{-# LANGUAGE DataKinds, TypeSynonymInstances, FlexibleContexts, FlexibleInstances, BangPatterns, ScopedTypeVariables #-}
module Math.Algebra.Polynomial.Univariate.Hermite
( hermiteH
, hermiteHe
, integralHermiteH
, integralHermiteHe
)
where
import Data.List
import Data.Ratio
import Data.Semigroup
import Data.Monoid
import GHC.TypeLits
import qualified Math.Algebra.Polynomial.FreeModule as ZMod
import Math.Algebra.Polynomial.FreeModule ( FreeMod , FreeModule(..) , ZMod , QMod )
import Math.Algebra.Polynomial.Univariate
import Math.Algebra.Polynomial.Class
import Math.Algebra.Polynomial.Pretty
import Math.Algebra.Polynomial.Misc
hermiteHe :: (Ring c, KnownSymbol v) => Int -> Univariate c v
hermiteHe = fromZUni . renameUniVar . integralHermiteHe
hermiteH :: (Ring c, KnownSymbol v) => Int -> Univariate c v
hermiteH = fromZUni . renameUniVar . integralHermiteH
x, twox :: Univariate Integer "x"
x = variableP ()
twox = monomP' (U 1) 2
integralHermiteHe :: Int -> Univariate Integer "x"
integralHermiteHe = intCache compute where
compute recur n = case n of
0 -> 1
1 -> x
n -> x * recur (n-1) - differentiateUni (recur (n-1))
integralHermiteH :: Int -> Univariate Integer "x"
integralHermiteH = intCache compute where
compute recur n = case n of
0 -> 1
1 -> twox
n -> twox * recur (n-1) - differentiateUni (recur (n-1))