Copyright	(C) 2012-2015 Edward Kmett
License	BSD-style (see the file LICENSE)
Maintainer	Edward Kmett <ekmett@gmail.com>
Stability	experimental
Portability	non-portable
Safe Haskell	Trustworthy
Language	Haskell98

Linear.Metric

Description

Free metric spaces

Synopsis

class Additive f => Metric f where
normalize :: (Floating a, Metric f, Epsilon a) => f a -> f a
project :: (Metric v, Fractional a) => v a -> v a -> v a

Documentation

class Additive f => Metric f where #

Free and sparse inner product/metric spaces.

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

dot :: (Foldable f, 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 [] #
Instance details Defined in Linear.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 Maybe #
Instance details Defined in Linear.Metric 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 ZipList #
Instance details Defined in Linear.Metric 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 Identity #
Instance details Defined in Linear.Metric 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 IntMap #
Instance details Defined in Linear.Metric 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 #
Instance details Defined in Linear.Metric 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 V1 #
Instance details Defined in Linear.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 #
Instance details Defined in Linear.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 #
Instance details Defined in Linear.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 #
Instance details Defined in Linear.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 V0 #
Instance details Defined in Linear.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 Quaternion #
Instance details Defined in Linear.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 #
Metric Plucker #
Instance details Defined in Linear.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 #
Ord k => Metric (Map k) #
Instance details Defined in Linear.Metric 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) #
Instance details Defined in Linear.Metric 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) #
Instance details Defined in Linear.Affine 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 n => Metric (V n) #
Instance details Defined in Linear.V Methods dot :: Num a => V n a -> V n a -> a # quadrance :: Num a => V n a -> a # qd :: Num a => V n a -> V n a -> a # distance :: Floating a => V n a -> V n a -> a # norm :: Floating a => V n a -> a # signorm :: Floating a => V n a -> V 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.