{-# LANGUAGE NoImplicitPrelude #-}
{- |
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.Additive as Additive
import qualified Algebra.ZeroTestable as ZeroTestable
import qualified Data.List.Match as Match
import Control.Monad (liftM2)
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]
addRoot x yt@(y:ys) =
y : (ys + Poly.scale x yt)
addRoot _ [] =
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 =
Match.take x (fromPS (op (toPS x)))
liftPowerSum2Gen :: ([a] -> [a]) -> ([a] -> [a]) ->
([a] -> [a] -> [a]) -> ([a] -> [a] -> [a])
liftPowerSum2Gen fromPS toPS op x y =
Match.take (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)
{- * Additive -}
{- Use binomial expansion of (x+y)^n -}
add :: (Field.C a, ZeroTestable.C a) => [a] -> [a] -> [a]
add = liftPowerSum2 PowerSum.add
addInt :: (Integral.C a, Eq a, ZeroTestable.C a) => [a] -> [a] -> [a]
addInt = liftPowerSumInt2 PowerSum.add
instance (Field.C a, ZeroTestable.C a) => Additive.C (T a) where
zero = const zero
(+) = lift2 add
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)