Data.Num.Linear

This module provides a linear Num class with instances. Import this module to use linear versions of (+), (-), etc, on numeric types like Int and Double.

The Typeclass Hierarchy

The Num class is broken up into several instances. Here is the basic hierarchy:

• Additive ⊆ AddIdentity ⊆ AdditiveGroup
• MultIdentity ⊆ MultIdentity
• (AddIdentity ∩ MultIdentity) ⊆ Semiring
• (AdditiveGroup ∩ Semiring) ⊆ Ring
• (FromInteger ∩ Ring) ⊆ Num

Num and sub-classes

class (Ring a, FromInteger a) => Num a where

Methods

abs :: a %1 -> a

signum :: a %1 -> a

Instances

Num Double

Defined in Data.Num.Linear

Methods

abs :: Double %1 -> Double

signum :: Double %1 -> Double

Num Int

Defined in Data.Num.Linear

Methods

abs :: Int %1 -> Int

signum :: Int %1 -> Int

class Additive a where

A type that can be added linearly. The operation (+) is associative and commutative, i.e., for all a, b, c

(a + b) + c = a + (b + c)
a + b = b + c

Methods

(+) :: a %1 -> a %1 -> a infixl 6

Instances

Additive Double

Defined in Data.Num.Linear

Methods

(+) :: Double %1 -> Double %1 -> Double

Additive Int

Defined in Data.Num.Linear

Methods

(+) :: Int %1 -> Int %1 -> Int

class Additive a => AddIdentity a where

An Additive type with an identity on (+).

Methods

zero :: a

Instances

AddIdentity Double

Defined in Data.Num.Linear

Methods

zero :: Double

AddIdentity Int

Defined in Data.Num.Linear

Methods

zero :: Int

class AddIdentity a => AdditiveGroup a where

An AddIdentity with inverses that satisfies the laws of an abelian group

Minimal complete definition

negate | (-)

Methods

negate :: a %1 -> a

(-) :: a %1 -> a %1 -> a infixl 6

Instances

AdditiveGroup Double

Defined in Data.Num.Linear

Methods

negate :: Double %1 -> Double

(-) :: Double %1 -> Double %1 -> Double

AdditiveGroup Int

Defined in Data.Num.Linear

Methods

negate :: Int %1 -> Int

(-) :: Int %1 -> Int %1 -> Int

class Multiplicative a where

A numeric type with an associative (*) operation

Methods

(*) :: a %1 -> a %1 -> a infixl 7

Instances

Multiplicative Double

Defined in Data.Num.Linear

Methods

(*) :: Double %1 -> Double %1 -> Double

Multiplicative Int

Defined in Data.Num.Linear

Methods

(*) :: Int %1 -> Int %1 -> Int

class Multiplicative a => MultIdentity a where

A Multiplicative type with an identity for (*)

Methods

one :: a

Instances

MultIdentity Double

Defined in Data.Num.Linear

Methods

one :: Double

MultIdentity Int

Defined in Data.Num.Linear

Methods

one :: Int

class (AddIdentity a, MultIdentity a) => Semiring a

A semiring class. This is basically a numeric type with mutliplication, addition and with identities for each. The laws:

zero * x = zero
a * (b + c) = (a * b) + (a * c)

Instances

Semiring Double

Defined in Data.Num.Linear

Semiring Int

Defined in Data.Num.Linear

class (AdditiveGroup a, Semiring a) => Ring a

A Ring instance is a numeric type with (+), (-), (*) and all the following properties: a group with (+) and a MultIdentity with (*) along with distributive laws.

Instances

Ring Double

Defined in Data.Num.Linear

Ring Int

Defined in Data.Num.Linear

class FromInteger a where

A numeric type that Integers can be embedded into while satisfying all the typeclass laws Integers obey. That is, if there's some property like commutivity of integers x + y == y + x, then we must have:

fromInteger x + fromInteger y == fromInteger y + fromInteger x

For mathy folk: fromInteger should be a homomorphism over (+) and (*).

Methods

fromInteger :: Integer %1 -> a

Instances

FromInteger Double

Defined in Data.Num.Linear

Methods

fromInteger :: Integer %1 -> Double

FromInteger Int

Defined in Data.Num.Linear

Methods

fromInteger :: Integer %1 -> Int

Mechanisms for deriving instances

newtype Adding a

Deprecated: Use Sum (reexported as Sum) instead

A newtype wrapper to give the underlying monoid for an additive structure.

Deprecated because Sum (reexported as Sum) now has a linear Semigroup and Monoid instance.

Constructors

Adding a

Deprecated: Use Sum (reexported as Sum) instead

Instances

AddIdentity a => Monoid (Adding a)

Defined in Data.Num.Linear

Methods

mempty :: Adding a

mappend :: Adding a -> Adding a -> Adding a

mconcat :: [Adding a] -> Adding a

Additive a => Semigroup (Adding a)

Defined in Data.Num.Linear

Methods

(<>) :: Adding a -> Adding a -> Adding a

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

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

Show a => Show (Adding a)

Defined in Data.Num.Linear

Methods

showsPrec :: Int -> Adding a -> ShowS

show :: Adding a -> String

showList :: [Adding a] -> ShowS

Eq a => Eq (Adding a)

Defined in Data.Num.Linear

Methods

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

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

Ord a => Ord (Adding a)

Defined in Data.Num.Linear

Methods

compare :: Adding a -> Adding a -> Ordering

(<) :: Adding a -> Adding a -> Bool

(<=) :: Adding a -> Adding a -> Bool

(>) :: Adding a -> Adding a -> Bool

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

max :: Adding a -> Adding a -> Adding a

min :: Adding a -> Adding a -> Adding a

AddIdentity a => Monoid (Adding a)

Defined in Data.Num.Linear

Methods

mempty :: Adding a

Additive a => Semigroup (Adding a)

Defined in Data.Num.Linear

Methods

(<>) :: Adding a %1 -> Adding a %1 -> Adding a

getAdded :: Adding a %1 -> a

Deprecated: Use Sum (reexported as Sum) and pattern-match to extract the inner value linearly

newtype Multiplying a

Deprecated: Use Product (reexported as Product) instead

A newtype wrapper to give the underlying monoid for a multiplicative structure.

Deprecated because Product (reexported as Product) now has a linear Semigroup and Monoid instance.

Constructors

Multiplying a

Deprecated: Use Product (reexported as Product) instead

Instances

MultIdentity a => Monoid (Multiplying a)

Defined in Data.Num.Linear

Methods

mempty :: Multiplying a

mappend :: Multiplying a -> Multiplying a -> Multiplying a

mconcat :: [Multiplying a] -> Multiplying a

Multiplicative a => Semigroup (Multiplying a)

Defined in Data.Num.Linear

Methods

(<>) :: Multiplying a -> Multiplying a -> Multiplying a

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

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

Show a => Show (Multiplying a)

Defined in Data.Num.Linear

Methods

showsPrec :: Int -> Multiplying a -> ShowS

show :: Multiplying a -> String

showList :: [Multiplying a] -> ShowS

Eq a => Eq (Multiplying a)

Defined in Data.Num.Linear

Methods

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

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

Ord a => Ord (Multiplying a)

Defined in Data.Num.Linear

Methods

compare :: Multiplying a -> Multiplying a -> Ordering

(<) :: Multiplying a -> Multiplying a -> Bool

(<=) :: Multiplying a -> Multiplying a -> Bool

(>) :: Multiplying a -> Multiplying a -> Bool

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

max :: Multiplying a -> Multiplying a -> Multiplying a

min :: Multiplying a -> Multiplying a -> Multiplying a

MultIdentity a => Monoid (Multiplying a)

Defined in Data.Num.Linear

Methods

mempty :: Multiplying a

Multiplicative a => Semigroup (Multiplying a)

Defined in Data.Num.Linear

Methods

(<>) :: Multiplying a %1 -> Multiplying a %1 -> Multiplying a

getMultiplied :: Multiplying a %1 -> a

Deprecated: Use Product (reexported as Product) and pattern-match to extract the inner value linearly

Orphan instances

MultIdentity a => Monoid (Product a)

Methods

mempty :: Product a

AddIdentity a => Monoid (Sum a)

Methods

mempty :: Sum a

Multiplicative a => Semigroup (Product a)

Methods

(<>) :: Product a %1 -> Product a %1 -> Product a

Additive a => Semigroup (Sum a)

Methods

(<>) :: Sum a %1 -> Sum a %1 -> Sum a