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# Advanced Numerical Methods with Matlab 1: Function Approximation and System Resolution

ISBN: 978-1-119-51655-2 March 2018 Wiley-ISTE 238 Pages

## Description

Most physical problems can be written in the form of mathematical equations (differential, integral, etc.). Mathematicians have always sought to find analytical solutions to the equations encountered in the different sciences of the engineer (mechanics, physics, biology, etc.). These equations are sometimes complicated and much effort is required to simplify them. In the middle of the 20th century, the arrival of the first computers gave birth to new methods of resolution that will be described by numerical methods. They allow solving numerically as precisely as possible the equations encountered (resulting from the modeling of course) and to approach the solution of the problems posed. The approximate solution is usually computed on a computer by means of a suitable algorithm.

The objective of this book is to introduce and study the basic numerical methods and those advanced to be able to do scientific computation. The latter refers to the implementation of approaches adapted to the treatment of a scientific problem arising from physics (meteorology, pollution, etc.) or engineering (structural mechanics, fluid mechanics, signal processing, etc.) .

Preface xi

Part 1. Introduction 1

Chapter 1. Review of Linear Algebra  3

1.1. Vector spaces 3

1.1.1. General definitions  3

1.1.2. Free families, generating families and bases  4

1.2. Linear mappings 5

1.3. Matrices  7

1.3.1. Operations on matrices 7

1.3.2. Change-of-basis matrices 8

1.3.3. Matrix notations 9

1.4. Determinants 10

1.5. Scalar product 12

1.6. Vector norm 12

1.7. Matrix eigenvectors and eigenvalues   13

1.7.1. Definitions and properties 13

1.7.2. Matrix diagonalization 15

1.7.3. Triangularization of matrices  15

1.8. Using Matlab 16

Chapter 2. Numerical Precision 21

2.1. Introduction 21

2.2. Machine representations of numbers   22

2.3. Integers 23

2.3.1. External representation 23

2.3.2. Internal representation of positive integers 24

2.4. Real numbers 25

2.4.1. External representation 25

2.4.2. Internal encoding of real numbers 25

2.5. Representation errors 26

2.5.1. Properties of computer-based arithmetic 27

2.5.2. Operation of subtraction 28

2.5.3. Stability  29

2.6. Determining the best algorithm  29

2.7. Using Matlab 30

2.7.1. Definition of variables  30

2.7.2. Manipulating numbers 30

Part 2. Approximating Functions 35

Chapter 3. Polynomial Interpolation  37

3.1. Introduction 37

3.2. Interpolation problems  37

3.2.1. Linear interpolation  38

3.3. Polynomial interpolation techniques   38

3.4. Interpolation with the Lagrange basis 39

3.4.1. Polynomial interpolation error  43

3.4.2. Neville–Aitken method 46

3.5. Interpolation with the Newton basis   46

3.6. Interpolation using spline functions   48

3.6.1. Hermite interpolation  50

3.6.2. Spline interpolation error 55

3.7. Using Matlab 58

3.7.1. Operations on polynomials 58

3.7.2. Manipulating polynomials 59

3.7.3. Evaluation of polynomials 60

3.7.4. Linear and nonlinear interpolation 60

3.7.5. Lagrange function 63

3.7.6. Newton function 64

Chapter 4. Numerical Differentiation  67

4.1. First-order numerical derivatives and the truncation error  67

4.2. Higher-order numerical derivatives   70

4.3. Numerical derivatives and interpolation  71

4.4. Studying the differentiation error  73

4.5. Richardson extrapolation  77

4.6. Application to the heat equation  78

4.7. Using Matlab 81

Chapter 5. Numerical Integration 83

5.1. Introduction 83

5.2. Rectangle method 84

5.3. Trapezoidal rule 84

5.4. Simpson’s rule 87

5.5. Hermite’s rule 90

5.6. Newton–Côtes rules 91

5.7. Gauss–Legendre method  92

5.7.1. Problem statement 92

5.7.2. Legendre polynomials  94

5.7.3. Choosing the αi and xi (i = 0,  , n) 99

5.8. Using Matlab 100

5.8.1. Matlab functions for numerical integration 100

5.8.2. Trapezoidal rule 101

5.8.3. Simpson’s rule 103

Part 3. Solving Linear Systems 107

Chapter 6. Matrix Norm and Conditioning  109

6.1. Introduction 109

6.2. Matrix norm 109

6.3. Condition number of a matrix 113

6.3.1. Approximation of K(A) 116

6.4. Preconditioning 116

6.5. Using Matlab 117

6.5.1. Matrices and vectors  117

6.5.2. Condition number of a matrix  119

Chapter 7. Direct Methods  123

7.1. Introduction 123

7.2. Method of determinants or Cramer’s method 123

7.2.1. Matrix inversion by Cramer’s method  124

7.3. Systems with upper triangular matrices  124

7.4. Gaussian method 125

7.4.1. Solving multiple systems in parallel  129

7.5. Gauss–Jordan method  129

7.5.1. Underlying principle  129

7.5.2. Computing the inverse of a matrix with the Gauss–Jordan algorithm  131

7.6. LU decomposition 132

7.7. Thomas algorithm 133

7.8. Cholesky decomposition  134

7.9. Using Matlab 136

7.9.1. Matrix operations 136

7.9.2. Systems of linear equations 138

Chapter 8. Iterative Methods  147

8.1. Introduction 147

8.2. Classical iterative techniques 148

8.2.1. Jacobi method 149

8.2.2. Gauss–Seidel method  151

8.2.3. Relaxation method 152

8.2.4. Block forms of the Jacobi, Gauss–Seidel and relaxation methods 154

8.3. Convergence of iterative methods  155

8.4. Conjugate gradient method 157

8.5. Using Matlab 159

8.5.1. Jacobi method 159

8.5.2. Relaxation method 160

Chapter 9. Numerical Methods for Computing Eigenvalues and Eigenvectors 163

9.1. Introduction 163

9.2. Computing det (A − λI) directly  164

9.3. Krylov methods 166

9.4. LeVerrier method 167

9.5. Jacobi method 168

9.6. Power iteration method  171

9.6.1. Deflation algorithm  172

9.7. Inverse power method 173

9.8. Givens–Householder method 174

9.8.1. Givens algorithm 175

9.9. Using Matlab 176

9.9.1. Application to a buckling beam   177

Chapter 10. Least-squares Approximation  185

10.1. Introduction 185

10.2. Analytic formulation 185

10.3. Algebraic formulation  191

10.3.1. Standard results on orthogonality 191

10.3.2. Least-squares problem 191

10.3.3. Solving by orthogonalization  192

10.4. Numerically solving linear equations by QR factorization 193

10.4.1. Householder transformations  193

10.4.2. QR factorization 193

10.4.3. Application to the least-squares problem 193

10.5. Applications 194

10.5.1. Curve fitting 194

10.5.2. Approximations of derivatives   195

10.6. Using Matlab 195

Part 4. Appendices 199

Appendix 1. Introduction to Matlab  201

Appendix 2. Introduction to Optimization  209

Bibliography 215

Index 217