numeric-prelude-0.3: An experimental alternative hierarchy of numeric type classes

Algebra.PrincipalIdealDomain

Synopsis

Class

class (C a, C a) => C a whereSource

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

Methods

extendedGCD :: a -> a -> (a, (a, a))Source

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.

gcd :: a -> a -> aSource

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

lcm :: a -> a -> aSource

Least common multiple

Instances

 C Int C Int8 C Int16 C Int32 C Int64 C Integer C T (C a, C a) => C (T a) (Ord a, C a, C a) => C (T a) Integral a => C (T a) C a => C (T a)

coprime :: C a => a -> a -> BoolSource

Standard implementations for instances

euclid :: (C a, C a) => (a -> a -> a) -> a -> a -> aSource

extendedEuclid :: (C a, C a) => (a -> a -> (a, a)) -> a -> a -> (a, (a, a))Source

Algorithms

extendedGCDMulti :: C a => [a] -> (a, [a])Source

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

diophantine :: C a => a -> a -> a -> Maybe (a, a)Source

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.

diophantineMin :: C a => a -> a -> a -> Maybe (a, a)Source

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

diophantineMulti :: C a => a -> [a] -> Maybe [a]Source

chineseRemainder :: C a => (a, a) -> (a, a) -> Maybe (a, a)Source

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

chineseRemainderMulti :: C a => [(a, a)] -> Maybe (a, a)Source

For Just (b,n) = chineseRemainder [(a0,m0), (a1,m1), ..., (an,mn)] and all x with x = b mod n the congruences x=a0 mod m0, x=a1 mod m1, ..., x=an mod mn are fulfilled.

Properties

propMaximalDivisor :: C a => a -> a -> a -> PropertySource

propGCDDiophantine :: (Eq a, C a) => a -> a -> BoolSource

propExtendedGCDMulti :: (Eq a, C a) => [a] -> BoolSource

propDiophantine :: (Eq a, C a) => a -> a -> a -> a -> BoolSource

propDiophantineMin :: (Eq a, C a) => a -> a -> a -> a -> BoolSource

propDiophantineMulti :: (Eq a, C a) => [(a, a)] -> BoolSource

propDiophantineMultiMin :: (Eq a, C a) => [(a, a)] -> BoolSource

propChineseRemainder :: (Eq a, C a) => a -> a -> [a] -> PropertySource

propDivisibleGCD :: C a => a -> a -> BoolSource

propDivisibleLCM :: C a => a -> a -> BoolSource

propGCDIdentity :: (Eq a, C a) => a -> BoolSource

propGCDCommutative :: (Eq a, C a) => a -> a -> BoolSource

propGCDAssociative :: (Eq a, C a) => a -> a -> a -> BoolSource

propGCDHomogeneous :: (Eq a, C a) => a -> a -> a -> BoolSource

propGCD_LCM :: (Eq a, C a) => a -> a -> BoolSource