Algebra.Field

Class

class C a => C a

Field again corresponds to a commutative ring. Division is partially defined and satisfies

   not (isZero b)  ==>  (a * b) / b === a
   not (isZero a)  ==>  a * recip a === one

when it is defined. To safely call division, the program must take type-specific action; e.g., the following is appropriate in many cases:

safeRecip :: (Integral a, Eq a, Field.C a) => a -> Maybe a
safeRecip x =
    let (q,r) = one `divMod` x
    in  toMaybe (isZero r) q

Typical examples include rationals, the real numbers, and rational functions (ratios of polynomial functions). An instance should be typically declared only if most elements are invertible.

Actually, we have also used this type class for non-fields containing lots of units, e.g. residue classes with respect to non-primes and power series. So the restriction not (isZero a) must be better isUnit a.

Minimal definition: recip or (/)

Minimal complete definition

recip | (/)

Instances

C Double

Defined in Algebra.Field

Methods

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

recip :: Double -> Double

fromRational' :: Rational -> Double

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

C Float

Defined in Algebra.Field

Methods

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

recip :: Float -> Float

fromRational' :: Rational -> Float

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

C T

Defined in Number.GaloisField2p32m5

Methods

(/) :: T -> T -> T

recip :: T -> T

fromRational' :: Rational -> T

(^-) :: T -> Integer -> T

C T

Defined in Number.FixedPoint.Check

Methods

(/) :: T -> T -> T

recip :: T -> T

fromRational' :: Rational -> T

(^-) :: T -> Integer -> T

C T

Defined in Number.Positional.Check

Methods

(/) :: T -> T -> T

recip :: T -> T

fromRational' :: Rational -> T

(^-) :: T -> Integer -> T

Integral a => C (Ratio a)

Defined in Algebra.Field

Methods

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

recip :: Ratio a -> Ratio a

fromRational' :: Rational -> Ratio a

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

RealFloat a => C (Complex a)

Defined in Algebra.Field

Methods

(/) :: Complex a -> Complex a -> Complex a

recip :: Complex a -> Complex a

fromRational' :: Rational -> Complex a

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

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

Defined in Number.NonNegative

Methods

(/) :: T a -> T a -> T a

recip :: T a -> T a

fromRational' :: Rational -> T a

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

C a => C (T a)

Defined in Algebra.Field

Methods

(/) :: T a -> T a -> T a

recip :: T a -> T a

fromRational' :: Rational -> T a

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

Fractional a => C (T a)

Defined in MathObj.Wrapper.Haskell98

Methods

(/) :: T a -> T a -> T a

recip :: T a -> T a

fromRational' :: Rational -> T a

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

C a => C (T a)

Defined in Number.ResidueClass.Func

Methods

(/) :: T a -> T a -> T a

recip :: T a -> T a

fromRational' :: Rational -> T a

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

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

Defined in Number.ResidueClass.Check

Methods

(/) :: T a -> T a -> T a

recip :: T a -> T a

fromRational' :: Rational -> T a

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

C a => C (T a)

Defined in Number.PartiallyTranscendental

Methods

(/) :: T a -> T a -> T a

recip :: T a -> T a

fromRational' :: Rational -> T a

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

C a => C (T a)

Defined in MathObj.PowerSeries

Methods

(/) :: T a -> T a -> T a

recip :: T a -> T a

fromRational' :: Rational -> T a

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

C a => C (T a)

Defined in MathObj.PowerSeries2

Methods

(/) :: T a -> T a -> T a

recip :: T a -> T a

fromRational' :: Rational -> T a

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

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

Defined in MathObj.PowerSum

Methods

(/) :: T a -> T a -> T a

recip :: T a -> T a

fromRational' :: Rational -> T a

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

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

Defined in MathObj.RootSet

Methods

(/) :: T a -> T a -> T a

recip :: T a -> T a

fromRational' :: Rational -> T a

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

C a => C (T a)

Defined in Number.Complex

Methods

(/) :: T a -> T a -> T a

recip :: T a -> T a

fromRational' :: Rational -> T a

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

C a => C (T a)

Defined in Number.Quaternion

Methods

(/) :: T a -> T a -> T a

recip :: T a -> T a

fromRational' :: Rational -> T a

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

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

Defined in MathObj.LaurentPolynomial

Methods

(/) :: T a -> T a -> T a

recip :: T a -> T a

fromRational' :: Rational -> T a

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

C a => C (T a)

Defined in MathObj.Wrapper.NumericPrelude

Methods

(/) :: T a -> T a -> T a

recip :: T a -> T a

fromRational' :: Rational -> T a

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

C v => C (T a v)

Defined in Number.OccasionallyScalarExpression

Methods

(/) :: T a v -> T a v -> T a v

recip :: T a v -> T a v

fromRational' :: Rational -> T a v

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

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

Defined in Number.DimensionTerm

Methods

(/) :: T u a -> T u a -> T u a

recip :: T u a -> T u a

fromRational' :: Rational -> T u a

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

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

Defined in Number.Physical

Methods

(/) :: T i a -> T i a -> T i a

recip :: T i a -> T i a

fromRational' :: Rational -> T i a

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

C v => C (T a v)

Defined in Number.SI

Methods

(/) :: T a v -> T a v -> T a v

recip :: T a v -> T a v

fromRational' :: Rational -> T a v

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

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

recip :: C a => a -> a

fromRational' :: C a => Rational -> a

fromRational :: C a => Rational -> a

Needed to work around shortcomings in GHC.

(^-) :: C a => a -> Integer -> a infixr 8

Properties

propDivision :: (Eq a, C a, C a) => a -> a -> Property

the restriction on the divisor should be isUnit a instead of not (isZero a)

propReciprocal :: (Eq a, C a, C a) => a -> Property