . . . . . . . . . . . . . . . 68
3.3.1 Numerical differentiation algorithms . . . . . . . . . . 68
3.3.2 Central-point difference algorithm with MATLAB im-
plementation . . . . . . . . . . . . . . . . . . . . . . . . 69
3.3.3 Gradient computations of functions with two variables 71
3.4 Numerical Integration Problems . . . . . . . . . . . . . . . . 72
3.4.1 Numerical integration from given data using trapezoidal
method . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.4.2 Numerical integration of single variable functions . . . 74
3.4.3 Numerical solutions to double integrals . . . . . . . . . 78
3.4.4 Numerical solutions to triple integrals . . . . . . . . . . 79
3.5 Path Integrals and Line Integrals . . . . . . . . . . . . . . . . 80
3.5.1 Path integrals . . . . . . . . . . . . . . . . . . . . . . . 80
3.5.2 Line integrals . . . . . . . . . . . . . . . . . . . . . . . 81
3.6 Surface Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.6.1 Scalar surface integrals . . . . . . . . . . . . . . . . . . 83
3.6.2 Vector surface integrals . . . . . . . . . . . . . . . . . . 84
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4 Linear Algebra Problems 89
4.1 Inputting Special Matrices . . . . . . . . . . . . . . . . . . . 90
4.1.1 Numerical matrix input . . . . . . . . . . . . . . . . . . 90
4.1.2 Defining symbolic matrices . . . . . . . . . . . . . . . . 94
4.2 Fundamental Matrix Operations . . . . . . . . . . . . . . . . 95
4.2.1 Basic concepts and properties of matrices . . . . . . . . 95
4.2.2 Matrix inversion and generalized inverse of a matrix . 102
4.2.3 Matrix eigenvalue problems . . . . . . . . . . . . . . . 106
4.3 Fundamental Matrix Transformations . . . . . . . . . . . . . 109
4.3.1 Similarity transformations and orthogonal matrices . . 109
4.3.2 Triangular and Cholesky decompositions . . . . . . . . 111
4.3.3 Jordan transformations . . . . . . . . . . . . . . . . . . 114
4.3.4 Singular value decompositions . . . . . . . . . . . . . . 116
4.4 Solving Matrix Equations . . . . . . . . . . . . . . . . . . . . 118
Contents 9
4.4.1 Solutions to linear algebraic equations . . . . . . . . . 118
4.4.2 Solutions to Lyapunov equations . . . . . . . . . . . . 121
4.4.3 Solutions to Sylvester equations . . . . . . . . . . . . . 124
4.4.4 Solutions to Riccati equations . . . . . . . . . . . . . . 125
4.5 Nonlinear Functions and Matrix Function Evaluations . . . . 126
4.5.1 Element-by-element computations . . . . . . . . . . . . 126
4.5.2 Matrix function evaluations . . . . . . . . . . . . . . . 127
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5 Integral Transforms and Complex Variable Functions 137
5.1 Laplace Transforms and Their Inverses . . . . . . . . . . . . . 137
5.1.1 Definitions and properties . . . . . . . . . . . . . . . . 138
5.1.2 Computer solution to Laplace transform problems . . . 139
5.2 Fourier Transforms and Their Inverses . . . . . . . . . . . . . 142
5.2.1 Definitions and properties . . . . . . . . . . . . . . . . 142
5.2.2 Solving Fourier transform problems . . . . . . . . . . . 142
5.2.3 Fourier sine and cosine transforms . . . . . . . . . . . . 144
5.2.4 Discrete Fourier sine, cosine transforms . . . . . . . . . 147
5.3 Other Integral Transforms . . . . . . . . . . . . . . . . . . . . 147
5.3.1 Mellin transform . . . . . . . . . . . . . . . . . . . . . 148
5.3.2 Hankel transform solutions . . . . . . . . . . . . . . . . 149
5.4 Z Transforms and Their Inverses . . . . . . . . . . . . . . . . 150
5.4.1 Definitions and properties of Z transforms and inverses 150
5.4.2 Computations of Z transform . . . . . . . . . . . . . . 151
5.5 Solving Complex Variable Function Problems . . . . . . . . . 152
5.5.1 Complex Variable Functions and Mapping Visualization 152
5.5.2 Concept and computation of residues . . . . . . . . . . 152
5.5.3 Partial fraction expansion for rational functions . . . . 155
5.5.4 Inverse Laplace transform using partial fraction expan-
sions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.5.5 Computing closed-path integrals . . . . . . . . . . . . . 160
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6 Nonlinear Equations and Numerical Optimization Problems 165
6.1 Nonlinear Algebraic Equations . . . . . . . . . . . . . . . . . 166
6.1.1 Graphical method for solving nonlinear equations . . . 166
6.1.2 Quasi-analytical solutions to polynomial-type equations 168
6.1.3 Numerical solutions to general nonlinear equations . . 172
6.1.4 Nonlinear matrix equations . . . . . . . . . . . . . . . 174
6.2 Unconstrained Optimization Problems . . . . . . . . . . . . . 176
6.2.1 Analytical solutions and graphical solution methods . . 176
6.2.2 Numerical solution of unconstrained optimization using
MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.2.3 Global minimum and local minima . . . . . . . . . . . 179
6.2.4 Solving optimization problems using gradient informa-
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.2.5 Optimization problems with bounded constraints . . . 182
6.3 Constrained Optimization Problems . . . . . . . . . . . . . . 183
10 Solving Applied Mathematical Problems with MATLAB
6.3.1 Constraints and feasibility regions . . . . . . . . . . . . 183
6.3.2 Solving linear programming problems . . . . . . . . . . 185
6.3.3 Solving quadratic programming problems . . . . . . . . 187
6.3.4 Solving general nonlinear programming problems . . . 188
6.4 Mixed Integer Programming Problems . . . . . . . . . . . . . 190
6.4.1 Solving mixed integer programming problems . . . . . 191
6.4.2 Solving binary programming problems . . . . . . . . . 194
6.5 Linear Matrix Inequalities . . . . . . . . . . . . . . . . . . . . 195
6.5.1 A general introduction to LMIs . . . . . . . . . . . . . 196
6.5.2 Lyapunov inequalities . . . . . . . . . . . . . . . . . . . 196
6.5.3 Classification of LMI problems . . . . . . . . . . . . . . 198
6.5.4 LMI problem solutions with MATLAB . . . . . . . . . 199
6.5.5 Optimization of LMI problems by YALMIP Toolbox . 201
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
7 Differential Equation Problems 207
7.1 Analytical Solution Methods for Special Classes of ODEs . . 208
7.1.1 Mathematical descriptions . . . . . . . . . . . . . . . . 208
7.1.2 Analytical solution methods . . . . . . . . . . . . . . . 210
7.1.3 Applications of Laplace transforms . . . . . . . . . . . 212
7.1.4 Analytical solutions to LTI state-space equations . . . 214
7.1.5 Analytical solutions to special nonlinear differential equa-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
7.2 Numerical Solutions to ODEs . . . . . . . . . . . . . . . . . . 215
7.2.1 Overview of numerical solution algorithms . . . . . . . 216
7.2.2 Fixed-step Runge-Kutta algorithm and itsMATLAB im-
plementation . . . . . . . . . . . . . . . . . . . . . . . . 218
7.2.3 Numerical solution to first-order vector ODEs . . . . . 219
7.2.4 Transforms to standard ODEs . . . . . . . . . . . . . . 224
7.2.5 Validation of numerical solutions to ODEs . . . . . . . 231
7.3 Numerical Solutions to Special Ordinary Differential Equations 232
7.3.1 Solutions of stiff ODEs . . . . . . . . . . . . . . . . . . 232
7.3.2 Solutions of implicit differential equations . . . . . . . 235
7.3.3 Solutions to differential algebraic equations . . . . . . 239
7.3.4 Solutions to delay differential equations . . . . . . . . 241
7.4 Solving Boundary Value Problems . . . . . . . . . . . . . . . 243
7.4.1 Solutions to two-point boundary value problems . . . . 243
7.4.2 Solutions to general boundary value problems . . . . . 245
7.5 Introduction to Partial Differential Equations . . . . . . . . . 247
7.5.1 Solving a set of 1D PDEs . . . . . . . . . . . . . . . . . 248
7.5.2 Mathematical description to 2D PDEs . . . . . . . . . 249
7.5.3 The GUI for the PDE Toolbox ¿ an introduction . . . 251
7.6 Solving ODEs with Block Diagrams in Simulink . . . . . . . 258
7.6.1 A brief introduction to Simulink . . . . . . . . . . . . . 258
7.6.2 Simulink ¿ relevant blocks . . . . . . . . . . . . . . . 258
7.6.3 Using Simulink for modeling and simulation of ODEs . 260
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
Contents 11
8 Interpolation and Approximation Problems 269
8.1 Interpolation and Data Fitting . . . . . . . . . . . . . . . . . 270
8.1.1 One-dimensional data interpolation . . . . . . . . . . . 270
8.1.2 Definite integral evaluation from given samples . . . . 272
8.1.3 Two-dimensional grid data interpolation . . . . . . . . 274
8.1.4 Two-dimensional scattered data interpolation . . . . . 277
8.1.5 High-dimensional data interpolations . . . . . . . . . . 280
8.2 Spline Interpolation and Numerical Calculus . . . . . . . . . 281
8.2.1 Spline interpolation in MATLAB . . . . . . . . . . . . 281
8.2.2 Numerical differentiation and integration with splines 284
8.3 Data Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 287
8.3.1 Polynomial fitting . . . . . . . . . . . . . . . . . . . . . 287
8.3.2 Approximation by continued fraction expansions . . . . 290
8.3.3 Pad´e rational approximations . . . . . . . . . . . . . . 292
8.3.4 Curve fitting by linear combination of basis functions . 294
8.3.5 Least square curve fitting . . . . . . . . . . . . . . . . . 296
8.4 Signal Analysis and Digital Signal Processing . . . . . . . . . 298
8.4.1 Correlation analysis . . . . . . . . . . . . . . . . . . . . 298
8.4.2 Fast Fourier transforms . . . . . . . . . . . . . . . . . . 300
8.4.3 Filtering techniques and filter design . . . . . . . . . . 302
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
9 Probability and Mathematical Statistics Problems 309
9.1 Probability Distributions and Pseudo-random Number Gener-
ators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
9.1.1 Introduction to PDFs and CDFs . . . . . . . . . . . . 309
9.1.2 PDFs/CDFs of commonly used distributions . . . . . 310
9.1.3 Solving probability problems . . . . . . . . . . . . . . . 317
9.1.4 Random numbers and pseudo-random numbers . . . . 318
9.2 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
9.2.1 Mean and variance of random variables . . . . . . . . . 319
9.2.2 Moments of random variables . . . . . . . . . . . . . . 321
9.2.3 Covariance analysis of multivariate random variables . 322
9.2.4 Joint PDFs and CDFs of multivariate normal distribu-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
9.2.5 Monte Carlo solutions to mathematical problems . . . 324
9.3 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . 326
9.3.1 Parametric estimation and interval estimation . . . . . 326
9.3.2 Multivariable linear regression and interval estimation 328
9.3.3 Nonlinear least square parametric and interval estima-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
9.4 Statistic Hypothesis Tests . . . . . . . . . . . . . . . . . . . . 333
9.4.1 Basic concept and procedures for statistic hypothesis
test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
9.4.2 Solving hypothesis test problems in MATLAB . . . . . 334
9.5 Analysis of Variance and Its Computation . . . . . . . . . . . 337
9.5.1 One-way ANOVA . . . . . . . . . . . . . . . . . . . . . 337
9.5.2 Two-way ANOVA . . . . . . . . . . . . . . . . . . . . . 339
9.5.3 n-way ANOVA . . . . . . . . . . . . . . . . . . . . . . 341
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
10 Non-traditional Solution Methods 345
10.1 Fuzzy Logic and Fuzzy Inference . . . . . . . . . . . . . . . . 346
10.1.1 Classical set theory and fuzzy sets . . . . . . . . . . . . 346
10.1.2 Membership function and fuzzification . . . . . . . . . 349
10.1.3 An interactive membership function editor . . . . . . . 351
10.1.4 Building fuzzy inference systems . . . . . . . . . . . . . 351
10.1.5 Fuzzy rules and fuzzy inference . . . . . . . . . . . . . 353
10.2 Neural Network and Its Applications in Data Fitting Problems 356
10.2.1 Fundamentals of neural networks . . . . . . . . . . . . 357
10.2.2 Graphical user interface for neural networks . . . . . . 364
10.3 Evolution Algorithms and Their Applications in Optimization
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
10.3.1 Basic idea of genetic algorithms . . . . . . . . . . . . . 366
10.3.2 MATLAB solutions to optimization problems with ge-
netic algorithms . . . . . . . . . . . . . . . . . . . . . . 368
10.3.3 Particle swam optimizations . . . . . . . . . . . . . . . 373
10.3.4 Solving optimization problems with GADS Toolbox . . 374
10.3.5 Towards accurate global minimum solutions . . . . . . 377
10.4 Wavelet Transform and Its Applications in Data Processing . 378
10.4.1 Wavelet transform and waveforms of wavelet bases . . 378
10.4.2 Wavelet transform in signal processing problems . . . . 383
10.4.3 Graphical user interface in wavelets . . . . . . . . . . . 386
10.5 Rough Set Theory and Its Applications . . . . . . . . . . . . 388
10.5.1 Introduction to rough set theory . . . . . . . . . . . . . 388
10.5.2 Data processing problem solutions using rough sets . . 391
10.6 Fractional-order Calculus . . . . . . . . . . . . . . . . . . . . 395
10.6.1 Definitions of fractional-order calculus . . . . . . . . . 395
10.6.2 Evaluating fractional-order differentiation . . . . . . . 400
10.6.3 Solving fractional-order differential equations . . . . . 405
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
References and Bibliography 415
MATLAB Functions Index 419
Index 425