intervals-0.7.2: Interval Arithmetic

Numeric.Interval.Kaucher

Description

Directed Interval arithmetic

Synopsis

# Documentation

data Interval a Source

Constructors

 I !a !a

Instances

 Source Source Source Source Source Source 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 Source type Rep1 Interval Source type Rep (Interval a) Source

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

Create a directed interval.

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

Try to create a non-empty interval.

The whole real number line

>>> whole
-Infinity ... Infinity

An empty interval

>>> empty
NaN ... NaN

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

negation handles NaN properly

>>> 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 infinumum (lower bound) of an interval

>>> inf (1 ... 20)
1

sup :: Interval a -> a Source

The supremum (upper bound) of an interval

>>> sup (1 ... 20)
20

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
NaN

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
NaN

intersection :: (Fractional a, 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
(NaN ... NaN,NaN ... NaN)

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

Magnitude

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

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

"mignitude"

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

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

Hausdorff distance between non-empty intervals.

>>> distance (1 ... 7) (6 ... 10)
0
>>> distance (1 ... 7) (15 ... 24)
8
>>> distance (1 ... 7) (-10 ... -2)
3
>>> distance empty (1 ... 1)
NaN

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 ... 2

deflate :: Fractional 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)
1.0 ... -1.0

scale :: Fractional 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)
2.0 ... -2.0

symmetric :: Num 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 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?

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

The nearest value to that supplied which is contained in the interval.

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