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