Introduction to MATLAB with Numerical Preliminaries

Introduction to MATLAB with Numerical Preliminaries
اسم المؤلف
Alexander Stanoyevitch
التاريخ
3 مارس 2023
المشاهدات
518
التقييم
(لا توجد تقييمات)
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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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