| Copyright | (c) 2019 Andrew Lelechenko |
|---|---|
| License | BSD3 |
| Maintainer | Andrew Lelechenko <andrew.lelechenko@gmail.com> |
| Safe Haskell | None |
| Language | Haskell98 |
Data.Euclidean
Description
Synopsis
- class GcdDomain a => Euclidean a where
- class (Euclidean a, Ring a) => Field a
- class Semiring a => GcdDomain a where
- newtype WrappedIntegral a = WrapIntegral {
- unwrapIntegral :: a
- newtype WrappedFractional a = WrapFractional {
- unwrapFractional :: a
- gcdExt :: (Eq a, Euclidean a, Ring a) => a -> a -> (a, a)
Documentation
class GcdDomain a => Euclidean a where #
Informally speaking, Euclidean is a superclass of Integral,
lacking toInteger, which allows to define division with remainder
for a wider range of types, e. g., complex integers
and polynomials with rational coefficients.
Euclidean represents a
Euclidean domain
endowed by a given Euclidean function degree.
No particular rounding behaviour is expected of quotRem. E. g.,
it is not guaranteed to truncate towards zero or towards negative
infinity (cf. divMod), and remainders are not guaranteed to be non-negative.
For a faithful representation of residue classes one can use
mod package instead.
Methods
Division with remainder.
\x y -> y == 0 || let (q, r) = x `quotRem` y in x == q * y + r
quot :: a -> a -> a infixl 7 #
Division. Must match its default definition:
\x y -> quot x y == fst (quotRem x y)
Remainder. Must match its default definition:
\x y -> rem x y == snd (quotRem x y)
Euclidean (aka degree, valuation, gauge, norm) function on a. Usually fromIntegral . abs
degree is rarely used by itself. Its purpose
is to provide an evidence of soundness of quotRem
by testing the following property:
\x y -> y == 0 || let (q, r) = x `quotRem` y in (r == 0 || degree r < degree y)
Instances
class (Euclidean a, Ring a) => Field a #
Instances
| Field CDouble # | |
Defined in Data.Euclidean | |
| Field CFloat # | |
Defined in Data.Euclidean | |
| Field Mod2 # | |
Defined in Data.Euclidean | |
| Field () # | |
Defined in Data.Euclidean | |
| Field Double # | |
Defined in Data.Euclidean | |
| Field Float # | |
Defined in Data.Euclidean | |
| Field a => Field (Complex a) # | |
Defined in Data.Euclidean | |
| Integral a => Field (Ratio a) # | |
Defined in Data.Euclidean | |
| Fractional a => Field (WrappedFractional a) # | |
Defined in Data.Euclidean | |
class Semiring a => GcdDomain a where #
GcdDomain represents a
GCD domain.
This is a domain, where GCD can be defined,
but which does not necessarily allow a well-behaved
division with remainder (as in Euclidean domains).
For example, there is no way to define rem over
polynomials with integer coefficients such that
remainder is always "smaller" than divisor. However,
gcd is still definable, just not by means of
Euclidean algorithm.
All methods of GcdDomain have default implementations
in terms of Euclidean. So most of the time
it is enough to write:
instance GcdDomain Foo instance Euclidean Foo where quotRem = ... degree = ...
Minimal complete definition
Nothing
Methods
divide :: a -> a -> Maybe a infixl 7 #
Division without remainder.
\x y -> (x * y) `divide` y == Just x
\x y -> maybe True (\z -> x == z * y) (x `divide` y)
Greatest common divisor. Must satisfy
\x y -> isJust (x `divide` gcd x y) && isJust (y `divide` gcd x y)
\x y z -> isJust (gcd (x * z) (y * z) `divide` z)
Lowest common multiple. Must satisfy
\x y -> isJust (lcm x y `divide` x) && isJust (lcm x y `divide` y)
\x y z -> isNothing (z `divide` x) || isNothing (z `divide` y) || isJust (z `divide` lcm x y)
Test whether two arguments are coprime. Must match its default definition:
\x y -> coprime x y == isJust (1 `divide` gcd x y)
Instances
newtype WrappedIntegral a #
Constructors
| WrapIntegral | |
Fields
| |
Instances
newtype WrappedFractional a #
Wrapper around Fractional
with trivial GcdDomain
and Euclidean instances.
Constructors
| WrapFractional | |
Fields
| |