F19NB - Numerical Analysis B

Course leader(s):

Sebastien Loisel (Edinburgh, January)

Aims

The course aims to provide an introduction to function approximation and interpolation methods in 1- and 2-D; to continue the study of numerical integration methods; to study the techniques required to apply and analyse numerical methods for solving linear systems of equations and eigenvalue problems. By the end of the course, students should be able to apply the methods and algorithms listed below, and carry out the associated analysis of errors, convergence and operations counts.

Syllabus

1. Basic Linear Algebra (1.1 1 Revision of Year 2 Linear Algebra, 1.2 2 Python and Jupyter, 1.3 3 Calculus of sums, 1.4 4 Calculus of FLOPS, 1.5 5 Matrix product, 1.6 6 Gaussian elimination, 1.7 7 Matrix inverse, 1.8 8 Eigenvalues and functional calculus, 1.9 9 The discrete 1-dimensional Laplacian, 1.10 10 The 2d discrete Laplacian, 1.11 11 Power and shifted inverse power method)

2. Matrix decompositions (2.1 12 LU decomposition, 2.2 13 LDL∗ and LL∗ Cholesky decompositions, 2.3 14 QR decomposition)

3. Classical iterative solvers (3.1 15 Classical iterations, 3.2 16 Convergence of Jacobi, 3.3 17 Convergence of Gauss-Seidel and SOR)

4. Advanced topics (4.1 18 Low Dimensional Approximation LDA, 4.2 19 GMRES, 4.3 20 Francis QR iteration)

Learning outcomes

By the end of the course, students should be able to do the following:

use the calculus of FLOPS to compute the FLOPS of numerical algorithms.
use functional calculus to compute matrix functions and estimate eigenvalues, especially for use in proofs.
define the 1d and 2d discrete Laplacians and assemble their sparse matrices.
use eigenvalue algorithms to approximate eigenvalues of a matrix, including proofs of convergence.
use matrix decompositions to solve linear and least-squares problems, including the proof for least-squares problems.
use classical iterations to approximately solve linear problems, including proofs of convergence.
use Krylov space methods to approximately solve linear algebra problems.

Further details

Curriculum explorer: Click here

SCQF Level: 9

Credits: 15