import Graphics.Dynamic.Plot.R2
import Data.Default.Class
import Data.VectorSpace
import Math.Polynomial
import Math.Polynomial.Legendre

type ℝ = Double

sawtooth :: ℝ -> ℝ
sawtooth x = x - fromIntegral (round x)

l2 :: (ℝ->ℝ) -> (ℝ->ℝ) -> ℝ
l2 f g = (f (-1)*g (-1) / 2
          + sum [ f x*g x
                | x <- take (n-1) [2/n-1, 4/n-1 ..] ]
          + f 1*g 1 / 2
         ) / n
 where n :: Num n=>n
       n = 256

[l2 (evalLegendre i) (evalLegendre i) * (2*fromIntegral i+1) | i<-[0..5]]

coeffs :: [ℝ]
coeffs = [l2 sawtooth (evalLegendre i) * (2*fromIntegral i+1) | i<-[0..] ]

approxSaw :: Poly ℝ
approxSaw = (/10) . fromIntegral . round . (*10)
           <$> sumV [legendre i ^* c
                    | (i,c) <- zip [0..20] coeffs ]

plotPrerender def [ continFnPlot sawtooth, continFnPlot $ evalPoly approxSaw
                  , forceXRange (-1,1), forceYRange (-3/4,3/4) ]

plotPrerender def [ continFnPlot sawtooth, continFnPlot $ evalPoly approxSaw
                  , forceXRange (-0.7,1.3), forceYRange (-1,0.4) ]

{-# LANGUAGE OverloadedStrings #-}
import Text.LaTeX
import Text.LaTeX.Packages.AMSMath
mathDisplay (sum [ fromRational (fromInteger (round $ c*10)/10) * "x"**fromIntegral i
                 | (i,c) <- zip [0..] $ polyCoeffs LE approxSaw
                 , abs c > 0.1 ]
              :: LaTeX)