{-# LANGUAGE DataKinds, FlexibleContexts, FlexibleInstances, GeneralizedNewtypeDeriving, MultiParamTypeClasses, NoImplicitPrelude, RebindableSyntax, ScopedTypeVariables, StandaloneDeriving, TemplateHaskell, TypeFamilies, UndecidableInstances #-} -- | Data type, functions, and instances for complex numbers. module Crypto.Lol.Types.Complex ( Complex , roundComplex , cis, real, imag, fromReal ) where import Algebra.Additive as Additive (C) import Algebra.Field as Field (C) import Algebra.IntegralDomain as IntegralDomain import Algebra.Ring as Ring (C) import Algebra.ZeroTestable as ZeroTestable (C) import qualified Number.Complex as C hiding (exp, signum) import Crypto.Lol.Types.Numeric as LP import Control.DeepSeq import Data.Array.Repa.Eval as R import Data.Vector.Storable (Storable) import Data.Vector.Unboxed (Unbox) import Data.Vector.Unboxed.Deriving import System.Random import Test.QuickCheck -- | Newtype wrapper (with slightly different instances) for -- <https://hackage.haskell.org/package/numeric-prelude-0.4.2/docs/Number-Complex.html numeric-prelude Complex>. newtype Complex a = Complex (C.T a) deriving (Additive.C, Ring.C, ZeroTestable.C, Field.C, Storable, Eq, Show, Arbitrary) derivingUnbox "Complex" [t| forall a . (Unbox a) => Complex a -> (a, a) |] [| \ (Complex x) -> (C.real x, C.imag x) |] [| \ (r, i) -> Complex $ r C.+: i |] -- a custom IntegralDomain instance, replacing the one provided by NP. -- it always returns 0 as the remainder of a division. If we were to -- use the NP instance, sometimes precision issues yield nonzero -- remainders, which makes, e.g., 'divGPow' think that division has -- failed, when it has not. This in turn causes 'divGCRT' to yield -- Nothing, among other problems. instance (Field a) => IntegralDomain.C (Complex a) where (Complex a) `divMod` (Complex b) = (Complex $ a / b, LP.zero) -- we can't use Generics for NFData because NP doesn't export the -- (deep) constructor for Complex.T instance (NFData a) => NFData (Complex a) where rnf (Complex x) = let r = C.real x i = C.imag x in rnf r `seq` rnf i `seq` () instance (Random a) => Random (Complex a) where random g = let (a,g') = random g (b,g'') = random g' in (Complex $ a C.+: b, g'') randomR = error "randomR not defined for (Complex t)" instance (R.Elt a) => R.Elt (Complex a) where touch (Complex c) = do touch $ C.real c touch $ C.imag c zero = Complex $ R.zero C.+: R.zero one = Complex $ R.one C.+: R.zero -- | Rounds the real and imaginary components to the nearest integer. roundComplex :: (RealRing a, ToInteger b) => Complex a -> (b,b) roundComplex (Complex x) = (round $ C.real x, round $ C.imag x) -- | 'cis' @t@ is a complex value with magnitude 1 and phase t (modulo @2*Pi@). cis :: Transcendental a => a -> Complex a cis = Complex . C.cis -- | Real component of a complex number. real :: Complex a -> a real (Complex a) = C.real a -- | Imaginary component of a complex number. imag :: Complex a -> a imag (Complex a) = C.imag a -- | Embeds a scalar as the real component of a complex number. fromReal :: Additive a => a -> Complex a fromReal = Complex . C.fromReal