4 Year GTA - Validated numerics for matrix functions

About the Project

Open to UK Applicants only

Mathematics are offering 3 fully-funded Graduate Teaching Assistant (GTA) PhD studentships available for UK applicants, starting in September 2026.

Graduate Teaching Assistantships allow research students to fund their PhD through part-time teaching work with the University.

A Graduate Teaching Assistant is responsible to the Head of School and is expected to undertake teaching or other duties within the School - not normally exceeding 8-10 contact hours per week - while undertaking research leading to a PhD.

Approximately 80% of their time will be spent on their doctoral research and 20% on their GTA responsibilities. Training is provided to help Graduate Teaching Assistants develop their teaching related skills and enhance their professional competencies.

Project Highlights

• Development of new verified algorithms for matrix functions to monitor the accuracy of solutions obtained by standard floating-point algorithms
• Computing rigorous a posteriori componentwise forward error bounds for matrix functions
• Development of open-source software package for computing guaranteed error bounds for matrix functions

Project Description

Matrix functions arise in numerous applications, including the solution of ordinary and partial differential equations, dynamical systems, probability theory, network science, control theory and particle physics.

Practical computation of matrix functions relies on floating-point arithmetic, but this requires truncation errors as well as rounding to numbers representable on the computer. These errors result in solutions that are only approximately correct. Understanding and controlling errors is essential for the secure application of these methods in real world contexts. James Wilkinson introduced what is now the standard framework to analyse rounding errors. This is known as backward error analysis and Nick Higham has comprehensively applied it in numerical linear algebra. However, backward error analysis does not yield rigorous error bounds and obtaining forward error bounds --- the primary measure of accuracy of computed solutions--- still requires a separate analysis of the problem’s condition number, which is not a straightforward automatic process.

In this project, we instead employ verified computing based on interval analysis. A major advantage of this approach over backward error analysis is that it is performed a posteriori and can automate forward error analysis: verified algorithms, implemented using machine interval arithmetic with directed rounding, compute rigorous error bounds alongside the approximate solution, and these bounds come with a mathematical guarantee of correctness. This certification makes the approach also well suited to computer-assisted mathematical proofs. Moreover, the algorithms developed in this project will also enhance existing standard floating-point algorithms in terms of the accuracy of the computed approximate solutions.

Project enquiries to Dr. Behnam Hashemi bh241@le.ac.uk

Application enquiries to pgrapply@le.ac.uk

To apply please refer to the application advice and use the application link at https://le.ac.uk/study/research-degrees/funded-opportunities/maths-gta

Start 21 September 2026

Funding Notes

The 4 year GTA funded studentships provide:

• A combined teaching and stipend payment, currently. for 2026/7 this will be £21,805 per year, paid in monthly instalments
• Tuition fees at UK rates

References

[1] A. Frommer, B. Hashemi, Computing enclosures for the matrix exponential, SIAM Journal on Matrix Analysis and Applications 41 (2020) 1674-1703.
[2] N. J. Higham, Functions of Matrices: Theory and Computation, SIAM, Philadelphia, 2008.