put back normal precision, inserted the rec Vnk in robust Pkn and imp…

…roved readme

README.jmd

@@ -121,11 +121,11 @@ p = R"$to_plot %>%
       theme_bw() + 
       scale_fill_gradient(name = 'Accuracy') + 
       ggtitle('Vnk')"
-R"png('Vnk_instability.png')
+R"png('figures/Vnk_instability.png')
   plot($p)
   dev.off()"
 ```
 
- ![](Vnk_instability.png)
+![](figures/Vnk_instability.png)
 
 Accuracy (bits) of the computed $V_{n,k}$ as a function of $n$ and $k$. The computations are carried out using 200 bits of precision. Light coloured areas correspond to where the precision decreases below 64 bits.

README.md

@@ -144,6 +144,7 @@ References:
 
 ## Instability of the $V_{n,k}$
 
+````julia
 
 using GibbsTypePriors, Nemo, DataFrames, DataFramesMeta, RCall
 β = 0.5
@@ -167,6 +168,9 @@ R"png('Vnk_instability.png')
   dev.off()"
 ````
 
- ![](Vnk_instability.png)
+
+
+
+![](Vnk_instability.png)
 
 Accuracy (bits) of the computed $V_{n,k}$ as a function of $n$ and $k$. The computations are carried out using 200 bits of precision. Light coloured areas correspond to where the precision decreases below 64 bits.

src/Pkn.jl

@@ -34,7 +34,7 @@ function Pkn_NGG_robust_in(k, n, β, σ; verbose = false)
         return arb_0
     else
         σ_arb = RR(σ)
-        Vnk_NGG(n, k, β, σ) // σ_arb^k * Cnk_robust(n, k, σ; verbose = verbose)
+        Vnk_NGG_rec(n, k, β, σ) // σ_arb^k * Cnk_robust(n, k, σ; verbose = verbose)
     end
 end
 

src/common_functions.jl

@@ -2,7 +2,7 @@ using SpecialFunctions, Nemo, StatsFuns, Memoization
 import Nemo.binom, Nemo.gamma
 
 "Session-wide precision"
-const prec = 200 ## Increase for better precision
+const prec = 5000 ## Increase for better precision
 
 "Converter from real to arbitrary precision real (Arb)"
 const RR = RealField(prec)