module Data.Polynomial.RootSeparation.Graeffe
( NthRoot (..)
, graeffesMethod
) where
import Control.Exception
import qualified Data.IntMap as IM
import Data.Polynomial
data NthRoot = NthRoot !Integer !Rational
deriving (Show)
graeffesMethod :: UPolynomial Rational -> Int -> [NthRoot]
graeffesMethod p v = xs !! (v 1)
where
xs = map (uncurry g) $ zip [1..] (tail $ iterate f $ associatedMonicPolynomial grlex p)
n = deg p
g :: Int -> UPolynomial Rational -> [NthRoot]
g v p = do
i <- [1::Int .. fromInteger n]
let yi = if i == 1 then (b i) else (b i / b (i1))
return $ NthRoot (2 ^ fromIntegral v) yi
where
bs = IM.fromList [(fromInteger i, b) | (b,ys) <- terms p, let i = n deg ys, i /= 0]
b i = IM.findWithDefault 0 i bs
f :: UPolynomial Rational -> UPolynomial Rational
f p = (1) ^ (deg p) *
fromTerms [ (c, mmFromList [assert (e `mod` 2 == 0) (x, e `div` 2) | (x,e) <- mmToList xs])
| (c,xs) <- terms (p * subst p (\_ -> var X)) ]
f' :: UPolynomial Rational -> UPolynomial Rational
f' p = fromTerms [(b k, mmFromList [(X, n k)]) | k <- [0..n]]
where
n = deg p
a :: Integer -> Rational
a k
| n >= k = coeff (mmFromList [(X, n k)]) p
| otherwise = 0
b :: Integer -> Rational
b k = (1)^k * (a k)^2 + 2 * sum [(1)^j * (a j) * (a (2*kj)) | j <- [0..k1]]
test v = graeffesMethod p v
where
x = var X
p = x^2 2
test2 v = graeffesMethod p v
where
x = var X
p = x^5 3*x 1