Advanced numerical methods with Matlab 1 : function approximation and system resolution / Abdelkhalak El Hami, Bouchaib Radi.

Chapter 1. Review of Linear Algebra 31.1. Vector spaces 31.1.1. General definitions 31.1.2. Free families, generating families and bases 41.2. Linear mappings 51.3. Matrices 71.3.1. Operations on matrices 71.3.2. Change-of-basis matrices 81.3.3. Matrix notations 91.4. Determinants 101.5. Scalar product 121.6. Vector norm 121.7. Matrix eigenvectors and eigenvalues 131.7.1. Definitions and properties 131.7.2. Matrix diagonalization 151.7.3. Triangularization of matrices 151.8. Using Matlab 16Chapter 2. Numerical Precision 212.1. Introduction 212.2. Machine representations of numbers 222.3. Integers 232.3.1. External representation 232.3.2. Internal representation of positive integers 242.4. Real numbers 252.4.1. External representation 252.4.2. Internal encoding of real numbers 252.5. Representation errors 262.5.1. Properties of computer-based arithmetic 272.5.2. Operation of subtraction 282.5.3. Stability 292.6. Determining the best algorithm 292.7. Using Matlab 302.7.1. Definition of variables 302.7.2. Manipulating numbers 30Part 2. Approximating Functions 35Chapter 3. Polynomial Interpolation 373.1. Introduction 373.2. Interpolation problems 373.2.1. Linear interpolation 383.3. Polynomial interpolation techniques 383.4. Interpolation with the Lagrange basis 393.4.1. Polynomial interpolation error 433.4.2. Neville-Aitken method 463.5. Interpolation with the Newton basis 463.6. Interpolation using spline functions 483.6.1. Hermite interpolation 503.6.2. Spline interpolation error 553.7. Using Matlab 583.7.1. Operations on polynomials 583.7.2. Manipulating polynomials 593.7.3. Evaluation of polynomials 603.7.4. Linear and nonlinear interpolation 603.7.5. Lagrange function 633.7.6. Newton function 64Chapter 4. Numerical Differentiation 674.1. First-order numerical derivatives and the truncation error 674.2. Higher-order numerical derivatives 704.3. Numerical derivatives and interpolation 714.4. Studying the differentiation error 734.5. Richardson extrapolation 774.6. Application to the heat equation 784.7. Using Matlab 81Chapter 5. Numerical Integration 835.1. Introduction 835.2. Rectangle method 845.3. Trapezoidal rule 845.4. Simpson's rule 875.5. Hermite's rule 905.6. Newton-Cotes rules 915.7. Gauss-Legendre method 925.7.1. Problem statement 925.7.2. Legendre polynomials 945.7.3. Choosing the i and xi (i = 0, , n) 995.8. Using Matlab 1005.8.1. Matlab functions for numerical integration 1005.8.2. Trapezoidal rule 1015.8.3. Simpson's rule 103Part 3. Solving Linear Systems 107Chapter 6. Matrix Norm and Conditioning 1096.1. Introduction 1096.2. Matrix norm 1096.3. Condition number of a matrix 1136.3.1. Approximation of K(A) 1166.4. Preconditioning 1166.5. Using Matlab 1176.5.1. Matrices and vectors 1176.5.2. Condition number of a matrix 119Chapter 7. Direct Methods 1237.1. Introduction 1237.2. Method of determinants or Cramer's method 1237.2.1. Matrix inversion by Cramer's method 1247.3. Systems with upper triangular matrices 1247.4. Gaussian method 1257.4.1. Solving multiple systems in parallel 1297.5. Gauss-Jordan method 1297.5.1. Underlying principle 1297.5.2. Computing the inverse of a matrix with the Gauss-Jordan algorithm 1317.6. LU decomposition 1327.7. Thomas algorithm 1337.8. Cholesky decomposition 1347.9. Using Matlab 1367.9.1. Matrix operations 1367.9.2. Systems of linear equations 138Chapter 8. Iterative Methods 1478.1. Introduction 1478.2. Classical iterative techniques 1488.2.1. Jacobi method 1498.2.2. Gauss-Seidel method 1518.2.3. Relaxation method 1528.2.4. Block forms of the Jacobi, Gauss-Seidel and relaxation methods 1548.3. Convergence of iterative methods 1558.4. Conjugate gradient method 1578.5. Using Matlab 1598.5.1. Jacobi method 1598.5.2. Relaxation method 160Chapter 9. Numerical Methods for Computing Eigenvalues and Eigenvectors 1639.1. Introduction 1639.2. Computing det (A I) directly 1649.3. Krylov methods 1669.4. LeVerrier method 1679.5. Jacobi method 1689.6. Power iteration method 1719.6.1. Deflation algorithm 1729.7. Inverse power method 1739.8. Givens-Householder method 1749.8.1. Givens algorithm 1759.9. Using Matlab 1769.9.1. Application to a buckling beam 177Chapter 10. Least-squares Approximation 18510.1. Introduction 18510.2. Analytic formulation 18510.3. Algebraic formulation 19110.3.1. Standard results on orthogonality 19110.3.2. Least-squares problem 19110.3.3. Solving by orthogonalization 19210.4. Numerically solving linear equations by QR factorization 19310.4.1. Householder transformations 19310.4.2. QR factorization 19310.4.3. Application to the least-squares problem 19310.5. Applications 19410.5.1. Curve fitting 19410.5.2. Approximations of derivatives 19510.6. Using Matlab 195Part 4. Appendices 199Appendix 1. Introduction to Matlab 201.

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.).