NumHask.Algebra.Field

Field classes

Documentation

class (MultiplicativeInvertible a, Ring a) => Semifield a

A Semifield is a Field without Commutative Multiplication.

Instances

Semifield Double

Semifield Float

Semifield a => Semifield (Complex a)

class (AdditiveGroup a, MultiplicativeGroup a, Ring a) => Field a

A Field is a Ring plus additive invertible and multiplicative invertible operations.

A summary of the rules inherited from super-classes of Field

zero + a == a
a + zero == a
(a + b) + c == a + (b + c)
a + b == b + a
a - a = zero
negate a = zero - a
negate a + a = zero
a + negate a = zero
one * a == a
a * one == a
(a * b) * c == a * (b * c)
a * (b + c) == a * b + a * c
(a + b) * c == a * c + b * c
a * zero == zero
zero * a == zero
a * b == b * a
a / a = one
recip a = one / a
recip a * a = one
a * recip a = one

Instances

Field Double

Field Float

Field a => Field (Complex a)

class Field a => ExpField a where

A hyperbolic field class

sqrt . (**2) == identity
log . exp == identity
for +ive b, a != 0,1: a ** logBase a b ≈ b

Minimal complete definition

exp, log

Methods

exp :: a -> a

log :: a -> a

logBase :: a -> a -> a

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

sqrt :: a -> a

Instances

ExpField Double

Methods

exp :: Double -> Double

log :: Double -> Double

logBase :: Double -> Double -> Double

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

sqrt :: Double -> Double

ExpField Float

Methods

exp :: Float -> Float

log :: Float -> Float

logBase :: Float -> Float -> Float

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

sqrt :: Float -> Float

(TrigField a, ExpField a) => ExpField (Complex a)

todo: bottom is here somewhere???

Methods

exp :: Complex a -> Complex a

log :: Complex a -> Complex a

logBase :: Complex a -> Complex a -> Complex a

(**) :: Complex a -> Complex a -> Complex a

sqrt :: Complex a -> Complex a

class Field a => QuotientField a where

quotient fields explode constraints if they allow for polymorphic integral types

a - one < floor a <= a <= ceiling a < a + one
round a == floor (a + one/(one+one))

Minimal complete definition

round, ceiling, floor, (^^)

Methods

round :: a -> Integer

ceiling :: a -> Integer

floor :: a -> Integer

(^^) :: a -> Integer -> a

Instances

QuotientField Double

Methods

round :: Double -> Integer

ceiling :: Double -> Integer

floor :: Double -> Integer

(^^) :: Double -> Integer -> Double

QuotientField Float

Methods

round :: Float -> Integer

ceiling :: Float -> Integer

floor :: Float -> Integer

(^^) :: Float -> Integer -> Float

class Field a => BoundedField a where

A bounded field includes the concepts of infinity and NaN, thus moving away from error throwing.

one / zero + infinity == infinity
infinity + a == infinity
isNaN (infinity - infinity)
isNaN (infinity / infinity)
isNaN (nan + a)
zero / zero != nan

Note the tricky law that, although nan is assigned to zero/zero, they are never-the-less not equal. A committee decided this.

Minimal complete definition

isNaN

Methods

maxBound :: a

minBound :: a

nan :: a

isNaN :: a -> Bool

Instances

BoundedField Double

Methods

maxBound :: Double

minBound :: Double

nan :: Double

isNaN :: Double -> Bool

BoundedField Float

Methods

maxBound :: Float

minBound :: Float

nan :: Float

isNaN :: Float -> Bool

BoundedField a => BoundedField (Complex a)

todo: work out boundings for complex as it stands now, complex is different eg

one / (zero :: Complex Float) == nan

Methods

maxBound :: Complex a

minBound :: Complex a

nan :: Complex a

isNaN :: Complex a -> Bool

infinity :: BoundedField a => a

prints as Infinity

neginfinity :: BoundedField a => a

prints as `-Infinity`

class (Ord a, Field a) => TrigField a where

Trigonometric Field

Minimal complete definition

pi, sin, cos, asin, acos, atan, sinh, cosh, asinh, acosh, atanh

Methods

pi :: a

sin :: a -> a

cos :: a -> a

tan :: a -> a

asin :: a -> a

acos :: a -> a

atan :: a -> a

sinh :: a -> a

cosh :: a -> a

tanh :: a -> a

asinh :: a -> a

acosh :: a -> a

atanh :: a -> a

atan2 :: a -> a -> a

Instances

TrigField Double

Methods

pi :: Double

sin :: Double -> Double

cos :: Double -> Double

tan :: Double -> Double

asin :: Double -> Double

acos :: Double -> Double

atan :: Double -> Double

sinh :: Double -> Double

cosh :: Double -> Double

tanh :: Double -> Double

asinh :: Double -> Double

acosh :: Double -> Double

atanh :: Double -> Double

atan2 :: Double -> Double -> Double

TrigField Float

Methods

pi :: Float

sin :: Float -> Float

cos :: Float -> Float

tan :: Float -> Float

asin :: Float -> Float

acos :: Float -> Float

atan :: Float -> Float

sinh :: Float -> Float

cosh :: Float -> Float

tanh :: Float -> Float

asinh :: Float -> Float

acosh :: Float -> Float

atanh :: Float -> Float

atan2 :: Float -> Float -> Float