====== Lecture plan ====== **This schedule is tentative, changes will appear.** Textbook: E. Süli and D. Mayers, //An introduction to Numerical Analysis//, Cambridge University Press (2003). ^ Week ^ Topics ^ Lecturer ^ Reading ^ Exercises ^ ^ 34 | Introduction to the course. \\ Taylor's theorem, big O-notation, rounding errors | LHO | [[http://www.math.ntnu.no/emner/TMA4215/2015h/notes/intro.pdf|Introduction]] | | ^ | Introduction to MATLAB | EHH | [[http://www.math.ntnu.no/emner/TMA4215/2015h/notes/intro_matlab.pdf|Introduction to MATLAB]] | Exercise 1 is posted [[tma4215:2015h:assignments|here]]. | ^ 35 | **Numerical solution of nonlinear equations** \\ Existence of solutions \\ Simple iterations \\ Fixed point theorem \\ Rate of convergence \\ Contraction mapping theorem | AM | 1.1-1.2 in S&M (up to Theorem 1.4)\\ Scalar equations: Convergence of Newton's method (Theorem 1.8) and max number of fixed point iterations (Theorem 1.4) with proofs. | | ^ | Rate of convergence, asymptotic error constant, order of convergence \\ Convergence of fixed point iterations. Bisection and Newton's method, Convergence analysis \\ Secant method | AM | 1.2 (from Theorem 1.4) - 1.8 in S&M \\ Slides: [[http://www.math.ntnu.no/emner/TMA4215/2015h/notes/nonlineq.pdf|Solving non-linear equations]] | Exercise 2 is posted [[tma4215:2015h:assignments|here]]. | ^ 36 | Multi-variate Taylor's expension. Newtons method for system of equations. Convergence of Newtons method (Theorem 4.4) (self study). \\ Fixed point iterations for systems of equations. The contraction mapping theorem in max-norm (Theorem 4.1 and 4.2). | AM | Notes:\\ - [[http://www.math.ntnu.no/emner/TMA4215/2015h/notes/nonlin.pdf|Solution of systems of nonlinear equations]] \\ - [[http://www.math.ntnu.no/emner/TMA4215/2015h/notes/newtons_method_for_systems.pdf|Newton's method for systems of non-linear equations]] | | ^ | **Numerical linear algebra** \\ Naiv Gauss elimination\\ LU factorization \\ | TK | 2.1-2.2 in S&M \\ | Project 1 is available [[tma4215:2015h:project|here]]. | ^ 37 | Gauss-elminiation with partial pivoting, Vector and matrix norms, sub-ordinate matrix norms,Stability of linear system, Condition number, Gershgorin’s theorem | TK | 2.2-2.6 in S&M\\ Notes: [[http://www.math.ntnu.no/emner/TMA4215/2015h/notes/Linalg.pdf|Numerical methods for linear algebra]] | | ^ | Special matrices: Symmetric, positive-definite, diagonally dominant, tridiagonal \\ Cholesky factorization, Iterative methods for linear systems, Jacobi method, Gauss-Seidel method, spectral radius. | TK | 3.1-3.3 in S&M \\ Section 5 in the note on Linear Algebra. | Exercise 3 is posted [[tma4215:2015h:assignments|here]]. | ^ 38 | **Numerical interpolation** \\ Lagrange interpolation. Existence and uniqueness of interpolation polynomials. Error formula. All with proofs. | AM | 6.1-6.3 in S&M | | ^ | Hermite interpolation with Lagrange polynomials. Numerical differentiation. | TK | 6.4 and 6.5 in S&M | Exercise 4 is posted [[tma4215:2015h:assignments|here]]. | ^ 39 | The max-norm of function spaces. Weierstrass approximation theorem. Minimax polynomials (existence, uniqueness, properties). | TK | 8.1-8.3 (no proofs required.) | | ^ | Chebyshev poynomials, their properties, why they are useful in the interpolation context. | TK | 8.4 and 8.5 (with proofs) | Exercise 5 is posted [[tma4215:2015h:assignments|here]]. | ^ 40 | ** Numerical integration ** \\ How Lagrange interpolation polynomials can be used to construct numerical quadrature.\\ Error estimates. | TK | 7.1-7.3 | | ^ | Composite formulas. The Euler-Maclaurin expansion. Extrapolation methods | TK | 7.4-7.7 | Exercise 6 is posted [[tma4215:2015h:assignments|here]]. | ^ 41 | ** Polynomial expansion in the 2-norm ** \\ Inner product space. Best approximation in the 2-norm. | TK | 9.1-9.3 | | ^ | Newton interpolation polynomial and divided differences. \\ Error for Lagrange interpolation. | AM | Slides: [[http://www.math.ntnu.no/~elenac/Interpolationsummary2.pdf| Divided differences and Newton interpolation polynomial]]\\ Reading material: [[http://www.math.ntnu.no/emner/TMA4215/2015h/notes/NewtonForm.pdf|Newton form]] | No new exercises this week. | ^ 42 | Adaptive Simpson.\\ Orthogonal polynomial. Comparisons. | TK | Notes: [[http://www.math.ntnu.no/emner/TMA4215/2015h/notes/AdaptiveSimpson.pdf|Adaptive Simpson]] with [[http://www.math.ntnu.no/emner/TMA4215/2015h/notes/simpson.m|Matlab code]]\\ 9.4-9.5 | | ^ | ** Gauss Quadrature ** | TK | Chapter 10.1-10.5 | Exercise 7 is posted [[tma4215:2015h:assignments|here]]. | ^ 43 | ** Splines ** | TK | Chapter 11 | | ^ | ** Ordinary differential equations ** \\ Eulers method: implementation, convergence proof, how to measure the order of a method. | TMO | Notes: [[http://www.math.ntnu.no/emner/TMA4215/2015h/notes/ak-odenote.pdf|Numerical solution of ordinary differential equations]] \\ Notes, section 1 and 2 | | ^ 44 | Numerical solution of ODE's | TMO | Notes, section 3. | | ^ | Order conditions for Runge-Kutta methods \\ Error estimates and stepsize selection \\ Embedded Runge-Kutta methods | TMO | Notes, section 4. | Exercise 8 is posted [[tma4215:2015h:assignments|here]]. | ^ 45 | **No lectures**. Project work | | | | ^ 46 | **No lectures**. Project work | | | | ^ 47 | Stiff ordinary differential equations \\ Linear Multistep Methods | TMO | Notes, section 5 and 7 | | === Lecturers: === * TK: Trond Kvamsdal * LHO: Lars Hov Odsæter * AM: Asif Mushtaq * EHH: Eirik Hoel Høiseth * TMO: Timo Matteo van Opstal