## Curriculum

The Chapters of the book by Süli and Mayers specified in the third column of the "Activity table" below.

The supplementary materials specified on the fourth column of the "Activity table" below.

All problems sets considered in the lectures.

All old exams sets.

All assignments.

#### Formulae, concepts and definitions you need to remember

This is a non exhaustive list of formulae and definitions you are encourage to remember for the exam.

• 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
• Error formula for the interpolation polynomial
• Newton form of the interpolation polynomial and divided differences
• Chebishev polynomials (definition and basic properties)
• 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.

Note on ODEs

Note on BVPs

## Activity

Week Dates Theme Süli and Mayers Extra Material Recommended exercises.
2 07.01 10.01 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. 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). exercise 4.7 in SM (for a solution see Problems lecture 3).
3 14.01 17.01 Convergence of Newton method. Newton for systems. Introduction to Python. Supervision of the first assignment. 1.4 (theorem 1.8 included), 4.1 Theorem 4.1.
4 21.01 24.01 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.
5 28.01 31.01 Least squares, condition numbers stability of linear systems, SVD 2.7, 2.9 (2.8 is not part of the curriculum).
6 04.02 Gaussian Elimination (no exercise lectures on February the 7th) 2
7 11.02 14.02 Polynomial interpolation 6.1, 6.2,6.2,6.3,6.4.
8 18.02 21.02 Polynomial interpolation. Divided differences (See problems Lecture 7). 8.1,8.2 (lemma 8.1, Theorem 8.1 only idea of the proof),8.3 (Theorems 8.2 and 8.5 included but without proof, theorems 8.3 and 8.4 excluded), 8.4, 8.5
9 25.02 28.02 Project second part
10 04.03 07.03 Numerical integration and differentiation. 6.5, 7.1,7.2,7.3,7.4.