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Introduction to numerical analysis-by Suli
1 Solution of equations by iteration 1
1.1 Introduction 1
1.2 Simple iteration 2
1.3 Iterative solution of equations 17
1.4 Relaxation and Newton’s method 19
1.5 The secant method 25
1.6 The bisection method 28
1.7 Global behaviour 29
1.8 Notes 32
Exercises 35
2 Solution of systems of linear equations 39
2.1 Introduction 39
2.2 Gaussian elimination 44
2.3 LU factorisation 48
2.4 Pivoting 52
2.5 Solution of systems of equations 55
2.6 Computational work 56
2.7 Norms and condition numbers 58
2.8 Hilbert matrix 72
2.9 Least squares method 74
2.10 Notes 79
Exercises 82
3 Special matrices 87
3.1 Introduction 87
3.2 Symmetric positive definite matrices 87
3.3 Tridiagonal and band matrices 93
3.4 Monotone matrices 98
3.5 Notes 101
Exercises 102
4 Simultaneous nonlinear equations 104
4.1 Introduction 104
4.2 Simultaneous iteration 106
4.3 Relaxation and Newton’s method 116
4.4 Global convergence 123
4.5 Notes 124
Exercises 126
5 Eigenvalues and eigenvectors of a symmetric matrix 133
5.1 Introduction 133
5.2 The characteristic polynomial 137
5.3 Jacobi’s method 137
5.4 The Gerschgorin theorems 145
5.5 Householder’s method 150
5.6 Eigenvalues of a tridiagonal matrix 156
5.7 The QR algorithm 162
5.7.1 The QR factorisation revisited 162
5.7.2 The definition of the QR algorithm 164
5.8 Inverse iteration for the eigenvectors 166
5.9 The Rayleigh quotient 170
5.10 Perturbation analysis 172
5.11 Notes 174
Exercises 175
6 Polynomial interpolation 179
6.1 Introduction 179
6.2 Lagrange interpolation 180
6.3 Convergence 185
6.4 Hermite interpolation 187
6.5 Differentiation 191
6.6 Notes 194
Exercises 195
7 Numerical integration – I 200
7.1 Introduction 200
7.2 Newton–Cotes formulae 201
7.3 Error estimates 204
7.4 The Runge phenomenon revisited 208
7.5 Composite formulae 209
7.6 The Euler–Maclaurin expansion 211
7.7 Extrapolation methods 215
7.8 Notes 219
Exercises 220
8 Polynomial approximation in the ∞ -norm 224
8.1 Introduction 224
8.2 Normed linear spaces 224
8.3 Best approximation in the ∞ -norm 228
8.4 Chebyshev polynomials 241
8.5 Interpolation 244
8.6 Notes 247
Exercises 248
9 Approximation in the 2-norm 252
9.1 Introduction 252
9.2 Inner product spaces 253
9.3 Best approximation in the 2-norm 256
9.4 Orthogonal polynomials 259
9.5 Comparisons 270
9.6 Notes 272
Exercises 273
10 Numerical integration – II 277
10.1 Introduction 277
10.2 Construction of Gauss quadrature rules 277
10.3 Direct construction 280
10.4 Error estimation for Gauss quadrature 282
10.5 Composite Gauss formulae 285
10.6 Radau and Lobatto quadrature 287
10.7 Note 2
Introduction to programming and numerical methods in MALAB-Otto
1. Simple Calculations with MATLAB .
1.1 Introduction and a Word of Warning .
1.2 Scalar Quantities and Variables .
1.2.1 Rules for Naming of Variables .
1.2.2 Precedence: The Order in Which Calculations Are Per-
formed .
1.2.3 Mathematical Functions
1.3 Format: The Way in Which Numbers Appear .
1.4 Vectors in MATLAB .
1.4.1 Initialising Vector Objects . .
1.4.2 Manipulating Vectors and Dot Arithmetic .
1.5 Setting Up Mathematical Functions .
1.6 Some MATLAB Specific Commands .
1.6.1 Looking at Variables and Their Sizes.
1.7 Accessing Elements of Arrays .
2. Writing Scripts and Functions .
2.1 Creating Scripts and Functions .
2.1.1 Functions
2.1.2 Brief Aside .
2.2 Plotting Simple Functions . .
2.2.1 Evaluating Polynomials and Plotting Curves .
2.2.2 More on Plotting .
2.3 Functions of Functions
2.4.1 Numerical Errors .
2.4.2 User
3. Loops and Conditional Statements
3.1 Introduction .
3.2 Loops Structures .
3.3 Summing Series . . . . . . . . . . . . . . . . . .
3.3.1 Sums of Series
3.3.2 Summing Infinite Series . .
3.3.3 Summing Series Using MATLAB Specific Commands
3.3.4 Loops Within Loops (Nested) .
3.4 Conditional Statements .
3.4.1 Constructing Logical Statements .
3.4.2 The MATLAB Command switch .
3.5 Conditional loops .
3.5.1 The break Command .
3.6 MATLAB Specific Commands
3.7 Error Checkin
3.8 Tasks
.4. Root Finding . . . . . . .
4.1 Introduction . .
4.2 Initial Estimates .
4.3 Fixed Point Iteration . . . .
4.4 Bisection . . .
4.5 Newton–Raphson and Secant Methods .
4.5.1 Derivation of the Newton–Raphson Method . .
4.6 Repeated Roots of Functions .
4.7 Zeros of Higher-Dimensional Functions(*) .
4.8 MATLAB Routines for Finding Zeros .
4.8.1 Roots of a Polynomial . . . .
4.8.2 The Command fzero . . . . .
5. Interpolation and Extrapolation . . . . .
5.1 Introduction .
5.2 Saving and Reading Data .
5.3 Which Points to Use? .
5.4 Newton Forward Differences and Lagrange Polynomials . . .
5.4.1 Linear Interpolation/Extrapolation . . .
5.5 Calculating Interpolated and Extrapolated Values . .
5.6 Splines .
5.7 Curves of Best Fit . .
5.8 Interpolation of Non-Smooth Data . .
5.8.1 Insufficient Data Points .
5.9 Minimisation of Functions and Parameter Retrieval .
5.9.1 Parameter Retrieval .
5.9.2 Using fmins for Parameter Retrieval
6. Matrices . . . . . .
6.1 Introduction . . . . .
6.1.1 Initialising Matrices Within MATLAB . . .
6.1.2 Matrix Operations . .
6.1.3 Operations on Elements of Matrices
6.1.4 More on Special Matrices . . .
6.1.5 Matrices Containing Strings
6.2 Properties of Matrices and Systems of Equations . . . .
6.2.1 Determinants of Matrices . .
6.3 Elementary Row Operations . . . . . .
6.3.1 Solving Many Equations at Once .
6.4 Matrix Decomposition . .
6.5 Eigenvalues and Eigenvectors .
6.6 Specific MATLAB Commands .
6.7 Characteristic Polynomials
6.8 Exponentials of Matrices .
7. Numerical Integration . .
7.1 Introduction
7.2 Integration Using Straight Lines .
7.2.1 Errors in the Trapezium Method . .
7.3 Integration Using Quadratics
7.4 Integration Using Cubic Polynomials
7.5 Integrating Using MATLAB Commands .
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