From bdb9f22c1bf5821697561e96bb1913469d160d5f Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 8 Feb 2019 16:01:02 -0500
Subject: initial commit

---
 01.06.hs | 70 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 70 insertions(+)
 create mode 100644 01.06.hs

(limited to '01.06.hs')

diff --git a/01.06.hs b/01.06.hs
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+++ b/01.06.hs
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+
+{-# LANGUAGE FlexibleInstances #-}
+
+import System.Random
+import Data.Random.Normal
+import Control.Monad.State
+import Numeric.LinearAlgebra as Lin
+
+import Control.Monad(void)
+import Graphics.Rendering.Chart
+import Graphics.Rendering.Chart.Easy
+import Graphics.Rendering.Chart.Grid
+import Graphics.Rendering.Chart.Backend.Cairo
+import Graphics.Rendering.Chart.Plot.Histogram(defaultNormedPlotHist)
+instance Default (PlotHist x Double) where
+    def = defaultNormedPlotHist
+
+-- | Evaluates the Winger surmise.
+ρWinger :: Floating t => t -> t
+ρWinger s = pi * s / 2 * exp(- pi * s^2 / 4)
+
+-- | Computes the splitting of the center eigenvalues of a symmetric matrix.
+eigenSplitting :: Herm Double -> Double
+eigenSplitting m = e2 - e1
+                   where
+                     e = sortVector $ eigenvaluesSH m
+                     i = (Lin.size e) `quot` 2
+                     e1 = e Lin.! (i - 1)
+                     e2 = e Lin.! i
+
+-- | Produces random GOE matrices.
+goeGen :: RandomGen g => Int -> State g (Herm Double)
+goeGen n = get >>= (\g -> let (g1, g2) = split g in 
+  (put g1) >> (return $ sym $ matrix n $ take (n^2) $ normals' (0.0, 2.0) g2))
+
+-- | Produces random ∓1 matrices.
+pm1Gen :: RandomGen g => Int -> State g (Herm Double)
+pm1Gen n = state (\g -> let (m, g2) = runState (goeGen n) g in (trustSym $ cmap signum (unSym m), g2))
+
+-- | Produces n eigenvalue splittings from the matrix ensemble m.
+eigenSplittings :: RandomGen g => Int -> State g (Herm Double) -> State g [Double]
+eigenSplittings n m = state (\g -> let (mats, g2) = runState (sequence $ replicate n m) g in (eigenSplitting <$> mats, g2))
+
+hsChart :: RandomGen g => g -> State g (Herm Double) -> Int -> Int -> String -> Layout Double Double
+hsChart g gen num bins tit = execEC $ do
+  plot $ fmap histToPlot $ liftEC $ do
+    plot_hist_title .= tit
+    plot_hist_bins .= bins
+    plot_hist_values .= scaledData
+    plot_hist_range .= Just (0.0, 5.0)
+    plot_hist_norm_func .= const (\x -> fromIntegral bins * fromIntegral x / fromIntegral num / 5.0)
+  plot (line "ρ_Winger" [ [(s,ρWinger s) | s <- [0,0.005..5]] ])
+  where
+    rawData = evalState (eigenSplittings num gen) g
+    mean = sum rawData / fromIntegral (length rawData)
+    scaledData = (\x -> x / mean) <$> rawData
+
+mkgrid g num bins = title `wideAbove` aboveN [ besideN [ layoutToGrid (hsChart g ((fst gen) n) num bins (snd gen Prelude.<> show n)) | gen <- gens] | n <- ns]
+  where
+    gens = [(goeGen, "GOE, N = "), (pm1Gen, "±1, N = ")]
+    ns = [2, 4, 10]
+    title = setPickFn nullPickFn $ label ls HTA_Centre VTA_Centre "Winger surmise compared with eigenvalue splitting of random matrices"
+    ls = def { _font_size = 15, _font_weight = FontWeightBold }
+    
+
+main = do
+  g <- newStdGen
+  void $ renderableToFile (def { _fo_format = PDF}) "01.06.pdf" $ fillBackground def $ gridToRenderable $ mkgrid g 1000 50
+
+
-- 
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