Mathematical Explorations With Matlab

by K. Chen , Peter J. Giblin , A. Irving

• ISBN13: 9780521630788
• ISBN10: 0521630789
• Format: Hardcover
• Copyright: 1999-05-13
• Publisher: Cambridge University Press

Summary

This 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.

Table of Contents

Preface	xi
Part one: Foundations	1	(98)
1 Introduction	3	(15)
1.1 First steps with MATLAB	3	(3)
1.2 Vectors and plots	6	(3)
1.3 Creating and editing script files	9	(6)
1.4 Getting hardcopy of things	15	(1)
Exercises	16	(2)
2 Matrices and Complex Numbers	18	(13)
2.1 Vectors and matrices	18	(5)
2.2 Complex numbers	23	(1)
2.3 Population dynamics: the Leslie matrix	24	(3)
Exercises	27	(4)
3 Whole Numbers	31	(13)
3.1 A loop to calculate Fibonacci numbers	31	(2)
3.2 A loop with conditionals: the 3n + 1 or hailstone problem	33	(1)
3.3 The euclidean algorithm for greatest common divisors	34	(2)
3.4 Fermat's theorem and the power algorithm	36	(3)
Exercises	39	(2)
3.5 Appendix	41	(3)
4 Graphs and Curves	44	(16)
4.1 Polynomials	44	(1)
4.2 Initial examples of drawing curves	45	(2)
4.3 Taylor polynomials	47	(2)
4.4 Approximations using the function polyfit	49	(1)
4.5 The goat problem	50	(1)
4.6 Envelopes of lines	51	(3)
Exercises	54	(4)
4.7 Appendix	58	(2)
5 Representation of Data	60	(15)
5.1 Data analysis	60	(3)
5.2 Least squares fitting	63	(7)
Exercises	70	(1)
5.3 Appendix	71	(4)
6 Probability and Random Numbers	75	(13)
6.1 Generating random numbers	75	(1)
6.2 Random integers	76	(2)
6.3 Simulating uniform distributions	78	(1)
6.4 Simulating normal distributions	78	(1)
6.5 Simulating negative exponential distributions	79	(3)
Exercises	82	(2)
6.6 Appendix	84	(4)
7 Differential and Difference Equations	88	(11)
7.1 Ordinary differential equations (ODEs)	88	(3)
7.2 Systems of differential equations	91	(3)
7.3 Difference equations	94	(2)
Exercises	96	(3)
Part two: Investigations	99	(144)
8 Magic Squares	101	(7)
8.1 Introduction	102	(1)
8.2 Magic squares size 3 X 3	102	(3)
8.3 Magic squares size 4 X 4	105	(2)
8.4 Magic squares size 5 X 5 (optional)	107	(1)
9 GCDs, Pseudoprimes and Miller's Test	108	(8)
A GCDs of random pairs and triples of numbers	108	(3)
B Pseudoprimes and Miller's test	111	(5)
10 Graphics: Curves and Envelopes	116	(12)
A Rose curves and epicycloids	116	(3)
B Envelopes	119	(3)
C Curves of constant width	122	(6)
11 Zigzags and Fast Curves	128	(18)
A Spirographs and zigzags	128	(7)
B Fast curves	135	(11)
12 Sequences of Real Numbers	146	(18)
12.1 Mobius sequences	146	(2)
12.2 Cobweb diagrams	148	(1)
12.3 Mobius functions and powers of matrices	149	(4)
A Investigation on Mobius sequences	153	(3)
B Attracting cycles	156	(3)
C Quadratic and exponential sequences; fixed points	159	(5)
13 Newton-Raphson Iteration and Fractals	164	(7)
13.1 Introduction	164	(1)
13.2 The equation z^(2) + 1 = 0	165	(1)
13.3 General quadratic equations	166	(2)
13.4 The cubic equation z^(3) - z = 0	168	(3)
14 Permutations	171	(18)
A Cycle decompositions	171	(6)
B Card shuffling	177	(1)
14.1 Introduction	177	(2)
14.2 Ins and outs	179	(6)
14.3 Cycles	185	(1)
14.4 Rough riffles (ruffles)	185	(3)
14.5 Appendix	188	(1)
15 Iterations for Nonlinear Equations	189	(18)
15.1 1D: Method 1 -- Newton-Raphson	190	(1)
15.2 1D: Method 2 -- Gauss-Jacobi	191	(1)
15.3 1D: Convergence analysis	191	(5)
15.4 2D: Iterations for nonlinear systems	196	(3)
15.5 2D: Contour plot and convergence history	199	(4)
Exercises	203	(4)
16 Matrices and Solution of Linear Systems	207	(14)
16.1 Operation counts	208	(1)
16.2 Dense linear systems	209	(3)
16.3 The iterative refinement algorithm	212	(1)
16.4 A perturbation analysis for Ax = b	213	(1)
16.5 Sparse matrices, graph ordering and permutations	214	(3)
Exercises	217	(4)
17 Function Interpolations and Approximation	221	(18)
17.1 1D: Introduction	222	(1)
17.2 The 1D example M-file intdemo1.m	223	(1)
17.3 1D data fitting	224	(3)
17.4 How accurate is my approximation?	227	(1)
17.5 Introduction to multi-variable approximation	228	(1)
17.6 The 2D M-file intdemo2.m	229	(1)
17.7 Contour plots, 3D plots and slicing	229	(4)
17.8 The `\' global method	233	(1)
17.9 The piecewise method	234	(1)
17.10 Comparison of approximations	234	(2)
Exercises	236	(3)
18 Ordinary Differential Equations	239	(4)
18.1 Strategy	239	(1)
Exercises	240	(3)
Part three: Modelling	243	(43)
19 Checkout Queues: Long or Short	245	(12)
19.1 Simulating queues	246	(4)
19.2 The motorway filling station	250	(1)
19.3 The Leo's cafeteria	251	(2)
Exercises	253	(4)
20 Fish Farming	257	(8)
20.1 Preliminary look at the problem	258	(1)
20.2 Models of fish growth	258	(3)
20.3 Designing the Leslie matrix	261	(1)
20.4 Fishing strategy	262	(1)
Exercises	263	(2)
21 Epidemics	265	(8)
21.1 Preliminary look at some data	266	(1)
21.2 The SIR model for the dynamics of an epidemic	267	(1)
21.3 Studying the behaviour analytically	268	(1)
21.4 Analysing the data	269	(2)
Exercises	271	(2)
22 Dynamics of Snowboating	273	(8)
22.1 Preliminary look at the problem	274	(1)
22.2 The equations of motion	275	(2)
22.3 Exploring the operating parameters	277	(1)
Exercises	278	(3)
23 Tides	281	(5)
23.1 Preliminary look at the tidal data	282	(1)
23.2 Fourier series and methods	282	(2)
23.3 Analysis of an electrical signal	284	(1)
23.4 Fourier analysis of the tidal data	284	(1)
Exercises	285	(1)
Appendix 1 MATLAB Command Summary	286	(4)
Appendix 2 Symbolic Calculations within MATLAB	290	(2)
Appendix 3 List of All M-files Supplied	292	(4)
Appendix 4 How to Get Solution M-files	296	(1)
Appendix 5 Selected MATLAB Resources on the Internet	297	(2)
References	299	(2)
Index	301