intervals-0.7.2: Interval Arithmetic

Copyright(c) Edward Kmett 2010-2013
LicenseBSD3
Maintainerekmett@gmail.com
Stabilityexperimental
PortabilityDeriveDataTypeable
Safe HaskellSafe
LanguageHaskell98

Numeric.Interval.NonEmpty

Description

Interval arithmetic

Synopsis

Documentation

data Interval a

Instances

Foldable Interval 
Generic1 Interval 
Eq a => Eq (Interval a) 
(RealFloat a, Ord a) => Floating (Interval a)

Transcendental functions for intervals.

conservative (exp :: Double -> Double) exp
conservativeExceptNaN (log :: Double -> Double) log
conservative (sin :: Double -> Double) sin
conservative (cos :: Double -> Double) cos
conservative (tan :: Double -> Double) tan
conservativeExceptNaN (asin :: Double -> Double) asin
conservativeExceptNaN (acos :: Double -> Double) acos
conservative (atan :: Double -> Double) atan
conservative (sinh :: Double -> Double) sinh
conservative (cosh :: Double -> Double) cosh
conservative (tanh :: Double -> Double) tanh
conservativeExceptNaN (asinh :: Double -> Double) asinh
conservativeExceptNaN (acosh :: Double -> Double) acosh
conservativeExceptNaN (atanh :: Double -> Double) atanh
>>> cos (0 ... (pi + 0.1))
-1.0 ... 1.0
(Fractional a, Ord a) => Fractional (Interval a)

Fractional instance for intervals.

ys /= singleton 0 ==> conservative2 ((/) :: Double -> Double -> Double) (/) xs ys
xs /= singleton 0 ==> conservative (recip :: Double -> Double) recip xs
Data a => Data (Interval a) 
(Num a, Ord a) => Num (Interval a)

Num instance for intervals.

conservative2 ((+) :: Double -> Double -> Double) (+)
conservative2 ((-) :: Double -> Double -> Double) (-)
conservative2 ((*) :: Double -> Double -> Double) (*)
conservative (abs :: Double -> Double) abs
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) 
Generic (Interval a) 
type Rep1 Interval 
type Rep (Interval a) 

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

Create a non-empty interval, turning it around if necessary

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

Try to create a non-empty interval.

whole :: Fractional a => Interval a

The whole real number line

>>> whole
-Infinity ... Infinity
(x :: Double) `elem` whole

singleton :: a -> Interval a

A singleton point

>>> singleton 1
1 ... 1
x `elem` (singleton x)
x /= y ==> y `notElem` (singleton x)

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

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

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

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

inf :: Interval a -> a

The infinumum (lower bound) of an interval

>>> inf (1 ... 20)
1
min x y == inf (x ... y)
inf x <= sup x

sup :: Interval a -> a

The supremum (upper bound) of an interval

>>> sup (1 ... 20)
20
sup x `elem` x
max x y == sup (x ... y)
inf x <= sup x

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

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

Calculate the width of an interval.

>>> width (1 ... 20)
19
>>> width (singleton 1)
0
0 <= width x

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

Nearest point to the midpoint of the interval.

>>> midpoint (10.0 ... 20.0)
15.0
>>> midpoint (singleton 5.0)
5.0
midpoint x `elem` (x :: Interval Double)

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

Hausdorff distance between intervals.

>>> distance (1 ... 7) (6 ... 10)
0
>>> distance (1 ... 7) (15 ... 24)
8
>>> distance (1 ... 7) (-10 ... -2)
3
commutative (distance :: Interval Double -> Interval Double -> Double)
0 <= distance x y

intersection :: Ord a => Interval a -> Interval a -> Maybe (Interval a)

Calculate the intersection of two intervals.

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

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

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
conservative2 const hull
conservative2 (flip const) hull

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

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)
let (a, b) = bisect (x :: Interval Double) in sup a == inf b
let (a, b) = bisect (x :: Interval Double) in inf a == inf x
let (a, b) = bisect (x :: Interval Double) in sup b == sup x

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

Magnitude

>>> magnitude (1 ... 20)
20
>>> magnitude (-20 ... 10)
20
>>> magnitude (singleton 5)
5
0 <= magnitude x

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

"mignitude"

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(clamp xs y) `elem` xs

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

Inflate an interval by enlarging it at both ends.

>>> inflate 3 (-1 ... 7)
-4 ... 10
>>> inflate (-2) (0 ... 4)
-2 ... 6
inflate x i `contains` i

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

Deflate an interval by shrinking it from both ends. Note that in cases that would result in an empty interval, the result is a singleton interval at the midpoint.

>>> deflate 3.0 (-4.0 ... 10.0)
-1.0 ... 7.0
>>> deflate 2.0 (-1.0 ... 1.0)
0.0 ... 0.0

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

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
abs x >= 1 ==> (scale (x :: Double) i) `contains` i
forAll (choose (0,1)) $ \x -> abs x <= 1 ==> i `contains` (scale (x :: Double) i)

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

Construct a symmetric interval.

>>> symmetric 3
-3 ... 3
>>> symmetric (-2)
-2 ... 2
x `elem` symmetric x
0 `elem` symmetric x

idouble :: Interval Double -> Interval Double

id function. Useful for type specification

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

ifloat :: Interval Float -> Interval Float

id function. Useful for type specification

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