Copyright	(c) Edward Kmett 2010-2013
License	BSD3
Maintainer	ekmett@gmail.com
Stability	experimental
Portability	DeriveDataTypeable
Safe Haskell	Safe
Language	Haskell98

Numeric.Interval.Internal

Description

Interval arithmetic

Synopsis

Documentation

data Interval a Source

Constructors

I !a !a
Empty

Instances

Foldable Interval Source
Generic1 Interval Source
Eq a => Eq (Interval a) Source
(RealFloat a, Ord a) => Floating (Interval a) Source
(Fractional a, Ord a) => Fractional (Interval a) Source
Data a => Data (Interval a) Source
(Num a, Ord a) => Num (Interval a) Source
Ord a => Ord (Interval a) Source
Real a => Real (Interval a) Source	`realToFrac` will use the midpoint
RealFloat a => RealFloat (Interval a) Source	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) Source
Show a => Show (Interval a) Source
Generic (Interval a) Source
type Rep1 Interval Source
type Rep (Interval a) Source

(...) :: Ord a => a -> a -> Interval a infix 3 Source

(+/-) :: (Num a, Ord a) => a -> a -> Interval a infixl 6 Source

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

whole :: Fractional a => Interval a Source

The whole real number line

>>> whole
-Infinity ... Infinity

empty :: Interval a Source

An empty interval

>>> empty
Empty

null :: Interval a -> Bool Source

Check if an interval is empty

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

>>> null (1 ... 1)
False

>>> null empty
True

singleton :: a -> Interval a Source

A singleton point

>>> singleton 1
1 ... 1

elem :: Ord a => a -> Interval a -> Bool Source

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 -> Bool Source

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 -> a Source

The infimum (lower bound) of an interval

>>> inf (1.0 ... 20.0)
1.0

>>> inf empty
*** Exception: empty interval

sup :: Interval a -> a Source

The supremum (upper bound) of an interval

>>> sup (1.0 ... 20.0)
20.0

>>> sup empty
*** Exception: empty interval

singular :: Ord a => Interval a -> Bool Source

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 -> a Source

Calculate the width of an interval.

>>> width (1 ... 20)
19

>>> width (singleton 1)
0

>>> width empty
0

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

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 :: Ord a => Interval a -> Interval a -> Interval a Source

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 a Source

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)

bisectIntegral :: Integral a => Interval a -> (Interval a, Interval a) Source

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

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 -> a Source

"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 -> a Source

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

inflate :: (Num a, Ord a) => a -> Interval a -> Interval a Source

Inflate an interval by enlarging it at both ends.

>>> inflate 3 (-1 ... 7)
-4 ... 10

>>> inflate (-2) (0 ... 4)
-2 ... 6

>>> inflate 1 empty
Empty

deflate :: (Fractional a, Ord a) => a -> Interval a -> Interval a Source

Deflate an interval by shrinking it from both ends.

>>> deflate 3.0 (-4.0 ... 10.0)
-1.0 ... 7.0

>>> deflate 2.0 (-1.0 ... 1.0)
Empty

>>> deflate 1.0 empty
Empty

scale :: (Fractional a, Ord a) => a -> Interval a -> Interval a Source

Scale an interval about its midpoint.

>>> scale 1.1 (-6.0 ... 4.0)
-6.5 ... 4.5

>>> scale (-2.0) (-1.0 ... 1.0)
Empty

>>> scale 3.0 empty
Empty

symmetric :: (Num a, Ord a) => a -> Interval a Source

Construct a symmetric interval.

>>> symmetric 3
-3 ... 3

>>> symmetric (-2)
-2 ... 2

contains :: Ord a => Interval a -> Interval a -> Bool Source

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 -> Bool Source

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 -> Bool Source

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

(<!) :: Ord a => Interval a -> Interval a -> Bool Source

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 -> Bool Source

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 -> Bool Source

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 -> Bool Source

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 -> Bool Source

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 -> Bool Source

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

(<?) :: Ord a => Interval a -> Interval a -> Bool Source

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

(<=?) :: Ord a => Interval a -> Interval a -> Bool Source

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

(==?) :: Ord a => Interval a -> Interval a -> Bool Source

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

(>=?) :: Ord a => Interval a -> Interval a -> Bool Source

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

(>?) :: Ord a => Interval a -> Interval a -> Bool Source

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

idouble :: Interval Double -> Interval Double Source

id function. Useful for type specification

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

ifloat :: Interval Float -> Interval Float Source

id function. Useful for type specification

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