Numerical Analysis

1 Introduction. - 2 Linear Systems. - 2. 1 Examples for Systems of Equations. - 2. 2 Gaussian Elimination. - 2. 3 LR Decomposition. - 2. 4 QR Decomposition. - Problems. - 3 Basic Functional Analysis. - 3. 1 Normed Spaces. - 3. 2 Scalar Products. - 3. 3 Bounded Linear Operators. - 3. 4 Matrix Norms. - 3. 5 Completeness. - 3. 6 The Banach Fixed Point Theorem. - 3. 7 Best Approximation. - Problems. - 4 Iterative Methods for Linear Systems. - 4. 1 Jacobi and GaussSeidel Iterations. - 4. 2 Relaxation Methods. - 4. 3 Two-Grid Methods. - Problems. - 5 Ill-Conditioned Linear Systems. - 5. 1 Condition Number. - 5. 2 Singular Value Decomposition. - 5. 3 Tikhonov Regularization. - Problems. - 6 Iterative Methods for Nonlinear Systems. - 6. 1 Successive Approximations. - 6. 2 Newton's Method. - 6. 3 Zeros of Polynomials. - 6. 4 Least Squares Problems. - Problems. - 7 Matrix Eigenvalue Problems. - 7. 1 Examples. - 7. 2 Estimates for the Eigenvalues. - 7. 3 The Jacobi Method. - 7. 4 The QR Algorithm. - 7. 5 Hessenberg Matrices. - Problems. - 8 Interpolation. - 8. 1 Polynomial Interpolation. - 8. 2 Trigonometric Interpolation. - 8. 3 Spline Interpolation. - 8. 4 Bézier Polynomials. - Problems. - 9 Numerical Integration. - 9. 1 Interpolatory Quadratures. - 9. 2 Convergence of Quadrature Formulae. - 9. 3 Gaussian Quadrature Formulae. - 9. 4 Quadrature of Periodic Functions. - 9. 5 Romberg Integration. - 9. 6 Improper Integrals. - Problems. - 10 Initial Value Problems. - 10. 1 The PicardLindelöf Theorem. - 10. 2 Euler's Method. - 10. 3 Single-Step Methods. - 10. 4 Multistep Methods. - Problems. - 11 Boundary Value Problems. - 11. 1 Shooting Methods. - 11. 2 Finite Difference Methods. - 11. 3 The Riesz and Lax-Milgram Theorems. - 11. 4 Weak Solutions. - 11. 5 The Finite Element Method. - Problems. - 12 Integral Equations. - 12. 1 The Riesz Theory. - 12. 2 Operator Approximations. - 12. 3 Nyström's Method. - 12. 4 The Collocation Method. - 12. 5 Stability. - Problems. - References. Language: English