NumHask.Algebra.Metric

Metric classes

Documentation

class (Additive a, Multiplicative a) => Signed a where

signum from base is not an operator name in numhask and is replaced by sign. Compare with Norm where there is a change in codomain.

\a -> abs a * sign a ~= a

abs zero == zero, so any value for sign zero is ok. We choose lawful neutral:

>>> sign zero == zero
True

>>> abs (-1)
1

>>> sign (-1)
-1

Methods

sign :: a -> a

abs :: a -> a

Instances

Signed Int16

Defined in NumHask.Algebra.Metric

Methods

sign :: Int16 -> Int16

abs :: Int16 -> Int16

Signed Int32

Defined in NumHask.Algebra.Metric

Methods

sign :: Int32 -> Int32

abs :: Int32 -> Int32

Signed Int64

Defined in NumHask.Algebra.Metric

Methods

sign :: Int64 -> Int64

abs :: Int64 -> Int64

Signed Int8

Defined in NumHask.Algebra.Metric

Methods

sign :: Int8 -> Int8

abs :: Int8 -> Int8

Signed Word16

Defined in NumHask.Algebra.Metric

Methods

sign :: Word16 -> Word16

abs :: Word16 -> Word16

Signed Word32

Defined in NumHask.Algebra.Metric

Methods

sign :: Word32 -> Word32

abs :: Word32 -> Word32

Signed Word64

Defined in NumHask.Algebra.Metric

Methods

sign :: Word64 -> Word64

abs :: Word64 -> Word64

Signed Word8

Defined in NumHask.Algebra.Metric

Methods

sign :: Word8 -> Word8

abs :: Word8 -> Word8

Signed Integer

Defined in NumHask.Algebra.Metric

Methods

sign :: Integer -> Integer

abs :: Integer -> Integer

Signed Natural

Defined in NumHask.Algebra.Metric

Methods

sign :: Natural -> Natural

abs :: Natural -> Natural

Signed Double

Defined in NumHask.Algebra.Metric

Methods

sign :: Double -> Double

abs :: Double -> Double

Signed Float

Defined in NumHask.Algebra.Metric

Methods

sign :: Float -> Float

abs :: Float -> Float

Signed Int

Defined in NumHask.Algebra.Metric

Methods

sign :: Int -> Int

abs :: Int -> Int

Signed Word

Defined in NumHask.Algebra.Metric

Methods

sign :: Word -> Word

abs :: Word -> Word

(Ord a, Signed a, Integral a, Ring a) => Signed (Ratio a)

Defined in NumHask.Data.Rational

Methods

sign :: Ratio a -> Ratio a

abs :: Ratio a -> Ratio a

class (Additive a, Multiplicative b, Additive b) => Norm a b | a -> b where

Norm is a slight generalisation of Signed. The class has the same shape but allows the codomain to be different to the domain.

\a -> norm a >= zero
\a -> norm zero == zero
\a -> a == norm a .* basis a
\a -> norm (basis a) == one

>>> norm (-0.5 :: Double) :: Double
0.5

>>> basis (-0.5 :: Double) :: Double
-1.0

Methods

norm :: a -> b

or length, or ||v||

basis :: a -> a

or direction, or v-hat

Instances

Norm Int16 Int16

Defined in NumHask.Algebra.Metric

Methods

norm :: Int16 -> Int16

basis :: Int16 -> Int16

Norm Int32 Int32

Defined in NumHask.Algebra.Metric

Methods

norm :: Int32 -> Int32

basis :: Int32 -> Int32

Norm Int64 Int64

Defined in NumHask.Algebra.Metric

Methods

norm :: Int64 -> Int64

basis :: Int64 -> Int64

Norm Int8 Int8

Defined in NumHask.Algebra.Metric

Methods

norm :: Int8 -> Int8

basis :: Int8 -> Int8

Norm Word16 Word16

Defined in NumHask.Algebra.Metric

Methods

norm :: Word16 -> Word16

basis :: Word16 -> Word16

Norm Word32 Word32

Defined in NumHask.Algebra.Metric

Methods

norm :: Word32 -> Word32

basis :: Word32 -> Word32

Norm Word64 Word64

Defined in NumHask.Algebra.Metric

Methods

norm :: Word64 -> Word64

basis :: Word64 -> Word64

Norm Word8 Word8

Defined in NumHask.Algebra.Metric

Methods

norm :: Word8 -> Word8

basis :: Word8 -> Word8

Norm Integer Integer

Defined in NumHask.Algebra.Metric

Methods

norm :: Integer -> Integer

basis :: Integer -> Integer

Norm Natural Natural

Defined in NumHask.Algebra.Metric

Methods

norm :: Natural -> Natural

basis :: Natural -> Natural

Norm Double Double

Defined in NumHask.Algebra.Metric

Methods

norm :: Double -> Double

basis :: Double -> Double

Norm Float Float

Defined in NumHask.Algebra.Metric

Methods

norm :: Float -> Float

basis :: Float -> Float

Norm Int Int

Defined in NumHask.Algebra.Metric

Methods

norm :: Int -> Int

basis :: Int -> Int

Norm Word Word

Defined in NumHask.Algebra.Metric

Methods

norm :: Word -> Word

basis :: Word -> Word

ExpField a => Norm (Complex a) a

A euclidean-style norm is strong convention for Complex.

Defined in NumHask.Data.Complex

Methods

norm :: Complex a -> a

basis :: Complex a -> Complex a

(Ord a, Signed a, Integral a, Ring a) => Norm (Ratio a) (Ratio a)

Defined in NumHask.Data.Rational

Methods

norm :: Ratio a -> Ratio a

basis :: Ratio a -> Ratio a

distance :: (Norm a b, Subtractive a) => a -> a -> b

Distance, which combines the Subtractive notion of difference, with Norm.

distance a b >= zero
distance a a == zero
distance a b .* basis (a - b) == a - b

class (Additive coord, Multiplicative coord, Additive dir, Multiplicative dir) => Direction coord dir | coord -> dir where

Convert between a "co-ordinated" or "higher-kinded" number and representations of an angle. Typically thought of as polar co-ordinate conversion.

See Polar coordinate system

ray . angle == basis
norm (ray x) == one

Methods

angle :: coord -> dir

ray :: dir -> coord

Instances

TrigField a => Direction (Complex a) a

Defined in NumHask.Data.Complex

Methods

angle :: Complex a -> a

ray :: a -> Complex a

data Polar mag dir

Something that has a magnitude and a direction.

Constructors

Polar

Fields

• magnitude :: !mag
• direction :: !dir

Instances

Generic (Polar mag dir)

Defined in NumHask.Algebra.Metric

Associated Types

type Rep (Polar mag dir) :: Type -> Type

Methods

from :: Polar mag dir -> Rep (Polar mag dir) x

to :: Rep (Polar mag dir) x -> Polar mag dir

(Show mag, Show dir) => Show (Polar mag dir)

Defined in NumHask.Algebra.Metric

Methods

showsPrec :: Int -> Polar mag dir -> ShowS

show :: Polar mag dir -> String

showList :: [Polar mag dir] -> ShowS

(Eq mag, Eq dir) => Eq (Polar mag dir)

Defined in NumHask.Algebra.Metric

Methods

(==) :: Polar mag dir -> Polar mag dir -> Bool

(/=) :: Polar mag dir -> Polar mag dir -> Bool

type Rep (Polar mag dir)

Defined in NumHask.Algebra.Metric

type Rep (Polar mag dir) = D1 ('MetaData "Polar" "NumHask.Algebra.Metric" "numhask-0.10.1.1-5se1IUyAn3UALQnNUDpTp7" 'False) (C1 ('MetaCons "Polar" 'PrefixI 'True) (S1 ('MetaSel ('Just "magnitude") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 mag) :*: S1 ('MetaSel ('Just "direction") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 dir)))

polar :: (Norm coord mag, Direction coord dir) => coord -> Polar mag dir

Convert from a number to a Polar.

coord :: (MultiplicativeAction coord mag, Direction coord dir) => Polar mag dir -> coord

Convert from a Polar to a (coordinated aka higher-kinded) number.

class (Eq a, Additive a) => Epsilon a where

A small number, especially useful for approximate equality.

Minimal complete definition

Nothing

Methods

epsilon :: a

nearZero :: a -> Bool

are we near enough?

>>> nearZero (epsilon :: Double)
True

default nearZero :: (Ord a, Subtractive a) => a -> Bool

aboutEqual :: a -> a -> Bool

Approximate equality

>>> aboutEqual zero (epsilon :: Double)
True

default aboutEqual :: Subtractive a => a -> a -> Bool

Instances

Epsilon Int16

Defined in NumHask.Algebra.Metric

Methods

epsilon :: Int16

nearZero :: Int16 -> Bool

aboutEqual :: Int16 -> Int16 -> Bool

Epsilon Int32

Defined in NumHask.Algebra.Metric

Methods

epsilon :: Int32

nearZero :: Int32 -> Bool

aboutEqual :: Int32 -> Int32 -> Bool

Epsilon Int64

Defined in NumHask.Algebra.Metric

Methods

epsilon :: Int64

nearZero :: Int64 -> Bool

aboutEqual :: Int64 -> Int64 -> Bool

Epsilon Int8

Defined in NumHask.Algebra.Metric

Methods

epsilon :: Int8

nearZero :: Int8 -> Bool

aboutEqual :: Int8 -> Int8 -> Bool

Epsilon Word16

Defined in NumHask.Algebra.Metric

Methods

epsilon :: Word16

nearZero :: Word16 -> Bool

aboutEqual :: Word16 -> Word16 -> Bool

Epsilon Word32

Defined in NumHask.Algebra.Metric

Methods

epsilon :: Word32

nearZero :: Word32 -> Bool

aboutEqual :: Word32 -> Word32 -> Bool

Epsilon Word64

Defined in NumHask.Algebra.Metric

Methods

epsilon :: Word64

nearZero :: Word64 -> Bool

aboutEqual :: Word64 -> Word64 -> Bool

Epsilon Word8

Defined in NumHask.Algebra.Metric

Methods

epsilon :: Word8

nearZero :: Word8 -> Bool

aboutEqual :: Word8 -> Word8 -> Bool

Epsilon Integer

Defined in NumHask.Algebra.Metric

Methods

epsilon :: Integer

nearZero :: Integer -> Bool

aboutEqual :: Integer -> Integer -> Bool

Epsilon Double

1e-14

Defined in NumHask.Algebra.Metric

Methods

epsilon :: Double

nearZero :: Double -> Bool

aboutEqual :: Double -> Double -> Bool

Epsilon Float

1e-6

Defined in NumHask.Algebra.Metric

Methods

epsilon :: Float

nearZero :: Float -> Bool

aboutEqual :: Float -> Float -> Bool

Epsilon Int

0

Defined in NumHask.Algebra.Metric

Methods

epsilon :: Int

nearZero :: Int -> Bool

aboutEqual :: Int -> Int -> Bool

Epsilon Word

Defined in NumHask.Algebra.Metric

Methods

epsilon :: Word

nearZero :: Word -> Bool

aboutEqual :: Word -> Word -> Bool

(Ord a, Signed a, Epsilon a, Subtractive a) => Epsilon (Complex a)

Defined in NumHask.Data.Complex

Methods

epsilon :: Complex a

nearZero :: Complex a -> Bool

aboutEqual :: Complex a -> Complex a -> Bool

(Ord a, Signed a, Integral a, Ring a, MeetSemiLattice a) => Epsilon (Ratio a)

Defined in NumHask.Data.Rational

Methods

epsilon :: Ratio a

nearZero :: Ratio a -> Bool

aboutEqual :: Ratio a -> Ratio a -> Bool

(~=) :: Epsilon a => a -> a -> Bool infixl 4

About equal operator.

>>> (1.0 + epsilon) ~= (1.0 :: Double)
True