Linear.Metric

Free metric spaces

Documentation

class Additive f => Metric f where

Free and sparse inner product/metric spaces.

Minimal complete definition

Nothing

Methods

dot :: Num a => f a -> f a -> a

Compute the inner product of two vectors or (equivalently) convert a vector f a into a covector f a -> a.

>>> V2 1 2 `dot` V2 3 4
11

quadrance :: Num a => f a -> a

Compute the squared norm. The name quadrance arises from Norman J. Wildberger's rational trigonometry.

qd :: Num a => f a -> f a -> a

Compute the quadrance of the difference

distance :: Floating a => f a -> f a -> a

Compute the distance between two vectors in a metric space

norm :: Floating a => f a -> a

Compute the norm of a vector in a metric space

signorm :: Floating a => f a -> f a

Convert a non-zero vector to unit vector.

Instances

Metric []

Methods

dot :: Num a => [a] -> [a] -> a

quadrance :: Num a => [a] -> a

qd :: Num a => [a] -> [a] -> a

distance :: Floating a => [a] -> [a] -> a

norm :: Floating a => [a] -> a

signorm :: Floating a => [a] -> [a]

Metric Identity

Methods

dot :: Num a => Identity a -> Identity a -> a

quadrance :: Num a => Identity a -> a

qd :: Num a => Identity a -> Identity a -> a

distance :: Floating a => Identity a -> Identity a -> a

norm :: Floating a => Identity a -> a

signorm :: Floating a => Identity a -> Identity a

Metric ZipList

Methods

dot :: Num a => ZipList a -> ZipList a -> a

quadrance :: Num a => ZipList a -> a

qd :: Num a => ZipList a -> ZipList a -> a

distance :: Floating a => ZipList a -> ZipList a -> a

norm :: Floating a => ZipList a -> a

signorm :: Floating a => ZipList a -> ZipList a

Metric Maybe

Methods

dot :: Num a => Maybe a -> Maybe a -> a

quadrance :: Num a => Maybe a -> a

qd :: Num a => Maybe a -> Maybe a -> a

distance :: Floating a => Maybe a -> Maybe a -> a

norm :: Floating a => Maybe a -> a

signorm :: Floating a => Maybe a -> Maybe a

Metric IntMap

Methods

dot :: Num a => IntMap a -> IntMap a -> a

quadrance :: Num a => IntMap a -> a

qd :: Num a => IntMap a -> IntMap a -> a

distance :: Floating a => IntMap a -> IntMap a -> a

norm :: Floating a => IntMap a -> a

signorm :: Floating a => IntMap a -> IntMap a

Metric Vector

Methods

dot :: Num a => Vector a -> Vector a -> a

quadrance :: Num a => Vector a -> a

qd :: Num a => Vector a -> Vector a -> a

distance :: Floating a => Vector a -> Vector a -> a

norm :: Floating a => Vector a -> a

signorm :: Floating a => Vector a -> Vector a

Metric V0

Methods

dot :: Num a => V0 a -> V0 a -> a

quadrance :: Num a => V0 a -> a

qd :: Num a => V0 a -> V0 a -> a

distance :: Floating a => V0 a -> V0 a -> a

norm :: Floating a => V0 a -> a

signorm :: Floating a => V0 a -> V0 a

Metric V1

Methods

dot :: Num a => V1 a -> V1 a -> a

quadrance :: Num a => V1 a -> a

qd :: Num a => V1 a -> V1 a -> a

distance :: Floating a => V1 a -> V1 a -> a

norm :: Floating a => V1 a -> a

signorm :: Floating a => V1 a -> V1 a

Metric V2

Methods

dot :: Num a => V2 a -> V2 a -> a

quadrance :: Num a => V2 a -> a

qd :: Num a => V2 a -> V2 a -> a

distance :: Floating a => V2 a -> V2 a -> a

norm :: Floating a => V2 a -> a

signorm :: Floating a => V2 a -> V2 a

Metric V3

Methods

dot :: Num a => V3 a -> V3 a -> a

quadrance :: Num a => V3 a -> a

qd :: Num a => V3 a -> V3 a -> a

distance :: Floating a => V3 a -> V3 a -> a

norm :: Floating a => V3 a -> a

signorm :: Floating a => V3 a -> V3 a

Metric V4

Methods

dot :: Num a => V4 a -> V4 a -> a

quadrance :: Num a => V4 a -> a

qd :: Num a => V4 a -> V4 a -> a

distance :: Floating a => V4 a -> V4 a -> a

norm :: Floating a => V4 a -> a

signorm :: Floating a => V4 a -> V4 a

Metric Plucker

Methods

dot :: Num a => Plucker a -> Plucker a -> a

quadrance :: Num a => Plucker a -> a

qd :: Num a => Plucker a -> Plucker a -> a

distance :: Floating a => Plucker a -> Plucker a -> a

norm :: Floating a => Plucker a -> a

signorm :: Floating a => Plucker a -> Plucker a

Metric Quaternion

Methods

dot :: Num a => Quaternion a -> Quaternion a -> a

quadrance :: Num a => Quaternion a -> a

qd :: Num a => Quaternion a -> Quaternion a -> a

distance :: Floating a => Quaternion a -> Quaternion a -> a

norm :: Floating a => Quaternion a -> a

signorm :: Floating a => Quaternion a -> Quaternion a

Ord k => Metric (Map k)

Methods

dot :: Num a => Map k a -> Map k a -> a

quadrance :: Num a => Map k a -> a

qd :: Num a => Map k a -> Map k a -> a

distance :: Floating a => Map k a -> Map k a -> a

norm :: Floating a => Map k a -> a

signorm :: Floating a => Map k a -> Map k a

(Hashable k, Eq k) => Metric (HashMap k)

Methods

dot :: Num a => HashMap k a -> HashMap k a -> a

quadrance :: Num a => HashMap k a -> a

qd :: Num a => HashMap k a -> HashMap k a -> a

distance :: Floating a => HashMap k a -> HashMap k a -> a

norm :: Floating a => HashMap k a -> a

signorm :: Floating a => HashMap k a -> HashMap k a

Metric f => Metric (Point f)

Methods

dot :: Num a => Point f a -> Point f a -> a

quadrance :: Num a => Point f a -> a

qd :: Num a => Point f a -> Point f a -> a

distance :: Floating a => Point f a -> Point f a -> a

norm :: Floating a => Point f a -> a

signorm :: Floating a => Point f a -> Point f a

Dim k n => Metric (V k n)

Methods

dot :: Num a => V k n a -> V k n a -> a

quadrance :: Num a => V k n a -> a

qd :: Num a => V k n a -> V k n a -> a

distance :: Floating a => V k n a -> V k n a -> a

norm :: Floating a => V k n a -> a

signorm :: Floating a => V k n a -> V k n a

normalize :: (Floating a, Metric f, Epsilon a) => f a -> f a

Normalize a Metric functor to have unit norm. This function does not change the functor if its norm is 0 or 1.

project :: (Metric v, Fractional a) => v a -> v a -> v a

project u v computes the projection of v onto u.