intervals-0.7: Interval Arithmetic

Portability DeriveDataTypeable experimental ekmett@gmail.com Safe-Inferred

Numeric.Interval

Description

Interval arithmetic

Synopsis

# Documentation

data Interval a Source

Instances

 Typeable1 Interval Foldable Interval Eq a => Eq (Interval a) (RealFloat a, Ord a) => Floating (Interval a) (Fractional a, Ord a) => Fractional (Interval a) Data a => Data (Interval a) (Num a, Ord a) => Num (Interval a) Ord a => Ord (Interval a) Real a => Real (Interval a) realToFrac will use the midpoint RealFloat a => RealFloat (Interval a) We have to play some semantic games to make these methods make sense. Most compute with the midpoint of the interval. RealFrac a => RealFrac (Interval a) Show a => Show (Interval a)

(...) :: Ord a => a -> a -> Interval aSource

interval :: Ord a => a -> a -> Maybe (Interval a)Source

The whole real number line

>>> whole
-Infinity ... Infinity

empty :: Ord a => Interval aSource

An empty interval

>>> empty
Empty

null :: Interval a -> BoolSource

Check if an interval is empty

>>> null (1 ... 5)
False
>>> null (1 ... 1)
False
>>> null empty
True

singleton :: a -> Interval aSource

A singleton point

>>> singleton 1
1 ... 1

elem :: Ord a => a -> Interval a -> BoolSource

Determine if a point is in the interval.

>>> elem 3.2 (1.0 ... 5.0)
True
>>> elem 5 (1.0 ... 5.0)
True
>>> elem 1 (1.0 ... 5.0)
True
>>> elem 8 (1.0 ... 5.0)
False
>>> elem 5 empty
False

notElem :: Ord a => a -> Interval a -> BoolSource

Determine if a point is not included in the interval

>>> notElem 8 (1.0 ... 5.0)
True
>>> notElem 1.4 (1.0 ... 5.0)
False

And of course, nothing is a member of the empty interval.

>>> notElem 5 empty
True

inf :: Interval a -> aSource

The infimum (lower bound) of an interval

>>> inf (1.0 ... 20.0)
1.0
>>> inf empty
*** Exception: empty interval

sup :: Interval a -> aSource

The supremum (upper bound) of an interval

>>> sup (1.0 ... 20.0)
20.0
>>> sup empty
*** Exception: empty interval

singular :: Ord a => Interval a -> BoolSource

Is the interval a singleton point? N.B. This is fairly fragile and likely will not hold after even a few operations that only involve singletons

>>> singular (singleton 1)
True
>>> singular (1.0 ... 20.0)
False

width :: Num a => Interval a -> aSource

Calculate the width of an interval.

>>> width (1 ... 20)
19
>>> width (singleton 1)
0
>>> width empty
0

midpoint :: Fractional a => Interval a -> aSource

Nearest point to the midpoint of the interval.

>>> midpoint (10.0 ... 20.0)
15.0
>>> midpoint (singleton 5.0)
5.0
>>> midpoint empty
*** Exception: empty interval

intersection :: (Fractional a, Ord a) => Interval a -> Interval a -> Interval aSource

Calculate the intersection of two intervals.

>>> intersection (1 ... 10 :: Interval Double) (5 ... 15 :: Interval Double)
5.0 ... 10.0

hull :: Ord a => Interval a -> Interval a -> Interval aSource

Calculate the convex hull of two intervals

>>> hull (0 ... 10 :: Interval Double) (5 ... 15 :: Interval Double)
0.0 ... 15.0
>>> hull (15 ... 85 :: Interval Double) (0 ... 10 :: Interval Double)
0.0 ... 85.0

bisect :: Fractional a => Interval a -> (Interval a, Interval a)Source

Bisect an interval at its midpoint.

>>> bisect (10.0 ... 20.0)
(10.0 ... 15.0,15.0 ... 20.0)
>>> bisect (singleton 5.0)
(5.0 ... 5.0,5.0 ... 5.0)
>>> bisect Empty
(Empty,Empty)

magnitude :: (Num a, Ord a) => Interval a -> aSource

Magnitude

>>> magnitude (1 ... 20)
20
>>> magnitude (-20 ... 10)
20
>>> magnitude (singleton 5)
5

throws EmptyInterval if the interval is empty.

>>> magnitude empty
*** Exception: empty interval

mignitude :: (Num a, Ord a) => Interval a -> aSource

"mignitude"

>>> mignitude (1 ... 20)
1
>>> mignitude (-20 ... 10)
0
>>> mignitude (singleton 5)
5

throws EmptyInterval if the interval is empty.

>>> mignitude empty
*** Exception: empty interval

distance :: (Num a, Ord a) => Interval a -> Interval a -> aSource

Hausdorff distance between intervals.

>>> distance (1 ... 7) (6 ... 10)
0
>>> distance (1 ... 7) (15 ... 24)
8
>>> distance (1 ... 7) (-10 ... -2)
3
>>> distance Empty (1 ... 1)
*** Exception: empty interval

contains :: Ord a => Interval a -> Interval a -> BoolSource

Check if interval X totally contains interval Y

>>> (20 ... 40 :: Interval Double) `contains` (25 ... 35 :: Interval Double)
True
>>> (20 ... 40 :: Interval Double) `contains` (15 ... 35 :: Interval Double)
False

isSubsetOf :: Ord a => Interval a -> Interval a -> BoolSource

Flipped version of contains. Check if interval X a subset of interval Y

>>> (25 ... 35 :: Interval Double) `isSubsetOf` (20 ... 40 :: Interval Double)
True
>>> (20 ... 40 :: Interval Double) `isSubsetOf` (15 ... 35 :: Interval Double)
False

certainly :: Ord a => (forall b. Ord b => b -> b -> Bool) -> Interval a -> Interval a -> BoolSource

For all x in X, y in Y. x op y

(<!) :: Ord a => Interval a -> Interval a -> BoolSource

For all x in X, y in Y. x < y

>>> (5 ... 10 :: Interval Double) <! (20 ... 30 :: Interval Double)
True
>>> (5 ... 10 :: Interval Double) <! (10 ... 30 :: Interval Double)
False
>>> (20 ... 30 :: Interval Double) <! (5 ... 10 :: Interval Double)
False

(<=!) :: Ord a => Interval a -> Interval a -> BoolSource

For all x in X, y in Y. x <= y

>>> (5 ... 10 :: Interval Double) <=! (20 ... 30 :: Interval Double)
True
>>> (5 ... 10 :: Interval Double) <=! (10 ... 30 :: Interval Double)
True
>>> (20 ... 30 :: Interval Double) <=! (5 ... 10 :: Interval Double)
False

(==!) :: Eq a => Interval a -> Interval a -> BoolSource

For all x in X, y in Y. x == y

Only singleton intervals or empty intervals can return true

>>> (singleton 5 :: Interval Double) ==! (singleton 5 :: Interval Double)
True
>>> (5 ... 10 :: Interval Double) ==! (5 ... 10 :: Interval Double)
False

(>=!) :: Ord a => Interval a -> Interval a -> BoolSource

For all x in X, y in Y. x >= y

>>> (20 ... 40 :: Interval Double) >=! (10 ... 20 :: Interval Double)
True
>>> (5 ... 20 :: Interval Double) >=! (15 ... 40 :: Interval Double)
False

(>!) :: Ord a => Interval a -> Interval a -> BoolSource

For all x in X, y in Y. x > y

>>> (20 ... 40 :: Interval Double) >! (10 ... 19 :: Interval Double)
True
>>> (5 ... 20 :: Interval Double) >! (15 ... 40 :: Interval Double)
False

possibly :: Ord a => (forall b. Ord b => b -> b -> Bool) -> Interval a -> Interval a -> BoolSource

Does there exist an x in X, y in Y such that x op y?

(<?) :: Ord a => Interval a -> Interval a -> BoolSource

Does there exist an x in X, y in Y such that x < y?

(<=?) :: Ord a => Interval a -> Interval a -> BoolSource

Does there exist an x in X, y in Y such that x <= y?

(==?) :: Ord a => Interval a -> Interval a -> BoolSource

Does there exist an x in X, y in Y such that x == y?

(>=?) :: Ord a => Interval a -> Interval a -> BoolSource

Does there exist an x in X, y in Y such that x >= y?

(>?) :: Ord a => Interval a -> Interval a -> BoolSource

Does there exist an x in X, y in Y such that x > y?

id function. Useful for type specification

>>> :t idouble (1 ... 3)
idouble (1 ... 3) :: Interval Double

id function. Useful for type specification

>>> :t ifloat (1 ... 3)
ifloat (1 ... 3) :: Interval Float