Safe Haskell	None
Language	Haskell2010

Numeric.Algebra.Normed

Description

Provides the Normed typeclass.

Since: 0.1

Synopsis

class Normed s where
- norm :: s -> s
- sgn :: s -> s

Documentation

class Normed s where Source #

Defines a "norm" on a given structure. We have the following laws:

Positive definiteness: norm x > 0 for x /= 0, norm 0 = 0.
Subadditivity: norm (x + y) <= norm x + norm y.
Inversion : norm x = norm (-x).
Sign : sgn x = x / norm x.

Notice these laws require various algebraic assumptions, from a semigroup up to a group structure. It is expected that types that implement the relevant typeclasses will follow the respective laws.

We do not require any of these typeclasses as constraints as it is possible that we may want to abstract over "positive" types with a type that does not have an additive semigroup instance (consider multiplicative groups).

Since: 0.1

Methods

norm :: s -> s Source #

Returns the norm.

Since: 0.1

sgn :: s -> s Source #

Sign function. Notice that when norm is abs, this is the familiar signum function i.e.

sgn x
  | x < 0 = -1
  | x == 0 = 0
  | x > 0 = 1

Since: 0.1

Instances

Instances details

Normed Int16 Source #	Since: 0.1
Instance details Defined in Numeric.Algebra.Normed Methods norm :: Int16 -> Int16 Source # sgn :: Int16 -> Int16 Source #
Normed Int32 Source #	Since: 0.1
Instance details Defined in Numeric.Algebra.Normed Methods norm :: Int32 -> Int32 Source # sgn :: Int32 -> Int32 Source #
Normed Int64 Source #	Since: 0.1
Instance details Defined in Numeric.Algebra.Normed Methods norm :: Int64 -> Int64 Source # sgn :: Int64 -> Int64 Source #
Normed Int8 Source #	Since: 0.1
Instance details Defined in Numeric.Algebra.Normed Methods norm :: Int8 -> Int8 Source # sgn :: Int8 -> Int8 Source #
Normed Word16 Source #	Since: 0.1
Instance details Defined in Numeric.Algebra.Normed Methods norm :: Word16 -> Word16 Source # sgn :: Word16 -> Word16 Source #
Normed Word32 Source #	Since: 0.1
Instance details Defined in Numeric.Algebra.Normed Methods norm :: Word32 -> Word32 Source # sgn :: Word32 -> Word32 Source #
Normed Word64 Source #	Since: 0.1
Instance details Defined in Numeric.Algebra.Normed Methods norm :: Word64 -> Word64 Source # sgn :: Word64 -> Word64 Source #
Normed Word8 Source #	Since: 0.1
Instance details Defined in Numeric.Algebra.Normed Methods norm :: Word8 -> Word8 Source # sgn :: Word8 -> Word8 Source #
Normed Integer Source #	Since: 0.1
Instance details Defined in Numeric.Algebra.Normed Methods norm :: Integer -> Integer Source # sgn :: Integer -> Integer Source #
Normed Natural Source #	Since: 0.1
Instance details Defined in Numeric.Algebra.Normed Methods norm :: Natural -> Natural Source # sgn :: Natural -> Natural Source #
Normed Double Source #	Since: 0.1
Instance details Defined in Numeric.Algebra.Normed Methods norm :: Double -> Double Source # sgn :: Double -> Double Source #
Normed Float Source #	Since: 0.1
Instance details Defined in Numeric.Algebra.Normed Methods norm :: Float -> Float Source # sgn :: Float -> Float Source #
Normed Int Source #	Since: 0.1
Instance details Defined in Numeric.Algebra.Normed Methods norm :: Int -> Int Source # sgn :: Int -> Int Source #
Normed Word Source #	Since: 0.1
Instance details Defined in Numeric.Algebra.Normed Methods norm :: Word -> Word Source # sgn :: Word -> Word Source #
RealFloat a => Normed (Complex a) Source #	Since: 0.1
Instance details Defined in Numeric.Algebra.Normed Methods norm :: Complex a -> Complex a Source # sgn :: Complex a -> Complex a Source #
Normed (Ratio Integer) Source #	Since: 0.1
Instance details Defined in Numeric.Algebra.Normed Methods norm :: Ratio Integer -> Ratio Integer Source # sgn :: Ratio Integer -> Ratio Integer Source #
Normed (Ratio Natural) Source #	Since: 0.1
Instance details Defined in Numeric.Algebra.Normed Methods norm :: Ratio Natural -> Ratio Natural Source # sgn :: Ratio Natural -> Ratio Natural Source #