Note: Cover -- Half Title -- Title Page -- Copyright Page -- Contents -- Preface -- Preface of the First Edition -- 1 Computer Mathematics Languages - An Overview -- 1.1 Computer Solutions to Mathematics Problems -- 1.1.1 Why should we study computer mathematics language -- 1.1.2 Analytical solutions versus numerical solutions -- 1.1.3 Mathematics software packages: an overview -- 1.1.4 Limitations of conventional computer languages -- 1.2 Summary of Computer Mathematics Languages -- 1.2.1 A brief historic review of MATLAB -- 1.2.2 Three widely used computer mathematics languages -- 1.2.3 Introduction to free scientific open-source softwares -- 1.3 Outline of the Book -- 1.3.1 The organization of the book -- 1.3.2 How to learn and use MATLAB -- 1.3.3 The three-phase solution methodology -- Exercises -- Bibliography -- 2 Fundamentals of MATLAB Programming and Scientific Visualization -- 2.1 Essentials in MATLAB Programming -- 2.1.1 Variables and constants in MATLAB -- 2.1.2 Data structures -- 2.1.3 Basic statement structures of MATLAB -- 2.1.4 Colon expressions and sub-matrices extraction -- 2.2 Fundamental Mathematical Calculations -- 2.2.1 Algebraic operations of matrices -- 2.2.2 Logic operations of matrices -- 2.2.3 Relationship operations of matrices -- 2.2.4 Simplifications and presentations of analytical results -- 2.2.5 Basic number theory computations -- 2.3 Flow Control Structures of MATLAB Language -- 2.3.1 Loop control structures -- 2.3.2 Conditional control structures -- 2.3.3 Switch structure -- 2.3.4 Trial structure -- 2.4 Writing and Debugging MATLAB Functions -- 2.4.1 Basic structure of MATLAB functions -- 2.4.2 Programming of functions with variable numbers of arguments in inputs and outputs -- 2.4.3 Inline functions and anonymous functions -- 2.4.4 Pseudo code and source code protection -- 2.5 Two-dimensional Graphics. , 2.5.1 Basic statements of two-dimensional plotting -- 2.5.2 Plotting with multiple horizontal or vertical axes -- 2.5.3 Other two-dimensional plotting functions -- 2.5.4 Plots of implicit functions -- 2.5.5 Graphics decorations -- 2.5.6 Data file access with MATLAB -- 2.6 Three-dimensional Graphics -- 2.6.1 Plotting of three-dimensional curves -- 2.6.2 Plotting of three-dimensional surfaces -- 2.6.3 Viewpoint settings in 3D graphs -- 2.6.4 Surface plots of parametric equations -- 2.6.5 Spheres and cylinders -- 2.6.6 Drawing 2D and 3D contours -- 2.6.7 Drawing 3D implicit functions -- 2.7 Four-dimensional Visualization -- Exercises -- Bibliography -- 3 Calculus Problems -- 3.1 Analytical Solutions to Limit Problems -- 3.1.1 Limits of univariate functions -- 3.1.2 Limits of interval functions -- 3.1.3 Limits of multivariate functions -- 3.2 Analytical Solutions to Derivative Problems -- 3.2.1 Derivatives and high-order derivatives -- 3.2.2 Partial derivatives of multivariate functions -- 3.2.3 Jacobian matrix of multivariate functions -- 3.2.4 Hessian partial derivative matrix -- 3.2.5 Partial derivatives of implicit functions -- 3.2.6 Derivatives of parametric equations -- 3.2.7 Gradients, divergences and curls of fields -- 3.3 Analytical Solutions to Integral Problems -- 3.3.1 Indefinite integrals -- 3.3.2 Computing definite, infinite and improper integrals -- 3.3.3 Computing multiple integrals -- 3.4 Series Expansions and Finite-term Series Approximations -- 3.4.1 Taylor series expansion -- 3.4.2 Fourier series expansion -- 3.5 Infinite Series and Products -- 3.5.1 Series -- 3.5.2 Product of sequences -- 3.5.3 Convergence test of infinite series -- 3.6 Path Integrals and Line Integrals -- 3.6.1 Path integrals -- 3.6.2 Line integrals -- 3.7 Surface Integrals -- 3.7.1 Scalar surface integrals -- 3.7.2 Vector surface integrals. , 3.8 Numerical Differentiation -- 3.8.1 Numerical differentiation algorithms -- 3.8.2 Central-point difference algorithm with MATLAB implementation -- 3.8.3 Gradient computations of functions with two variables -- 3.9 Numerical Integration Problems -- 3.9.1 Numerical integration from given data using trapezoidal method -- 3.9.2 Numerical integration of univariate functions -- 3.9.3 Numerical infinite integrals -- 3.9.4 Evaluating integral functions -- 3.9.5 Numerical solutions to double integrals -- 3.9.6 Numerical solutions to triple integrals -- 3.9.7 Multiple integral evaluations -- Exercises -- Bibliography -- 4 Linear Algebra Problems -- 4.1 Inputting Special Matrices -- 4.1.1 Numerical matrix input -- 4.1.2 Defining symbolic matrices -- 4.1.3 Sparse matrix input -- 4.2 Fundamental Matrix Operations -- 4.2.1 Basic concepts and properties of matrices -- 4.2.2 Matrix inversion -- 4.2.3 Generalized matrix inverse -- 4.2.4 Matrix eigenvalue problems -- 4.3 Fundamental Matrix Transformations -- 4.3.1 Similarity transformations and orthogonal matrices -- 4.3.2 Triangular and Cholesky factorizations -- 4.3.3 Companion, diagonal and Jordan transformations -- 4.3.4 Singular value decompositions -- 4.4 Solving Matrix Equations -- 4.4.1 Solutions to linear algebraic equations -- 4.4.2 Solutions to Lyapunov equations -- 4.4.3 Solutions to Sylvester equations -- 4.4.4 Solutions of Diophantine equations -- 4.4.5 Solutions to Riccati equations -- 4.5 Nonlinear Functions and Matrix Function Evaluations -- 4.5.1 Element-by-element computations -- 4.5.2 Computations of matrix exponentials -- 4.5.3 Trigonometric functions of matrices -- 4.5.4 General matrix functions -- 4.5.5 Power of a matrix -- Exercises -- Bibliography -- 5 Integral Transforms and Complex-valued Functions -- 5.1 Laplace Transforms and Their Inverses -- 5.1.1 Definitions and properties. , 5.1.2 Computer solution to Laplace transform problems -- 5.1.3 Numerical solutions of Laplace transforms -- 5.2 Fourier Transforms and Their Inverses -- 5.2.1 Definitions and properties -- 5.2.2 Solving Fourier transform problems -- 5.2.3 Fourier sinusoidal and cosine transforms -- 5.2.4 Discrete Fourier sine, cosine transforms -- 5.2.5 Fast Fourier transforms -- 5.3 Other Integral Transforms -- 5.3.1 Mellin transform -- 5.3.2 Hankel transform solutions -- 5.4 z Transforms and Their Inverses -- 5.4.1 Definitions and properties of z transforms and inverses -- 5.4.2 Computations of z transform -- 5.4.3 Bilateral z transforms -- 5.4.4 Numerical inverse z transform of rational functions -- 5.5 Essentials of Complex-valued Functions -- 5.5.1 Complex matrices and their manipulations -- 5.5.2 Mapping of complex-valued functions -- 5.5.3 Riemann surfaces -- 5.6 Solving Complex-valued Function Problems -- 5.6.1 Concept and computation of poles and residues -- 5.6.2 Partial fraction expansion for rational functions -- 5.6.3 Inverse Laplace transform using PFEs -- 5.6.4 Laurent series expansions -- 5.6.5 Computing closed-path integrals -- 5.7 Solutions of Difference Equations -- 5.7.1 Analytical solutions of linear difference equations -- 5.7.2 Numerical solutions of linear time varying difference equations -- 5.7.3 Solutions of linear time-invariant difference equations -- 5.7.4 Numerical solutions of nonlinear difference equations -- Exercises -- Bibliography -- 6 Nonlinear Equations and Numerical Optimization Problems -- 6.1 Nonlinear Algebraic Equations -- 6.1.1 Graphical method for solving nonlinear equations -- 6.1.2 Quasi-analytic solutions to polynomial-type equations -- 6.1.3 Numerical solutions to general nonlinear equations -- 6.2 Nonlinear Equations with Multiple Solutions -- 6.2.1 Numerical solutions -- 6.2.2 Finding high-precision solutions. , 6.2.3 Solutions of under determined equations -- 6.3 Unconstrained Optimization Problems -- 6.3.1 Analytical solutions and graphical solution methods -- 6.3.2 Solution of unconstrained optimization using MATLAB -- 6.3.3 Global minimum and local minima -- 6.3.4 Solving optimization problems with gradient information -- 6.4 Constrained Optimization Problems -- 6.4.1 Constraints and feasibility regions -- 6.4.2 Solving linear programming problems -- 6.4.3 Solving quadratic programming problems -- 6.4.4 Solving general nonlinear programming problems -- 6.5 Mixed Integer Programming Problems -- 6.5.1 Enumerate method in integer programming problems -- 6.5.2 Solutions of linear integer programming problems -- 6.5.3 Solutions of nonlinear integer programming problems -- 6.5.4 Solving binary programming problems -- 6.5.5 Assignment problems -- 6.6 Linear Matrix Inequalities -- 6.6.1 A general introduction to LMIs -- 6.6.2 Lyapunov in equalities -- 6.6.3 Classification of LMI problems -- 6.6.4 LMI problem solutions with MATLAB -- 6.6.5 Optimization of LMI problems by YALMIP Toolbox -- 6.7 Solutions of Multi-objective Programming Problems -- 6.7.1 Multi-objective optimization model -- 6.7.2 Least squares solutions of unconstrained multi-objective programming problems -- 6.7.3 Converting multi-objective problems into single-objective ones -- 6.7.4 Pareto front of multi-objective programming problems -- 6.7.5 Solutions of minimax problems -- 6.7.6 Solutions of multi-objective goal attainment problems -- 6.8 Dynamic Programming and Shortest Path Planning -- 6.8.1 Matrix representation of graphs -- 6.8.2 Optimal path planning of oriented graphs -- 6.8.3 Optimal path planning of undigraphs -- 6.8.4 Optimal path planning for graphs described by coordinates -- Exercises -- Bibliography -- 7 Differential Equation Problems. , 7.1 Analytical Solution Methods for Some Ordinary Differential Equations.