1.. _relaxation: 2 3Relaxation 4========== 5 6The (unscaled, see :ref:`scaling`) SCS update equations are: 7 8.. math:: 9 \begin{align} 10 \tilde u^{k+1} &= (I + \mathcal{Q})^{-1} w^k \\ 11 u^{k+1} &= (I + N_{\mathcal{C}_+})^{-1} (2 \tilde u^{k+1} - w^k) \\ 12 w^{k+1} &= w^k + \alpha (u^{k+1} - \tilde u^{k+1}) \\ 13 \end{align} 14 15where :math:`\alpha \in (0,2)` is the *relaxation* parameter. Vanilla 16Douglas-Rachford corresponds to setting :math:`\alpha = 1`. If :math:`\alpha < 171` it is referred to as under-relaxation, if :math:`\alpha > 1` it is 18over-relaxation. Typically values of :math:`\alpha \approx 1.5` work well. It 19is controlled by the :code:`alpha` :ref:`setting <settings>`. If :math:`\alpha 20= 2` the method reduces to Peaceman-Rachford splitting, which is not guaranteed 21to converge in general (but will if, for example, the operators in the problem 22are both maximal strongly monotone). 23 24Thankfully, there is no interaction between :math:`\alpha` and the scaling 25described in :ref:`scaling`, and they can be combined immediately. 26