module ToySolver.Data.AlgebraicNumber.Graeffe
( NthRoot (..)
, graeffesMethod
) where
import Control.Exception
import qualified Data.IntMap as IM
import ToySolver.Data.Polynomial (UPolynomial, X (..))
import qualified ToySolver.Data.Polynomial as P
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 $ P.toMonic P.nat p)
n = P.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 (i-1))
return $ NthRoot (2 ^ fromIntegral v) yi
where
bs = IM.fromList [(fromInteger i, b) | (b,ys) <- P.terms p, let i = n - P.deg ys, i /= 0]
b i = IM.findWithDefault 0 i bs
f :: UPolynomial Rational -> UPolynomial Rational
f p = (-1) ^ (P.deg p) *
P.fromTerms [ (c, assert (P.deg xs `mod` 2 == 0) (P.var X `P.mpow` (P.deg xs `div` 2)))
| (c, xs) <- P.terms (p * P.subst p (\X -> - P.var X)) ]
f' :: UPolynomial Rational -> UPolynomial Rational
f' p = P.fromTerms [(b k, P.var X `P.mpow` (n - k)) | k <- [0..n]]
where
n = P.deg p
a :: Integer -> Rational
a k
| n >= k = P.coeff (P.var X `P.mpow` (n - k)) p
| otherwise = 0
b :: Integer -> Rational
b k = (-1)^k * (a k)^2 + 2 * sum [(-1)^j * (a j) * (a (2*k-j)) | j <- [0..k-1]]
test v = graeffesMethod p v
where
x = P.var X
p = x^2 - 2
test2 v = graeffesMethod p v
where
x = P.var X
p = x^5 - 3*x - 1