Mathematical Explorations with MATLAB

Mathematical Explorations with MATLAB examines the mathematics most frequently encountered in first-year university courses. A key feature of the book is its use of MATLAB, a popular and powerful software package. The book's emphasis is on understanding and investigating the mathematics by putting the mathematical tools into practice in a wide variety of modeling situations. Even readers who have no prior experience with MATLAB will gain fluency. The book covers a wide range of material: matrices, whole numbers, complex numbers, geometry of curves and families of lines, data analysis, random numbers and simulations, and differential equations from the basic mathematics. These lessons are applied to a rich variety of investigations and modeling problems, from sequences of real numbers to cafeteria queues, from card shuffling to models of fish growth. All extras to the standard MATLAB package are supplied on the World Wide Web. Contents......Page 5 Preface......Page 11 Acknowledgments......Page 14 Part one: Foundations......Page 15 1.1 First steps with MATLAB......Page 17 1.2 Vectors and plots......Page 20 1.3 Creating and editing script files......Page 23 1.4 Getting hardcopy of things......Page 29 Exercises......Page 30 2.1 Vectors and matrices......Page 32 2.2 Complex numbers......Page 37 2.3 Population dynamics: the Leslie matrix......Page 38 Exercises......Page 41 3.1 A loop to calculate Fibonacci numbers......Page 45 3.2 A loop with conditionals: the 3n + 1 or hailstone problem......Page 47 3.3 The euclidean algorithm for greatest common divisors......Page 48 3.4 Fermat's theorem and the power algorithm......Page 50 Exercises......Page 53 3.5 Appendix......Page 55 4.1 Polynomials......Page 58 4.2 Initial examples of drawing curves......Page 59 4.3 Taylor polynomials......Page 61 4.4 Approximations using the function polyfit......Page 63 4.5 The goat problem......Page 64 4.6 Envelopes of lines......Page 65 Exercises......Page 68 4.7 Appendix......Page 72 5.1 Data analysis......Page 74 5.2 Least squares fitting......Page 77 Exercises......Page 84 5.3 Appendix......Page 85 6.1 Generating random numbers......Page 89 6.2 Random integers......Page 90 6.4 Simulating normal distributions......Page 92 6.5 Simulating negative exponential distributions......Page 93 Exercises......Page 96 6.6 Appendix......Page 98 7.1 Ordinary differential equations (ODEs)......Page 102 7.2 Systems of differential equations......Page 105 7.3 Difference equations......Page 108 Exercises......Page 110 Part two: Investigations......Page 113 8 Magic Squares......Page 115 8.2 Magic squares size 3 x 3......Page 116 8.3 Magic squares size 4 x 4......Page 119 8.4 Magic squares size 5 x 5 (optional)......Page 121 A GCDs of random pairs and triples of numbers......Page 122 B Pseudoprimes and Miller's test......Page 125 A Rose curves and epicycloids......Page 130 B Envelopes......Page 133 C Curves of constant width......Page 136 A Spirographs and zigzags......Page 142 B Fast curves......Page 149 12.1 Mobius sequences......Page 160 12.2 Cobweb diagrams......Page 162 12.3 Mobius functions and powers of matrices......Page 163 A Investigation on Mobius sequences......Page 167 B Attracting cycles......Page 170 C Quadratic and exponential sequences; fixed points......Page 173 13.1 Introduction......Page 178 13.2 The equation z2 + 1 = 0......Page 179 13.3 General quadratic equations......Page 180 13.4 The cubic equation z3 - z = 0......Page 182 A Cycle decompositions......Page 185 14.1 Introduction......Page 191 14.2 Ins and outs......Page 193 14.4 Rough riffles (ruffles)......Page 199 14.5 Appendix......Page 202 15 Iterations for Nonlinear Equations......Page 203 15.1 1D: Method 1 - Newton-Raphson......Page 204 15.3 1D: Convergence analysis......Page 205 15.4 2D: Iterations for nonlinear systems......Page 210 15.5 2D: Contour plot and convergence history......Page 213 Exercises......Page 217 16 Matrices and Solution of Linear Systems......Page 221 16.1 Operation counts......Page 222 16.2 Dense linear systems......Page 223 16.3 The iterative refinement algorithm......Page 226 16.4 A perturbation analysis for Ax = b......Page 227 16.5 Sparse matrices, graph ordering and permutations......Page 228 Exercises......Page 231 17 Function Interpolations and Approximation......Page 235 17.1 1D: Introduction......Page 236 17.2 The 1D example M-file intdemo 1. m......Page 237 17.3 1D data fitting......Page 238 17.4 How accurate is my approximation?......Page 241 17.5 Introduction to multi-variable approximation......Page 242 17.7 Contour plots, 3D plots and slicing......Page 243 17.8 The `\' global method......Page 247 17.10 Comparison of approximations......Page 248 Exercises......Page 250 18.1 Strategy......Page 253 Exercises......Page 254 Part three: Modelling......Page 257 19 Checkout Queues: Long or Short......Page 259 19.1 Simulating queues......Page 260 19.2 The motorway filling station......Page 264 19.3 The Leo's cafeteria......Page 265 Exercises......Page 267 20 Fish Farming......Page 271 20.2 Models of fish growth......Page 272 20.3 Designing the Leslie matrix......Page 275 20.4 Fishing strategy......Page 276 Exercises......Page 277 21 Epidemics......Page 279 21.1 Preliminary look at some data......Page 280 21.2 The SIR model for the dynamics of an epidemic......Page 281 21.3 Studying the behaviour analytically......Page 282 21.4 Analysing the data......Page 283 Exercises......Page 285 22 Dynamics of Snowboating......Page 287 22.1 Preliminary look at the problem......Page 288 22.2 The equations of motion......Page 289 22.3 Exploring the operating parameters......Page 291 Exercises......Page 292 23 Tides......Page 295 23.2 Fourier series and methods......Page 296 23.4 Fourier analysis of the tidal data......Page 298 Exercises......Page 299 Appendix 1 MATLAB Command Summary......Page 300 Appendix 2 Symbolic Calculations within MATLAB......Page 304 Appendix 3 List of All M-files Supplied......Page 306 Appendix 4 How to Get Solution M-files......Page 310 Appendix 5 Selected MATLAB Resources on the Internet......Page 311 References......Page 313 Index......Page 315 This 1999 book is about the kind of mathematics usually encountered in first year university courses. A key feature of the book is that this mathematics is explored in depth using the popular and powerful package MATLAB. The emphasis is on understanding and investigating the mathematics, and putting it into practice in a wide variety of modelling situations. In the process, the reader will gain some fluency with MATLAB, no starting knowledge of the package being assumed. The range of material is wide: matrices, whole numbers, complex numbers, geometry of curves and families of lines, data analysis, random numbers and simulations, and differential equations form the basic mathematics. This is applied to a large number of investigations and modelling problems, from sequences of real numbers to cafeteria queues, from card shuffling to models of fish growth. All extras to the standard MATLAB package are supplied on the World Wide Web

this Book Is Explores The Mathematics Encountered In First Year University Courses Using The Popular Package Matlab.