# Lecture plan

Will be updated throughout the semester.

Week | Themes | Curriculum | Notes |
---|---|---|---|

2 | Introduction to MATLAB, absolute and relative error, Taylor series, O(h) notation, floating point numbers, computer artithmetic, and roundoff errors. | Ch. 1.1, 1.2, 1.3 | |

3 | Loss of significance (cancellation errors), linear systems, Gaussian elimination, Matrix and vector norms, ill-conditioned systems | Ch. 1.4, 2.1, 8.4 (part about norms and the condition number) | |

4 | Row exchanges and scaled partial pivoting in Gaussian elimination, numerical complexity analysis, Gaussian elimination for tridiagonal linear systems, matrix factorizations, connection between LU factorization and Gaussian elimination | Ch. 2.2, 2.3 (not pentadiagonal systems), 8.1 | |

5 | LDL^{T} factorization for symmetric matrices, Cholesky factorization. Non-linear equations: bisection method. | Ch. 8.1, 3.1 | |

6 | Non-linear equations: Method of false position, fixed-point iterations, Newton's method. | Ch 3.1, 3.2 . | note |

7 | Newton's method for nonlinear systems, the secant method. Eigenvalues. | Ch. 3.2, 3.3, 8.2 (up to SVD) | |

8 | Power method (including inverse and shifted power methods), iterative solvers for linear systems (meaning: (damped) Richardson, Jacobi, Gauss-Seidel, and SOR), a short discussion on their convergence properties. Polynomial interpolation (Lagrangian interpolation) | Ch. 8.3, 8.4 (not conjugate gradient), 4.1 | |

9 | Newton interpolation and divided differences, Project work | Ch. 4.1 | |

10 | Interpolation errors, minimization of interpolation errors: Chebyshev points, estimation of derivatives, Richardson extrapolation. | Ch. 4.2, 4.3 | |

11 | Numerical integration: Trapezoid rule, Simpson rule, Newton-Cotes rules, approximation errors and convergence rates. | Ch. 5.1, 5.2, 5.3. | |

12 | Numerical integration: Romberg integration, adaptive quadrature, Gaussian quadrature & Legendre polynomials, singularities. | Ch. 5.3, 5.4. | |

13 | Splines: Linear splines, natural cubic splines. | Ch. 6.1, 6.2. (Not specific material on quadratic splines). | |

15 | Least squares, linear regression, and normal equations. Initial value problems: Euler's method, Runge-Kutta methods, | Ch. 9.1, 9.3, 7.1, 7.2 | |

16 | Initial value problems: Multi-step methods, implicit methods and stiff ODEs. | Ch. 7.3, 7.4, 7.5 |

There will also be a double lecture with repetition of desired topics and questions (we might also go through some relevant former exam problems), closer to the final exam.