In this paper we study numerical stability of path-based algorithms for the traffic assignment problem. This type of methods are based on decomposition of the original problem into smaller sub-problems which are optimised sequentially. Previously, path-based algorithms were numerically tested only in the setting of moderate requirements to the level of solution precision. In this study we analyse convergence of these methods when convergence measure
approaches machine epsilon of IEEE double precision format. In particular, we demonstrate that the straightforward implementation of one of the algorithms of this group (projected gradient) suffers from loss of precision and is not able to converge to highly precise solution. We propose a way to solve this problem and test the proposed fixed version of the algorithm on various benchmark instances.