Activity/Tentative plan
| Week | Dates | Theme | Süli and Mayers | Extra Material | Recommended exercises. | Lecturer | ||
|---|---|---|---|---|---|---|---|---|
| 2 | 11.01 12.01 | Introduction to the course, principles of computational mathematics, learning outcome of the course. Floating point numbers, roundoff error, stability of problems and algorithms. Bisection method and Newton method. | Note on floating point numbers. Pendulum code | exercise 4.7 in SM. | KEF | |||
| 3 | 18.01 19.01 | Convergence of fixed point iterations. Brouwer's Theorem. Contraction mapping Theorem. | Sects. 1.1.-1.4. Sects. 4.1.-4.2. | Note on Newton methods for systems. Also included in note on ODEs chapter 9. Bisection code. | Exercise 4 in exam 2017. | KEF | ||
| 4 | 25.01 26.01 | Newton's method for systems. Solution of systems of linear equations with iterative methods. | Sect. 2.7. | Linear algebra note part 1. Code simple use of autograd, Newton method for systems. | Problem 2 c and problem 4 in exam from 2015. Recommended execises. | KEF | ||
| 5 | 01.02 02.02 | Least squares, condition numbers stability of linear systems, SVD | Sects. 2.7., 2.9. | KEF | ||||
| 6 | 08.02 09.02 | Computing SVD. Householder transformations, computing QR. Gaussian Elimination with pivoting. | Sections 2, 5.5 | Linear algebra note part 2. | Computing QR and SVD, exercises | KEF | ||
| 7 | 15.02 16.02 | Polynomial interpolation. | Sects. 6.1, 6.2, 6.3. | Animation code for Runge phenomenon gif animation | Exercises on Interpolation. | VT | ||
| 8 | 22.02 24.02 | Polynomial approximation. | Sects. 6.4, 8.1-8.5 | Lecture summary - Interpolation | Divided differences-Hermite interpolation | VT | ||
| 9 | 01.03 02.03 | Numerical integration (Newton-Cotes and its error, composite trapezoidal and Simpson with error estimates, Bernoulli polynomials and Euler-Maclaurin expansion with proof) | Sects. 7.1., 7.2., 7.3., 7.4., 7.5., 7.6., 7.7. | Exercises on quadrature. | VT | |||
| 10 | 08.03 09.03 | Numerical Integration (extrapolation, Romberg quadrature, Gaussian quadrature). Numerical differentiation. | Sects. 10.2., 10.3., 10.4., 6.5. | VT | ||||
| 11 | 15.03 16.03 | Initial value problems for ODEs | Note on ODEs. Chapters 1-6, except Sect. 4.2, Chapt. 5, Sect. 6.4 and Sect. 6.5. | Lecture summary-ODEs. Convergence of one-step methods | Problem 7, exam June 2016. Problem 6 exsam 2015. Problem 6 exsam 2014. Problem 6 retake exam 2013. Problem 6 exsam 2013. Problem 2, exam 2012. Problem 2, exam 2011. | KEF | ||
| 12 | 22.03 23.03 | Initial value problems for ODEs | Note on ODEs. Chapters 1-6, except Sect. 4.2, Chapt. 5, Sect. 6.4 and Sect. 6.5. | KEF | ||||
| 13 | 29.03 30.03 | Finalising, assignments. | ||||||
| 14 | Easter holidays - no teaching | |||||||
| 15 | 12.04 13.04 | Definition of finite differences approximations of the first and second derivatives. Boundary value problems | [Note on BVPs Ch 2 (section 2.3 excluded) | VT | ||||
| 16 | 19.04 20.04 | Boundary value problems. | [Note on BVPs Ch 2 (section 2.3 excluded) [Note on finite difference for heat equation, briefly, page 28-34 and 55-56 | VT | ||||
| 17 | 26.04 27.04 | Boundary value problems. Concluding remarks. | VT |
Formulae, concepts and definitions you need to remember
This is a non exhaustive tentative list of formulae and definitions you are encourage to remember for the exam. This list will be updated during the semester.
- Definition of contraction
- Lipschitz condition for functions
- Definition of convergence of order p and quadratic convergence
- Newton method for scalar equations and for systems
- Fixed point equation and fixed point iteration
- Contractions and convergence of fixed point iteration.
- Definition of spectral radius of a matrix
- Definition of eigenvalues and eigenvectors
- Definition of condition number
- Definition of 1- 2- and infinity-norm for vectors, matrices, and functions of one variable
- Cauchy-Schwartz inequality.
- Symmetric matrices
- Symmetric and positive definite matrix
- LU factorisation, pivoting
- QR factorisation and how to use Householder transformations to obtain the QR factorisation.
- Gauss-Seidel Jacobi and SOR iterative methods and conditions for their convergence
- Lagrange interpolation polynomial
- Hermite interpolation polynomial
- Error formula for the interpolation polynomial for Lagrange and Hermite interpolation
- Basic concepts of orthogonal polynomials
- Newton form of the interpolation polynomial and divided differences
- Newton-Cotes quadrature formulae.
- Chebyshev polynomials (definition and basic properties)
- Gaussian quadrature
- Euler method, improved Euler method, modified Euler method, trapezoidal method for ODEs
- Need to know how to apply a Runge-Kutta method to an ODE, when you are given the parameters of the Runge-Kutta method (Butcher tableau).
- Consistency of one-step methods.
- Need to know how to apply a linear multistep method to an ODE when you are given the parameters of the linear multi-step method.
- Zero-stability and consistency for linear multistep methods.
- Familiarity with finite difference formulae, forward differences, backward differences and central differences.
- Stability and consistency for finite differences approximations of boundary value problems.
- How to prove convergence of a consistent and stable finite difference discretisation of a boundary value problem.