Preface	7
	Why Do We Need Numerical Methods?	8
	The Structure of This Text	8
	Contents	11
	1 Simple Calculations with MATLAB	16
	1.1 Introduction and a Word of Warning	16
	1.2 Scalar Quantities and Variables	17
	1.2.1 Rules for Naming of Variables	19
	1.2.2 Precedence: The Order in Which Calculations Are Performed	20
	1.2.3 Mathematical Functions	23
	1.3 Format: The Way in Which Numbers Appear	27
	1.4 Vectors in MATLAB	28
	1.4.1 Initialising Vector Objects	28
	1.4.2 Manipulating Vectors and Dot Arithmetic	29
	1.5 Setting Up Mathematical Functions	32
	1.6 Some MATLAB Speci.c Commands	35
	1.6.1 Looking at Variables and Their Sizes	37
	1.7 Accessing Elements of Arrays	38
	1.8 Tasks	39
	2 Writing Scripts and Functions	42
	2.1 Creating Scripts and Functions	42
	2.1.1 Functions	45
	2.1.2 Brief Aside	50
	2.2 Plotting Simple Functions	51
	2.2.1 Evaluating Polynomials and Plotting Curves	56
	2.2.2 More on Plotting	59
	2.3 Functions of Functions	64
	2.4 Errors	66
	2.4.1 Numerical Errors	66
	2.4.2 User Error	69
	2.5 Tasks	72
	3 Loops and Conditional Statements	78
	3.1 Introduction	78
	3.2 Loops Structures	78
	3.3 Summing Series	83
	3.3.1 Sums of Series of the Form	88
	3.3.2 Summing In.nite Series	91
	3.3.3 Summing Series Using MATLAB Speci.c Commands	94
	3.3.4 Loops Within Loops (Nested)	97
	3.4 Conditional Statements	98
	3.4.1 Constructing Logical Statements	100
	3.4.2 The MATLAB Command	103
	3.5 Conditional loops	105
	3.5.1 The break Command	107
	3.6 MATLAB Speci.c Commands	107
	3.7 Error Checking	109
	3.8 Tasks	112
	4 Root Finding	118
	4.1 Introduction	118
	4.2 Initial Estimates	119
	4.3 Fixed Point Iteration	124
	4.4 Bisection	128
	4.5 Newton–Raphson and Secant Methods	132
	4.5.1 Derivation of the Newton–Raphson Method	132
	4.6 Repeated Roots of Functions	138
	4.7 Zeros of Higher-Dimensional Functions(*)	140
	4.8 MATLAB Routines for Finding Zeros	143
	4.8.1 Roots of a Polynomial	143
	4.8.2 The Command	143
	4.9 Tasks	145
	5 Interpolation and Extrapolation	148
	5.1 Introduction	148
	5.2 Saving and Reading Data	149
	5.3 Which Points to Use?	154
	5.4 Newton Forward Differences and Lagrange Polynomials	156
	5.4.1 Linear Interpolation/Extrapolation	162
	5.5 Calculating Interpolated and Extrapolated Values	163
	5.6 Splines	165
	5.7 Curves of Best Fit	167
	5.8 Interpolation of Non-Smooth Data	170
	5.8.1 Insufficient Data Points	173
	5.9 Minimisation of Functions and Parameter Retrieval	176
	5.9.1 Parameter Retrieval	178
	5.9.2 Using fmins for Parameter Retrieval	179
	5.10 Tasks	181
	6 Matrices	184
	6.1 Introduction	184
	6.1.1 Initialising Matrices Within MATLAB	184
	6.1.2 Matrix Operations	189
	6.1.3 Operations on Elements of Matrices	195
	6.1.4 More on Special Matrices	197
	6.1.5 Matrices Containing Strings	200
	6.2 Properties of Matrices and Systems of Equations	201
	6.2.1 Determinants of Matrices	205
	6.3 Elementary Row Operations	206
	6.3.1 Solving Many Equations at Once	213
	6.4 Matrix Decomposition	214
	6.5 Eigenvalues and Eigenvectors	219
	6.6 Speci.c MATLAB Commands	223
	6.7 Characteristic Polynomials	227
	6.8 Exponentials of Matrices	229
	6.9 Tasks	232
	7 Numerical Integration	240
	7.1 Introduction	240
	7.2 Integration Using Straight Lines	241
	7.2.1 Errors in the Trapezium Method	244
	7.3 Integration Using Quadratics	245
	7.4 Integration Using Cubic Polynomials	250
	7.5 Integrating Using MATLAB Commands	252
	7.6 Speci.c Examples of Integrals	253
	7.6.1 Infinite Integrals and Removable Singularities	253
	7.6.2 Indefinite Integrals	255
	7.7 Tasks	257
	8 Solving Differential Equations	262
	8.1 Introduction	262
	8.2 Euler’s Method and Crank–Nicolson	262
	8.2.1 Analytical Comparisons	268
	8.3 Banded Matrices	274
	8.4 Runge–Kutta Methods	278
	8.5 Higher-Order Systems	281
	8.5.1 Second-Order Systems	281
	8.5.2 Bessel’s equation	285
	8.5.3 Airy’s Equation	288
	8.5.4 Shooting Methods	289
	8.6 Boundary-Value Problems	293
	8.7 Population Dynamics	297
	8.8 Eigenvalues of Di.erential Systems	300
	8.9 Tasks	301
	9 Simulations and Random Numbers	306
	9.1 Introduction	306
	9.2 Statistical quantities	306
	9.2.1 Averages	306
	9.2.2 Other Statistical Measures	308
	9.3 Random Numbers and Distributions	310
	9.3.1 Normal Distribution	313
	9.3.2 Calculating Probabilities	314
	9.3.3 Permutations	315
	9.4 Maps and White Noise	315
	9.4.1 Modelling Discrete Systems	322
	9.4.2 Periodicity and Chaos	324
	9.4.3 Random Motion	331
	9.5 Tasks	334
	A Mathematical Introduction to Matrices	338
	A.1 Special Matrices	345
	A.2 Inverses of Matrices	346
	B Glossary of Useful Terms	350
	B.1 Arithmetic and Logical Operators	350
	B.2 Symbols	358
	B.3 Plotting Commands	362
	B.4 General MATLAB Commands	380
	C Solutions to Tasks	404
	C.1 Solutions for Tasks from Chapter 1	404
	C.2 Solutions for Tasks from Chapter 2	409
	C.3 Solutions for Tasks from Chapter 3	415
	C.4 Solutions for Tasks from Chapter 4	423
	C.5 Solutions for Tasks from Chapter 5	429
	C.6 Solutions for Tasks from Chapter 6	433
	C.7 Solutions for Tasks from Chapter 7	444
	C.8 Solutions for Tasks from Chapter 8	451
	C.9 Solutions for Tasks from Chapter 9	469
	Index	474
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