ریاضیات برنامه‌نویسی با استفاده از MATLAB®

Providing an alternative to engineering-focused resources in the area, __Programming Mathematics Using MATLAB®__ introduces the basics of programming and of using MATLAB® by highlighting many mathematical examples. Emphasizing mathematical concepts through the visualization of programming throughout the book, this useful resource utilizes examples that may be familiar to math students (such as numerical integration) and others that may be new (such as fractals). Additionally, the text uniquely offers a variety of MATLAB® projects, all of which have been class-tested thoroughly, and which enable students to put MATLAB® programming into practice while expanding their comprehension of concepts such as Taylor polynomials and the Gram–Schmidt process. Programming Mathematics Using MATLAB® is appropriate for readers familiar with sophomore-level mathematics (vectors, matrices, multivariable calculus), and is useful for math courses focused on MATLAB® specifically and those focused on mathematical concepts which seek to utilize MATLAB® in the classroom. Contents 7 Preface 12 Supplements 12 Introduction 13 1 Introduction to MATLAB® 14 1.1 Basic MATLAB® information 14 1.1.1 Starting MATLAB 14 1.1.2 Good commands to know 14 1.2 Basic mathematics 15 1.2.1 Built-in mathematical functions 16 1.2.2 Precedence rules 17 1.2.3 Formats 19 1.3 Variables 20 1.4 Diaries and script files 21 1.5 Exercises 23 2 Vectors and Matrices (Arrays) 26 2.1 One-dimensional arrays (vectors) 26 2.1.1 Constant spaced vectors 26 2.1.2 Equally spaced vectors 27 2.2 Two-dimensional arrays (matrices) 28 2.3 Addressing elements of vectors/arrays 29 2.4 Component-wise calculations 33 2.5 Random numbers 36 2.6 Exercises 39 3 Plotting in MATLAB® 44 3.1 Basic 2D plots 44 3.2 Bad domain examples 45 3.3 Axis settings 46 3.4 Multiple plots 51 3.5 Color, line, and marker modifications 54 3.5.1 Clf/close all 57 3.5.2 Subplots 57 3.6 Other 2D plots 60 3.6.1 Parametric curves 60 3.6.2 Polar curves 61 3.7 Exercises 63 4 Three-Dimensional Plots 69 4.1 Vector functions or space curves 69 4.2 Plotting surfaces 72 4.2.1 The meshgrid command 73 4.2.2 Domain issues 74 4.2.3 Level curves 75 4.2.4 Multiple plots and modifying colors 76 4.3 View command 78 4.4 Axis settings, revisited 80 4.5 Other coordinate systems and 3D graphs 82 4.5.1 The sphere and cylinder commands 82 4.5.2 Cylindrical coordinates 85 4.5.3 Spherical coordinates 87 4.6 Exercises 88 5 Functions 92 5.1 The [basicstyle=]|lookfor| and [basicstyle=]|help| commands 92 5.2 File format 93 5.3 Function examples 95 5.3.1 Basic function examples 95 5.3.2 More function examples - multiple inputs 95 5.3.3 Multiple outputs 96 5.3.4 Bad examples 98 5.4 Exercises 99 6 Control Flow 102 6.1 Relational and logical operators 102 6.2 If statements 106 6.3 Switch/case 108 6.4 Use of characteristic functions 108 6.5 For loops 109 6.6 While loops 111 6.7 Useful commands break, continue, return, and error 112 6.8 Optional inputs and outputs of functions 113 6.9 Exercises 116 7 Miscellaneous Commands and Code Improvement 123 7.1 Miscellaneous commands 123 7.1.1 The fprintf command 123 7.1.2 The sprintf command 125 7.1.3 Formats revisited 126 7.1.4 The save/load commands 126 7.1.5 The tic/toc commands 127 7.1.6 The fill command 127 7.1.7 The command alpha 130 7.1.8 The syms, diff, int, and subs commands 133 7.1.9 Commands for polynomials 135 7.2 Code improvement 137 7.2.1 Vectorization of code 138 7.2.2 Preallocation 139 8 Transformations and Fern Fractals 141 8.1 Linear transformations 141 8.2 Affine transformations 145 8.3 Fern fractals 146 8.4 Exercises 147 9 Complex Numbers and Fractals 153 9.1 Complex numbers 153 9.1.1 Adding complex numbers 153 9.1.2 Multiplication by a real numbers (scalars) 153 9.1.3 Multiplication and de Moivre's theorem/formula 153 9.1.4 Plotting complex numbers in MATLAB® 156 9.1.5 Creating line segments with complex numbers 157 9.2 The Chaos Game 159 9.3 Line replacement fractals 160 9.3.1 Snowflake fractals 160 9.3.2 Gosper Island 161 9.4 Geometric series 162 9.5 Exercises 164 10 Series and Taylor Polynomials 172 10.1 Review of series 172 10.2 Power series 174 10.3 Taylor polynomials and Taylor series 178 10.4 Exercises 182 11 Numerical Integration 187 11.1 Approximating integrals/numerical integration 187 11.2 Riemann sums 187 11.3 Error bounds 189 11.4 Simpson's rule 190 11.5 Exercises 193 12 The Gram-Schmidt Process 196 12.1 General vector spaces and subspaces 196 12.1.1 Vector spaces 196 12.1.2 Subspaces 197 12.2 Linear combinations of vectors 198 12.3 Linear independence and bases 199 12.3.1 Linear independence 199 12.3.2 Bases 200 12.4 Rank 203 12.5 Orthonormal vectors and the Gram-Schmidt process 204 12.5.1 Orthogonal and orthonormal vectors 204 12.5.2 The Gram-Schmidt process 207 12.6 Answers to example problems 213 12.7 Exercises 214 A Publishing and Live Scripts 217 A.1 Live scripts 217 A.2 Basic scripts or M-files 217 A.3 Publishing M-files 218 A.4 Using sections 218 A.4.1 Using sections for publishing 219 A.4.2 Using sections for running/debugging files 224 A.5 Formatting text 225 A.5.1 Basic text formatting 225 A.5.2 Lists 226 A.5.3 HTML links 227 A.5.4 Inserting images 227 A.5.5 Pre-formatted text 228 A.5.6 Inserting HTML code 229 A.5.7 Inserting LaTeX equations 229 B Final Projects 230 B.1 Ciphers 230 B.1.1 Substitution cipher 230 B.1.2 Columnar transposition cipher 231 B.2 Game of Pig 232 B.3 Linearization and Newton's method 233 B.3.1 Linearization 233 B.3.2 Newton's method 233 B.4 Disk and Shell method 235 B.5 Power ball data 236 C Linear Algebra Projects 237 C.1 Matrix calculations and linear systems 237 C.1.1 First handout 237 C.1.2 Exercises 239 C.2 The Hill cipher 243 C.2.1 Useful commands 245 C.2.2 Exercises 251 C.3 Least-squares solutions 252 C.3.1 Brief overview 252 C.3.2 Curve fitting 253 C.3.3 Exercises 254 C.4 Markov matrices 256 C.4.1 Brief overview 256 C.4.2 Exercises 256 D Multivariable Calculus Projects 260 D.1 Lines and planes 260 D.2 Vector functions 261 D.2.1 2D example plots 261 D.2.2 3D example plot 261 D.2.3 Bad domain example 262 D.2.4 Adjusting the view 262 D.2.5 Sphere command 262 D.2.6 Multiple plots on one figure 264 D.2.7 Exercises 264 D.3 Applications of double integrals 265 D.3.1 Calculating integrals and viewing regions 265 D.3.2 Exercises 267 References 269 Index 271 Providing An Alternative To Engineering-focused Resources In The Area, Programming Mathematics Using Matlab® Introduces The Basics Of Programming And Of Using Matlab® By Highlighting Many Mathematical Examples. Emphasizing Mathematical Concepts Through The Visualization Of Programming Throughout The Book, This Useful Resource Utilizes Examples That May Be Familiar To Math Students (such As Numerical Integration) And Others That May Be New (such As Fractals). Additionally, The Text Uniquely Offers A Variety Of Matlab® Projects, All Of Which Have Been Class-tested Thoroughly, And Which Enable Students To Put Matlab® Programming Into Practice While Expanding Their Comprehension Of Concepts Such As Taylor Polynomials And The Gram-schmidt Process. Programming Mathematics Using Matlab® Is Appropriate For Readers Familiar With Sophomore-level Mathematics (vectors, Matrices, Multivariable Calculus), And Is Useful For Math Courses Focused On Matlab® Specifically And Those Focused On Mathematical Concepts Which Seek To Utilize Matlab® In The Classroom. Provides Useful Visual Examples Throughout For Student Comprehension Includes Valuable, Class-tested Projects To Reinforce Both Familiarity With Matlab® And A Deeper Understanding Of Mathematical Principles Offers Downloadable Matlab® Scripts To Supplement Practice And Provide Useful Example