Data.ExtendedReal

Extension of real numbers with positive/negative infinities (±∞). It is useful for describing various limiting behaviors in mathematics.

Remarks:

• ∞ - ∞ is left undefined as usual, but we define 0 × ∞ = 0 × -∞ = 0 by following the convention of probability or measure theory.

References:

• Wikipedia contributors, "Extended real number line," Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Extended_real_number_line (accessed September 1, 2014).

Documentation

data Extended r

Extended r is an extension of r with positive/negative infinity (±∞).

Constructors

NegInf

negative infinity (-∞)

Finite !r

finite value

PosInf

positive infinity (+∞)

Instances

Foldable Extended

Since: 0.2.6.0

Defined in Data.ExtendedReal

Methods

fold :: Monoid m => Extended m -> m

foldMap :: Monoid m => (a -> m) -> Extended a -> m

foldMap' :: Monoid m => (a -> m) -> Extended a -> m

foldr :: (a -> b -> b) -> b -> Extended a -> b

foldr' :: (a -> b -> b) -> b -> Extended a -> b

foldl :: (b -> a -> b) -> b -> Extended a -> b

foldl' :: (b -> a -> b) -> b -> Extended a -> b

foldr1 :: (a -> a -> a) -> Extended a -> a

foldl1 :: (a -> a -> a) -> Extended a -> a

toList :: Extended a -> [a]

null :: Extended a -> Bool

length :: Extended a -> Int

elem :: Eq a => a -> Extended a -> Bool

maximum :: Ord a => Extended a -> a

minimum :: Ord a => Extended a -> a

sum :: Num a => Extended a -> a

product :: Num a => Extended a -> a

Traversable Extended

Since: 0.2.6.0

Defined in Data.ExtendedReal

Methods

traverse :: Applicative f => (a -> f b) -> Extended a -> f (Extended b)

sequenceA :: Applicative f => Extended (f a) -> f (Extended a)

mapM :: Monad m => (a -> m b) -> Extended a -> m (Extended b)

sequence :: Monad m => Extended (m a) -> m (Extended a)

Functor Extended

Defined in Data.ExtendedReal

Methods

fmap :: (a -> b) -> Extended a -> Extended b

(<$) :: a -> Extended b -> Extended a

Lift r => Lift (Extended r :: Type)

Since: 0.2.6.0

Defined in Data.ExtendedReal

Methods

lift :: Quote m => Extended r -> m Exp

liftTyped :: forall (m :: Type -> Type). Quote m => Extended r -> Code m (Extended r)

Data r => Data (Extended r)

Defined in Data.ExtendedReal

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Extended r -> c (Extended r)

gunfold :: (forall b r0. Data b => c (b -> r0) -> c r0) -> (forall r1. r1 -> c r1) -> Constr -> c (Extended r)

toConstr :: Extended r -> Constr

dataTypeOf :: Extended r -> DataType

dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (Extended r))

dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Extended r))

gmapT :: (forall b. Data b => b -> b) -> Extended r -> Extended r

gmapQl :: (r0 -> r' -> r0) -> r0 -> (forall d. Data d => d -> r') -> Extended r -> r0

gmapQr :: forall r0 r'. (r' -> r0 -> r0) -> r0 -> (forall d. Data d => d -> r') -> Extended r -> r0

gmapQ :: (forall d. Data d => d -> u) -> Extended r -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> Extended r -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> Extended r -> m (Extended r)

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Extended r -> m (Extended r)

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Extended r -> m (Extended r)

Bounded (Extended r)

Defined in Data.ExtendedReal

Methods

minBound :: Extended r

maxBound :: Extended r

Generic (Extended r)

Defined in Data.ExtendedReal

Associated Types

type Rep (Extended r) :: Type -> Type

Methods

from :: Extended r -> Rep (Extended r) x

to :: Rep (Extended r) x -> Extended r

(Num r, Ord r) => Num (Extended r)

Note that Extended r is not a field, nor a ring.

PosInf + NegInf is left undefined as usual, but we define 0 * PosInf = 0 * NegInf = 0 by following the convention of probability or measure theory.

Defined in Data.ExtendedReal

Methods

(+) :: Extended r -> Extended r -> Extended r

(-) :: Extended r -> Extended r -> Extended r

(*) :: Extended r -> Extended r -> Extended r

negate :: Extended r -> Extended r

abs :: Extended r -> Extended r

signum :: Extended r -> Extended r

fromInteger :: Integer -> Extended r

Read r => Read (Extended r)

Defined in Data.ExtendedReal

Methods

readsPrec :: Int -> ReadS (Extended r)

readList :: ReadS [Extended r]

readPrec :: ReadPrec (Extended r)

readListPrec :: ReadPrec [Extended r]

(Fractional r, Ord r) => Fractional (Extended r)

Note that Extended r is not a field, nor a ring.

Defined in Data.ExtendedReal

Methods

(/) :: Extended r -> Extended r -> Extended r

recip :: Extended r -> Extended r

fromRational :: Rational -> Extended r

Show r => Show (Extended r)

Defined in Data.ExtendedReal

Methods

showsPrec :: Int -> Extended r -> ShowS

show :: Extended r -> String

showList :: [Extended r] -> ShowS

NFData r => NFData (Extended r)

Defined in Data.ExtendedReal

Methods

rnf :: Extended r -> ()

Eq r => Eq (Extended r)

Defined in Data.ExtendedReal

Methods

(==) :: Extended r -> Extended r -> Bool

(/=) :: Extended r -> Extended r -> Bool

Ord r => Ord (Extended r)

Defined in Data.ExtendedReal

Methods

compare :: Extended r -> Extended r -> Ordering

(<) :: Extended r -> Extended r -> Bool

(<=) :: Extended r -> Extended r -> Bool

(>) :: Extended r -> Extended r -> Bool

(>=) :: Extended r -> Extended r -> Bool

max :: Extended r -> Extended r -> Extended r

min :: Extended r -> Extended r -> Extended r

Hashable r => Hashable (Extended r)

Defined in Data.ExtendedReal

Methods

hashWithSalt :: Int -> Extended r -> Int

hash :: Extended r -> Int

type Rep (Extended r)

Since: 0.2.6.0

Defined in Data.ExtendedReal

type Rep (Extended r) = D1 ('MetaData "Extended" "Data.ExtendedReal" "extended-reals-0.2.6.0-Dg3SlKF58uu9KRRfiGdf5p" 'False) (C1 ('MetaCons "NegInf" 'PrefixI 'False) (U1 :: Type -> Type) :+: (C1 ('MetaCons "Finite" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 r)) :+: C1 ('MetaCons "PosInf" 'PrefixI 'False) (U1 :: Type -> Type)))

inf :: Extended r

Infinity (∞)

isFinite :: Extended r -> Bool

isFinite x = not (isInfinite x).

isInfinite :: Extended r -> Bool

isInfinite x returns True iff x is PosInf or NegInf.

fromRealFloat :: RealFloat r => r -> Extended r

Helper to convert Double and Float to Extended, taking care of infinite values automatically.

>>> fromRealFloat (1 / 0)
PosInf
>>> fromRealFloat (-1 / 0)
NegInf
>>> fromRealFloat (0 / 0)
*** Exception: fromRealFloat: argument should not be NaN

Since: 0.2.5.0