# Schedule

**This schedule is now final, and it constitutes the curriculum of TMA4215.**

SM = Süli and Mayers, An introduction to Numerical Analysis

N = Notes

Week | Date | SM | N | Subject | Handout notes which are part of the curriculum. Links. |
---|---|---|---|---|---|

34 | 21.08, 23.08 | Ch. 2.7, Appendix A | Background. Floatingpoint numbers, rounding errors, stability of problems and algorithms. Condition numbers. Taylor's theorem and Big O notation. Vector and matrix norms. Cauchy-Schwarz inequality. | slides (part of the curriculum). Floating point numbers. Interval arithmetic. Computer assisted proofs. Taylor's theorem. Big O notation. | |

35 | 28.08, 30.08 | Ch.2 p 70-72, Ch. 1, p12 def 1.4, p22-23 p 25, p 28 | Condition number for linear systems. Numerical solution of nonlinear equations. | Cramer's rule. | |

36 | 04.09, 06.09 | Ch.1, 4 | Nonlinear systems of equations. Iterative solution of linear systems. | Note about nonlinear equations (part of the curriculum). | |

37 | 11.09, 13.09 | Ch. 2.1-2.6 | Krylov subspace methods. Preconditioning. LU- Cholesky and other factorizations. Gershgoring's theorem. | Note on linear algebra with some examples (with small matrices) (part of the curriculum). Ch. 2.1 is not covered in the lectures but it is part of the curriculum. | |

38 | 18.09, 20.09 | Ch. 6.1-6.3 8.1-8.5 | Interpolation. Lagrange interpolation polynomial and error. Normed linear spaces. Max-norm and 2-norm of continuous functions on [a,b].Theorems of Faber and Weierstrass. Best approximation in the max-norm. Chebishev points and near-optimality of the interpolation error. | Summary of chapter 6 and 8. Matlab codes interpolation.m and lk.m. Can be used for comparison of interpolation on equidistant and Chebyshev nodes. | |

39 | 25.09, 27.09 | Ch. 6.4, 6.5, 9.1, 9.2. | Hermite interpolation. Numerical differentiation. Best approximation in the 2-norm. Divided differences and Newton form of the interpolation polynomial. | Slides on divided differences and Newton interpolation polynomial (part of the curriculum). der.pdf Error in the numerical approximation of the derivative of cos(x) for x=3,(-sin(3)=-0.14112000805987), by difference approximations (Taylor theorem) and for smaller and smaller values of h. When h is too small the rounding error starts propagating. | |

40 | 02.10, 04.10 | Ch. 9.3,9.4, 11.1, 11.2,11.3, 11.4 | Best approximation in the 2-norm. Splines. | Approximation in the 2 norm. Splines | |

41 | 09.10, 11.10 | Semester project supervision | |||

42 | 16.10, 18.10 | Ch. 7.1-7.6 | Quadrature: Newton-Cotes formulae, composite formulae. Romberg algorithm. Gauss quadrature. | Note on Euler-Mclaurin formula and Bernoulli polynomials. Summary on Euler-Maclaurin and Romberg algorithm. | |

43 | 23.10, 25.10 | Ch. 7.6-7.7, 10.-10-4 | Gauss quadrature. Adaptive quadrature. ODEs. | Gauss and Adaptive quadrature gvst.m Matlab file comparing Gauss uqadrature on 2 nodes with the composite Trapezium rule for a chosen function. | |

44 | 30.10, 01.11 | Ch. 12 | odes | Exercise. Note on odes, introduction.Note on odes. Supervision of the second project. Euler method, convergence. Implementation of explicit and implicit Euler. Checking the order of a method. Matlab files for odes: test1.m eulerstep.m beulerstep.m fode.m fodeb.m fLV.m LV.m | |

45 | 06.11, 08.11 | Ch. 12 | odes | Runge-Kutta methods. RK-methods http://www.math.ntnu.no/emner/TMA4215/2009h/notater/lecture3.pdf order conditions | |

46 | 13.11, 15.11 | Ch. 12 | odes | Linear multi-step methods. | |

47 | 20.10 | Ch. 12 | ODEs | Summary on linear multistep methods. |