```{-# OPTIONS -fno-implicit-prelude #-}
{- |
Copyright   :  (c) Henning Thielemann 2004-2005

Maintainer  :  numericprelude@henning-thielemann.de
Stability   :  provisional
Portability :  requires multi-parameter type classes

Computations on the set of roots of a polynomial.
These are represented as the list of their elementar symmetric terms.
The difference between a polynomial and the list of elementar symmetric terms
is the reversed order and the alternated signs.

Cf. /MathObj.PowerSum/ .
-}
module MathObj.RootSet where

import qualified MathObj.Polynomial  as Poly
import qualified MathObj.PowerSum    as PowerSum

import qualified Algebra.Algebraic    as Algebraic
import qualified Algebra.IntegralDomain as Integral
import qualified Algebra.Field        as Field
import qualified Algebra.Ring         as Ring
import qualified Algebra.ZeroTestable as ZeroTestable

import NumericPrelude.List (takeMatch)

import PreludeBase as P hiding (const)
import NumericPrelude as NP

newtype T a = Cons {coeffs :: [a]}

{- * Conversions -}

lift0 :: [a] -> T a
lift0 = Cons

lift1 :: ([a] -> [a]) -> (T a -> T a)
lift1 f (Cons x0) = Cons (f x0)

lift2 :: ([a] -> [a] -> [a]) -> (T a -> T a -> T a)
lift2 f (Cons x0) (Cons x1) = Cons (f x0 x1)

const :: (Ring.C a) => a -> T a
const x = Cons [1,x]

toPolynomial :: Poly.T a -> T a
toPolynomial xs = Cons (reverse (Poly.coeffs xs))

fromPolynomial :: T a -> Poly.T a
fromPolynomial (Cons xs) = Poly.fromCoeffs (reverse xs)

toPowerSums :: (Field.C a, ZeroTestable.C a) => [a] -> [a]
toPowerSums = PowerSum.fromElemSymDenormalized

fromPowerSums :: (Field.C a, ZeroTestable.C a) => [a] -> [a]
fromPowerSums = PowerSum.toElemSym

{- | cf. 'MathObj.Polynomial.mulLinearFactor' -}
addRoot :: Ring.C a => a -> [a] -> [a]
y : (ys + Poly.scale x yt)
error "addRoot: list of elementar symmetric terms must consist at least of a 1"

fromRoots :: Ring.C a => [a] -> [a]
fromRoots = foldl (flip addRoot) [1]

liftPowerSum1Gen :: ([a] -> [a]) -> ([a] -> [a]) ->
([a] -> [a]) -> ([a] -> [a])
liftPowerSum1Gen fromPS toPS op x =
takeMatch x (fromPS (op (toPS x)))

liftPowerSum2Gen :: ([a] -> [a]) -> ([a] -> [a]) ->
([a] -> [a] -> [a]) -> ([a] -> [a] -> [a])
liftPowerSum2Gen fromPS toPS op x y =
takeMatch (undefined : liftM2 (,) (tail x) (tail y))
(fromPS (op (toPS x) (toPS y)))

liftPowerSum1 :: (Field.C a, ZeroTestable.C a) =>
([a] -> [a]) -> ([a] -> [a])
liftPowerSum1 = liftPowerSum1Gen fromPowerSums toPowerSums

liftPowerSum2 :: (Field.C a, ZeroTestable.C a) =>
([a] -> [a] -> [a]) -> ([a] -> [a] -> [a])
liftPowerSum2 = liftPowerSum2Gen fromPowerSums toPowerSums

liftPowerSumInt1 :: (Integral.C a, Eq a, ZeroTestable.C a) =>
([a] -> [a]) -> ([a] -> [a])
liftPowerSumInt1 = liftPowerSum1Gen PowerSum.toElemSymInt PowerSum.fromElemSym

liftPowerSumInt2 :: (Integral.C a, Eq a, ZeroTestable.C a) =>
([a] -> [a] -> [a]) -> ([a] -> [a] -> [a])
liftPowerSumInt2 = liftPowerSum2Gen PowerSum.toElemSymInt PowerSum.fromElemSym

{- * Show -}

appPrec :: Int
appPrec  = 10

instance (Show a) => Show (T a) where
showsPrec p (Cons xs) =
showParen (p >= appPrec)
(showString "RootSet.Cons " . shows xs)

{- Use binomial expansion of (x+y)^n -}
add :: (Field.C a, ZeroTestable.C a) => [a] -> [a] -> [a]

addInt :: (Integral.C a, Eq a, ZeroTestable.C a) => [a] -> [a] -> [a]

instance (Field.C a, ZeroTestable.C a) => Additive.C (T a) where
zero   = const zero
negate = lift1 Poly.alternate

{- * Ring -}

mul :: (Field.C a, ZeroTestable.C a) => [a] -> [a] -> [a]
mul = liftPowerSum2 PowerSum.mul

mulInt :: (Integral.C a, Eq a, ZeroTestable.C a) => [a] -> [a] -> [a]
mulInt = liftPowerSumInt2 PowerSum.mul

pow :: (Field.C a, ZeroTestable.C a) => Integer -> [a] -> [a]
pow n = liftPowerSum1 (PowerSum.pow n)

powInt :: (Integral.C a, Eq a, ZeroTestable.C a) => Integer -> [a] -> [a]
powInt n = liftPowerSumInt1 (PowerSum.pow n)

instance (Field.C a, ZeroTestable.C a) => Ring.C (T a) where
one           = const one
fromInteger n = const (fromInteger n)
(*)           = lift2 mul
x^n           = lift1 (pow n) x

{- * Field.C -}

instance (Field.C a, ZeroTestable.C a) => Field.C (T a) where
recip = lift1 reverse

{- * Algebra -}

instance (Field.C a, ZeroTestable.C a) => Algebraic.C (T a) where
root n = lift1 (PowerSum.root n)
```