The RANSAC framework is often used to estimate the relative pose of two cameras from outlier-contaminated point correspondences, via the essential matrix, however this is computationally expensive due the cost of computing essential matrices from many sets of five to seven correspondences. The leading contemporary 5-point solver (Nister, 2004) is slow because of the expensive linear algebra decompositions and polynomial solve which are required. To avoid these costs we propose to use Levenberg-Marquardt optimisation on a manifold to find a subset of the compatible essential matrices. The proposed algorithm finds essential matrices at a higher rate than closed-form approaches, and reduces the time needed to find relative poses using RANSAC by 25%. The second contribution of this paper is to apply the optimisations used in 5-point solvers to the classic 7-point algorithm. RANSAC using the optimised 7-point algorithm is considerably faster than 5-point RANSAC (unless planar point configurations are common), despite the increased number of iterations necessary.