Finite Recovery by Extrapolation for the Douglas--Rachford Method on Affine Subspaces

We completely describe the dynamics of the Douglas--Rachford (DR) method for solving affine subspace feasibility. We characterise the affine hull of the resulting iterates, and we show how the corresponding search space's dimension is an explicit function of the number of distinct nontrivial principal angles. This dimension can be much smaller than the ambient dimension, helping to explain why extrapolation can be so effective. Using this characterisation, we extend the Lyapunov surrogate method (LSM) to higher extrapolation orders and prove that it recovers the DR limit exactly from finitely many iterates. We show that LSM and a least-squares extrapolation (LeSE) share the affine search spaces used by the generalized minimal residual method (GMRES) applied to the DR fixed-point equation (DR-GMRES). The dimension characterisation gives bounds on the number of steps required for exact recovery by all three methods. We develop adaptive implementations that estimate the search-space dimension from differences between successive DR iterates and use this estimate to select the extrapolation order. Numerical experiments show how ill-conditioning affects recovery accuracy, and we identify cases where adaptive extrapolation runs faster than DR-GMRES.