GTA Funded – Validated numerics for matrix functions

Project Highlights:

1. Development of new verified algorithms for matrix functions to monitor the accuracy of solutions obtained by standard floating point algorithms 

2. Computing rigorous a posteriori componentwise forward error bounds for matrix functions

3. Development of open-source software package for computing guaranteed error bounds for matrix functions

Project Summary:

Matrix functions appear in numerous applications including solution of ODEs and PDEs, probability theory, network science, control theory and particle physics, (see [3]). For instance, assuming A is a square matrix, the initial value problem 

(d^2 y)/(dt^2 )+Ay=0,    y(0)=y_0,    y^’ (0)=〖y’〗_0

has the following unique solution

y(t)=cos(√A t) y_0+〖(√A )〗^(-1) sin(√A t) 〖y’〗_0

which involves the matrix square root, inverse square root and the matrix sine and cosine. Practical computation of matrix functions relies on floating point arithmetic whose main limitation is the presence of rounding errors resulting in solutions which are only approximately correct. Indeed, analysis of errors committed in the course of a computation is a central part of numerical computing. The error analysis tool to be employed in this project is verified computing based on interval analysis; another well-known approach to error analysis is backward error analysis (developed mostly by the British mathematician and computer scientist, James Wilkinson).

While most verified algorithms for matrix functions explore algebraic functions [2], the aim of this project is to use ideas from approximation theory [4] which potentially open the door to a whole new class of verification algorithms applicable to non-algebraic matrix functions [1]. Our plan is to derive new rigorous error bounds for matrix functions which are subsequently used to develop algorithms with automatic result verification. We will then implement, test and compare our algorithms on a wide range of numerical experiments.

Entry requirements

Applicants are required to hold/or expect to obtain a UK Bachelor Degree 2:1 or better in a relevant subject or overseas equivalent. 

A Master’s degree is preferred but not essential. Experience with research or teaching would be a plus. 

The University of Leicester English language requirements apply where applicable.

Eligibility

UK and International applicants are welcome to apply.

* EU applicants who hold EU settled or EU pre-settled status please provide PGR Admissions with a share code (the one that starts with S) so we can verify your fee status email to [email protected]

Informal enquiries

Project and funding enquiries [email protected]

Project supervisor: Dr Behnam Hashemi – [email protected]

How to Apply

Please refer to the application advice and use the application link on our web page

https://le.ac.uk/study/research-degrees/funded-opportunities/maths-gta