Algebra.Classes

Documentation

type Natural = Integer

newtype Sum a

Constructors

Sum

Fields

• fromSum :: a

Instances

Generic (Sum a)

Defined in Algebra.Classes

Associated Types

type Rep (Sum a) :: Type -> Type

Methods

from :: Sum a -> Rep (Sum a) x

to :: Rep (Sum a) x -> Sum a

Additive a => Semigroup (Sum a)

Defined in Algebra.Classes

Methods

(<>) :: Sum a -> Sum a -> Sum a

sconcat :: NonEmpty (Sum a) -> Sum a

stimes :: Integral b => b -> Sum a -> Sum a

Additive a => Monoid (Sum a)

Defined in Algebra.Classes

Methods

mempty :: Sum a

mappend :: Sum a -> Sum a -> Sum a

mconcat :: [Sum a] -> Sum a

Binary a => Binary (Sum a)

Defined in Algebra.Classes

Methods

put :: Sum a -> Put

get :: Get (Sum a)

putList :: [Sum a] -> Put

type Rep (Sum a)

Defined in Algebra.Classes

type Rep (Sum a) = D1 (MetaData "Sum" "Algebra.Classes" "gasp-1.2.0.0-60xnh1c7HhuLy98Md0lIBW" True) (C1 (MetaCons "Sum" PrefixI True) (S1 (MetaSel (Just "fromSum") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 a)))

newtype Product a

Constructors

Product

Fields

• fromProduct :: a

Instances

Multiplicative a => Semigroup (Product a)

Defined in Algebra.Classes

Methods

(<>) :: Product a -> Product a -> Product a

sconcat :: NonEmpty (Product a) -> Product a

stimes :: Integral b => b -> Product a -> Product a

Multiplicative a => Monoid (Product a)

Defined in Algebra.Classes

Methods

mempty :: Product a

mappend :: Product a -> Product a -> Product a

mconcat :: [Product a] -> Product a

newtype Exponential a

Constructors

Exponential

Fields

• fromExponential :: a

Instances

Group a => Division (Exponential a)

Defined in Algebra.Classes

Methods

recip :: Exponential a -> Exponential a

(/) :: Exponential a -> Exponential a -> Exponential a

Additive a => Multiplicative (Exponential a)

Defined in Algebra.Classes

Methods

(*) :: Exponential a -> Exponential a -> Exponential a

one :: Exponential a

(^) :: Exponential a -> Natural -> Exponential a

class Additive a where

Additive monoid

Minimal complete definition

(+), zero

Methods

(+) :: a -> a -> a infixl 6

zero :: a

times :: Natural -> a -> a

Instances

Additive Double

Defined in Algebra.Classes

Methods

(+) :: Double -> Double -> Double

zero :: Double

times :: Natural -> Double -> Double

Additive Float

Defined in Algebra.Classes

Methods

(+) :: Float -> Float -> Float

zero :: Float

times :: Natural -> Float -> Float

Additive Int

Defined in Algebra.Classes

Methods

(+) :: Int -> Int -> Int

zero :: Int

times :: Natural -> Int -> Int

Additive Integer

Defined in Algebra.Classes

Methods

(+) :: Integer -> Integer -> Integer

zero :: Integer

times :: Natural -> Integer -> Integer

Additive Word8

Defined in Algebra.Classes

Methods

(+) :: Word8 -> Word8 -> Word8

zero :: Word8

times :: Natural -> Word8 -> Word8

Additive Word16

Defined in Algebra.Classes

Methods

(+) :: Word16 -> Word16 -> Word16

zero :: Word16

times :: Natural -> Word16 -> Word16

Additive Word32

Defined in Algebra.Classes

Methods

(+) :: Word32 -> Word32 -> Word32

zero :: Word32

times :: Natural -> Word32 -> Word32

Additive CInt

Defined in Algebra.Classes

Methods

(+) :: CInt -> CInt -> CInt

zero :: CInt

times :: Natural -> CInt -> CInt

Additive InitialAdditive

Defined in Algebra.Classes

Methods

(+) :: InitialAdditive -> InitialAdditive -> InitialAdditive

zero :: InitialAdditive

times :: Natural -> InitialAdditive -> InitialAdditive

Integral a => Additive (Ratio a)

Defined in Algebra.Classes

Methods

(+) :: Ratio a -> Ratio a -> Ratio a

zero :: Ratio a

times :: Natural -> Ratio a -> Ratio a

(Ord k, Additive v) => Additive (Map k v)

Defined in Algebra.Classes

Methods

(+) :: Map k v -> Map k v -> Map k v

zero :: Map k v

times :: Natural -> Map k v -> Map k v

(Applicative f, Additive a) => Additive (Euclid f a)

Defined in Algebra.Linear

Methods

(+) :: Euclid f a -> Euclid f a -> Euclid f a

zero :: Euclid f a

times :: Natural -> Euclid f a -> Euclid f a

(Applicative f, Applicative g, Additive a) => Additive (Mat a f g)

Defined in Algebra.Linear

Methods

(+) :: Mat a f g -> Mat a f g -> Mat a f g

zero :: Mat a f g

times :: Natural -> Mat a f g -> Mat a f g

add :: (Foldable t, Additive a) => t a -> a

class Additive r => DecidableZero r where

Methods

isZero :: r -> Bool

Instances

DecidableZero Double

Defined in Algebra.Classes

Methods

isZero :: Double -> Bool

DecidableZero Float

Defined in Algebra.Classes

Methods

isZero :: Float -> Bool

DecidableZero Int

Defined in Algebra.Classes

Methods

isZero :: Int -> Bool

DecidableZero Integer

Defined in Algebra.Classes

Methods

isZero :: Integer -> Bool

DecidableZero Word8

Defined in Algebra.Classes

Methods

isZero :: Word8 -> Bool

DecidableZero Word16

Defined in Algebra.Classes

Methods

isZero :: Word16 -> Bool

DecidableZero Word32

Defined in Algebra.Classes

Methods

isZero :: Word32 -> Bool

DecidableZero CInt

Defined in Algebra.Classes

Methods

isZero :: CInt -> Bool

(Ord k, DecidableZero v) => DecidableZero (Map k v)

Defined in Algebra.Classes

Methods

isZero :: Map k v -> Bool

class Additive a => AbelianAdditive a

Instances

AbelianAdditive Double

Defined in Algebra.Classes

AbelianAdditive Float

Defined in Algebra.Classes

AbelianAdditive Int

Defined in Algebra.Classes

AbelianAdditive Integer

Defined in Algebra.Classes

AbelianAdditive CInt

Defined in Algebra.Classes

Integral a => AbelianAdditive (Ratio a)

Defined in Algebra.Classes

(Ord k, AbelianAdditive v) => AbelianAdditive (Map k v)

Defined in Algebra.Classes

(Applicative f, AbelianAdditive a) => AbelianAdditive (Euclid f a)

Defined in Algebra.Linear

(Applicative f, Applicative g, AbelianAdditive a) => AbelianAdditive (Mat a f g)

Defined in Algebra.Linear

class Additive a => Group a where

Minimal complete definition

(negate | (-))

Methods

(-) :: a -> a -> a infixl 6

negate :: a -> a

mult :: Integer -> a -> a

Instances

Group Double

Defined in Algebra.Classes

Methods

(-) :: Double -> Double -> Double

negate :: Double -> Double

mult :: Integer -> Double -> Double

Group Float

Defined in Algebra.Classes

Methods

(-) :: Float -> Float -> Float

negate :: Float -> Float

mult :: Integer -> Float -> Float

Group Int

Defined in Algebra.Classes

Methods

(-) :: Int -> Int -> Int

negate :: Int -> Int

mult :: Integer -> Int -> Int

Group Integer

Defined in Algebra.Classes

Methods

(-) :: Integer -> Integer -> Integer

negate :: Integer -> Integer

mult :: Integer -> Integer -> Integer

Group Word8

Defined in Algebra.Classes

Methods

(-) :: Word8 -> Word8 -> Word8

negate :: Word8 -> Word8

mult :: Integer -> Word8 -> Word8

Group Word16

Defined in Algebra.Classes

Methods

(-) :: Word16 -> Word16 -> Word16

negate :: Word16 -> Word16

mult :: Integer -> Word16 -> Word16

Group Word32

Defined in Algebra.Classes

Methods

(-) :: Word32 -> Word32 -> Word32

negate :: Word32 -> Word32

mult :: Integer -> Word32 -> Word32

Group CInt

Defined in Algebra.Classes

Methods

(-) :: CInt -> CInt -> CInt

negate :: CInt -> CInt

mult :: Integer -> CInt -> CInt

Integral a => Group (Ratio a)

Defined in Algebra.Classes

Methods

(-) :: Ratio a -> Ratio a -> Ratio a

negate :: Ratio a -> Ratio a

mult :: Integer -> Ratio a -> Ratio a

(Ord k, Group v) => Group (Map k v)

Defined in Algebra.Classes

Methods

(-) :: Map k v -> Map k v -> Map k v

negate :: Map k v -> Map k v

mult :: Integer -> Map k v -> Map k v

(Applicative f, Group a) => Group (Euclid f a)

Defined in Algebra.Linear

Methods

(-) :: Euclid f a -> Euclid f a -> Euclid f a

negate :: Euclid f a -> Euclid f a

mult :: Integer -> Euclid f a -> Euclid f a

(Applicative f, Applicative g, Group a) => Group (Mat a f g)

Defined in Algebra.Linear

Methods

(-) :: Mat a f g -> Mat a f g -> Mat a f g

negate :: Mat a f g -> Mat a f g

mult :: Integer -> Mat a f g -> Mat a f g

class (AbelianAdditive a, PreRing scalar) => Module scalar a where

Module

Methods

(*^) :: scalar -> a -> a infixr 7

Instances

Module Double Double

Defined in Algebra.Classes

Methods

(*^) :: Double -> Double -> Double

Module Float Float

Defined in Algebra.Classes

Methods

(*^) :: Float -> Float -> Float

Module Int Int

Defined in Algebra.Classes

Methods

(*^) :: Int -> Int -> Int

Module Integer Integer

Defined in Algebra.Classes

Methods

(*^) :: Integer -> Integer -> Integer

Module Rational Double

Defined in Algebra.Classes

Methods

(*^) :: Rational -> Double -> Double

Module CInt CInt

Defined in Algebra.Classes

Methods

(*^) :: CInt -> CInt -> CInt

(Ord k, Module v v) => Module v (Map k v)

Defined in Algebra.Classes

Methods

(*^) :: v -> Map k v -> Map k v

(Applicative f, Module s a) => Module s (Euclid f a)

Defined in Algebra.Linear

Methods

(*^) :: s -> Euclid f a -> Euclid f a

(Applicative f, Applicative g, Module s a) => Module s (Mat a f g)

Defined in Algebra.Linear

Methods

(*^) :: s -> Mat a f g -> Mat a f g

Integral a => Module (Ratio a) (Ratio a)

Defined in Algebra.Classes

Methods

(*^) :: Ratio a -> Ratio a -> Ratio a

class Multiplicative a where

Multiplicative monoid

Minimal complete definition

(*), one

Methods

(*) :: a -> a -> a infixl 7

one :: a

(^) :: a -> Natural -> a

Instances

Multiplicative Double

Defined in Algebra.Classes

Methods

(*) :: Double -> Double -> Double

one :: Double

(^) :: Double -> Natural -> Double

Multiplicative Float

Defined in Algebra.Classes

Methods

(*) :: Float -> Float -> Float

one :: Float

(^) :: Float -> Natural -> Float

Multiplicative Int

Defined in Algebra.Classes

Methods

(*) :: Int -> Int -> Int

one :: Int

(^) :: Int -> Natural -> Int

Multiplicative Integer

Defined in Algebra.Classes

Methods

(*) :: Integer -> Integer -> Integer

one :: Integer

(^) :: Integer -> Natural -> Integer

Multiplicative Word8

Defined in Algebra.Classes

Methods

(*) :: Word8 -> Word8 -> Word8

one :: Word8

(^) :: Word8 -> Natural -> Word8

Multiplicative Word16

Defined in Algebra.Classes

Methods

(*) :: Word16 -> Word16 -> Word16

one :: Word16

(^) :: Word16 -> Natural -> Word16

Multiplicative Word32

Defined in Algebra.Classes

Methods

(*) :: Word32 -> Word32 -> Word32

one :: Word32

(^) :: Word32 -> Natural -> Word32

Multiplicative CInt

Defined in Algebra.Classes

Methods

(*) :: CInt -> CInt -> CInt

one :: CInt

(^) :: CInt -> Natural -> CInt

Integral a => Multiplicative (Ratio a)

Defined in Algebra.Classes

Methods

(*) :: Ratio a -> Ratio a -> Ratio a

one :: Ratio a

(^) :: Ratio a -> Natural -> Ratio a

Additive a => Multiplicative (Exponential a)

Defined in Algebra.Classes

Methods

(*) :: Exponential a -> Exponential a -> Exponential a

one :: Exponential a

(^) :: Exponential a -> Natural -> Exponential a

(Ring s, Applicative v, Traversable v) => Multiplicative (OrthoMat v s)

Defined in Algebra.Linear

Methods

(*) :: OrthoMat v s -> OrthoMat v s -> OrthoMat v s

one :: OrthoMat v s

(^) :: OrthoMat v s -> Natural -> OrthoMat v s

multiply :: (Multiplicative a, Foldable f) => f a -> a

type SemiRing a = (Multiplicative a, AbelianAdditive a)

type PreRing a = (SemiRing a, Group a)

fromIntegerDefault :: PreRing a => Integer -> a

class (Module a a, PreRing a) => Ring a where

Minimal complete definition

Nothing

Methods

fromInteger :: Integer -> a

Instances

Ring Double

Defined in Algebra.Classes

Methods

fromInteger :: Integer -> Double

Ring Float

Defined in Algebra.Classes

Methods

fromInteger :: Integer -> Float

Ring Int

Defined in Algebra.Classes

Methods

fromInteger :: Integer -> Int

Ring Integer

Defined in Algebra.Classes

Methods

fromInteger :: Integer -> Integer

Ring CInt

Defined in Algebra.Classes

Methods

fromInteger :: Integer -> CInt

Integral a => Ring (Ratio a)

Defined in Algebra.Classes

Methods

fromInteger :: Integer -> Ratio a

class Multiplicative a => Division a where

Minimal complete definition

(recip | (/))

Methods

recip :: a -> a

(/) :: a -> a -> a infixl 7

Instances

Division Double

Defined in Algebra.Classes

Methods

recip :: Double -> Double

(/) :: Double -> Double -> Double

Division Float

Defined in Algebra.Classes

Methods

recip :: Float -> Float

(/) :: Float -> Float -> Float

Integral a => Division (Ratio a)

Defined in Algebra.Classes

Methods

recip :: Ratio a -> Ratio a

(/) :: Ratio a -> Ratio a -> Ratio a

Group a => Division (Exponential a)

Defined in Algebra.Classes

Methods

recip :: Exponential a -> Exponential a

(/) :: Exponential a -> Exponential a -> Exponential a

(Ring s, Applicative v, Traversable v) => Division (OrthoMat v s)

Defined in Algebra.Linear

Methods

recip :: OrthoMat v s -> OrthoMat v s

(/) :: OrthoMat v s -> OrthoMat v s -> OrthoMat v s

class (Ring a, Division a) => Field a where

Minimal complete definition

Nothing

Methods

fromRational :: Rational -> a

Instances

Field Double

Defined in Algebra.Classes

Methods

fromRational :: Rational -> Double

Field Float

Defined in Algebra.Classes

Methods

fromRational :: Rational -> Float

Integral a => Field (Ratio a)

Defined in Algebra.Classes

Methods

fromRational :: Rational -> Ratio a

type VectorSpace scalar a = (Field scalar, Module scalar a)

class Ring a => EuclideanDomain a where

Minimal complete definition

(stdUnit | normalize), (divMod | div, mod)

Methods

stdAssociate :: a -> a

stdUnit :: a -> a

normalize :: a -> (a, a)

div :: a -> a -> a infixl 7

mod :: a -> a -> a infixl 7

divMod :: a -> a -> (a, a)

Instances

EuclideanDomain Int

Defined in Algebra.Classes

Methods

stdAssociate :: Int -> Int

stdUnit :: Int -> Int

normalize :: Int -> (Int, Int)

div :: Int -> Int -> Int

mod :: Int -> Int -> Int

divMod :: Int -> Int -> (Int, Int)

EuclideanDomain Integer

Defined in Algebra.Classes

Methods

stdAssociate :: Integer -> Integer

stdUnit :: Integer -> Integer

normalize :: Integer -> (Integer, Integer)

div :: Integer -> Integer -> Integer

mod :: Integer -> Integer -> Integer

divMod :: Integer -> Integer -> (Integer, Integer)

EuclideanDomain CInt

Defined in Algebra.Classes

Methods

stdAssociate :: CInt -> CInt

stdUnit :: CInt -> CInt

normalize :: CInt -> (CInt, CInt)

div :: CInt -> CInt -> CInt

mod :: CInt -> CInt -> CInt

divMod :: CInt -> CInt -> (CInt, CInt)

class (Real a, Enum a, EuclideanDomain a) => Integral a where

Minimal complete definition

toInteger

Methods

quot :: a -> a -> a

rem :: a -> a -> a

quotRem :: a -> a -> (a, a)

toInteger :: a -> Integer

Instances

Integral Integer

Defined in Algebra.Classes

Methods

quot :: Integer -> Integer -> Integer

rem :: Integer -> Integer -> Integer

quotRem :: Integer -> Integer -> (Integer, Integer)

toInteger :: Integer -> Integer

data Ratio a

Constructors

!a :% !a

Instances

Eq a => Eq (Ratio a)

Defined in Algebra.Classes

Methods

(==) :: Ratio a -> Ratio a -> Bool

(/=) :: Ratio a -> Ratio a -> Bool

type MyRational = Ratio Integer

gcd :: Integral a => a -> a -> a

ifThenElse :: Bool -> t -> t -> t

data InitialAdditive

Constructors

InitialAdditive :+ InitialAdditive

Zero

Instances

Show InitialAdditive

Defined in Algebra.Classes

Methods

showsPrec :: Int -> InitialAdditive -> ShowS

show :: InitialAdditive -> String

showList :: [InitialAdditive] -> ShowS

Additive InitialAdditive

Defined in Algebra.Classes

Methods

(+) :: InitialAdditive -> InitialAdditive -> InitialAdditive

zero :: InitialAdditive

times :: Natural -> InitialAdditive -> InitialAdditive