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My research focuses on the solution of large sparse systems of linear equations and large-scale linear and non linear optimization problems. An important orientation is the development of underlying theory and algorithms that are then realised as state-of-the-art mathematical software. For the last 30 years, I have been a major contributor to the internationally renowned ; I had overall responsibility for HSL from 2007 to 2018.

My key scientific contributions, which involve studying the underlying mathematics, new analysis and theoretical developments, leading to new library software, include:

* Sparse matrix ordering algorithms (bandwidth and profile minimisation and algorithms for challenging symmetric indefinite matrices that limit the need for numerical pivoting while maintaining sparsity);

* Algorithms to improve the efficiency and applicability of sparse direct solvers (including novel out-of-core techniques, the introduction of multiple front algorithms for coarse-grained parallelism, the use of single precision within a double precision solver, and sparse scaling algorithms);

*  Robust limited memory incomplete factorization preconditioners for sparse linear systems;

* Novel techniques for solving large-scale sparse and mixed sparse-dense least squares problems.