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

Numeric.Interval.NonEmpty

Description

Interval arithmetic

Synopsis

Documentation

data Interval a Source

Instances

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

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

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

Try to create a non-empty interval.

whole :: Fractional a => Interval a Source

The whole real number line

>>> whole
-Infinity ... Infinity

(x :: Double) `elem` whole

singleton :: a -> Interval a Source

A singleton point

>>> singleton 1
1 ... 1

x `elem` (singleton x)

x /= y ==> y `notElem` (singleton x)

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

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

inf :: Interval a -> a Source

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 Source

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 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

0 <= width x

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 x `elem` (x :: Interval Double)

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

commutative (distance :: Interval Double -> Interval Double -> Double)

0 <= distance x y

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

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 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

conservative2 const hull

conservative2 (flip const) hull

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)

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

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

0 <= magnitude x

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

"mignitude"

>>> mignitude (1 ... 20)
1

>>> mignitude (-20 ... 10)
0

>>> mignitude (singleton 5)
5

0 <= mignitude x

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?

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

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 Source

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 Source

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 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

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 Source

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 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