Broyden-algorithm.md

+# Broyden algorithm - general info
+Broyden algorithm is implemented according prescription provided in the paper:  
+[A. Baran, A. Bulgac, M. M. Forbes, G. Hagen, W. Nazarewicz, N. Schunck, and M. V. Stoitsov, Broyden’s method in nuclear structure calculations, Phys. Rev. C 78, 014318 (2008)](https://link.aps.org/doi/10.1103/PhysRevC.78.014318), ([see here for preprint](https://arxiv.org/abs/0805.4446))  
+
+`st-wslda` codes support two types of updating schemes of densities:
+* linear mixing, according to formulas (3),
+* Broyden mixing, according to prescription given in sec III.B.
+
+# Controlling Broyden algorithm
+In order to activate the Broyden mixing user must activate following flag in the *input* file: 
+ 
+```bash
+broyden                 1        # activate broyden mixing, default=0 (linear mixing)
+```
+There are two parameters that control mixing parameter (alpha parameter)
+```bash
+linearmixing            0.5      # mixing parameter (alpha) for linear mixing, default=0.5
+broydenmixing           0.75     # mixing parameter (alpha) for Broyden mixing, default=0.75
+```
+We separate these two parameters as in general they may be different. The remaining parameters controlling the Broyden algorithm are:
+```bash
+Mbroyden                5       # number of previous iterations used by Broyden, default=5 
+
+startbroyden            5       # iteration number (it) for which Broyden should be activated,
+                                # for it<startbroyden algorithm proceeds with linear mixing 
+                                # and used linearmixing value for mixing
+                                # NOTE: for it in [startbroyden, startbroyden+Mbroyden] the algorithm
+                                #       still continue with linear mixing, and it stores only data
+                                # Broyden starts to operate for iteration it = startbroyden+Mbroyden+1
+                                # and used broydenmixing value for mixing
+                                # default=0
+stopbroyden             100     # Broyden is deactivated for iteration it=stopbroyden
+                                # starting from iteration it = stopbroyden+1 the algorithm again proceeds
+                                # with linear mixing and used linearmixing value for mixing
+                                # default=999999
+# Recommendation: Use  startbroyden and stopbroyden only if Broyden fails to converge. 
+#                 According experience of our collaborators, there are cases (physical problems)
+#                 where activating Broyden only for some number of iteration provides the best performance 
+
+# For advanced users 
+omega0broyden           0.01    # weight omega_0, see eqn. (6) 
+omeganbroyden           1.0     # weight omega_n, see eqn. (6) 
+omegakbroyden           1.0     # weight omega_k, see eqn. (6) 
+# No significant dependence was found between Broyden performance and values of weights. Default values are recommended
+```
+# Contribution
+The Broyden algorithm was implemented by:  
+* Wojciech Pudełko, Warsaw University of Technology,<br/> Implementation of Broyden algorithm (engineer thesis, 2020) 
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