Computational Techniques

Module aims

To improve the analytic skills to the level required for engineering and financial applications.

To develop the students' abilities needed for problem solving in linear algebra.

Learning outcomes

At the end of course the students would know

1- How to repersent a linear map in different bases, how to diagonalise or provide the Jordan normal form of a matrix.

2. How to obtain the range space and the null space of a matrix.

3. How to decompose a vector space into the direct sum of a the range space and the null space of a matrix.

4. How to determine the geometric and the algebraic multiplicity of an eigenvalue of a matrix.

5. How to construct the spectral decomposition of a symmetruc matrix

6. How to find the singular value decomposition of an arbitrary real matrix

7. How to solve the least square method

8. How to find the cholesky fatorisation of  a symmetric positive semi-definite matrix

Module syllabus

Contents:


1. Computational linear algebra:

Vectors, matrices, special matrices, operations and algorithms. Vector and matrix norms Vector spaces, linear maps,  linear dependence, bases, null space, rank space. Eigenvalues, eigenvectors, algebraic and geometric multiplicity of eigenvalues, generalised eigenvectors, Jordan normal form, singular values, spectral decomposition of symmtric matrices, singular value decomposition. Solution of systems of linear equations: Triangular systems, Gaussian and Gauss-Jordan elimination, computational implications, LU decomposition, Cholesky factorisation, computational form. Least squares problems.  Condition of a mathematical problem. Basics of sparse computing.

2. Convergent sequences: Metric spaces. Limits. Cauchy sequences. Fixed-point theorem for contractions. Iterative methods.

3. Functions of several variables: Partial differentiation, the gradient, the Hessian. Taylor expansion. Newton's method for min f (x). Quadratic forms and linear systems. Method of conjugate gradients.

Pre-requisites

 1st year mathematical methods

Teaching methods

 Lectures and Tutorials

Assessments

Two assessed course works and a written exam in April/May

Reading list

Module leaders

Professor Peter Harrison