Copyright | (c) Masahiro Sakai 2011-2013 |
---|---|
License | BSD-style |
Maintainer | masahiro.sakai@gmail.com |
Stability | provisional |
Portability | non-portable (CPP, ScopedTypeVariables, DeriveDataTypeable) |
Safe Haskell | Safe |
Language | Haskell2010 |
Interval datatype and interval arithmetic.
Unlike the intervals package (http://hackage.haskell.org/package/intervals),
this module provides both open and closed intervals and is intended to be used
with Rational
.
For the purpose of abstract interpretation, it might be convenient to use
Lattice
instance. See also lattices package
(http://hackage.haskell.org/package/lattices).
Synopsis
- data Interval r
- module Data.ExtendedReal
- data Boundary
- interval :: Ord r => (Extended r, Boundary) -> (Extended r, Boundary) -> Interval r
- (<=..<=) :: Ord r => Extended r -> Extended r -> Interval r
- (<..<=) :: Ord r => Extended r -> Extended r -> Interval r
- (<=..<) :: Ord r => Extended r -> Extended r -> Interval r
- (<..<) :: Ord r => Extended r -> Extended r -> Interval r
- whole :: Ord r => Interval r
- empty :: Ord r => Interval r
- singleton :: Ord r => r -> Interval r
- null :: Ord r => Interval r -> Bool
- isSingleton :: Ord r => Interval r -> Bool
- member :: Ord r => r -> Interval r -> Bool
- notMember :: Ord r => r -> Interval r -> Bool
- isSubsetOf :: Ord r => Interval r -> Interval r -> Bool
- isProperSubsetOf :: Ord r => Interval r -> Interval r -> Bool
- isConnected :: Ord r => Interval r -> Interval r -> Bool
- lowerBound :: Interval r -> Extended r
- upperBound :: Interval r -> Extended r
- lowerBound' :: Interval r -> (Extended r, Boundary)
- upperBound' :: Interval r -> (Extended r, Boundary)
- width :: (Num r, Ord r) => Interval r -> r
- (<!) :: Ord r => Interval r -> Interval r -> Bool
- (<=!) :: Ord r => Interval r -> Interval r -> Bool
- (==!) :: Ord r => Interval r -> Interval r -> Bool
- (>=!) :: Ord r => Interval r -> Interval r -> Bool
- (>!) :: Ord r => Interval r -> Interval r -> Bool
- (/=!) :: Ord r => Interval r -> Interval r -> Bool
- (<?) :: Ord r => Interval r -> Interval r -> Bool
- (<=?) :: Ord r => Interval r -> Interval r -> Bool
- (==?) :: Ord r => Interval r -> Interval r -> Bool
- (>=?) :: Ord r => Interval r -> Interval r -> Bool
- (>?) :: Ord r => Interval r -> Interval r -> Bool
- (/=?) :: Ord r => Interval r -> Interval r -> Bool
- (<??) :: (Real r, Fractional r) => Interval r -> Interval r -> Maybe (r, r)
- (<=??) :: (Real r, Fractional r) => Interval r -> Interval r -> Maybe (r, r)
- (==??) :: (Real r, Fractional r) => Interval r -> Interval r -> Maybe (r, r)
- (>=??) :: (Real r, Fractional r) => Interval r -> Interval r -> Maybe (r, r)
- (>??) :: (Real r, Fractional r) => Interval r -> Interval r -> Maybe (r, r)
- (/=??) :: (Real r, Fractional r) => Interval r -> Interval r -> Maybe (r, r)
- intersection :: forall r. Ord r => Interval r -> Interval r -> Interval r
- intersections :: Ord r => [Interval r] -> Interval r
- hull :: forall r. Ord r => Interval r -> Interval r -> Interval r
- hulls :: Ord r => [Interval r] -> Interval r
- mapMonotonic :: (Ord a, Ord b) => (a -> b) -> Interval a -> Interval b
- pickup :: (Real r, Fractional r) => Interval r -> Maybe r
- simplestRationalWithin :: RealFrac r => Interval r -> Maybe Rational
Interval type
The intervals (i.e. connected and convex subsets) over real numbers R.
Instances
module Data.ExtendedReal
Boundary of an interval may be open (excluding an endpoint) or closed (including an endpoint).
Since: 2.0.0
Instances
Bounded Boundary Source # | |
Enum Boundary Source # | |
Eq Boundary Source # | |
Data Boundary Source # | |
Defined in Data.Interval.Internal gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Boundary -> c Boundary # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c Boundary # toConstr :: Boundary -> Constr # dataTypeOf :: Boundary -> DataType # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c Boundary) # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c Boundary) # gmapT :: (forall b. Data b => b -> b) -> Boundary -> Boundary # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Boundary -> r # gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Boundary -> r # gmapQ :: (forall d. Data d => d -> u) -> Boundary -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> Boundary -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> Boundary -> m Boundary # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Boundary -> m Boundary # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Boundary -> m Boundary # | |
Ord Boundary Source # | |
Defined in Data.Interval.Internal | |
Read Boundary Source # | |
Show Boundary Source # | |
Generic Boundary Source # | |
NFData Boundary Source # | |
Defined in Data.Interval.Internal | |
Hashable Boundary Source # | |
Defined in Data.Interval.Internal | |
type Rep Boundary Source # | |
Construction
:: Ord r | |
=> (Extended r, Boundary) | lower bound and whether it is included |
-> (Extended r, Boundary) | upper bound and whether it is included |
-> Interval r |
smart constructor for Interval
closed interval [l
,u
]
left-open right-closed interval (l
,u
]
left-closed right-open interval [l
, u
)
open interval (l
, u
)
Query
isSubsetOf :: Ord r => Interval r -> Interval r -> Bool Source #
Is this a subset?
(i1 `
tells whether isSubsetOf
` i2)i1
is a subset of i2
.
isProperSubsetOf :: Ord r => Interval r -> Interval r -> Bool Source #
Is this a proper subset? (i.e. a subset but not equal).
isConnected :: Ord r => Interval r -> Interval r -> Bool Source #
Does the union of two range form a connected set?
Since 1.3.0
lowerBound :: Interval r -> Extended r Source #
Lower endpoint (i.e. greatest lower bound) of the interval.
lowerBound
of the empty interval isPosInf
.lowerBound
of a left unbounded interval isNegInf
.lowerBound
of an interval may or may not be a member of the interval.
upperBound :: Interval r -> Extended r Source #
Upper endpoint (i.e. least upper bound) of the interval.
upperBound
of the empty interval isNegInf
.upperBound
of a right unbounded interval isPosInf
.upperBound
of an interval may or may not be a member of the interval.
width :: (Num r, Ord r) => Interval r -> r Source #
Width of a interval. Width of an unbounded interval is undefined
.
Universal comparison operators
(/=!) :: Ord r => Interval r -> Interval r -> Bool infix 4 Source #
For all x
in X
, y
in Y
. x
?/=
y
Since 1.0.1
Existential comparison operators
(<?) :: Ord r => Interval r -> Interval r -> Bool infix 4 Source #
Does there exist an x
in X
, y
in Y
such that x
?<
y
(<=?) :: Ord r => Interval r -> Interval r -> Bool infix 4 Source #
Does there exist an x
in X
, y
in Y
such that x
?<=
y
(==?) :: Ord r => Interval r -> Interval r -> Bool infix 4 Source #
Does there exist an x
in X
, y
in Y
such that x
?==
y
Since 1.0.0
(>=?) :: Ord r => Interval r -> Interval r -> Bool infix 4 Source #
Does there exist an x
in X
, y
in Y
such that x
?>=
y
(>?) :: Ord r => Interval r -> Interval r -> Bool infix 4 Source #
Does there exist an x
in X
, y
in Y
such that x
?>
y
(/=?) :: Ord r => Interval r -> Interval r -> Bool infix 4 Source #
Does there exist an x
in X
, y
in Y
such that x
?/=
y
Since 1.0.1
Existential comparison operators that produce witnesses (experimental)
(<??) :: (Real r, Fractional r) => Interval r -> Interval r -> Maybe (r, r) infix 4 Source #
Does there exist an x
in X
, y
in Y
such that x
?<
y
Since 1.0.0
(<=??) :: (Real r, Fractional r) => Interval r -> Interval r -> Maybe (r, r) infix 4 Source #
Does there exist an x
in X
, y
in Y
such that x
?<=
y
Since 1.0.0
(==??) :: (Real r, Fractional r) => Interval r -> Interval r -> Maybe (r, r) infix 4 Source #
Does there exist an x
in X
, y
in Y
such that x
?==
y
Since 1.0.0
(>=??) :: (Real r, Fractional r) => Interval r -> Interval r -> Maybe (r, r) infix 4 Source #
Does there exist an x
in X
, y
in Y
such that x
?>=
y
Since 1.0.0
(>??) :: (Real r, Fractional r) => Interval r -> Interval r -> Maybe (r, r) infix 4 Source #
Does there exist an x
in X
, y
in Y
such that x
?>
y
Since 1.0.0
(/=??) :: (Real r, Fractional r) => Interval r -> Interval r -> Maybe (r, r) infix 4 Source #
Does there exist an x
in X
, y
in Y
such that x
?/=
y
Since 1.0.1
Combine
intersection :: forall r. Ord r => Interval r -> Interval r -> Interval r Source #
intersection of two intervals
intersections :: Ord r => [Interval r] -> Interval r Source #
intersection of a list of intervals.
Since 0.6.0
hull :: forall r. Ord r => Interval r -> Interval r -> Interval r Source #
convex hull of two intervals
Map
mapMonotonic :: (Ord a, Ord b) => (a -> b) -> Interval a -> Interval b Source #
mapMonotonic f i
is the image of i
under f
, where f
must be a strict monotone function.
Operations
pickup :: (Real r, Fractional r) => Interval r -> Maybe r Source #
pick up an element from the interval if the interval is not empty.
simplestRationalWithin :: RealFrac r => Interval r -> Maybe Rational Source #
simplestRationalWithin
returns the simplest rational number within the interval.
A rational number y
is said to be simpler than another y'
if
, andabs
(numerator
y) <=abs
(numerator
y')
.denominator
y <=denominator
y'
(see also approxRational
)
Since 0.4.0
Orphan instances
(Real r, Fractional r) => Fractional (Interval r) Source # | |
(Num r, Ord r) => Num (Interval r) Source # | |
(Ord r, Read r) => Read (Interval r) Source # | |
(Ord r, Show r) => Show (Interval r) Source # | |
Ord r => Lattice (Interval r) Source # | |
Ord r => BoundedJoinSemiLattice (Interval r) Source # | |
Ord r => BoundedMeetSemiLattice (Interval r) Source # | |