====== 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. | [[http://www.math.ntnu.no/emner/TMA4215/2011h/notes/motivasjonnumMat11.pdf| slides]] (part of the curriculum). [[http://en.wikipedia.org/wiki/Floating_point|Floating point numbers.]] [[http://en.wikipedia.org/wiki/Interval_arithmetic|Interval arithmetic.]] [[http://en.wikipedia.org/wiki/Computer-assisted_proof|Computer assisted proofs.]] [[http://en.wikipedia.org/wiki/Taylor%27s_theorem|Taylor's theorem.]] [[http://en.wikipedia.org/wiki/Big_O_notation|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.| [[http://en.wikipedia.org/wiki/Cramer%27s_rule| Cramer's rule.]]| |36 | 06.09, 08.09 | Ch.1, 4 | | Nonlinear systems of equations. | [[http://www.math.ntnu.no/emner/TMA4215/2010h/notes/nonlin.pdf|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. | [[http://www.math.ntnu.no/~elenac/Linalg.pdf|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. | [[http://www.math.ntnu.no/~elenac/Interpolationsummary.pdf|Summary of chapter 6 and 8.]] Matlab codes {{:tma4215:2010h:interpolation.m| interpolation.m}} and {{:tma4215:2010h:lk.m|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. | [[http://www.math.ntnu.no/~elenac/Interpolationsummary2.pdf|Slides on divided differences and Newton interpolation polynomial]] (part of the curriculum). {{:tma4215:2010h:der.pdf| 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 [[http://www.math.ntnu.no/emner/TMA4215/2010h/notes/EulerMcLaurinBernoulli.pdf|Euler-Mclaurin]] formula and Bernoulli polynomials.| |43 | 25.10, 27.10 | Ch. 7.6-7.7, 10.-10-4 | | Extrapolation. Romberg algorithm. Gauss quadrature. |[[http://www.math.ntnu.no/~elenac/EulerMaclaurin.pdf|Summary on Euler-Maclaurin and Romberg algorithm.]] [[http://www.math.ntnu.no/~elenac/Gaussquadrature.pdf|Summary on Gauss quadrature.]] [[http://www.math.ntnu.no/~elenac/gvst.m|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| [[http://www.math.ntnu.no/~elenac/Exercise.pdf|Exercise.]] Note on [[http://www.math.ntnu.no/emner/TMA4215/2010h/notes/odes1.pdf|odes]], introduction.Note on [[http://www.math.ntnu.no/emner/TMA4215/2010h/notes/odes2.pdf|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: [[http://www.math.ntnu.no/~elenac/matlab/test1.m|test1.m]] [[http://www.math.ntnu.no/~elenac/matlab/eulerstep.m|eulerstep.m]] [[http://www.math.ntnu.no/~elenac/matlab/beulerstep.m|beulerstep.m]] [[http://www.math.ntnu.no/~elenac/matlab/fode.m|fode.m]] [[http://www.math.ntnu.no/~elenac/matlab/fodeb.m|fodeb.m]] [[http://www.math.ntnu.no/~elenac/matlab/fLV.m|fLV.m]] [[http://www.math.ntnu.no/~elenac/matlab/LV.m|LV.m]]| |46 | 13.11, 15.11 | Ch. 12 | | odes | Runge-Kutta methods. RK-methods [[http://www.math.ntnu.no/emner/TMA4215/2009h/notater/lecture3.pdf|order conditions]], [[http://www.math.ntnu.no/emner/TMA4215/2009h/notater/lecture4.pdf|step-size selection and linear stability]]. | |47 | 20.11, 22.11 | Ch. 12 | | odes | [[http://www.math.ntnu.no/emner/TMA4215/2009h/notater/lecture6.pdf|Linear multi-step methods]].|