Update README.md

README.md

@@ -3,11 +3,33 @@ Software for automated data analysis of Wilson_tm fermions with GEVP techniques.
 The GEVP is solved as outlined in 1010.0202, i.e., given a correlator f(t) the following matrix is constructed:
 
 ```math
-C(t) = \begin{matrix} f(t) & f(t+\tau) \\ f(t+\tau) & f(t+2\tau)  \end{matrix}
+C(t) = \bigg[\begin{matrix} f(t) & f(t+\tau) \\ f(t+\tau) & f(t+2\tau)  \end{matrix} \bigg]
 ```
 
+The code automatically builds these matrices for PP and PA1 correlators  at each time slice and for each \mu sector outlined by the user in the sector dictionary (ll-hl-sh etc).
+After this procedure, the generalised eigenvalues are extracted by solving the equation 
+```math
+C(t,\tau)v(t,t_0,\tau)  = \lambda(t,t_0, \tau) C(t_0, \tau)v(t,t_0,\tau) 
+```
+The two time parameters \tau and t_0 can be choosen by the user before compiling the code in the setting options.
+The  effective energy spectrum is determined from:
+```math
+E_n^{eff}(t, t_0, \tau) = \log( frac{\lambda_n(t, t_0, \tau)}{\lambda_n(t+a, t_0, \tau)}) = E_n + O(e^{-(E_m-e_n)t})  
+```
+Finally the energy levels are determined by fitting the effective energy with the functional form 
+```math
+E_n^{eff}(t, t_0, \tau) = E_n + p_1*e^{-\Delta E * t}  
+```
+The fit is performed in the time extent 
+```math
+t \in [\text{range_t_fit}[1]:\text{range_t_fit}[2], end-5]
+```
+where range_t_fit is an array of integers set by default to [2,5], but this can be changed by the user.
+Final results for the energy spectrum is given by the weighthed average of the fit parameters determined by the above outlined procedure.
+N.B. at the moment the code does not extract matrix element, hence decay constants, from the gevp.
+
 To run the code in the juliaREPL type:
-julia> include("/path_to/analysis.jl")
+julia> include("/path_to/gevp.jl")
 Go to SET UP VARIABLES section in the analysis.jl code to properly configure before running.
 You have to up set the following variables:
 
@@ -15,9 +37,10 @@ path_data where data are stored
 path_plat where a plat.txt file is stored with plateaus for all ensembles
 path_result where you want to save results.txt and plots
 ensembles to analyse
+sector you are interested in: light-light, light-heavy, heavy-heavy...
 tau value to use for matrix construction 
 _t0 value to use for solving the GEVP
-sector you are interested in: light-light, light-heavy, heavy-heavy...
+range_t_fit array of integers with the fit range value for the extractions of the energies 
 rwf set to true if you want to include reweighting factors
 compute_t0 set to true if you want to extract t0 from ms.dat file. Otherwise, t0 is taken from 1608.08900 in const.jl
 mass_shift set to true if you want to include the mass shift