| Copyright | (c) Masahiro Sakai 2011-2013 Andrew Lelechenko 2020 |
|---|---|
| License | BSD-style |
| Maintainer | masahiro.sakai@gmail.com |
| Stability | provisional |
| Portability | non-portable (CPP, ScopedTypeVariables, DeriveDataTypeable) |
| Safe Haskell | Safe |
| Language | Haskell2010 |
Data.Interval
Description
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
- extractSingleton :: Ord r => Interval r -> Maybe r
- 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
- relate :: Ord r => Interval r -> Interval r -> Relation
Interval type
The intervals ( i.e. connected and convex subsets) over real numbers R .
Instances
| Eq r => Eq ( Interval r) Source # | |
| ( RealFrac r, Floating r) => Floating ( Interval r) Source # |
When results of
|
|
Defined in Data.Interval Methods exp :: Interval r -> Interval r Source # log :: Interval r -> Interval r Source # sqrt :: Interval r -> Interval r Source # (**) :: Interval r -> Interval r -> Interval r Source # logBase :: Interval r -> Interval r -> Interval r Source # sin :: Interval r -> Interval r Source # cos :: Interval r -> Interval r Source # tan :: Interval r -> Interval r Source # asin :: Interval r -> Interval r Source # acos :: Interval r -> Interval r Source # atan :: Interval r -> Interval r Source # sinh :: Interval r -> Interval r Source # cosh :: Interval r -> Interval r Source # tanh :: Interval r -> Interval r Source # asinh :: Interval r -> Interval r Source # acosh :: Interval r -> Interval r Source # atanh :: Interval r -> Interval r Source # log1p :: Interval r -> Interval r Source # expm1 :: Interval r -> Interval r Source # |
|
| ( Real r, Fractional r) => Fractional ( Interval r) Source # |
|
| ( Ord r, Data r) => Data ( Interval r) Source # | |
|
Defined in Data.Interval.Internal Methods gfoldl :: ( forall d b. Data d => c (d -> b) -> d -> c b) -> ( forall g. g -> c g) -> Interval r -> c ( Interval r) Source # gunfold :: ( forall b r0. Data b => c (b -> r0) -> c r0) -> ( forall r1. r1 -> c r1) -> Constr -> c ( Interval r) Source # toConstr :: Interval r -> Constr Source # dataTypeOf :: Interval r -> DataType Source # dataCast1 :: Typeable t => ( forall d. Data d => c (t d)) -> Maybe (c ( Interval r)) Source # dataCast2 :: Typeable t => ( forall d e. ( Data d, Data e) => c (t d e)) -> Maybe (c ( Interval r)) Source # gmapT :: ( forall b. Data b => b -> b) -> Interval r -> Interval r Source # gmapQl :: (r0 -> r' -> r0) -> r0 -> ( forall d. Data d => d -> r') -> Interval r -> r0 Source # gmapQr :: forall r0 r'. (r' -> r0 -> r0) -> r0 -> ( forall d. Data d => d -> r') -> Interval r -> r0 Source # gmapQ :: ( forall d. Data d => d -> u) -> Interval r -> [u] Source # gmapQi :: Int -> ( forall d. Data d => d -> u) -> Interval r -> u Source # gmapM :: Monad m => ( forall d. Data d => d -> m d) -> Interval r -> m ( Interval r) Source # gmapMp :: MonadPlus m => ( forall d. Data d => d -> m d) -> Interval r -> m ( Interval r) Source # gmapMo :: MonadPlus m => ( forall d. Data d => d -> m d) -> Interval r -> m ( Interval r) Source # |
|
| ( Num r, Ord r) => Num ( Interval r) Source # |
When results of
|
|
Defined in Data.Interval Methods (+) :: Interval r -> Interval r -> Interval r Source # (-) :: Interval r -> Interval r -> Interval r Source # (*) :: Interval r -> Interval r -> Interval r Source # negate :: Interval r -> Interval r Source # abs :: Interval r -> Interval r Source # signum :: Interval r -> Interval r Source # fromInteger :: Integer -> Interval r Source # |
|
| ( Ord r, Read r) => Read ( Interval r) Source # | |
| ( Ord r, Show r) => Show ( Interval r) Source # | |
| ( Storable r, Ord r) => Storable ( Interval r) Source # | |
|
Defined in Data.Interval.Internal Methods sizeOf :: Interval r -> Int Source # alignment :: Interval r -> Int Source # peekElemOff :: Ptr ( Interval r) -> Int -> IO ( Interval r) Source # pokeElemOff :: Ptr ( Interval r) -> Int -> Interval r -> IO () Source # peekByteOff :: Ptr b -> Int -> IO ( Interval r) Source # pokeByteOff :: Ptr b -> Int -> Interval r -> IO () Source # |
|
| NFData r => NFData ( Interval r) Source # | |
|
Defined in Data.Interval.Internal |
|
| Hashable r => Hashable ( Interval r) Source # | |
| Ord r => Lattice ( Interval r) Source # | |
| Ord r => BoundedJoinSemiLattice ( Interval r) Source # | |
|
Defined in Data.Interval |
|
| Ord r => BoundedMeetSemiLattice ( Interval r) Source # | |
|
Defined in Data.Interval |
|
module Data.ExtendedReal
Boundary of an interval may be open (excluding an endpoint) or closed (including an endpoint).
Since: 2.0.0
Instances
Construction
Arguments
| :: 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
extractSingleton :: Ord r => Interval r -> Maybe r Source #
If the interval is a single point, return this point.
Since: 2.1.0
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.
-
lowerBoundof the empty interval isPosInf. -
lowerBoundof a left unbounded interval isNegInf. -
lowerBoundof 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.
-
upperBoundof the empty interval isNegInf. -
upperBoundof a right unbounded interval isPosInf. -
upperBoundof 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
(<=!) :: Ord r => Interval r -> Interval r -> Bool infix 4 Source #
For all
x
in
X
,
y
in
Y
.
x
?
<=
y
(==!) :: Ord r => Interval r -> Interval r -> Bool infix 4 Source #
For all
x
in
X
,
y
in
Y
.
x
?
==
y
(>=!) :: Ord r => Interval r -> Interval r -> Bool infix 4 Source #
For all
x
in
X
,
y
in
Y
.
x
?
>=
y
(>!) :: Ord r => Interval r -> Interval r -> Bool infix 4 Source #
For all
x
in
X
,
y
in
Y
.
x
?
>
y
(/=!) :: 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
hulls :: Ord r => [ Interval r] -> Interval r Source #
convex hull of a list of intervals.
Since 0.6.0
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,
preserving negative and positive infinities.
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
-
abs(numeratory) <=abs(numeratory') -
denominatory <=denominatory'
(see also
approxRational
)
Since 0.4.0
Intervals relation
relate :: Ord r => Interval r -> Interval r -> Relation Source #
Computes how two intervals are related according to the
classification
Relation