# Lecture log

Here follows a short overview of what has been lectured.

Date | Lecturer | Topic | Litterature |
---|---|---|---|

06.01 – 19.02 | YL | See plan | |

24.02 | GS | Numerical computations in general: "computers and numbers", floating point systems. (In K. 19.1, the sections "error" and "error propagation" should be studied on your own). Iteration methods: Fixed-point iteration (beware incorrect theorem on fixed-point iteration convergence in Kreyszig!), Newton's method (not done). | K. 19.1 – 19.2 |

26.02 | GS | Newton's method and its convergence order, with Heron's method for computing \(\sqrt{2}\) as an example. Secant method. Newton's method for systems of equations, with example. | K. 19.2, Notes on Newton's method for systems of equations |

03.03 | GS | Polynomial interpolation: Uniqueness, computation using Lagrange's method, computation using Newton's divided differences, and examples. | K. 19.3 |

06.03 | GS | Error in polynomial interpolation. Polynomial interpolation gives us numerical integration: The rectangle rule (0th degree polynomials), the trapezoid rule (1st degree polynomials), Simpson's rule (2nd degree polynomials). Error in numerical integration. | K. 19.3, K. 19.5 |

10.03 | GS | Numerical linear algebra in general. Partial pivoting in Gaussian elimination gives increased resilience to rounding error. Computational complexity of Gaussian elimination. LU-factorization writes \(A=LU\) for \(L\) lower- and \(U\) upper-triangular. Solving \(Ax=b\) becomes solving \(Ly=b\) and \(Ux=y\) (which is easy since the matrices are triangular). Computational complexity of solving by LU-factorization. Examples. Cholesky's method, \(A=LL^\top\) when \(A\) is positive definite, mentioned. | K. 20.1 – 20.2 |

12.03 | GS | Solving linear systems by iteration; Gauss–Seidel and Jacobi method. Gauss–Seidel implemented Matlab, Jacobi example on blackboard. Matrix norms and sufficient conditions for convergence of Gauss–Seidel and Jacobi (note: Kreyszig does not cover the very useful condition "strict diagonal dominance!) | K. 20.3 |

17.03 | GS | Numerical methods for solving first-order ODEs: Euler's method, improved Euler's (Heun's) method, RK4, Runge–Kutta methods in general (described by Butcher tables). Stability considerations. Explicit vs implicit mentioned, but first implicit method postponed for next lecture. | K. 21.1, notes on stability of numerical methods for ODEs and notes on Butcher tables |

19.03 | GS | Backwards Euler, an implicit method. Transforming higher-order ODEs to systems of first-order ODEs (with example). All the RK methods work for systems as well; in particular, wrote down Euler, improved Euler, RK4 and backwards Euler for systems. Examples of a few of them. Example of using RK4 to solve a very complicated system (if you can't play the video, try this alternative format). | K. 21.1, K. 21.3 |

24.03 | GS | PDEs and their numerical solution in general. Parabolic type illustrated with heat equation. Spatial central differences reduces the heat equation to systems of temporal ODEs, which we can solve with the methods we have. Forward and backward Euler written out. The Crank–Nicolson method (which comes from an implicit ODE method we haven't seen before) also written out. Examples next time. | K. 21.4, K. 21.6, notes on CN |

26.03 | GS | Crank–Nicolson method repeated, with example of use. Five-point scheme for hyperbolic PDEs (wave equation as prototype). Five-point scheme for elliptic PDEs (Poisson equation as prototype). | K. 21.4, K. 21.6 – 21.7, notes on CN |

09.04 | GS | Five-point scheme example of use. Quick summary. Taste of Finite Element Methods, a different way to treat PDEs numerically (not in the curriculum). | |

13.05 | GS | Exam repetition. Problems chosen by popular request: 4M/4N, spring 2005, problem 5; 4M/4N, retake 2006, problem 6 (didn't fully write out the solution); 4M/4N, spring 2011, problem 1; 4D, fall 2014, problem 6 |

# Things displayed during the lectures

Various slides, animations and code have been displayed during the lectures. Not all of them make sense out of context, but here they are.

## Slides

** Note:** Some of the slides make little sense on their own, and shouldn't be used without knowledge of the corresponding comments made during lectures.

- Yurii's slides from the first parts of the curriculum. The slides contain the lecture plan, they list the key points of the course. I also reccommend to use them when preparing to the exam: try to recover the content of each item without textbook first.

## Animations

- Illustration of convolution (not shown during any lectures)