Schedule
This schedule is not yet final. Changes will be made during the semester. At the end of the course this page will constitute 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 | 23.08, 25.08 | 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 | 30.09, 01.09 | 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 | 06.09, 08.09 | Ch.1, 4 | Nonlinear systems of equations. | Note about nonlinear equations (part of the curriculum). | |
| 37 | 13.09, 15.09 | Ch. 2.1-2.6 | Iterative solution of linear systems. 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 | 20.09, 22.09 | Supervision of the first project. | |||
| 39 | 27.09, 29.10 | 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. | |
| 40 | 04.10, 06.10 | 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. | |
| 41 | 11.10, 13.10 | Ch. 9.3,9.4, 11.1, 11.2,11.3, 11.4 | Best approximation in the 2-norm. Splines. | ||
| 42 | 18.10, 20.10 | Ch. 7.1-7.6 | Quadrature: Newton-Cotes formulae, composite formulae. | Note on Euler-Mclaurin formula and Bernoulli polynomials. | |
| 43 | 25.10, 27.10 | Ch. 7.6-7.7, 10.-10-4 | Extrapolation. Romberg algorithm. Gauss quadrature. | Summary on Euler-Maclaurin and Romberg algorithm. Summary on Gauss 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. | |
| 45 | 06.11, 08.11 | Ch. 12 | 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 | |
| 46 | 13.11, 15.11 | Ch. 12 | odes | Runge-Kutta methods. RK-methods order conditions, step-size selection and linear stability. | |
| 47 | 20.11, 22.11 | Ch. 12 | odes | Linear multi-step methods. |