Numeric.Domain.Euclidean

Documentation

class (Ring r, DecidableZero r, DecidableUnits r, Domain r) => Euclidean r where

Minimal complete definition

splitUnit, degree, divide

Methods

splitUnit :: r -> (r, r)

splitUnit r calculates its leading unit and normal form.

let (u, n) = splitUnit r in r == u * n && fst (splitUnit n) == one && isUnit u

degree :: r -> Maybe Natural

Euclidean (degree) function on r.

divide infix 7

Arguments

:: r

elements divided by

-> r

divisor

-> (r, r)

quotient and remin

Division algorithm. a divide b calculates quotient and reminder of a divided by b.

let (q, r) = divide a p in p*q + r == a && degree r < degree q

quot :: r -> r -> r infixl 7

rem :: r -> r -> r infixl 7

gcd :: r -> r -> r

gcd a b calculates greatest common divisor of a and b.

euclid :: r -> r -> [(r, r, r)]

Extended euclidean algorithm.

euclid f g == xs ==> all (\(r, s, t) -> r == f * s + g * t) xs

Instances

Euclidean Integer

prs :: Euclidean r => r -> r -> [(r, r, r)]

normalize :: Euclidean r => r -> r

gcd' :: Euclidean r => [r] -> r

leadingUnit :: Euclidean r => r -> r

chineseRemainder

Arguments

:: Euclidean r

=> [(r, r)]

List of (m_i, v_i)

-> r

f with f = v_i (mod v_i)