- class (C a, C a) => C a where
- euclid :: (C a, C a) => (a -> a -> a) -> a -> a -> a
- extendedEuclid :: (C a, C a) => (a -> a -> (a, a)) -> a -> a -> (a, (a, a))
- extendedGCDMulti :: C a => [a] -> (a, [a])
- diophantine :: C a => a -> a -> a -> Maybe (a, a)
- diophantineMin :: C a => a -> a -> a -> Maybe (a, a)
- diophantineMulti :: C a => a -> [a] -> Maybe [a]
- chineseRemainder :: C a => (a, a) -> (a, a) -> Maybe (a, a)
- chineseRemainderMulti :: C a => [(a, a)] -> Maybe (a, a)
- propMaximalDivisor :: C a => a -> a -> a -> Property
- propGCDDiophantine :: (Eq a, C a) => a -> a -> Bool
- propExtendedGCDMulti :: (Eq a, C a) => [a] -> Bool
- propDiophantine :: (Eq a, C a) => a -> a -> a -> a -> Bool
- propDiophantineMin :: (Eq a, C a) => a -> a -> a -> a -> Bool
- propDiophantineMulti :: (Eq a, C a) => [(a, a)] -> Bool
- propDiophantineMultiMin :: (Eq a, C a) => [(a, a)] -> Bool
- propChineseRemainder :: (Eq a, C a) => a -> a -> [a] -> Property
- propDivisibleGCD :: C a => a -> a -> Bool
- propDivisibleLCM :: C a => a -> a -> Bool
- propGCDIdentity :: (Eq a, C a) => a -> Bool
- propGCDCommutative :: (Eq a, C a) => a -> a -> Bool
- propGCDAssociative :: (Eq a, C a) => a -> a -> a -> Bool
- propGCDHomogeneous :: (Eq a, C a) => a -> a -> a -> Bool
- propGCD_LCM :: (Eq a, C a) => a -> a -> Bool
A principal ideal domain is a ring in which every ideal
(the set of multiples of some generating set of elements)
every element can be written as the multiple of some generating element.
gcd a b gives a generator for the ideal generated by
The algorithm above works whenever
mod x y is smaller
(in a suitable sense) than both
otherwise the algorithm may run forever.
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
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
Standard implementations for instances
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
a*x+b*y = z.
It is required that
a is minimal
with respect to the measure function of the Euclidean algorithm.
Not efficient enough, because GCD/LCM is computed twice.
Just (b,n) = chineseRemainder [(a0,m0), (a1,m1), ..., (an,mn)]
x = b mod n the congruences
x=a0 mod m0, x=a1 mod m1, ..., x=an mod mn