Introduction to MATLAB with Numerical Preliminaries

Introduction to MATLAB with Numerical Preliminaries
Alexander Stanoyevitch
Contents
Preface
Chapter 1:
Chapter 2:
Chapter 3 :
Chapter 4:
Chapter 5:
M A T L A B Basics 1
What Is MATLAB?
Starting and Ending a MATLAB Session
A First MATLAB Tutorial
Vectors and an Introduction to MATLAB Graphics
A Tutorial Introduction to Recursion on MATLAB
Basic Concepts of Numerical Analysis 23
with Taylor’s T h e o r e m
What Is Numerical Analysis?
Taylor Polynomials
Taylor’s Theorem
Introduction to M-Files 45
What Are M-files?
Creating an M-file for a Mathematical Function
P r o g r a m m i n g in M A T L A B 57
Some Basic Logic
Logical Control Flow in MATLAB
Writing Good Programs
Floating Point Arithmetic and Error Analysis 85
Floating Point Numbers
Floating Point Arithmetic: The Basics
Floating Point Arithmetic: Further Examples and Details
C h a p t e r 6: Rootfinding 107
6.1: A Brief Account of the History of Rootfinding
6.2: The Bisection Method
6.3: Newton’s Method
*6.4: The Secant Method
*6.5: Error Analysis and Comparison of Rootfinding Methods
C h a p t e r 7: M a t r i c e s a n d L i n e a r S y s t e m s 143
7.1: Matrix Operations and Manipulations with MATLAB
*7.2: Introduction to Computer Graphics and Animation
7.3: Notations and Concepts of Linear Systems
7.4: Solving General Linear Systems with MATLAB
7.5: Gaussian Elimination, Pivoting, and LU Factorization
*7.6: Vector and Matrix Norms, Error Analysis, and Eigendata
7.7: Iterative Methods
A p p e n d i x A : M A T L A B ‘ s S y m b o l i c T o o l b o x 2 8 5
A p p e n d i x B: Solutions to All Exercises for the R e a d e r 2 9 3
R e f e r e n c e s 3 2 3
M A T L A B C o m m a n d I n d e x 3 2 7
I n d e x 3 2 9
General Index
Abel, Niels Henrik, 108,109
Actual error, 24
Affine transformation, 163
Algebraic multiplicity, 243
Approximation, 24
Associated matrix norm, 226
Asymptotic error constant, 133
Augmented matrix, 195
Back substitution, 206
Banded matrix, 152
Base, 85, 94
Basis theorem, 193
Binary arithmetic, 85
Birthday problem, 70
Bisection method, 110
Bracket, 116
Cantor, Georg F.L.P., 169
Cantor square, 184
Cardano, Girolamo, 108
Central difference formula, 43
Characteristic polynomial, 241
Chopped arithmetic, 89
Clay Foundation, 68
Cofactor expansion, 76
Collatz, Lothar, 77
Collatz conjecture, 67, 68
Column, 143
Combinatorics: alternating power sums, 202
Combinatorics: power sums, 202
Component-wise operation, 9
Computed solution, 231
Computer graphics, 157
Condition number, 228-230
Convergence order, 132
Convergence theorem, 262,264,266
Counter, 60
Cramer, Gabriel, 203
Cramer’s rule, 203
Cycling, 122
Determinant, 75, 222
Degree, 25
Diagonal matrix, 147
Diameter, 71
Dilation, 172
Dimension, 171
Direct method, 252
Divided difference, 131
Dot product, 22, 144
Double root, 130
Eigendata, 240
Eigenspace, 243
Eigenvalue, 240
Eigenvector, 240
Elementary matrix, 208
Elementary row operation (ERO), 207
Epicycloids, 13
Equivalent linear system, 195
Error bound via residual, 233
Error function, 42
Error term, 231
Essentially disjoint, 172
Euclidean length, 224
Exact answer, 24,231
Expected value, 83
False, 57
Fern leaf fractal, 185
First generation, 170
Fixed point iteration, 140
Floating point number, 85
Flop, 74
Flop counts (for Gaussian elimination), 226
Fontana, Niccolo 108
Forward substitution, 207
Forward difference formula, 43
Fractals (fractal sets), 169
Future value annuities, 72,73
Galois, Evariste, 109
Gauss, Carl Friedrich, 204
Gaussian elimination, 203-213
Gauss-Seidel iteration, 256
Generalized minimum residual method, 273
Geometric multiplicity, 243
Global variables, 46
Gosper island fractal, 185,186
Hilbert, David, 193
Hilbert matrix, 192
Homogeneous coordinates, 163,164
Hyper convergence of order a, 133
Identity matrix, 148
IEEE double precision standard, 86
Ill-conditioned, 102
Ill-posed, 187
Inline function, 51
Infinite loop, 16
Infinity matrix norm, 227
Infinity (vector) norm, 225
Input-output analysis, 200,201
329 330 General Index
Internal demand matrix, 201
Inverse of a matrix, 148
Invertible (nonsingular), 148
Iterative, 109
Iterative method, 252
Iterative refinement, 249
Jacobi-Gauss Convergence Theorem, 262
Jacobi iteration, 253
Julia, Gaston, 169
Leading one, 195
Leontief, Wassily, 200
Linear convergence, 133
Linear transformation, 160
Local variables, 46
Logic, 57
Logical operators, 58
Lower triangular, 205
LU decomposition (or factorization), 213
Machin, John, 43
Machine epsilon, 86
Maclaurin, Colin, 39
Maclaurin series, 38
Mandelbrot, Benoit, 170
Mantissa, 87
Matrix, 143
Matrix arithmetic, 144
Max norm, 225
M-file, 45

• Function M-files, 45
• Script M-files, 45
  Monte-Carlo method, 173
  Mother loop, 62
  Multiple root, 125
  Multiplicity 1, 55
  Nearly singular (poorly conditioned), 228
  Nested loop, 61
  Newton’s method, 118,119
  Nonsingular (invertible), 148
  Numerical differentiation, 43
  Order, 130
  Output matrix, 201
  Overflow, 88
  Parallel, 239
  Parametric equations, 11
  Partial pivoting, 211
  Path (MATLAB’s), 45
  Peano, Guiseppe, 169
  Perfect number, 81
  Pivot, 211
  Polynomial, 25
  Polynomial interpolation, 189,197-199
  Poorly conditioned matrix, 150,228
  Positive definite matrix, 265
  Preconditioned conjugate gradient method, 273
  Preconditioning, 273
  Prime number, 81
  Prompt, 2
  Quadratic convergence, 139
  Quadrature, 51
  Quartic, 108
  Quintic, 108
  Random integer matrix generator, 152
  Random walk, 82
  Recursion formulas, 15
  Reduced row echelon form, 195
  Reflection, 162
  Relative error bound (via residual), 233
  Relaxation parameter, 258
  Remainder (Taylor’s), 35
  Residual, 116
  Residual matrix, 250
  Residual vector, 232
  Rhind Mathematical Papyrus, 107
  Root, 110
  Rotation, 161
  Rounded arithmetic, 89
  Row, 143
  Scalar, 240
  Scalar multiplication, 144
  Scaling, 161
  Secant method, 128,129
  Self-similarity property, 169
  Shearing, 181
  Shift transformation, 162
  Sierpinski, Waclaw, 170
  Sierpinski carpet fractal, 184,185
  Sierpinski gasket fractal, 170
  Significant digits, 85
  Similarity transformation, 172
  Simple root, 125
  Simulation, 79
  SOR (successive over relaxation), 258
  SOR convergence theorem, 264
  Sparse matrix, 151,269-278
  Special function, 289
  Spectrum, 251
  Statement, 57
  Strictly diagonally-dominant (by columns), 221
  Strictly dagonally-dominant (by rows), 264
  Submatrix, 76
  Symbolic computation, 285
  Symmetric matrix, 243
  Tartaglia, 108
  Taylor, Brook, 34
  Taylor polynomial, 25
  Taylor series, 38
  Taylor’s theorem, 35
  Technology matrix, 201 General Index
  Tessellation, 186
  Thomas, Llewellyn H., 220
  Thomas method, 220
  Tolerance, 24
  Traffic logistics, 199,200
  Transpose, 7
  Tridiagonal matrix, 150
  Triple root, 126
  True, 57
  Truth value, 57
  Underflow, 88
  Unit roundoff, 86
  Upper triangular matrix, 204
  Vandermonde matrix, 197
  Variable precision arithmetic, 285
  Vector, 7
  Vector norm, 225
  von Koch, Niels F.H., 179
  von Koch snowflake, 179
  Weierstrass, Karl, 179
  Well-conditioned, 102
  Well-posed, 187
  Zero divisors, 105
  Zeroth generation, 170

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