# Lecture plan

**This schedule is tentative, changes will appear.**

Textbook: E. Süli and D. Mayers, *An introduction to Numerical Analysis*, Cambridge University Press (2003).

Week | Topics | Lecturer | Reading | Exercises |
---|---|---|---|---|

34 | Introduction to the course. Taylor's theorem, big O-notation, rounding errors | TK | Introduction | |

Numerical solution of nonlinear equations Existence of solutions Simple iterations Fixed point theorem Rate of convergence Contraction mapping theorem | TK | 1.1-1.2 in S&M (up to Theorem 1.4) Scalar equations: Convergence of Newton's method (Theorem 1.8) and max number of fixed point iterations (Theorem 1.4) with proofs. | ||

35 | Rate of convergence, asymptotic error constant, order of convergence Convergence of fixed point iterations. Bisection and Newton's method, Convergence analysis Secant method | TK | 1.2 (from Theorem 1.4) - 1.8 in S&M Slides: Solving non-linear equations | Exercise 2 is posted here. |

Multi-variate Taylor's expension. Newtons method for system of equations. Convergence of Newtons method (Theorem 4.4) (self study). Fixed point iterations for systems of equations. The contraction mapping theorem in max-norm (Theorem 4.1 and 4.2). | TK | Notes: - Solution of systems of nonlinear equations - Newton's method for systems of non-linear equations - Examples of nonlinear iteration | ||

36 | Numerical linear algebra Naiv Gauss elimination LU factorization | TK | 2.1-2.2 in S&M | Project 1 is available here. |

Gauss-elminiation with partial pivoting, Vector and matrix norms, sub-ordinate matrix norms,Stability of linear system, Condition number, Gershgorin’s theorem | TK | 2.2-2.6 in S&M Notes: Numerical methods for linear algebra | ||

37 | Special matrices: Symmetric, positive-definite, diagonally dominant, tridiagonal Cholesky factorization, Iterative methods for linear systems, Jacobi method, Gauss-Seidel method, spectral radius. | TK | 3.1-3.3 in S&M Section 5 in the note on Linear Algebra. | Exercise 3 is posted here. |

Numerical interpolation Lagrange interpolation. Existence and uniqueness of interpolation polynomials. Error formula. All with proofs. | TK | 6.1-6.3 in S&M | ||

38 | Hermite interpolation with Lagrange polynomials. Numerical differentiation. | TK | 6.4 and 6.5 in S&M | Exercise 4 is posted here. |

39 | The max-norm of function spaces. Weierstrass approximation theorem. Minimax polynomials (existence, uniqueness, properties). | TK | 8.1-8.3 (no proofs required.) | |

Chebyshev poynomials, their properties, why they are useful in the interpolation context. | TK | 8.4 and 8.5 (with proofs) | Exercise 5 is posted here. | |

40 | Numerical integration How Lagrange interpolation polynomials can be used to construct numerical quadrature. Error estimates. | TK | 7.1-7.3 | |

Composite formulas. The Euler-Maclaurin expansion. Extrapolation methods | TK | 7.4-7.7 | Exercise 6 is posted here. | |

41 | Polynomial expansion in the 2-norm Inner product space. Best approximation in the 2-norm. | TK | 9.1-9.3 | |

Newton interpolation polynomial and divided differences. Error for Lagrange interpolation. | TK | Slides: Divided differences and Newton interpolation polynomial Reading material: Newton form | No new exercises this week. | |

42 | Adaptive Simpson. Orthogonal polynomial. Comparisons. | TK | Notes: Adaptive Simpson with Matlab code 9.4-9.5 | |

43 | Gauss Quadrature | TK | Chapter 10.1-10.5 | Exercise 7 is posted here. |

44 | Splines | TK | Chapter 11 | |

Ordinary differential equations Eulers method: implementation, convergence proof, how to measure the order of a method. | TK | Notes: Numerical solution of ordinary differential equations Notes, section 1 and 2 | ||

45 | Numerical solution of ODE's | TK | Notes, section 3. | |

Order conditions for Runge-Kutta methods Error estimates and stepsize selection Embedded Runge-Kutta methods | TK | Notes, section 4. | Exercise 8 is posted here. | |

46 | Stiff ordinary differential equations Linear Multistep Methods | TK | Notes, section 5 and 7 | |

47 | No lectures. Finishing the project work |

#### Lecturers:

- TK: Trond Kvamsdal