{-# OPTIONS -fno-implicit-prelude #-} module Algebra.RealField where import qualified Algebra.Field as Field import qualified Algebra.PrincipalIdealDomain as PID import qualified Algebra.Real as Real import qualified Algebra.Ring as Ring import qualified Algebra.ToRational as ToRational import qualified Algebra.ToInteger as ToInteger import Algebra.Field ((/)) import Algebra.RealIntegral (quotRem, ) import Algebra.IntegralDomain (divMod, even, ) import Algebra.Ring ((*), fromInteger, ) import Algebra.Additive ((+), (-), negate, ) import Algebra.ZeroTestable (isZero, ) import Algebra.ToInteger (fromIntegral, ) import qualified Number.Ratio as Ratio import Number.Ratio (T((:%)), Rational) import qualified GHC.Float as GHC import Prelude(Int,Float,Double) import qualified Prelude as P import PreludeBase {- | Minimal complete definition: 'splitFraction' or 'floor' There are probably more laws, but some laws are > (fromInteger.fst.splitFraction) a + (snd.splitFraction) a === a > ceiling (toRational x) === ceiling x :: Integer > truncate (toRational x) === truncate x :: Integer > floor (toRational x) === floor x :: Integer If there wouldn't be @Real.C a@ and @ToInteger.C b@ constraints, we could also use this class for splitting ratios of polynomials. As an aside, let me note the similarities between @splitFraction x@ and @x divMod 1@ (if that were defined). In particular, it might make sense to unify the rounding modes somehow. IEEEFloat-specific calls are removed here (cf. 'Prelude.RealFloat') so probably nobody will actually use this default definition. Henning: New function 'fraction' doesn't return the integer part of the number. This also removes a type ambiguity if the integer part is not needed. The new methods 'fraction' and 'splitFraction' differ from 'Prelude.properFraction' semantics. They always round to 'floor'. This means that the fraction is always non-negative and is always smaller than 1. This is more useful in practice and can be generalised to more than real numbers. Since every 'Number.Ratio.T' denominator type supports 'Algebra.IntegralDomain.divMod', every 'Number.Ratio.T' can provide 'fraction' and 'splitFraction', e.g. fractions of polynomials. However the ''integral'' part would not be of type class 'ToInteger.C'. Can there be a separate class for 'fraction', 'splitFraction', 'floor' and 'ceiling' since they do not need reals and their ordering? -} class (Real.C a, Field.C a) => C a where splitFraction :: (ToInteger.C b) => a -> (b,a) fraction :: a -> a ceiling, floor :: (ToInteger.C b) => a -> b truncate, round :: (ToInteger.C b) => a -> b splitFraction x = (floor x, fraction x) fraction x = x - fromInteger (floor x) floor x = fromInteger (fst (splitFraction x)) ceiling x = - floor (-x) -- truncate x = signum x * floor (abs x) truncate x = if x>=0 then floor x else ceiling x round x = let (n,r) = splitFraction x in case compare r (1/2) of LT -> n EQ -> if even n then n else n+1 GT -> n+1 instance (ToInteger.C a, PID.C a) => C (Ratio.T a) where splitFraction (x:%y) = (fromIntegral q, r:%y) where (q,r) = divMod x y instance C Float where splitFraction = preludeSplitFraction fraction = fractionTrunc (GHC.int2Float . GHC.float2Int) -- preludeFraction floor = fromInteger . P.floor ceiling = fromInteger . P.ceiling round = fromInteger . P.round truncate = fromInteger . P.truncate instance C Double where splitFraction = preludeSplitFraction fraction = fractionTrunc (GHC.int2Double . GHC.double2Int) floor = fromInteger . P.floor ceiling = fromInteger . P.ceiling round = fromInteger . P.round truncate = fromInteger . P.truncate preludeSplitFraction :: (P.RealFrac a, Ring.C a, ToInteger.C b) => a -> (b,a) preludeSplitFraction x = let (n,f) = P.properFraction x -- if x>=0 || f==0 in if f>=0 then (fromInteger n, f) else (fromInteger n-1, f+1) preludeFraction :: (P.RealFrac a, Ring.C a) => a -> a preludeFraction x = let second :: (Int, a) -> a second = snd in fixFraction (second (P.properFraction x)) fractionTrunc :: (Ring.C a, Ord a) => (a -> a) -> a -> a fractionTrunc trunc x = fixFraction (x - trunc x) fixFraction :: (Ring.C a, Ord a) => a -> a fixFraction y = if y>=0 then y else y+1 {- | TODO: Should be moved to a continued fraction module. -} approxRational :: (ToRational.C a, C a) => a -> a -> Rational approxRational rat eps = simplest (rat-eps) (rat+eps) where simplest x y | y < x = simplest y x | x == y = xr | x > 0 = simplest' n d n' d' | y < 0 = - simplest' (-n') d' (-n) d | otherwise = 0 :% 1 where xr@(n:%d) = ToRational.toRational x (n':%d') = ToRational.toRational y simplest' n d n' d' -- assumes 0 < n%d < n'%d' | isZero r = q :% 1 | q /= q' = (q+1) :% 1 | otherwise = (q*n''+d'') :% n'' where (q,r) = quotRem n d (q',r') = quotRem n' d' (n'':%d'') = simplest' d' r' d r