## Curriculum and Lecture plan

The Chapters of the book by Süli and Mayers (SM) specified in the third column of the 'Activity' table below are part of the curriculum.

The supplementary materials specified in the column 'Extra Material' of the 'Activity' table below are part of the curriculum.

All problem sets considered in the lectures including those suggested and published in the last column of the table below (supervision of these exercises has been offered in the two weeks after February 16 and in the weeks from April 1st to the end of the course) are part of the curriculum.

All old exams sets are part of the curriculum (those excluded will be specified).

All assignments are part of the curriculum.

## Activity/Tentative plan

Week | Dates | Theme | Süli and Mayers | Extra Material | Recommended exercises. | Lecturer | |
---|---|---|---|---|---|---|---|

2 | 12.01 14.01 | 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 | 19.01 21.01 | Convergence of fixed point iterations. Brouwer's Theorem. Contraction mapping Theorem. | 1.1-1.2 (theorems 1.4, 1.5,1.6 excluded) 1.3,1.4 (theorem 1.8 included). 4.1 Theorem 4.1. Newton for systems. | Note on Newton methods for systems. Also included in note on ODEs chapter 9. Bisection code. | Exercise 4 in exam 2017. | KEF | |

4 | 26.01 28.01 | Newton for systems. Solution of systems of linear equations with iterative methods. | 2.7 (Theorems 2.4,2.5,2.6 excluded). | See iterative methods in ch. 13 of Finite difference schemes and partial differential equations, John C. Strikwerda, SIAM, (second edition). 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 | 02.02 04.02 | Least squares, condition numbers stability of linear systems, SVD | 2.7, 2.9 (2.8 is not part of the curriculum). | KEF | |||

6 | 09.02 11.02 | Computing SVD. Householder transformations, computing QR. Gaussian Elimination with pivoting. | 2, 5.5 | Linear algebra note part 2. | Computing QR and SVD, exercises | KEF | |

7 | 16.02 18.02 | Polynomial interpolation. | 6.1, 6.2,6.3. | Animation code for Runge phenomenon gif animation | Exercises on Interpolation. | YS | |

8 | 23.02 25.02 | Polynomial approximation. | 6.4, 8.1 to 8.5 | Lecture summary - Interpolation | Divided differences-Hermite interpolation | YS | |

9 | 02.03 04.03 | Numerical integration (Newton-Cotes and its error, composite trapezoidal and Simpson with error estimates, Bernoulli polynomials and Euler-Maclaurin expansion with proof) | 7.1,7.2,7.3,7.4. 7.5, 7.6, 7.7. | Exercises on quadrature. | YS | ||

10 | 09.03 11.03 | Numerical Integration (extrapolation, Romberg quadrature, Gaussian quadrature). Numerical differentiation. | 10.2 10.3, 10.4, 6.5 | YS | |||

11 | 16.03 18.03 | Initial value problems for ODEs | Note on ODEs. Chapters 1-6 except 4.2 and 5, and 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 | 23.03 25.03 | Initial value problems for ODEs | Note on ODEs. Chapters 1-6 except 4.2 and 5, and 6.5. | KEF | |||

13 | 30.03 01.04 | Finalising, assignments. | |||||

14 | 06.04 08.04 | Definition of finite differences approximations of the first and second derivatives. Boundary value problems | [Note on BVPs Ch 2 (section 2.3 excluded) | YS | |||

15 | Easter holidays - no teaching | ||||||

16 | 20.04 22.04 | 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 | YS | |||

17 | 27.04 29.04 | Boundary value problems. Concluding remarks. | YS |

#### 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
- Lipscitz 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 (see problems lecture 4 ).
- 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
- Forward Euler method, Backward Euler 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.