Safe Haskell	None
Language	Haskell2010

NumHask.Algebra.Module

Description

Algebra for Modules

Synopsis

class Additive a => AdditiveModule r a where
class (AdditiveGroup a, AdditiveModule r a) => AdditiveGroupModule r a where
class Multiplicative a => MultiplicativeModule r a where
class (MultiplicativeGroup a, MultiplicativeModule r a) => MultiplicativeGroupModule r a where
class (ExpField a, Normed (r a) a, MultiplicativeGroupModule r a) => Banach r a where
class Semiring a => Hilbert r a where
type family (a :: k1) >< (b :: k2) :: *
class TensorProduct a where

Documentation

class Additive a => AdditiveModule r a where #

Additive Module Laws

(a + b) .+ c == a + (b .+ c)
(a + b) .+ c == (a .+ c) + b
a .+ zero == a
a .+ b == b +. a

Minimal complete definition

(.+), (+.)

Methods

(.+) :: r a -> a -> r a infixl 6 #

(+.) :: a -> r a -> r a infixl 6 #

class (AdditiveGroup a, AdditiveModule r a) => AdditiveGroupModule r a where #

Subtraction Module Laws

(a + b) .- c == a + (b .- c)
(a + b) .- c == (a .- c) + b
a .- zero == a
a .- b == negate b +. a

Minimal complete definition

(.-), (-.)

Methods

(.-) :: r a -> a -> r a infixl 6 #

(-.) :: a -> r a -> r a infixl 6 #

class Multiplicative a => MultiplicativeModule r a where #

Multiplicative Module Laws

a .* one == a
(a + b) .* c == (a .* c) + (b .* c)
c *. (a + b) == (c *. a) + (c *. b)
a .* zero == zero
a .* b == b *. a

Minimal complete definition

(.*), (*.)

Methods

(.*) :: r a -> a -> r a infixl 7 #

(*.) :: a -> r a -> r a infixl 7 #

class (MultiplicativeGroup a, MultiplicativeModule r a) => MultiplicativeGroupModule r a where #

Division Module Laws

nearZero a || a ./ one == a
b == zero || a ./ b == recip b *. a

Minimal complete definition

(./), (/.)

Methods

(./) :: r a -> a -> r a infixl 7 #

(/.) :: a -> r a -> r a infixl 7 #

class (ExpField a, Normed (r a) a, MultiplicativeGroupModule r a) => Banach r a where #

Banach (with Norm) laws form rules around size and direction of a number, with a potential crossing into another codomain.

a == singleton zero || normalizeL2 a *. normL2 a == a

Methods

normalizeL1 :: r a -> r a #

normalizeL2 :: r a -> r a #

normalizeLp :: a -> r a -> r a #

class Semiring a => Hilbert r a where #

the inner product of a representable over a semiring

a <.> b == b <.> a
a <.> (b +c) == a <.> b + a <.> c
a <.> (s *. b + c) == s * (a <.> b) + a <.> c

(s0 *. a) . (s1 *. b) == s0 * s1 * (a . b)

Minimal complete definition

(<.>)

Methods

(<.>) :: r a -> r a -> a infix 8 #

type family (a :: k1) >< (b :: k2) :: * infix 8 #

tensorial type

Instances

type (r a :: ) >< (b :: ) #
Instance details Defined in NumHask.Algebra.Module type (r a :: ) >< (b :: )
type Double >< Double #
Instance details Defined in NumHask.Algebra.Module type Double >< Double = Double
type Float >< Float #
Instance details Defined in NumHask.Algebra.Module type Float >< Float = Float
type Int >< Int #
Instance details Defined in NumHask.Algebra.Module type Int >< Int = Int
type Int8 >< Int8 #
Instance details Defined in NumHask.Algebra.Module type Int8 >< Int8 = Int8
type Int16 >< Int16 #
Instance details Defined in NumHask.Algebra.Module type Int16 >< Int16 = Int16
type Int32 >< Int32 #
Instance details Defined in NumHask.Algebra.Module type Int32 >< Int32 = Int32
type Int64 >< Int64 #
Instance details Defined in NumHask.Algebra.Module type Int64 >< Int64 = Int64
type Integer >< Integer #
Instance details Defined in NumHask.Algebra.Module type Integer >< Integer = Integer
type Natural >< Natural #
Instance details Defined in NumHask.Algebra.Module type Natural >< Natural = Natural
type Word >< Word #
Instance details Defined in NumHask.Algebra.Module type Word >< Word = Word
type Word8 >< Word8 #
Instance details Defined in NumHask.Algebra.Module type Word8 >< Word8 = Word8
type Word16 >< Word16 #
Instance details Defined in NumHask.Algebra.Module type Word16 >< Word16 = Word16
type Word32 >< Word32 #
Instance details Defined in NumHask.Algebra.Module type Word32 >< Word32 = Word32
type Word64 >< Word64 #
Instance details Defined in NumHask.Algebra.Module type Word64 >< Word64 = Word64

class TensorProduct a where #

generalised outer product

a><b + c><b == (a+c) >< b
a><b + a><c == a >< (b+c)

todo: work out why these laws down't apply > a *. (b>== (a<b) .* c > (a>.* c == a *. (b<c)

Minimal complete definition

(><), timesleft, timesright

Methods

(><) :: a -> a -> a >< a infix 8 #

outer :: a -> a -> a >< a #

timesleft :: a -> (a >< a) -> a #

timesright :: (a >< a) -> a -> a #