Algebra.PrincipalIdealDomain

A principal ideal domain is a ring in which every ideal (the set of multiples of some generating set of elements) is principal: That is, every element can be written as the multiple of some generating element. gcd a b gives a generator for the ideal generated by a and b. The algorithm above works whenever mod x y is smaller (in a suitable sense) than both x and y; otherwise the algorithm may run forever.

Laws:

  divides x (lcm x y)
  x `gcd` (y `gcd` z) == (x `gcd` y) `gcd` z
  gcd x y * z == gcd (x*z) (y*z)
  gcd x y * lcm x y == x * y

(etc: canonical)

Minimal definition: * nothing, if the standard Euclidean algorithm work * if extendedGCD is implemented customly, gcd and lcm make use of it

Compute the greatest common divisor and solve a respective Diophantine equation.

  (g,(a,b)) = extendedGCD x y ==>
       g==a*x+b*y   &&  g == gcd x y

TODO: This method is not appropriate for the PID class, because there are rings like the one of the multivariate polynomials, where for all x and y greatest common divisors of x and y exist, but they cannot be represented as a linear combination of x and y. TODO: The definition of extendedGCD does not return the canonical associate.

The Greatest Common Divisor is defined by:

  gcd x y == gcd y x
  divides z x && divides z y ==> divides z (gcd x y)   (specification)
  divides (gcd x y) x

Least common multiple

Compute the greatest common divisor for multiple numbers by repeated application of the two-operand-gcd.

A variant with small coefficients.

Just (a,b) = diophantine z x y means a*x+b*y = z. It is required that gcd(y,z) divides x.

Like diophantine, but a is minimal with respect to the measure function of the Euclidean algorithm.

Not efficient enough, because GCD/LCM is computed twice.

For Just (n,b) = chineseRemainderMulti [(m0,a0), (m1,a1), ..., (mk,ak)] and all x with x = b mod n, the congruences x=a0 mod m0, x=a1 mod m1, ..., x=ak mod mk are fulfilled. Also, n is the least common multiplier of all mi.

>>> PID.chineseRemainderMulti [(100,21), (10000,2021::Integer)]Just (10000,2021)>>> PID.chineseRemainderMulti [(97,90),(99,10),(100,0::Integer)]Just (960300,100000)>>> PID.chineseRemainderMulti [(95,30),(97,27),(98,8),(99,1::Integer)]Just (89403930,1000000)

prop> QC.listOf genResidueClass /\ \xs -> case PID.chineseRemainderMulti xs of Nothing -> True; Just (n,b) -> abs n == abs (foldl lcm 1 (map fst xs)) && map snd xs == map (mod b . fst) xs

prop> \(QC.NonEmpty ms) b -> let xs = map (\(QC.NonZero m) -> (m, mod b m)) ms in case PID.chineseRemainderMulti xs of Nothing -> False; Just (n,c) -> abs n == abs (foldl lcm 1 (map QC.getNonZero ms)) && mod b n == (c::Integer)