Algebra.Additive

Class

class C a where

Additive a encapsulates the notion of a commutative group, specified by the following laws:

          a + b === b + a
    (a + b) + c === a + (b + c)
       zero + a === a
   a + negate a === 0

Typical examples include integers, dollars, and vectors.

Minimal definition: +, zero, and (negate or '(-)')

Minimal complete definition

zero, (+), ((-) | negate)

Methods

zero :: a

zero element of the vector space

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

add and subtract elements

negate :: a -> a

inverse with respect to +

Instances

C Double

C Float

C Int

C Int8

C Int16

C Int32

C Int64

C Integer

C Word

C Word8

C Word16

C Word32

C Word64

C T

C T

C T

C T

C v => C [v]

Integral a => C (Ratio a)

RealFloat a => C (Complex a)

(Ord a, C a) => C (T a)

C a => C (T a)

C a => C (T a)

(C a, C a, C a) => C (T a)

C a => C (T a)

C a => C (T a)

C a => C (T a)

(Eq a, C a) => C (T a)

(Eq a, C a) => C (T a)

Num a => C (T a)

C a => C (T a)

C a => C (T a)

(C a, C a) => C (T a)

C a => C (T a)

C a => C (T a)

C a => C (T a)

C a => C (T a)

C a => C (T a)

C a => C (T a)

C v => C (b -> v)

(C v0, C v1) => C (v0, v1)

(Ord i, Eq v, C v) => C (Map i v)

(Ord a, C b) => C (T a b)

(C u, C a) => C (T u a)

C v => C (T a v)

(Ord i, C a) => C (T i a)

C v => C (T a v)

(C v0, C v1, C v2) => C (v0, v1, v2)

subtract :: C a => a -> a -> a

subtract is (-) with swapped operand order. This is the operand order which will be needed in most cases of partial application.

Complex functions

sum :: C a => [a] -> a

Sum up all elements of a list. An empty list yields zero.

This function is inappropriate for number types like Peano. Maybe we should make sum a method of Additive. This would also make lengthLeft and lengthRight superfluous.

sum1 :: C a => [a] -> a

Sum up all elements of a non-empty list. This avoids including a zero which is useful for types where no universal zero is available.

sumNestedAssociative :: C a => [a] -> a

Sum the operands in an order, such that the dependencies are minimized. Does this have a measurably effect on speed?

Requires associativity.

sumNestedCommutative :: C a => [a] -> a

Instance definition helpers

elementAdd :: C x => (v -> x) -> T (v, v) x

Instead of baking the add operation into the element function, we could use higher rank types and pass a generic uncurry (+) to the run function. We do not do so in order to stay Haskell 98 at least for parts of NumericPrelude.

elementSub :: C x => (v -> x) -> T (v, v) x

elementNeg :: C x => (v -> x) -> T v x

(<*>.+) :: C x => T (v, v) (x -> a) -> (v -> x) -> T (v, v) a infixl 4

addPair :: (Additive.C a, Additive.C b) => (a,b) -> (a,b) -> (a,b)
addPair = Elem.run2 $ Elem.with (,) <*>.+  fst <*>.+  snd

(<*>.-) :: C x => T (v, v) (x -> a) -> (v -> x) -> T (v, v) a infixl 4

(<*>.-$) :: C x => T v (x -> a) -> (v -> x) -> T v a infixl 4

Instances for atomic types

propAssociative :: (Eq a, C a) => a -> a -> a -> Bool

propCommutative :: (Eq a, C a) => a -> a -> Bool

propIdentity :: (Eq a, C a) => a -> Bool

propInverse :: (Eq a, C a) => a -> Bool