The Computational Mathematics Group writes and maintains a number of software libraries and test suites:

The HSL Mathematical Software Library was started in 1963 as a general purposes library, but has since evolved into a highly respected Sparse Linear Algebra Library, with a focus on solving linear equations and eigenvalue problems.
The GALAHAD Optimization Library solves nonlinear optimization problems. It includes packages to solve unconstrained and bound-constrained optimization, quadratic programming, nonlinear programming, systems of nonlinear equations and inequalities, and nonlinear least squares problems.
CUTEst is a test suite aimed at testing constrained and unconstrained optimization solvers, and is the latest incarnation of CUTE
The Sparse Parallel Robust Algorithms Library (SPRAL) is an open-source library for sparse linear algebra and associated algorithms.
NLAFET is a developing open source software library for parallel numerical linear algebra for future extreme systems.