numeric-prelude-0.4.0.3: An experimental alternative hierarchy of numeric type classes

Algebra.IntegralDomain

Contents

Synopsis

Class

class C a => C a whereSource

IntegralDomain corresponds to a commutative ring, where a mod b picks a canonical element of the equivalence class of a in the ideal generated by b. div and mod satisfy the laws

a * b === b * a
(a `div` b) * b + (a `mod` b) === a
(a+k*b) `mod` b === a `mod` b
0 `mod` b === 0

Typical examples of IntegralDomain include integers and polynomials over a field. Note that for a field, there is a canonical instance defined by the above rules; e.g.,

instance IntegralDomain.C Rational where
divMod a b =
if isZero b
then (undefined,a)
else (a\/b,0)

It shall be noted, that div, mod, divMod have a parameter order which is unfortunate for partial application. But it is adapted to mathematical conventions, where the operators are used in infix notation.

Minimal definition: divMod or (div and mod)

Methods

div, mod :: a -> a -> aSource

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

Instances

 C Int C Int8 C Int16 C Int32 C Int64 C Integer C Word C Word8 C Word16 C Word32 C Word64 C T (Ord a, C a) => C (T a) (C a, C a) => C (T a) Integral a => C (T a) (C a, C a) => C (T a) The C instance is intensionally built from the C structure of the polynomial coefficients. If we would use Integral.C a superclass, then the Euclidean algorithm could not determine the greatest common divisor of e.g. [1,1] and . C a => C (T a) C a => C (T a) (Ord a, C a, C a) => C (T a) divMod is implemented in terms of divModStrict. If it is needed we could also provide a function that accesses the divisor first in a lazy way and then uses a strict divisor for subsequent rounds of the subtraction loop. This way we can handle the cases "dividend smaller than divisor" and "dividend greater than divisor" in a lazy and efficient way. However changing the way of operation within one number is also not nice.

Derived functions

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

Allows division by zero. If the divisor is zero, then the dividend is returned as remainder.

divides :: (C a, C a) => a -> a -> BoolSource

sameResidueClass :: (C a, C a) => a -> a -> a -> BoolSource

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

Returns the result of the division, if divisible. Otherwise undefined.

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

Returns the result of the division, if divisible. Otherwise undefined.

even :: (C a, C a) => a -> BoolSource

odd :: (C a, C a) => a -> BoolSource

divUp :: C a => a -> a -> aSource

divUp n m is similar to div but it rounds up the quotient, such that divUp n m * m = roundUp n m.

roundDown :: C a => a -> a -> aSource

roundDown n m rounds n down to the next multiple of m. That is, roundDown n m is the greatest multiple of m that is at most n. The parameter order is consistent with div and friends, but maybe not useful for partial application.

roundUp :: C a => a -> a -> aSource

roundUp n m rounds n up to the next multiple of m. That is, roundUp n m is the greatest multiple of m that is at most n.

Algorithms

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

decomposeVarPositional [b0,b1,b2,...] x decomposes x into a positional representation with mixed bases x0 + b0*(x1 + b1*(x2 + b2*x3)) E.g. decomposeVarPositional (repeat 10) 123 == [3,2,1]

decomposeVarPositionalInf :: C a => [a] -> a -> [a]Source

Properties

propInverse :: (Eq a, C a, C a) => a -> a -> PropertySource

propMultipleDiv :: (Eq a, C a, C a) => a -> a -> PropertySource

propMultipleMod :: (Eq a, C a, C a) => a -> a -> PropertySource

propProjectAddition :: (Eq a, C a, C a) => a -> a -> a -> PropertySource

propProjectMultiplication :: (Eq a, C a, C a) => a -> a -> a -> PropertySource

propUniqueRepresentative :: (Eq a, C a, C a) => a -> a -> a -> PropertySource

propSameResidueClass :: (Eq a, C a, C a) => a -> a -> a -> PropertySource