{-# LANGUAGE NoImplicitPrelude #-}
T, coeffs, fromCoeffs,
fromPolynomial,
toPolynomial,
toPolynomialFast,
refinePolynomial,
convolvePolynomial,
convolveTruncatedPowerPolynomials,
) where

import qualified MathObj.Polynomial as Poly
import qualified MathObj.Polynomial.Core as PolyCore
import qualified Algebra.RealField as RealField
import qualified Algebra.Field  as Field
import qualified Algebra.Ring   as Ring
import qualified Algebra.Vector as Vector

import qualified Data.List as List
import qualified Data.List.HT as ListHT
import qualified Data.List.Match as Match
import Data.Maybe (fromMaybe, )

import qualified Test.QuickCheck as QC

import qualified NumericPrelude.List.Generic as NPList
import NumericPrelude.Base
import NumericPrelude.Numeric

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

{-# INLINE fromCoeffs #-}
fromCoeffs :: [a] -> T a
fromCoeffs = lift0

{-# INLINE lift0 #-}
lift0 :: [a] -> T a
lift0 = Cons

{-
{-# INLINE lift1 #-}
lift1 :: ([a] -> [a]) -> (T a -> T a)
lift1 f (Cons x0) = Cons (f x0)

{-# INLINE lift2 #-}
lift2 :: ([a] -> [a] -> [a]) -> (T a -> T a -> T a)
lift2 f (Cons x0) (Cons x1) = Cons (f x0 x1)
-}

{-
Functor instance is e.g. useful for converting number types,
say 'Rational' to 'Double'.
-}

instance Functor T where
fmap f (Cons xs) = Cons (map f xs)

{-# INLINE appPrec #-}
appPrec :: Int
appPrec  = 10

instance (Show a) => Show (T a) where
showsPrec p (Cons xs) =
showParen (p >= appPrec)
(showString "RefinementMask2.fromCoeffs " . shows xs)

instance (QC.Arbitrary a, Field.C a) => QC.Arbitrary (T a) where
arbitrary =
liftM2
(\degree body ->
let s = sum body
in  Cons \$ map ((2 ^- degree - s) / NPList.lengthLeft body +) body)
(QC.choose (-5,0)) QC.arbitrary

{- |

R - alternating binomial coefficients
L - differences of translated polynomials in columns

p2 = L * R^(-1) * m

R * L^(-1) * p2 = m
-}
fromPolynomial ::
(Field.C a) => Poly.T a -> T a
fromPolynomial poly =
fromCoeffs \$
foldr (\p ps ->
[] \$
foldr (\(db,dp) cs ->
ListHT.switchR
(error "RefinementMask2.fromPolynomial: polynomial should be non-empty")
(\dps dpe ->
cs ++ [(db - Ring.scalarProduct dps cs) / dpe])
dp) [] \$
zip
(Poly.coeffs \$ Poly.dilate 2 poly)
(List.transpose \$
Match.take (Poly.coeffs poly) \$
map Poly.coeffs \$
iterate polynomialDifference poly)

polynomialDifference ::
(Ring.C a) => Poly.T a -> Poly.T a
polynomialDifference poly =
Poly.fromCoeffs \$ init \$ Poly.coeffs \$
Poly.translate 1 poly - poly

{- |
If the mask does not sum up to a power of @1/2@
then the function returns 'Nothing'.
-}
toPolynomial ::
(RealField.C a) => T a -> Maybe (Poly.T a)
toPolynomial (Cons []) = Just \$ Poly.fromCoeffs []
let s = sum \$ coeffs mask
ks = reverse \$ takeWhile (<=1) \$ iterate (2*) s
in  case ks of
1:ks0 ->
Just \$
foldl
(\p k ->
let ip = Poly.integrate zero p
in  ip + Poly.const (correctConstant (fmap (k/s*) mask) ip))
(Poly.const 1) ks0
_ -> Nothing
{-
> fmap (6 Vector.*>) \$ toPolynomial (Cons [0.1, 0.02, 0.005::Rational])
Just (Polynomial.fromCoeffs [-12732 % 109375, 272 % 625, -18 % 25, 1 % 1])
-}

{-
The constant term must be zero,
higher terms must already satisfy the refinement constraint.
-}
correctConstant ::
(Field.C a) => T a -> Poly.T a -> a
let refined = refinePolynomial mask poly

toPolynomialFast ::
(RealField.C a) => T a -> Maybe (Poly.T a)
let s = sum \$ coeffs mask
ks = reverse \$ takeWhile (<=1) \$ iterate (2*) s
in  case ks of
1:ks0 ->
Just \$
foldl
(\p k ->
let ip = Poly.integrate zero p
in  ip + Poly.const (c*k / ((1-k)*s)))
(Poly.const 1) ks0
_ -> Nothing

refinePolynomial ::
(Ring.C a) => T a -> Poly.T a -> Poly.T a
Poly.shrink 2 .
iterate (Poly.translate 1)
{-
> mapM_ print \$ take 50 \$ iterate (refinePolynomial (Cons [0.1, 0.02, 0.005])) (Poly.fromCoeffs [0,0,0,1::Double])
...
Polynomial.fromCoeffs [-0.11640685714285712,0.4351999999999999,-0.7199999999999999,1.0]
-}

convolve ::
(Ring.C a) => T a -> T a -> T a
convolve x y =
fromCoeffs \$
PolyCore.mul (coeffs x) (coeffs y)

{- |

-}
convolvePolynomial ::
(RealField.C a) =>
Poly.T a -> Poly.T a -> Poly.T a
convolvePolynomial x y =
fromMaybe
toPolynomial \$ fmap (/2) \$
convolve (fromPolynomial x) (fromPolynomial y)

{-
This function interprets all monomials as truncated power functions,
that is power functions that are set to zero for negative arguments.
However the convolution implied by this interpretation
cannot be represented by means of mask convolution.
See for instance:

*MathObj.RefinementMask2> let x = Poly.fromCoeffs [1,1] :: Poly.T Rational
*MathObj.RefinementMask2> fromPolynomial \$ convolvePolynomial2 x x
RefinementMask2.fromCoeffs [1 % 3,-1 % 8,-1 % 8,1 % 24]

The obtained mask cannot be factored,
thus it is not a complete square.
But maybe it becomes a square if we add u*(-1,1)^4.
However this mask has sum 1/8 and the added term has sum 0,
thus the sum of the modified mask is still 1/8 and thus not a square.
-}
convolveTruncatedPowerPolynomials ::
(RealField.C a) =>
Poly.T a -> Poly.T a -> Poly.T a
convolveTruncatedPowerPolynomials x y =
let facs = scanl (*) 1 \$ iterate (1+) 1
xl = Poly.coeffs x
yl = Poly.coeffs y
in  Poly.integrate 0 \$
Poly.fromCoeffs \$
zipWith (flip (/)) facs \$
PolyCore.mul
(zipWith (*) facs xl)
(zipWith (*) facs yl)