Safe Haskell	Safe
Language	Haskell2010

Data.Semifield

Contents

Semifields
Fields

Synopsis

type SemifieldLaw a = ((Additive - Monoid) a, (Multiplicative - Group) a)
class (Semiring a, SemifieldLaw a) => Semifield a
anan :: Semifield a => a
pinf :: Semifield a => a
(/) :: (Multiplicative - Group) a => a -> a -> a
(\\) :: (Multiplicative - Group) a => a -> a -> a
recip :: (Multiplicative - Group) a => a -> a
type FieldLaw a = ((Additive - Group) a, (Multiplicative - Group) a)
class (Ring a, Semifield a, FieldLaw a) => Field a
class Field a => Real a
ninf :: Field a => a
(^^) :: (Multiplicative - Group) a => a -> Integer -> a

Semifields

type SemifieldLaw a = ((Additive - Monoid) a, (Multiplicative - Group) a) Source #

class (Semiring a, SemifieldLaw a) => Semifield a Source #

A semifield, near-field, or division ring.

Instances needn't have commutative multiplication or additive inverses, however addition must be commutative, and addition and multiplication must be associative as usual.

See also the wikipedia definitions of:

Instances

Semifield Double Source #
Instance details Defined in Data.Semifield
Semifield Float Source #
Instance details Defined in Data.Semifield
Semifield Rational Source #
Instance details Defined in Data.Semifield
Semifield () Source #
Instance details Defined in Data.Semifield
Semifield Uni Source #
Instance details Defined in Data.Semifield
Semifield Deci Source #
Instance details Defined in Data.Semifield
Semifield Centi Source #
Instance details Defined in Data.Semifield
Semifield Milli Source #
Instance details Defined in Data.Semifield
Semifield Micro Source #
Instance details Defined in Data.Semifield
Semifield Nano Source #
Instance details Defined in Data.Semifield
Semifield Pico Source #
Instance details Defined in Data.Semifield
Semifield CFloat Source #
Instance details Defined in Data.Semifield
Semifield CDouble Source #
Instance details Defined in Data.Semifield
Semifield (Ratio Natural) Source #
Instance details Defined in Data.Semifield

anan :: Semifield a => a Source #

The NaN value of the semifield.

 anan = zero / zero

pinf :: Semifield a => a Source #

The positive infinity of the semifield.

 pinf = one / zero

(/) :: (Multiplicative - Group) a => a -> a -> a infixl 7 Source #

Right division by a multiplicative group element.

(\\) :: (Multiplicative - Group) a => a -> a -> a infixl 7 Source #

Left division by a multiplicative group element.

When * is commutative we must have:

 x \\ y = y / x

recip :: (Multiplicative - Group) a => a -> a Source #

Reciprocal of a multiplicative group element.

x / y = x * recip y
x \\ y = recip x * y

>>> recip (3 :+ 4) :: Complex Rational
3 % 25 :+ (-4) % 25
>>> recip (3 :+ 4) :: Complex Double
0.12 :+ (-0.16)
>>> recip (3 :+ 4) :: Complex Pico
0.120000000000 :+ -0.160000000000

Fields

type FieldLaw a = ((Additive - Group) a, (Multiplicative - Group) a) Source #

class (Ring a, Semifield a, FieldLaw a) => Field a Source #

A field.

Instances

Field Double Source #
Instance details Defined in Data.Semifield
Field Float Source #
Instance details Defined in Data.Semifield
Field Rational Source #
Instance details Defined in Data.Semifield
Field () Source #
Instance details Defined in Data.Semifield
Field Uni Source #
Instance details Defined in Data.Semifield
Field Deci Source #
Instance details Defined in Data.Semifield
Field Centi Source #
Instance details Defined in Data.Semifield
Field Milli Source #
Instance details Defined in Data.Semifield
Field Micro Source #
Instance details Defined in Data.Semifield
Field Nano Source #
Instance details Defined in Data.Semifield
Field Pico Source #
Instance details Defined in Data.Semifield
Field CFloat Source #
Instance details Defined in Data.Semifield
Field CDouble Source #
Instance details Defined in Data.Semifield

class Field a => Real a Source #

A type modeling the real numbers.

Instances

Real Double Source #
Instance details Defined in Data.Semifield
Real Float Source #
Instance details Defined in Data.Semifield
Real Rational Source #
Instance details Defined in Data.Semifield
Real Uni Source #
Instance details Defined in Data.Semifield
Real Deci Source #
Instance details Defined in Data.Semifield
Real Centi Source #
Instance details Defined in Data.Semifield
Real Milli Source #
Instance details Defined in Data.Semifield
Real Micro Source #
Instance details Defined in Data.Semifield
Real Nano Source #
Instance details Defined in Data.Semifield
Real Pico Source #
Instance details Defined in Data.Semifield
Real CFloat Source #
Instance details Defined in Data.Semifield
Real CDouble Source #
Instance details Defined in Data.Semifield

ninf :: Field a => a Source #

The one / zero value of the field.

 ninf = negate one / zero

(^^) :: (Multiplicative - Group) a => a -> Integer -> a infixr 8 Source #

Integral power of a multiplicative group element.

 one == a ^^ 0

>>> 8 ^^ 0 :: Double
1.0
>>> 8 ^^ 0 :: Pico
1.000000000000