Copyright	(c) The University of Glasgow 2001
License	BSD-style (see the file LICENSE)
Maintainer	libraries@haskell.org
Stability	stable
Portability	portable
Safe Haskell	None
Language	Haskell2010

Data.YAP.Ratio

Description

Standard functions on rational numbers.

This is a replacement for Data.Ratio, using the same Ratio type, but with components generalized from Integral to StandardAssociate + Euclidean.

Using the same type means we have the old instances. The Ord and Read instances could be relaxed to:

instance (Ord a, Semiring a) => Ord (Ratio a)
instance (Eq a, StandardAssociate a, Euclidean a, Read a) => Read (Ratio a)

Synopsis

data Ratio a
type Rational = Ratio Integer
(%) :: (Eq a, StandardAssociate a, Euclidean a) => a -> a -> Ratio a
numerator :: Ratio a -> a
denominator :: Ratio a -> a
approxRational :: ToRational a => a -> a -> Rational

Documentation

data Ratio a #

Rational numbers, with numerator and denominator of some Integral type.

Note that Ratio's instances inherit the deficiencies from the type parameter's. For example, Ratio Natural's Num instance has similar problems to Natural's.

Instances

Instances details

(Data a, Integral a) => Data (Ratio a)	Since: base-4.0.0.0
Instance details Defined in Data.Data Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Ratio a -> c (Ratio a) # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Ratio a) # toConstr :: Ratio a -> Constr # dataTypeOf :: Ratio a -> DataType # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (Ratio a)) # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Ratio a)) # gmapT :: (forall b. Data b => b -> b) -> Ratio a -> Ratio a # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Ratio a -> r # gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Ratio a -> r # gmapQ :: (forall d. Data d => d -> u) -> Ratio a -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> Ratio a -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> Ratio a -> m (Ratio a) # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Ratio a -> m (Ratio a) # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Ratio a -> m (Ratio a) #
(Storable a, Integral a) => Storable (Ratio a)	Since: base-4.8.0.0
Instance details Defined in Foreign.Storable Methods sizeOf :: Ratio a -> Int # alignment :: Ratio a -> Int # peekElemOff :: Ptr (Ratio a) -> Int -> IO (Ratio a) # pokeElemOff :: Ptr (Ratio a) -> Int -> Ratio a -> IO () # peekByteOff :: Ptr b -> Int -> IO (Ratio a) # pokeByteOff :: Ptr b -> Int -> Ratio a -> IO () # peek :: Ptr (Ratio a) -> IO (Ratio a) # poke :: Ptr (Ratio a) -> Ratio a -> IO () #
Integral a => Enum (Ratio a)	Since: base-2.0.1
Instance details Defined in GHC.Real Methods succ :: Ratio a -> Ratio a # pred :: Ratio a -> Ratio a # toEnum :: Int -> Ratio a # fromEnum :: Ratio a -> Int # enumFrom :: Ratio a -> [Ratio a] # enumFromThen :: Ratio a -> Ratio a -> [Ratio a] # enumFromTo :: Ratio a -> Ratio a -> [Ratio a] # enumFromThenTo :: Ratio a -> Ratio a -> Ratio a -> [Ratio a] #
Integral a => Num (Ratio a)	Since: base-2.0.1
Instance details Defined in GHC.Real Methods (+) :: Ratio a -> Ratio a -> Ratio a # (-) :: Ratio a -> Ratio a -> Ratio a # (*) :: Ratio a -> Ratio a -> Ratio a # negate :: Ratio a -> Ratio a # abs :: Ratio a -> Ratio a # signum :: Ratio a -> Ratio a # fromInteger :: Integer -> Ratio a #
(Integral a, Read a) => Read (Ratio a)	Since: base-2.1
Instance details Defined in GHC.Read Methods readsPrec :: Int -> ReadS (Ratio a) # readList :: ReadS [Ratio a] # readPrec :: ReadPrec (Ratio a) # readListPrec :: ReadPrec [Ratio a] #
Integral a => Fractional (Ratio a)	Since: base-2.0.1
Instance details Defined in GHC.Real Methods (/) :: Ratio a -> Ratio a -> Ratio a # recip :: Ratio a -> Ratio a # fromRational :: Rational -> Ratio a #
Integral a => Real (Ratio a)	Since: base-2.0.1
Instance details Defined in GHC.Real Methods toRational :: Ratio a -> Rational #
Integral a => RealFrac (Ratio a)	Since: base-2.0.1
Instance details Defined in GHC.Real Methods properFraction :: Integral b => Ratio a -> (b, Ratio a) # truncate :: Integral b => Ratio a -> b # round :: Integral b => Ratio a -> b # ceiling :: Integral b => Ratio a -> b # floor :: Integral b => Ratio a -> b #
Show a => Show (Ratio a)	Since: base-2.0.1
Instance details Defined in GHC.Real Methods showsPrec :: Int -> Ratio a -> ShowS # show :: Ratio a -> String # showList :: [Ratio a] -> ShowS #
Eq a => Eq (Ratio a)	Since: base-2.1
Instance details Defined in GHC.Real Methods (==) :: Ratio a -> Ratio a -> Bool # (/=) :: Ratio a -> Ratio a -> Bool #
Integral a => Ord (Ratio a)	Since: base-2.0.1
Instance details Defined in GHC.Real Methods compare :: Ratio a -> Ratio a -> Ordering # (<) :: Ratio a -> Ratio a -> Bool # (<=) :: Ratio a -> Ratio a -> Bool # (>) :: Ratio a -> Ratio a -> Bool # (>=) :: Ratio a -> Ratio a -> Bool # max :: Ratio a -> Ratio a -> Ratio a # min :: Ratio a -> Ratio a -> Ratio a #
(Eq a, StandardAssociate a, Euclidean a, Ring a) => AbelianGroup (Ratio a) Source #
Instance details Defined in Data.YAP.Algebra.Internal Methods (-) :: Ratio a -> Ratio a -> Ratio a Source # negate :: Ratio a -> Ratio a Source # gtimes :: (AbelianGroup b, ToInteger b) => b -> Ratio a -> Ratio a Source #
(Eq a, StandardAssociate a, Euclidean a) => AdditiveMonoid (Ratio a) Source #
Instance details Defined in Data.YAP.Algebra.Internal Methods (+) :: Ratio a -> Ratio a -> Ratio a Source # zero :: Ratio a Source # atimes :: ToInteger b => b -> Ratio a -> Ratio a Source #
(Eq a, StandardAssociate a, Euclidean a, Ring a) => DivisionRing (Ratio a) Source #
Instance details Defined in Data.YAP.Algebra.Internal
(Eq a, StandardAssociate a, Euclidean a) => DivisionSemiring (Ratio a) Source #
Instance details Defined in Data.YAP.Algebra.Internal Methods recip :: Ratio a -> Ratio a Source #
(Eq a, StandardAssociate a, Euclidean a, Ring a) => Field (Ratio a) Source #
Instance details Defined in Data.YAP.Algebra.Internal
Integral a => Fractional (Ratio a) Source #	As in Data.Ratio.
Instance details Defined in Data.YAP.Algebra.Internal
Integral a => FromRational (Ratio a) Source #
Instance details Defined in Data.YAP.Algebra.Internal Methods fromRational :: Rational -> Ratio a Source #
Integral a => Num (Ratio a) Source #	As in Data.Ratio.
Instance details Defined in Data.YAP.Algebra.Internal Methods abs :: Ratio a -> Ratio a Source # signum :: Ratio a -> Ratio a Source #
(Integral a, Integral a) => Real (Ratio a) Source #	As in Data.Ratio. The original version of `Integral` is required here by the old instance `Ord (Ratio a)`. Ideally there would be only one.
Instance details Defined in Data.YAP.Algebra.Internal
(Integral a, Integral a) => RealFrac (Ratio a) Source #	As in Data.Ratio. The original version of `Integral` is required here by the old instance `Ord (Ratio a)`. Ideally there would be only one.
Instance details Defined in Data.YAP.Algebra.Internal Methods properFraction :: Integral b => Ratio a -> (b, Ratio a) Source # truncate :: Integral b => Ratio a -> b Source # round :: Integral b => Ratio a -> b Source # ceiling :: Integral b => Ratio a -> b Source # floor :: Integral b => Ratio a -> b Source #
(Eq a, StandardAssociate a, Euclidean a, Ring a) => Ring (Ratio a) Source #
Instance details Defined in Data.YAP.Algebra.Internal Methods fromInteger :: Integer -> Ratio a Source #
(Eq a, StandardAssociate a, Euclidean a) => Semifield (Ratio a) Source #
Instance details Defined in Data.YAP.Algebra.Internal Methods (/) :: Ratio a -> Ratio a -> Ratio a Source #
(Eq a, StandardAssociate a, Euclidean a) => Semiring (Ratio a) Source #
Instance details Defined in Data.YAP.Algebra.Internal Methods (*) :: Ratio a -> Ratio a -> Ratio a Source # one :: Ratio a Source # fromNatural :: Natural -> Ratio a Source # rescale :: Ratio a -> Ratio a -> (Ratio a, Ratio a, Ratio a -> Ratio a) Source #
(ToInteger a, Integral a) => ToRational (Ratio a) Source #	The original version of `Integral` is required here by the old instance `Ord (Ratio a)`. Ideally this would be replaced with `Ord a`.
Instance details Defined in Data.YAP.Algebra.Internal Methods toRational :: Ratio a -> Rational Source #

type Rational = Ratio Integer #

Arbitrary-precision rational numbers, represented as a ratio of two Integer values. A rational number may be constructed using the % operator.

(%) :: (Eq a, StandardAssociate a, Euclidean a) => a -> a -> Ratio a infixl 7 Source #

Forms the ratio of two values in a Euclidean domain (e.g. Integer).

numerator :: Ratio a -> a Source #

Extract the numerator of the ratio in reduced form: the numerator and denominator have no common factor and the denominator is positive.

denominator :: Ratio a -> a Source #

Extract the denominator of the ratio in reduced form: the numerator and denominator have no common factor and the denominator is positive.

approxRational :: ToRational a => a -> a -> Rational Source #

approxRational, applied to two real fractional numbers x and epsilon, returns the simplest rational number within epsilon of x. A rational number y is said to be simpler than another y' if

abs (numerator y) <= abs (numerator y'), and
denominator y <= denominator y'.

Any real interval contains a unique simplest rational; in particular, note that 0/1 is the simplest rational of all.