The Computational Mathematics Theme 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.

An open source tool for comparing different minimizers/fitting frameworks.

A non-linear least squares solver with Python, Fortran, and C interfaces.

GOFit is a package of C++ algorithms with python interfaces designed for the global optimization of parameters in curve fitting, i.e. for nonlinear least-squares problems arising from curve fitting.