# Module descriptors - Advanced Computational Methods

The module descriptors for this programme can be found below.

Modules shown are for the current academic year and are subject to change depending on your year of entry.

Please note that the curriculum of this programme is currently being reviewed as part of a College-wide process to introduce a standardised modular structure. As a result, the content and assessment structures of this course may change for your year of entry. We therefore recommend that you check this course page before finalising your application and after submitting it as we will aim to update this page as soon as any changes are ratified by the College.

Find out more about the limited circumstances in which we may need to make changes to or in relation to our courses, the type of changes we may make and how we will tell you about changes we have made.

## Computational Linear Algebra

### Module aims

Part 1- To provide the student with both a theoretical and practical understanding of the standard algorithms for solving simultaneous linear equations (Ax=b), as well as of the challenges involved.

Part 2- To provide the student with both a theoretical and practical understanding of the standard algorithms for efficient solution of eigenvalue problem and singular value decomposition and their applications in data compression and Principal Component Analysis (PCA).

### Learning outcomes

• Introduce state-of-the-art iterative methods.Learning to use Sensitivity analysis of Eigenvalues and Eigenvectors to adjust response of a system.

• Understand ill-posed problems and ill-condition matrices.

• Regularisation and application of SVD to ill-posed problems

• Understand and utilize PCA and its application in data reduction and image processing.

• Understand Chebyshev acceleration, conjugate gradient method and GMRES as optimisation algorithms.

• Understand the effect of inexact arithmetic on Gaussian elimination

• Understand and exploit the properties of symmetric positive-definite and banded matrices

• Understanding the orthogonal basis of a matrix

• Understanding the physical and mathematical concept of Eigenvalue and Eigenvector

• Define nd implement the classical iterative algorithms

• Describe Gaussian elimination with pivoting as a numerical algorithm

• Understand how the LU  and Cholesky factorisations arise from Gaussian elimination

### Module syllabus

Part 1

1 Introduction to the course; review of vector spaces, vector and matrix norms; floating point numbers and arithmetic.

2 Backward stability, accuracy of approximate solutions to linear systems, condition number; introduction to direct methods for linear systems; forward & backward substitution for triangular systems.

3 Gaussian elimination and LU factorisation; pivoting; stability of LU factorization.

4 LU factorisation for banded matrices; Cholesky factorisation; summary of direct methods and examples.

5 Splitting iterative methods for linear systems: general theory and convergence criteria, classical iterative methods, examples and comparison with direct methods.

Part 2

6 Eigenvalue problems, algorithms.

7 Applications of Eigenvalue analysis: Sensitivity analysis

8 The Singular Value Decomposition, algorithms.

9 Application of SVD : Regularization, Noise reduction

10 PCA analysis and data reduction

11 Chebyshev acceleration, conjugate gradient method, GMRES.

12 Parallel algorithms and multigrid methods

### Pre-requisites

Basic linear algebra (basis vectors, eigenvectors).

### Teaching methods

Lectures/tutorials

Online/blended learning.

### Assessments

Formative assessment: self and peer-assessed problems.

Summative assessment: 2  pieces of assessed coursework (50% each)

### Supplementary

• #### Numerical linear algebra

Lloyd N. Trefethen

SIAM

• #### Applied numerical linear algebra

James W. Demmel

SIAM

• #### Matrix computations

Golub, Gene H.

4th, Johns Hopkins University Press

• #### Matrix computation

Jennings, Alan.

2nd ed., Chichester : Wiley

• #### The algebraic eigenvalue problem

Wilkinson, J. H.

Oxford : Clarendon Press

• #### Generalized inverses [electronic resource] : theory and applications

2nd ed., Springer

• #### Machine learning : the new AI

Alpaydin, Ethem,

The MIT Press; IEEE Xplore

• #### Principal Component Analysis Handbook

Rebecca Cross

ML Books International - IPS