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 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.
2012-11-21, Elena Celledoni